Contrasting the fuzzball and wormhole paradigms
for black holes

Bin<EMAIL_ADDRESS>Marcel R. R<EMAIL_ADDRESS>Samir
D<EMAIL_ADDRESS>and Madhur<EMAIL_ADDRESS>

Department of Physics,

The Ohio State University,

Columbus, OH 43210, USA

###### Abstract

We examine an interesting set of recent proposals describing a ‘wormhole
paradigm’ for black holes. These proposals require that in some effective
variables, semiclassical low-energy dynamics emerges at the horizon. We prove
the ‘effective small corrections theorem’ to show that such an effective
horizon behavior is not compatible with the requirement that the black hole
radiate like a piece of coal as seen from outside. This theorem thus
concretizes the fact that the proposals within the wormhole paradigm require
some nonlocality linking the hole and its distant radiation. We try to
illustrate various proposals for nonlocality by making simple bit models to
encode the nonlocal effects. In each case, we find either nonunitarity of
evolution in the black hole interior or a nonlocal Hamiltonian interaction
between the hole and infinity; such an interaction is not present for burning
coal. We examine recent arguments about the Page curve and observe that the
quantity that is argued to follow the Page curve of a normal body is not the
entanglement entropy but a different quantity. It has been suggested that this
replacement of the quantity to be computed arises from the possibility of
topology change in gravity which can generate replica wormholes. We examine
the role of topology change in quantum gravity but do not find any source of
connections between different replica copies in the path integral for the
Rényi entropy. We also contrast the wormhole paradigm with the fuzzball
paradigm, where the fuzzball does radiate like a piece of coal. Just as in the
case of a piece of coal, the fuzzball does not have low-energy semiclassical
dynamics at its surface at energies $E\sim T$ (effective dynamics at energies
$E\gg T$ is possible under the conjecture of fuzzball complementarity, but
these $E\gg T$ modes have no relevance to the Page curve or the information
paradox).

###### Contents

  1. 1 Introduction
    1. 1.1 Different models for radiating objects
      1. 1.1.1 Burning coal
      2. 1.1.2 The semiclassical black hole
      3. 1.1.3 Fuzzballs
      4. 1.1.4 An impossibility
      5. 1.1.5 Wormhole model - I: Nonlocal construction of effective variables
      6. 1.1.6 Wormhole model - II: Nonlocal Hamiltonian interactions
      7. 1.1.7 Wormhole model - III: Identifying bits between the hole and infinity
      8. 1.1.8 The Page curve
      9. 1.1.9 Investigating nonlocalities
    2. 1.2 Summary
    3. 1.3 The plan of the paper
  2. 2 The effective small corrections theorem
    1. 2.1 The proof of the effective small corrections theorem
  3. 3 Some definitions
    1. 3.1 The classical black hole
    2. 3.2 The semiclassical black hole
    3. 3.3 Some definitions
    4. 3.4 The Euclidean hole
    5. 3.5 (1+1)-dimensional gravity
  4. 4 The Page curve - I: What topology change can and cannot do
    1. 4.1 Topology change in (1+1)-dimensional gravity: the Lorentzian theory
    2. 4.2 The small corrections theorem and topology change
    3. 4.3 Using Euclidean path integrals for (1+1)-dimensional gravity
    4. 4.4 Summary
  5. 5 The Page curve - II: The prescription of replacing Rényi entropies by new quantities
    1. 5.1 Entanglement entropies: review of notation
    2. 5.2 Computing the Rényi entropies
    3. 5.3 The ‘prescription’
    4. 5.4 Topology change
  6. 6 The Page curve - III: Path integrals and the difference between Rényi and Gibbons-Hawking type computations
    1. 6.1 Expressing states through path integrals
    2. 6.2 The Gibbons-Hawking computation
    3. 6.3 Wormholes that represent entanglement
    4. 6.4 Modeling the evaporation of coal
    5. 6.5 Summary
  7. 7 Postulating nonlocalities
    1. 7.1 Nonlocal definition of effective variables
      1. 7.1.1 How can we differentiate such a black hole from coal?
      2. 7.1.2 The kinematics of effective bits
      3. 7.1.3 Using the dynamics of the bits at $r<10\,r_{h}$
      4. 7.1.4 Using dynamics of the radiation bits at infinity
      5. 7.1.5 Summary
    2. 7.2 The experiment
    3. 7.3 Nonlocal effects between black hole interiors: baby universes
    4. 7.4 Nonlocal effects between different regions near spatial infinity
    5. 7.5 Nonlocal effects between the hole and its radiation
      1. 7.5.1 The difficulty with invoking holography
      2. 7.5.2 Using small nonlocal interactions
      3. 7.5.3 Identifying bits between the hole and the radiation region
  8. 8 The requirements for a bit model of the wormhole paradigm
  9. 9 Discussion
  10. A Some details of the fuzzball paradigm
    1. A.1 How fuzzballs differ from the traditional hole
    2. A.2 Construction of fuzzball microstates
    3. A.3 Understanding the fuzzball resolution to the information paradox
    4. A.4 Fuzzball complementarity
  11. B A bit model for Hawking pair creation at the horizon
    1. B.1 The divergence of trajectories at the horizon
    2. B.2 The state for $t<0$
    3. B.3 Evolution for $t>0$
    4. B.4 Matching at $t=0$
    5. B.5 The entangled nature of the final state

##  1 Introduction

In 1974 Hawking discovered that black holes evaporate by producing entangled
pairs at the horizon [1, 2]. The two members of the pair $\\{b,c\\}$ are in an
entangled state which can be schematically written as

$|\psi\rangle_{pair}={1\over\sqrt{2}}\Big{(}|0\rangle_{b}|0\rangle_{c}+|1\rangle_{b}|1\rangle_{c}\Big{)}+O(\epsilon)\
.$ (1.1)

Here the $O(\epsilon)$ correction takes into account any small quantum gravity
corrections not captured by the leading order ‘quantum fields on curved space’
computation done by Hawking. This pair creation process leads to a
monotonically rising entanglement entropy $S_{ent}$ of the hole with its
radiation over time. We then get a violation of quantum unitarity when the
hole evaporates away; an aspect of a problem known as the black hole
information paradox. The graph of $S_{ent}$ with time is called the Page
curve; thus another way of stating the problem is that the Page curve for
Hawking’s computation does not come down to zero at the end of the evaporation
process.

Over the past two decades a resolution to this puzzle has emerged in string
theory; this resolution is called the fuzzball paradigm [3, 4, 5, 6, 7, 8, 9,
10, 11, 12]. In string theory, we must make a black hole by taking a bound
state of the strings and branes in the theory. In each case where such a bound
state has been made, it has been found that the bound state is a ‘fuzzball’: a
horizon-sized quantum object with no horizon. If all black hole microstates
are assumed to have this behavior, then the black hole is an object no
different from a planet or a star; it radiates from its surface like any
normal body, not by the production of entangled pairs. In other words, there
is _no_ analogue of (1.1). In this way the fuzzball paradigm resolves the
information paradox (for an overview of the fuzzball paradigm, see appendix
A).

Recently, there has been interest in looking for an alternative resolution of
the information paradox; we will call this attempt the ‘wormhole paradigm’.
The central aspect of the wormhole paradigm is the requirement that, in some
effective variables, we _do_ have an approximation to low-energy semiclassical
dynamics at the horizon. This low-energy effective dynamics will lead to the
creation of entangled pairs in the effective variables

$|\psi_{ef\\!f}\rangle_{pair}={1\over\sqrt{2}}\Big{(}|0\rangle_{b,ef\\!f}|0\rangle_{c,ef\\!f}+|1\rangle_{b,ef\\!f}|1\rangle_{c,ef\\!f}\Big{)}+O(\epsilon)\
.$ (1.2)

This pair creation again leads to a monotonically rising entanglement of the
hole with its radiation. The problem is then argued to be resolved by nonlocal
effects in the gravity theory that connect the hole to its radiation. In a
rough picture we can imagine this connection to be in the form of a wormhole
extending from the hole to each radiation quantum [13]; hence the term
‘wormhole paradigm’ for this class of models.

There has been quite some confusion about the wormhole paradigm. What does it
assume and what does it show? One reason for this confusion is that there are
several different lines of thought that come under the rough umbrella of the
wormhole paradigm. The common feature shared by all these approaches is that
there should be an emergence of effective semiclassical behavior at the
horizon, i.e. we should get vacuum pair production in the state (1.2). In this
manner the wormhole approaches all differ from the fuzzball paradigm, where we
do not have any such pair production. However, the arguments for why the Page
curve should come down (in spite of having the pair creation (1.2)) differ
between different formulations of the wormhole paradigm. As noted above, the
Page curve will be argued to come down using some postulate involving
nonlocality. This postulate, however, is not explicit in many of the
approaches. One of our main goals will be to make such nonlocality explicit.

Some confusion is also caused by the fact that there are several different
aspects of these arguments, whose roles are sometimes not sufficiently
clarified:

  1. (A1)

The existence of nonlocal Hamiltonian interactions between the hole and its
radiation.

  2. (A2)

The idea that the effective bits appearing in (1.2) can be made as
combinations of bits in the hole as well as bits in the radiation at infinity.

  3. (A3)

The idea that if nonlocal effects are ‘small’ then they could be somehow
consistent with the notion that physics far from the hole should be ‘normal’.

  4. (A4)

The idea that degrees of freedom (bits) describing quanta at infinity are not
independent of the degrees of freedom in the hole.

Note that the nonlocality needed in the wormhole paradigm is not the
‘nonlocality’ of entanglement. In ordinary quantum mechanics two quanta with
an arbitrarily large separation can be in an entangled state. Such an
entanglement is present between the hole and its radiation in Hawking’s
original computation of radiation. But this entanglement does not create any
interaction between the hole and its radiation. This entanglement will not by
itself bring the Page curve down; rather, this entanglement is the basic cause
of the information paradox.

Our goal in this article is to try to clarify the above confusions by
detailing what we understand about the various proposals. In seeking this
clarity we make simple bit models to explain what we think the different
proposals are saying. Our hope is that these models will prompt a discussion
in the field that will make precise what is assumed, what is claimed and what
is proved in the different approaches to the wormhole paradigm. It may be that
proponents of the wormhole paradigm have other models in mind; in that case it
would be very useful to have these other models expressed in the same bit
model language that we use here so that the underlying ideas behind the
paradigm become clear.

Let us now summarize the various issues arising in the fuzzball and wormhole
paradigms; in the process we will keep in mind how points (A1)-(A4) above
appear in the various arguments.

###  1.1 Different models for radiating objects

Let us list the various kinds of radiating objects that appear in the
discussion of the fuzzball and wormhole paradigms. In what follows, the reader
should assume the following: (a) the exact theory has a unitary evolution,
unless stated otherwise, (b) the bits being described are bits of the exact
theory, unless they are explicitly termed effective bits and, (c) the
radiation quanta at infinity are always the exact bits that are observed by an
apparatus placed at infinity.

####  1.1.1 Burning coal

First consider how a normal object like a piece of coal burns away. The state
of the emitted quanta depend on the state of the coal they are emitted from,
since the emitted quanta are produced by interactions between the quanta
making up the coal. But these interactions are short ranged; once a radiated
quantum has left the vicinity of the coal, _its state is no longer affected by
the coal_. The Page curve for the coal at first rises and then falls back down
to zero.

One might say that there could be some small long-ranged interactions between
the emitted quanta and the remaining coal. But the point is that such
interactions are not the reason that the Page curve drops down to zero for a
piece of burning coal. We could perfectly well take a model of the coal where
the interactions between the radiated quantum and the remaining coal fall to
zero outside some radius $R_{max}$ and in such a model we will still find that
the Page curve comes down to zero at the end of evaporation.

A third fact that is of relevance in view of point (A4) above is that degrees
of freedom at infinity are independent of the degrees of freedom in the coal.
The ‘bits’ at infinity are made by excitation of say, a scalar field $\phi(x)$
near infinity, while bits in the coal are made from fields in the region of
the coal. These are independent degrees of freedom. To summarize, the
properties of burning coal are:

  1. (C1)

There are no relevant interactions between the radiated quanta and the
remaining coal once these radiated quanta have left the vicinity of the coal.

  2. (C2)

The bits at infinity which describe the radiation are independent of the bits
that make up the remaining coal.

  3. (C3)

The Page curve first rises and then falls back to zero at the end of the
burning process.

####  1.1.2 The semiclassical black hole

The semiclassical black hole radiates quanta by pulling pairs out of the
_vacuum_. Thus these quanta have no information about the details of the
matter which made the hole. The state of the created pair $\\{b,c\\}$ can be
modelled as

$|\psi\rangle_{pair}={1\over\sqrt{2}}\Big{(}|0\rangle_{b}|0\rangle_{c}+|1\rangle_{b}|1\rangle_{c}\Big{)}\
.$ (1.3)

One can imagine that there may well be small quantum gravity effects that
modify the state of the created pair by the $O(\epsilon)$ corrections noted in
(1.1); here the correction to any pair can depend on the matter which made the
hole and the state of the quanta that fell into the hole at earlier steps of
pair creation. Some people had originally hoped that small $O(\epsilon)$
corrections to (1.1) would somehow introduce suitable correlations among the
radiated quanta and bring the entanglement down to zero. However, the small
corrections theorem [14] showed that this is not possible; the entanglement
entropy $S_{ent}(N)$ after $N$ emissions will continue to rise as

$S_{ent}(N+1)>S_{ent}(N)+\ln 2-2\epsilon\ .$ (1.4)

Thus we need an order _unity_ correction to the low-energy dynamics at the
horizon in order to resolve the information paradox.

To summarize, the semiclassical hole is defined as one where we study quantum
fields on the fixed geometry of the classical black hole and include the
possibility of small corrections arising out of nonperturbative quantum
gravity processes. For this semiclassical hole, we have the following
properties:

  1. (SC1)

The horizon is a vacuum region to leading order. Thus semiclassical dynamics
holds to leading order in this region. The metric can be taken as

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\ ,$ (1.5)

with $\bar{g}_{\mu\nu}$ being the classical black hole metric and $h_{\mu\nu}$
is small. For a scalar field on this background, we will have

$\square\phi\approx 0\ ,$ (1.6)

around the horizon. This dynamics will give rise to the creation of entangled
pairs of the form (1.1).

  2. (SC2)

There are no relevant interactions between the radiated quanta and the
remaining hole once these radiated quanta have left the vicinity of the hole.

  3. (SC3)

The bits at infinity which describe the radiation are independent of the bits
that make up the remaining hole.

  4. (SC4)

The Page curve will keep rising monotonically in the form (1.4).

####  1.1.3 Fuzzballs

A fuzzball behaves just like a piece of coal. The nontrivial step here is the
demonstration that brane bound states in string theory do not generate the
geometry of the semiclassical hole; instead they generate extended objects
that have no horizon or singularity. In [15] it was shown how the traditional
no-hair theorems are violated by specific features of string theory.

A fuzzball radiates from its surface just like a piece of coal radiates
photons from its surface. For a piece of coal we do not have any effective
pair creation of the states (1.1); similarly, for a fuzzball we will not have
any effective variables where we get (1.1). This issue is very important and
will be discussed in more detail below. Thus for a fuzzball we have the
following behavior:

  1. (F1)

There are no relevant interactions between the radiated quanta and the
remaining fuzzball once these radiated quanta have left the vicinity of the
fuzzball.

  2. (F2)

The bits at infinity which describe the radiation are independent of the bits
that make up the remaining fuzzball.

  3. (F3)

The Page curve first rises and then falls back to zero at the end of the
burning process.

  4. (F4)

There are no effective variables in which we get (1.2).

  5. (F5)

The full structure of string theory is required to obtain ‘fuzzballs’; i.e. to
obtain objects that do not collapse to the traditional semiclassical hole.
Thus a simple theory like (1+1)-dimensional JT gravity will not have
fuzzballs; in such a theory we will just get the traditional semiclassical
hole.

####  1.1.4 An impossibility

The discovery that brane bound states in string theory generate fuzzballs with
no horizon gives a simple resolution to the information paradox. In the
initial days of the fuzzball paradigm, some people thought that this change to
the geometry of the hole was too radical; they hoped that small corrections to
the traditional black hole could somehow encode enough correlations in the
radiated quanta to bring the Page curve down to zero. The small corrections
theorem (1.4) showed that this hope could not be realized; one needs an order
unity correction to horizon dynamics, so the fuzzball paradigm was a _natural_
resolution to the puzzle rather than a radical one.

At this point some people felt that the following might be possible. Suppose
it was true that in the exact quantum gravity theory the microstates of the
hole behaved like pieces of coal. But this description of the microstates
would be very complicated. Suppose that there was some choice of effective
variables describing the complicated degrees of freedom of the black hole, in
which an approximation to the semiclassical dynamics emerged. In that case
getting an exact description of the hole in terms of fuzzballs would be
correct, but it may be that the effective variables would give a simpler and
more useful description of the dynamics. To make this suggestion more precise,
suppose we require the effective description to have the following properties:

  1. (EFF1)

There are no relevant interactions in the exact theory between the radiated
quanta and the remaining hole once these radiated quanta have left the
vicinity of the coal. (This is just like (C1) for burning coal.)

  2. (EFF2)

The bits at infinity which describe the radiation are independent of the bits
that make up the remaining hole. (This is just like (C2) for burning coal.)

  3. (EFF3)

The effective degrees of freedom describing the hole are obtained (in some
possibly very complicated way) from all the degrees of freedom in the region
of the hole; say in the region $r<10\,r_{h}$, where $r_{h}$ is the radius of
the hole.111In the rest of this paper we will use this region $r<10\,r_{h}$ as
describing the region up to a few horizon radii from the center of the hole;
the reader could of course substitute the number $10$ by any other number of
his choice. Note that from property (EFF2), these degrees of freedom making
the effective bits are independent of the degrees of freedom near infinity.

  4. (EFF4)

We will be quite generous in how little we demand from these effective
variables:

    1. (i)

The semiclassical dynamics of the hole has to emerge only approximately with
these variables. Thus for a scalar field the equation $\square\phi=0$ in the
vicinity of the horizon can be relaxed to

$\square\phi_{ef\\!f}=O(\epsilon)\ ,\quad\epsilon\ll 1\ .$ (1.7)

    2. (ii)

This effective semiclassical dynamics has to only describe low-energy physics.
This low-energy physics includes modes with wavelengths
$r_{h}/100\lesssim\lambda\lesssim 10\,r_{h}$ since it is the stretching of
modes in this range which gives the Hawking pair production that we are
interested in. But the effective dynamics does not have to work for
wavelengths down to string length or Planck length.

    3. (iii)

The evolution (1.7) will yield the creation of approximate entangled pairs
(1.2) in the region of the hole. We do not require that the above effective
description describe the creation of _all_ the pairs emitted by the hole. We
merely ask that it describes the emission of a few pairs, after which we may
have to choose a new set of effective variables $\phi_{ef\\!f}(x)$ to continue
getting an effective semiclassical dynamics around the horizon.

Given the above conditions one can prove the following:

  5. (EFF5)

The Page curve for the _exact_ theory will keep rising monotonically; i.e. we
cannot get the analogue of property (C3) of burning coal.

This statement can be proved by a simple adaptation of the proof of the small
corrections theorem (1.4). In this adaptation we will use the pair creation
(1.2) for _effective_ variables in place of the pair creation (1.1) for
semiclassical excitations around the traditional black hole geometry. The
result (EFF5) will be termed the effective small corrections theorem; we will
see its derivation in section 2.

To summarize the content of (EFF1)-(EFF5): _we cannot require that the black
hole behaves like a piece of coal as seen from outside and also require that
the variables in the region $r<10\,r_{h}$ give rise to effective semiclassical
dynamics around the horizon._

####  1.1.5 Wormhole model - I: Nonlocal construction of effective variables

We have seen above that in the fuzzball paradigm the black hole behaves like
burning coal as seen from outside, thus properties (C1)-(C3) are reflected in
(F1)-(F3). Furthermore, for a piece of coal we do not have any effective
dynamics where we see pair creation at the horizon; likewise, for fuzzballs we
have (F4) which says that there is no such effective description of pair
creation. Thus if we did not wish to insist on getting (1.7) for low-energy
modes, then fuzzballs already provide a resolution of the information paradox.

As we have noted, the wormhole paradigm _does_ ask for effective low-energy
semiclassical dynamics around the horizon and so will have the effective pair
creation (1.2). As discussed in section 1.1.4 above, this effective pair
creation cannot be achieved while, (i) keeping all the properties of burning
coal (C1)-(C3) and, (ii) making the effective degrees of freedom only out of
degrees of freedom in the region $r<10\,r_{h}$.

The wormhole paradigm tries to get around this difficulty by a variety of
ways, which we will now discuss. We start with the idea that the effective
variables can be made using both the exact bits in the region of the hole and
the exact bits at infinity. We have the following postulates:

  1. (WI-1)

There are no relevant interactions between the radiated quanta and the
remaining hole once these radiated quanta have left the vicinity of the coal.
Here we are talking about the _exact_ bits of the theory (just like (C1) for
burning coal).

  2. (WI-2)

The bits at infinity which describe the radiation are independent of the bits
that make up the remaining hole. Again we are talking about the _exact_ bits
of the theory (just like (C2) for burning coal).

  3. (WI-3)

The effective degrees of freedom describing the hole are obtained (in some
possibly very complicated way) from the exact bits in the region of the hole
($r<10\,r_{h}$), _as well as_ the exact bits making up the radiation $R$ that
has been emitted from the hole.

  4. (WI-4)

We ask for the same effective dynamics that was listed in (EFF4) in section
1.1.4 above.

  5. (WI-5)

We ask that the Page curve for the _exact_ bits making up the radiation $R$
come down to zero at the end of the evaporation process.

This may look like an appealing set of properties to ask for, since we have
asked for the exact dynamics to be like that of coal and have also asked for
effective semiclassical dynamics around the horizon. However, the construction
(WI-3) runs into immediate difficulty with the postulate (WI-1) as follows.
Suppose we do have a set of effective bits which make the horizon region an
approximation to the semiclassical hole, say in the region $r<10\,r_{h}$.
Place an apparatus at $r=5\,r_{h}$ that sends a beam into the hole and another
at $r=5\,r_{h}$ that checks for a reflected beam. Since the effective
variables yield semiclassical dynamics to a first approximation, there will be
very little reflected beam; the incoming beam will fall into the hole. Now
take the exact bits in the radiation $R$ and change their state; for example
if they were spins, rotate the spins using a magnetic field applied in the
radiation region near infinity. Since the effective bits near the horizon
involved the exact bits at infinity, the state of the effective bits near the
horizon will change; for example by

${1\over\sqrt{2}}\Big{(}|0\rangle_{b,ef\\!f}|0\rangle_{c,ef\\!f}+|1\rangle_{b,ef\\!f}|1\rangle_{c,ef\\!f}\Big{)}\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\
{1\over\sqrt{2}}\Big{(}|0\rangle_{b,ef\\!f}|1\rangle_{c,ef\\!f}+|1\rangle_{b,ef\\!f}|0\rangle_{c,ef\\!f}\Big{)}\
.$ (1.8)

But if the state on the left-hand side of (1.8) was the local vacuum at the
horizon, the state on the right-hand side will _not_ be the vacuum. Thus the
beam will now reflect off the hole and be observed in the detection
apparatus.222For an explicit example of how waves reflect off a fuzzball see
[16]. Thus we see that manipulating the radiation bits $R$ at infinity in a
suitable way can change the structure of the hole in the region $r<10\,r_{h}$.
As a consequence, we should modify (WI-1) to read

  1. (WI-1’)

Modifying the radiation quanta at infinity will change the dynamics that is
observed by experimenters in the vicinity of the hole. This is different from
the behavior of a piece of burning coal, where manipulating the radiation
quanta at infinity does not change the observations of an experimenter
examining the coal. We will investigate models of this type in section 7.1.

Maldacena [17] has suggested a model where manipulating the bits at infinity
will lead to a modification of the bits in the _interior_ of the hole.333We
could regard this interior as the region inside of the QES, in the cases where
the QES is within the horizon. In particular, he has conjectured that if the
hole has evaporated away completely and its contents have pinched off into a
baby universe, then manipulating the radiation bits $R$ at infinity can
extract information from this baby universe.

As we will observe in section 8 however, manipulating the bits at infinity
will also force to a modification of the bits in the vicinity of the horizon
and this latter modification will lead to the above noted change in the
results of an experimenter who seeks to scatter light off the hole. It is very
important to understand why in any situation where the interior bits can be
manipulated from infinity, the horizon region must _also_ change under such
modifications. We have the following situation:

  1. (a)

Suppose that we take a bit model where the bits $b_{ef\\!f},c_{ef\\!f}$
emerging in the Hawking process are made by using the bits at infinity as well
as the bits in the region $r<10\,r_{h}$. In this case manipulating the bits at
infinity will change the observations of an experimenter outside the hole who
is trying to reflect a beam off the hole.

  2. (b)

Suppose we say that the effective bits around the horizon do _not_ involve the
bits at infinity. Then we find that the $b_{ef\\!f},c_{ef\\!f}$ are entangled
with each other just as in Hawking’s original computation. In this case the
effective small corrections theorem will tell us that the Page curve of the
exact theory will have to keep rising monotonically.

  3. (c)

Suppose we proceed as in case (b), but try to get the Page curve to come down
by requiring that the $c_{ef\\!f}$ quanta in the interior of the hole (i.e. in
the ‘island’ region) are made as some combinations of the bits at infinity as
well as the bits in the region $r<10\,r_{h}$. Such an attempt will lead to
nonunitarity of evolution in the region $r<10\,r_{h}$. The reason is the we
have required approximate semiclassical evolution in the region around the
horizon, and this semiclassical evolution takes the $c_{ef\\!f}$ created
around the horizon in (b) above and deposits them on the island. So we do not
have any freedom to choose what bits the $c_{ef\\!f}$ in the island are made
of; in particular, the $c_{ef\\!f}$ that end up on the island are maximally
entangled with the $b_{ef\\!f}$ the escape to infinity. If we try to
annihilate the $c_{ef\\!f}$ that are created at the horizon so that they do
not reach the island, then we get a nonunitarity of evolution in the region
$r<10\,r_{h}$.

In short, we have not been able to find a unitary bit model where, (i) the
horizon exhibits an effective semiclassical dynamics, (ii) The Page curve
comes down and, (iii) an experimenter outside the hole cannot see
modifications of horizon behavior when bits at infinity are manipulated.

####  1.1.6 Wormhole model - II: Nonlocal Hamiltonian interactions

Some people have argued that quantum gravity can have nonlocal Hamiltonian
interactions and that it is these interactions that resolve the puzzle. One
could then ask: why do we not see these nonlocal interactions in everyday
experiments? The idea of these proposals would then be that the nonlocal
effects are not large, in a sense that we explain below with the following
postulates:

  1. (WII-1)

There are nonlocal interactions between the radiation $R$ and the remaining
degrees of freedom in the region $r<10\,r_{h}$. Thus this is different from
postulate (C1) for burning coal.

  2. (WII-2)

These nonlocal effects can change, for example, the spin of any radiation
quantum at infinity through an interaction that depends on the state in the
region $r<10\,r_{h}$. But this change in the spin would be small if we look
at, (i) just a few radiation quanta and, (ii) look at these quanta over
timescales much shorter than the Hawking evaporation time.

  3. (WII-3)

With these interactions, the Page curve comes down to zero by the end of the
evaporation process.

We will give a toy model for such nonlocal interactions in section 7.5.2. We
do not believe that such nonlocal interactions actually arise in string
theory. Here we just observe that if one _does_ postulate such effects, then a
model along the lines of (WII-1)-(WII-3) is possible. It is important to note
that (WII-1) is different from (C1) for burning coal, so we cannot say that in
such models the hole behaves like a piece of burning coal as seen from
outside.

####  1.1.7 Wormhole model - III: Identifying bits between the hole and
infinity

Another set of arguments has taken the track of altering the condition (C2)
that we had for coal. That is, we postulate that the degrees of freedom far
from the hole are not independent of degrees of freedom in the region of the
hole $r<10\,r_{h}$. Thus the postulates would have the form

  1. (WIII-1)

There are no Hamiltonian terms giving an interaction between the bits in the
region $r<10\,r_{h}$ and the radiation region $R$.

  2. (WIII-2)

The degrees of freedom at infinity are not independent of the degrees of
freedom in the region $r<10\,r_{h}$. (This is different from (C2) for burning
coal.)

  3. (WIII-3)

We require the horizon have the semiclassical behavior of the traditional
hole; i.e. we have the creation of entangled pairs (1.1) at the horizon. (We
do not talk about effective bits, since the identification of bits is supposed
to resolve the puzzle while maintaining semiclassical dynamics (1.5) at the
horizon.)

  4. (WIII-4)

We require that the Page curve comes down to zero at the end of the
evaporation process.

However, there is an immediate issue with such a model. By (WIII-3), we create
entangled pairs at the horizon. Each of the excitations $\\{b,c\\}$ have two
states $\\{0,1\\}$, so we have a $4$-dimensional Hilbert space from these
excitations. The excitation $b$ moves off to infinity, while $c$ falls into
the hole; at this stage we still have a $4$-dimensional space of states for
this pair of quanta. Now suppose we wish to make an identification of bits, so
that the bit representing $b$ is identified with the bit representing $c$. We
can try this in two ways:

  1. (i)

We require that the state of the bit $b$ become the state of the bit $c$.
Since $b$ and $c$ have the same state, the Hilbert space spanned by them is
now $2$-dimensional. The reduction from $4$ to $2$ dimensions is a nonunitary
evolution of the system.

  2. (ii)

We keep all $4$ states of the $b,c$ pair but introduce a nonlocal Hamiltonian
between the $b,c$ quanta so that the states where $b,c$ are not identified
rise in energy to a level where they are not part of the low-energy space of
excitations. With this model we find that the $b$ quantum at infinity does not
behave like a normal quantum: it costs energy to change its state between
$|0\rangle$ and $|1\rangle$, while a similar bit radiated by a piece of coal
does not have such an energy increase.

The above are two very crude models of what happens if we try to identify
bits. One can try to include more complicated identifications, however, the
essential difficulty will remain the same: the continuous production of new
entangled pairs gives an enlargement of the Hilbert space of excitations and
if we try to bring the Page curve down by introducing identifications between
bits, then we either have nonunitarity or find that the bits at infinity
behave differently from bits radiated from a piece of coal. We will consider
models of this type in section 7.5.3.

####  1.1.8 The Page curve

It has been argued that one can obtain a Page curve for the black hole that
comes down to zero at the end of evaporation using general arguments of
semiclassical gravity (e.g. (1+1)-dimensional JT gravity) without knowing the
details of the quantum gravity theory. We will argue that one cannot get the
Page curve in this manner. In more detail, we do the following:

  1. (a)

It has been stated that such a semiclassical argument for the Page curve is
similar to the Gibbons-Hawking computation of black hole entropy. We will see,
however, that there is an important difference. With the Gibbons-Hawking
computation we start with a Euclidean path integral with time period $\beta$
which correctly counts the states for _any_ system with Hamiltonian $H$ and
spectrum $\\{E_{i}\\}$, through the partition function

$Z(\beta)=\operatorname{Tr}\big{[}e^{-\beta H}\big{]}=\sum_{i}e^{-\beta
E_{i}}\ .$ (1.9)

One then observes that there is a plausible classical saddle for this path
integral, with the only assumption made being that this saddle should give a
good approximation to the integral. For the Page curve computation we are not
able to cast the the starting path integral as a quantity that gives the
entanglement for an arbitrary system. For example, in computing the second
Rényi entropy $S_{2}(A)$ we need to compute
$\operatorname{Tr}[(\rho_{A})^{2}]$ where $\rho_{A}$ is the reduced density
matrix of the region $A$. In the recent Page curve computation one makes a
replacement of the type

$\operatorname{Tr}\big{[}(\rho_{A})^{2}\big{]}\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\
\Big{(}\operatorname{Tr}\big{[}(\rho_{A})^{2}\big{]}+C\big{(}\operatorname{Tr}[\rho_{A}]\big{)}^{2}\Big{)}\
,$ (1.10)

for some constant $C$. Thus we will be starting with a quantity that appears
to be _different_ from the entanglement entropy that we wished to compute.

  2. (b)

It has been argued that the prescription (1.10) is justified because it takes
into account the fact that there can be topology change in gravity. We
consider the role of topology changing processes in section 4 (using
(1+1)-dimensional gravity as an example) and find that topology change does
not allow for the prescription (1.10). The second Rényi entropy $S_{2}(A)$ is
still given by $\operatorname{Tr}[(\rho_{A})^{2}]$ on the full Hilbert space,
which now includes disconnected line segments arising from the possibility of
topology change.

  3. (c)

So what can give the prescription (1.10)? It is crucial to note that we cannot
make arbitrary ‘prescriptions’ for the behavior of the effective semiclassical
gravity theory and then use these to compute quantities like entanglement
entropy. Let the variables of the exact gravity theory be denoted by
$g_{exact}$ and of the approximate effective theory by $g_{ef\\!f}$. The
effective variables $g_{ef\\!f}$ need to be some functionals of the exact
variables $g_{exact}$; we write this symbolically as

$g_{ef\\!f}=F[g_{exact}]\ .$ (1.11)

The Lagrangian of the exact theory then determines, through (1.11), the
dynamics of the effective theory

${\mathcal{L}}_{exact}[g_{exact}]\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ {\mathcal{L}}_{ef\\!f}[g_{ef\\!f}]\ .$ (1.12)

Similarly, any quantity $Q_{exact}[g_{exact}]$ which is of interest in the
exact theory will map, through (1.11), to a quantity $Q_{ef\\!f}[g_{ef\\!f}]$
in the effective theory

$Q_{exact}[g_{exact}]\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
Q_{ef\\!f}[g_{ef\\!f}]\ .$ (1.13)

Thus any prescription like (1.10) for the semiclassical dynamics must have its
origins in the dynamics of the exact theory. To summarize, in our
investigations we have not been able to find any way that the effective theory
emerging from the exact theory will give rise to a wormhole that will connect
different replica copies.

####  1.1.9 Investigating nonlocalities

If the effective variables describing the black hole are made from the exact
variables in the region of the black hole $r<10\,r_{h}$ then we have seen that
the postulates (EFF1)-(EFF4) imply (EFF5); i.e. the Page curve does not come
down. So to look for what feature in the exact theory can give a prescription
like (1.10) we consider nonlocal effects. We consider three kinds of these
nonlocalities:

  1. (i)

_Nonlocal effects in the exact theory that connect the interior of one hole to
the interior of another hole._ The effective small correction theorem extends
in a straightforward way in this case to say that the Page curve cannot come
down. We look at a model where the interior regions of the holes disconnect to
give rise to baby universes, and this correlates the different interiors. We
find that in this case the evolution in the black hole interior violates
unitarity.

  2. (ii)

_Nonlocal effects in the exact theory that connect the interior of the hole to
the radiation region._ In this case we can have models like that in section
1.1.6 where the Page curve comes down to zero, but the hole does not look like
a piece of burning coal as seen from outside: (WII-1) differs from (C1). It is
also not clear how such nonlocal interactions lead to the prescription (1.10)
used in the Euclidean path integral.

  3. (iii)

_Nonlocal effects between the radiation near infinity from one hole and the
radiation near infinity from another hole._ It has been argued that such
effects can change the way we measure the entanglement of the radiation $R$.
This is because one needs to measure many identical copies of the radiation
$R_{1},R_{2},\dots$ produced from identically prepared holes in order to judge
the state of this radiation. If these different measurements interfered with
each other, then one would have a novel effect with radiation from a black
hole; i.e. we would have an effect that is not present when we check the
entanglement of radiation from normal quantum objects. Note that since we can
separate the different copies of the hole by an arbitrary distance, this
interaction between the radiation regions $R_{i}$ must not fall off with
distance. We do not believe that there are such nonlocal effects in string
theory; it is also not clear how exactly such effects would lead to the
prescription (1.10).

###  1.2 Summary

Let us return to our original issue: why has the wormhole paradigm been so
confusing? One reason is that the wormhole paradigm is not addressing the
information paradox itself, but a somewhat different question. The information
paradox arises from a combination of two observations:

  1. (i)

The no-hair results suggest that all matter in a black hole rushes to the
central singularity, leaving the vacuum state around the horizon.

  2. (ii)

Hawking’s computation shows that entangled pairs are created from such a
vacuum region, leading to a monotonically rising Page curve. The small
corrections theorem (1.4) makes this difficulty precise, since no small
corrections to Hawking’s computation can bring the Page curve down.

The fuzzball paradigm resolves the paradox by showing that in string theory
the no-hair theorems are violated: all microstates that have been constructed
are horizon-sized quantum fuzzballs with no horizon or singularity. But these
constructions need the full structure of string theory; there are no fuzzballs
in (1+1)-dimensional gravity, we simply get a monotonically rising Page curve
[18, 19, 20, 21].

The recent wormhole paradigm arguments do not seek to address the information
paradox as summarized in points (i) and (ii) above. Instead, these arguments
typically start with an _assumption_ that some hitherto unknown effects in the
quantum gravity theory makes the black hole behave like a piece of coal as
seen from outside; i.e. the Page curve comes down to zero at the end of
evaporation. The question that is then asked is: _given_ this behavior of the
Page curve, how can we recover some approximation to semiclassical dynamics
around the horizon? Note that this question is _different_ from the
information paradox.

There is an immediate difficulty in answering the above question about
semiclassical behavior at the horizon. As noted in section 1.1.4, one cannot
get this semiclassical behavior through _any_ combination of the degrees of
freedom in the black hole region $r<10\,r_{h}$. In the fuzzball paradigm, we
note this fact and observe that there will be no low-energy semiclassical
dynamics at the horizon (property (F4) in section 1.1.3). There is a
possibility of getting some effective classical dynamics for infalling objects
with _high_ energies $E\gg T$, where $T$ is the temperature of the hole; this
possibility is called the conjecture of fuzzball complementarity [22, 23].
However, this conjecture has no bearing on the discussions of the information
paradox and the Page curve since these discussions only involve the Hawking
quanta which have energy $E\sim T$. For such $E\sim T$ quanta, the fuzzball
paradigm says that there is no effective dynamics that yields (1.2).

The wormhole paradigm seeks to get the effective dynamics (1.2) through a
variety of postulates that involve _nonlocal_ effects connecting the hole to
its far away radiation. Since the question being asked is about effective low-
energy dynamics, the computations with the wormhole paradigm typically involve
simple (1+1)-dimensional theories like JT gravity, not the full structure of
string theory. But (1+1)-dimensional gravity has been well studied and here
one finds no resolution of the information puzzle: the Page curve keeps rising
monotonically. So what can we hope to learn by using such simple theories?
What one does in the wormhole paradigm is to add ‘prescriptions’ to the
behavior of the (1+1)-dimensional theory. These prescriptions can, for
example, be in the form of a modification (1.10) of how the Rényi entropy
should be written in terms of path integrals. With these new prescriptions for
the low-energy dynamics, it is then argued that one has found a Page curve
that comes down to zero.

Here we come to a crucial issue. One cannot make an arbitrary prescription for
low-energy effective dynamics. Instead, these low-energy effective variables
have to emerge from some map of the form (1.11) This map then determines the
low-energy effective dynamics and the rules for computing low-energy
quantities as in (1.12) and (1.13). The wormhole paradigm does not seek to
give us the map (1.11). But in that case, how do we know that the low-energy
prescriptions are correct? We have tried to list various prescriptions that
have been considered in the wormhole paradigm, and then ask what effect in the
exact theory these would emerge from. These effects in the exact theory must
take the form of nonlocal effects over long distances, since constructions of
effective variables that use only the degrees of freedom in the region of the
hole $r<10\,r_{h}$ cannot bring the Page curve down. We do not believe such
nonlocal effects are actually present in string theory. But in the present
article we will not discuss the existence of nonlocality in string theory; we
will take up this issue in a following article. Instead, we will elaborate on
the observations made in the sections above in an effort to concretize the
nonlocalities that are explicitly or implicitly part of the wormhole paradigm.

###  1.3 The plan of the paper

The plan of this paper is as follows.

In section 2 we derive the ‘effective small corrections theorem’. This theorem
extends the small corrections theorem of [14] to the case where we have only
approximate semiclassical behavior at the horizon in terms of effective
variables made out of all the degrees of freedom in the region of the hole.
Thus, we do not assume that the actual spacetime around the horizon is close
to the classical one. This theorem shows that we cannot have both, (i) the
black hole radiating like a piece of coal as seen from outside (conditions
(C1)-(C3) of section (1.1.1) and, (ii) some effective degrees of freedom in
the region $r<10\,r_{h}$ giving rise to a ‘code subspace’ where we have
approximate semiclassical behavior satisfying the weak requirements of (EFF4).
This theorem thus implies that the wormhole paradigm must have some kind of
nonlocal effects as an essential ingredient in getting the Page curve to come
down444To be precise, getting around the effective small corrections theorem
requires the violation of one of its assumptions, the least radical of which
is the introduction of non-locality. Otherwise, something like non-unitarity
would be required; something that is not very appealing..

This is followed by section 3, where some notation and background for black
holes is recalled: the Penrose diagram, good slices, baby universes etc.

In sections 4-6 we examine recent suggestions that the Page curve for a black
hole can be computed by adding certain prescriptions to how we use
semiclassical gravity. In these computations one finds the entanglement
entropy by taking a suitable limit of Rényi entropies, and argues that the
resulting Page curve will come down like the Page curve of a normal body. We
find that in these computations the Rényi entropies are replaced by a new
quantity that is _not_ the Rényi entropy, so the curve that is argued to come
down is not the Page curve, in the sense of entanglement entropy. It has
sometimes been argued that the replacement of the Rényi entropies by the new
quantities is dictated by the possibility of topology change in gravity. We
examine the role of topology change in the computation of entanglement
entropies via path integrals and find that, at least in the (1+1)-dimensional
quantum gravity example studied, topology change does _not_ imply a replica
wormhole connecting different copies in the Rényi entropy computation. We note
that one cannot make an arbitrary prescription for how semiclassical
geometries should behave in an effective theory, since this effective theory
must descend from the exact theory through the relations (1.11), (1.12) and
(1.13). We therefore argue that the recent Page curve computations differ from
the Gibbons-Hawking computation of black hole entropy in a fundamental way:
while the Gibbons-Hawking computation starts with a path integral that would
yield the entropy for _any_ physical system, the Page curve computation
modifies the starting path integral in a way that yields a quantity different
from the entanglement entropy.

We could not identify, in section 7, any clear nonlocality postulate for the
exact theory that could yield the prescriptions used for the effective theory
in the recent Page curve computations. We therefore proceed by examining
various kinds of nonlocalities that have been postulated and find that the
postulate that baby universes connect the interiors of different black holes
leads to a nonunitarity of evolution. Alternatively, nonlocalities that
connect the hole to infinity lead to an asymptotic observer finding different
behaviors for quanta radiated from coal and from a black hole.

In section 8 we give an explicit set of conditions that must be met by any bit
model for the wormhole paradigm. These conditions impose the requirement of an
effective low-energy semiclassical dynamics at the horizon and the requirement
of a Page curve that comes down.

We then conclude in section 9 by a summary. The appendix A contains a review
of some aspects of the fuzzball paradigm and appendix B details a bit model
for the process of Hawking pair creation for a classical black hole horizon.

In this paper we have just tried to isolate the nonlocality postulates that
are implied by the wormhole paradigm.555We do not discuss approaches like that
of [24] which require the exact theory to be an ensemble averaged theory; this
is because we have in mind string theory as our exact theory and we believe
that string theory is not an ensemble averaged theory. The literature of the
wormhole paradigm spans a large number of papers, but we have included very
few references. This is because we are not seeking to analyze in detail any
particular paper in this paradigm but to instead explore the different
categories of models that have been proposed. Thus where we do include
references, they just point to the kind of model that we are analyzing. As we
already noted above, it is possible that the proponents of some wormhole
models have ideas in mind different from those we cover and it is the hope of
this paper that these authors would explain their work in the bit model
language used in the present paper and thus clarify the physics that leads to
the Page curve coming down in their model.

In a follow-up paper we plan to present a discussion of why we believe that
such nonlocalities do not exist in string theory. In particular some confusion
has been caused by suggestions that AdS/CFT duality implies a nonlocality in
gravity. But such is not the case; the CFT and the gravity theory are both
completely local, and the nonlocality of the map between the two cannot be
used to argue for a nonlocality in gravity itself.

##  2 The effective small corrections theorem

The small corrections theorem, proved in [14], shows that any small
corrections to semiclassical horizon dynamics will not change Hawking’s
conclusion that the Page curve monotonically rises. We are now interested in a
situation where the actual state of the hole is not necessarily close to the
semiclassical geometry; in fact this state can be a very complicated mess of
the quantum gravitational degrees of freedom in a region which for
concreteness we take to be $r<10\,r_{h}$. We then ask that in some effective
variables made out of these complicated degrees of freedom, we get the low-
energy semiclassical dynamics described by the conditions (EFF4) listed in
section 1.1.4. In this situation we can immediately extend the small
corrections theorem to an ‘effective small corrections theorem’; here the term
effective denotes the fact that instead of the exact bits $\\{b,c\\}$ we now
have effective bits $\\{b_{ef\\!f},c_{ef\\!f}\\}$.

The effective small corrections theorem provides a strong constraint which
relates the exact theory to any effective theory. In short, the result of the
theorem is the following. Suppose that in the effective theory the dynamics of
low-energy modes around the horizon is the traditional semiclassical dynamics,
then in this effective description, we will find the production of entangled
pairs of the form (1.2). Suppose further that far from the hole (say for
$r>100\,r_{h}$) the physics decouples from the physics of the hole, by which
we mean: (i) the effective degrees of freedom $g_{ef\\!f}$ describing the hole
do not involve the degrees of freedom $g_{exact}$ at $r>100\,r_{h}$ and, (ii)
quanta in the exact theory that reach $r>100\,r_{h}$ are no longer influenced
significantly by the hole. _Then the entanglement entropy $S_{ent}$ will keep
rising monotonically in the exact theory, i.e. the Page curve of the exact
theory will not come down._ We will outline the derivation of this theorem
below, setting it up in the context of our present discussion. As we proceed
with this outline we will make clear the assumptions that go into the proof.

###  2.1 The proof of the effective small corrections theorem

We proceed in the following steps.

  1. (A)

The exact theory: First consider the exact theory. The black hole has a mass
$M$ as seen from infinity; let the classical black hole for this mass have
Schwarzschild radius $r_{h}$. We assume that far from the hole the exact
theory is just given by standard string theory around gently curved space
(‘normal physics’). For pedagogical convenience, let us say that this far-away
region is $r>100\,r_{h}$. We assume that the degrees of freedom in this far
region are independent of the degrees of freedom in the hole, just as is the
case in normal quantum field theory.

Figure 1: The gravitational mess. The blue region depicts the classical black
hole (within $r<r_{h}$) and the green region depicts ‘the region of the hole’
($r<10\,r_{h}$) where complicated quantum gravity effects may occur. Also
shown is the region far away ($r>100\,r_{h}$) where ‘normal physics’ occurs.

In and around the black hole the classical metric has a low curvature; this
may suggest that we should take the traditional picture of the hole where the
metric is given by (1.5) with $\bar{g}_{\mu\nu}$ the classical black hole
metric. However, we will not limit ourselves to such a semiclassical picture,
instead, allowing for the possibility that due to some unknown quantum gravity
effects, the entire region of the hole is a complicated quantum gravitational
mess; we depict this symbolically in fig. 1. For pedagogical concreteness we
let the ‘region of the hole’ be the region $r<10\,r_{h}$.

  2. (B)

Requiring an effective semiclassical description: Now consider the effective
theory. In the distant region $r>100\,r_{h}$ we will not define any effective
theory, since we have already assumed that the low-energy physics in the far
region is just low-energy string theory in gently curved space (‘normal
physics’). So the exact theory in the far region already has the behavior we
would want for any effective theory. In the region of the hole ($r<10\,r_{h}$)
the exact theory is very complicated. We assume that from the large number of
degrees of freedom describing the exact theory in this region, a small subset
can be used to describe dynamics that approximate the dynamics expected from
the semiclassical black hole. This subset is described by effective variables
$g_{ef\\!f}$ which are some complicated functionals of the exact degrees of
freedom $g_{exact}$

$g_{ef\\!f}=F[g_{exact}]\ .$ (2.1)

This small subset $g_{ef\\!f}$ is sometimes called the ‘code subspace’ which
captures semiclassical dynamics from all the complicated degrees of freedom in
the region. For pedagogical concreteness, we assume that the degrees of
freedom $g_{ef\\!f}$ describe the metric and a scalar field $\phi$ satisfying

$\square\phi\approx 0\ .$ (2.2)

The approximation sign here indicates that the effective semiclassical
description is only required to be approximate; we will be more explicit about
the accuracy of this approximation below. We will be quite generous in the
freedom which we allow for this effective theory. We require the effective
behavior (2.2) only for low-energy modes. Thus for a hole with
$r_{h}=3\,\rm{km}$, we can ask that (2.2) need only hold for wavelengths
between, say, $1\,\rm{cm}$ and $20\,\rm{km}$, the range where we need to
follow vacuum modes to see the emergence of particle pairs in the Hawking
computation. Further, we require that the effective variables $g_{ef\\!f}$
given by (2.1) describe the semiclassical dynamics (2.2) only for the duration
of production of a few Hawking pairs; after which, one may need a different
choice of effective variables $\tilde{g}_{ef\\!f}=\tilde{F}[g_{exact}]$ to get
the semiclassical dynamics. These are the minimum conditions that we need to
describe the requirement that there be some kind of effective semiclassical
behavior. Note that the only assumption we have made in these conditions is
that the effective degrees of freedom describing the region of the hole emerge
from the exact degrees of freedom in the region of the hole; thus in
particular they do not involve the exact degrees of freedom in the far region
$r>100\,r_{h}\,$.

  3. (C)

Pair production in the effective description: Given (2.2), we will have the
production of entangled pairs in the effective theory (fig. 2). We lose no
generality of the argument by taking a simple form for the state of the pair

$|\psi_{ef\\!f}\rangle_{pair}={1\over\sqrt{2}}\Big{(}|0\rangle_{b,ef\\!f}|0\rangle_{c,ef\\!f}+|1\rangle_{b,ef\\!f}|1\rangle_{c,ef\\!f}\Big{)}+O(\epsilon)\
.$ (2.3)

Here the $O(\epsilon)$ corrections encode the fact that the evolution (2.2)
was only approximate; we will specify the magnitude of these corrections more
precisely below. In the region immediately around the hole (say the region
$r<2\,r_{h}$) spacetime is curved and the definition of particles is somewhat
ambiguous. Once the quantum $b_{ef\\!f}$ gets far from the hole, the
definition of particles becomes well defined. We will use this latter fact to
remove part of the ambiguity from $b_{ef\\!f}$ in the step below. (There will
be no need to remove any ambiguity in the definition of particles from
$c_{ef\\!f}$.)

Figure 2: The left-hand figure depicts an entangled pair $\\{c,b\\}$ in the
exact theory. The right-hand side depicts the entangled degrees in the
effective theory, where the entangled pair emerges in the state
$|\psi_{ef\\!f}\rangle_{pair}$ as shown above (also mentioned in (2.3)).

  4. (D)

The movement of $b_{ef\\!f}$ away from the hole: The Hawking pair (2.3) is
created in the vicinity of the hole, say in the region $r\lesssim 10\,r_{h}$,
with the degrees of freedom in $b_{ef\\!f}$ then moving from this region
towards infinity. When these degrees of freedom reach the region
$r>100\,r_{h}$, they must be described as excitations of standard string
theory around gently curved space by the assumptions in (A) above (this is
depicted in fig. 3). That is, the effective degrees of freedom become some set
of degrees of freedom of the _exact_ theory. Let these exact degrees of
freedom resulting from $b_{ef\\!f}$ be denoted by $b$.

This step in the argument is very important, since it connects the effective
theory to the exact theory. If there were no such connection, then the
effective theory would likely be an irrelevant figment of our imagination.
Note that in step (A) we have assumed that the degrees of freedom in the far
region $r>100\,r_{h}$ are independent of the degrees of freedom in the region
of the hole $r<10\,r_{h}$. Thus the degrees of freedom making up $b$ will be
independent of the degrees of freedom in $r<10\,r_{h}$. We use this fact to
partially fix the ambiguity in the definition of the quantum $b_{ef\\!f}$,
which was noted in (C) above. We do this by choosing the definition of
$b_{ef\\!f}$ such that the _exact_ degrees of freedom giving rise to
$b_{ef\\!f}$ are independent of the _exact_ degrees of freedom making up
$c_{ef\\!f}$. This can always be done, since $b_{ef\\!f}$ turns into the
excitations of the exact theory that reach the far region, while $c_{ef\\!f}$
remains in the hole. We will denote by $c$ the exact bit that $b$ is entangled
with.

Figure 3:  Movement of $b_{eff}$ away from the hole. When these degrees of
freedom reach the region $r>100\,r_{h}$, they must be described as excitations
of standard string theory around gently curved space by the assumptions in
(A). This is shown in the figure by there being no difference between the
exact and effective theory for $r>100\,r_{h}$.

  5. (E)

Entanglements: We label the successive emissions from the hole by steps
$1,2,3,\dots$ and denote the quanta emitted at emission steps $1,2,\dots,N$ as
$\\{b_{1},b_{2},\dots,b_{N}\\}\equiv\\{b\\}$ (an ‘emitted’ quantum here is one
that has reached the region $r>100\,r_{h}$). Consider any subset $A$ of the
degrees of freedom for the _exact_ theory. We write $S(A)$ for the
entanglement entropy of the degrees of freedom in $A$ with the rest of the
degrees of freedom of the exact theory. Then the entanglement entropy $S_{N}$
of the radiation with the remaining hole in the exact theory after $N$
emissions is

$S_{N}=S(\\{b\\})\ .$ (2.4)

We are now interested in the next step of the emission (the $(N+1)$th step).
It is convenient to break this emission process into two steps: (E)(a) the
process in (C) where a pair is created in the effective theory in the region
of the hole $r<10\,r_{h}$; (E)(b) the process in (D) where the degrees of
freedom in $b_{ef\\!f}$ move to the far region $r>100\,r_{h}$. Let us consider
the entanglement entropy in these steps.

    1. (a)

Here the pair (2.3) is created in the region of the hole $r<10\,r_{h}$ and it
is assumed that the far region decouples from the region of the hole. Thus
this process of pair creation must be given by a unitary transformation of the
exact degrees of freedom in the region $r<10\,r_{h}$. The entanglement of the
far region with the region of the hole cannot change in this process.666If two
parts of a system are entangled, and we make a unitary action on one part, the
entanglement between the two parts does not change. Thus after this $(N+1)$th
step of pair creation in the region of the hole the entanglement of the exact
degrees of freedom in the far region with exact degrees of freedom in the
region of the hole is still $S_{N}$. The entanglement of the newly created
quanta $\\{b_{N+1},c_{N+1}\\}$ are such that the leading order part of the
state (2.3) of the effective quanta gives

$S(b_{N+1,eff}+c_{N+1,eff})=0\ ,\quad S(c_{N+1,eff})=\ln 2\ .$ (2.5)

The entanglement of the exact degrees of freedom corresponding to these
excitations must then satisfy

$S(b_{N+1}+c_{N+1})<\epsilon_{1}\ ,\quad S(c_{N+1})>\ln 2-\epsilon_{2}\ ,$
(2.6)

for some $\epsilon_{1},\epsilon_{2}\ll 1$. The relation (2.6) finally
specifies the magnitude of the small corrections that we have mentioned in the
above steps.777The steps relating the corrections to the state of the pair to
$\epsilon_{1}$ and $\epsilon_{2}$ are discussed in [14].

    2. (b)

The degrees of freedom that give rise to the new radiation quantum $b_{N+1}$
move out to the far region. The value of the entanglement entropy at this
emission step is then, by definition, given by

$S_{N+1}=S(\\{b\\}+b_{N+1})\ ,$ (2.7)

since now $b_{N+1}$ has joined the earlier quanta $\\{b\\}$ in the outer
region $r>100\,r_{h}$.

  6. (F)

The inequality: We now recall the strong subadditivity relation

$S(A+B)+S(B+C)\geq S(A)+S(C)\ .$ (2.8)

Here $A,B,C$ are three subspaces made from degrees of freedom that are
independent of each other. We set

$A=\\{b\\}\ ,\quad B=b_{N+1}\ ,\quad C=c_{N+1}\ ,$ (2.9)

and from (2.8) we get

$S(\\{b\\}+b_{N+1})+S(b_{N+1}+c_{N+1})\geq S(\\{b\\})+S(c_{N+1})\ .$ (2.10)

From (2.6) and (2.7) this can be written as

$S_{N+1}>S_{N}+\ln 2-(\epsilon_{1}+\epsilon_{2})\ .$ (2.11)

Thus for $\epsilon_{1},\epsilon_{2}\ll 1$, the entanglement entropy keeps
growing monotonically with the number of emission steps; it does not behave
like the entanglement for a normal body which rises till the halfway point of
evaporation and then falls back to zero.

In brief, as a result of the steps (A)-(F), the effective small corrections
theorem says the following. Suppose that the far region decouples from the
region of the hole (as is the case for the burning away of a piece of coal),
then if semiclassical dynamics (2.2) emerges in any effective description, it
will force the Page curve of the _exact_ theory to keep rising monotonically.

##  3 Some definitions

In this section we summarize the meaning of some terms that will be used in
the discussions of later sections.

###  3.1 The classical black hole

The classical Schwarzschild hole in 3+1 dimensions is given by the metric

$ds^{2}=-\Big{(}1-{r_{h}\over r}\Big{)}dt^{2}+{dr^{2}\over 1-{r_{h}\over
r}}+r^{2}\big{(}d\theta^{2}+\sin^{2}\\!\theta\,d\phi^{2}\big{)}\ ,$ (3.1)

with $r_{h}=2GM$. These Schwarzschild coordinates describe only the exterior
of the horizon $r>r_{h}$. To see both the outside and inside of the horizon in
a common coordinate patch, we can use the Eddington-Finkelstein coordinate

$u=t+r^{*}=t+r+r_{h}\log\Big{(}{r\over r_{h}}-1\Big{)}\ ,$ (3.2)

in which the metric (3.1) becomes

$ds^{2}=-\Big{(}1-{r_{h}\over
r}\Big{)}du^{2}+2dudr+r^{2}\big{(}d\theta^{2}+\sin^{2}\\!\theta\,d\phi^{2}\big{)}\
.$ (3.3)

We can start from flat space and form the black hole by sending in a shell of
energy $M$ composed of radially infalling massless particles. The Penrose
diagram for the corresponding classical hole is given in fig. 4. If we just
analytically continue the metric (3.1) as far as it can be continued, we find
the eternal hole whose Penrose diagram is depicted in fig. 4. This ‘eternal’
hole has a singularity in the past (bottom) quadrant and a second asymptotic
infinity in the left quadrant. Thus the eternal hole does not correspond to a
physical situation that we can create in the lab.

Figure 4: The Penrose diagrams for a classical black hole forming from an
infalling null shell and for an eternal black hole. The blue dotted line
represents the classical horizon. Each asymptotic region of the eternal black
hole sees a future and past horizon.

###  3.2 The semiclassical black hole

We can study the process of black hole formation and evaporation using ‘good
slices’; i.e. slices that are smooth and pass only through regions where the
curvature is low (i.e. the Ricci scalar ${\mathcal{R}}\ll l_{p}^{-2}$). This
is important because if we were forced to have slices that passed through a
singularity, then we could not be sure of how quantum fields evolve past this
singularity. The good slices in Eddington-Finkelstein coordinates are depicted
in fig. 5.

Figure 5: Good slices of a classical black hole. Asymptotically these are
constant-time slices, while inside the horizon time and space switch roles, so
they become constant-$r$ slices. These segments are then linked smoothly. The
good slices do not pass close to the singularity.

Outside the horizon $r=r_{h}=2GM$ a spacelike slice can be taken as
$t=\bar{t}$ for some constant $\bar{t}$. Inside the horizon space and time
interchange roles and a spacelike slice can be taken as $r=\bar{r}$ for some
constant $\bar{r}$. As a concrete example we may take $\bar{r}={r_{h}\over
2}=GM$, so that this part of the slice is neither near the horizon nor near
the singularity. The inside and outside parts of this spacelike slice can be
joined by a smooth ‘connector’ segment. To move to a later slice we can
advance the $t=\bar{t}$ part of the slice to $t=\bar{t}+\Delta\bar{t}$. We
keep the shape of the connector part the same. We then join up with the
$r=\bar{r}$ part of the slice by making this part _longer_. This shows the
stretching of hypersurfaces that leads to the creation of particle pairs in
the region around the horizon (see appendix B for a bit model description of
this. Each time we advance the slice by $\Delta\bar{t}\sim r_{h}$, we create
$\sim 1$ particle pairs whose entangled state can be schematically represented
by (1.1).

The essential feature of a classical black hole is the existence of a horizon.
It is this horizon that allows a constant $r$ segment $r=\bar{r}$ to be
_spacelike_ rather than timelike. There are two aspects of this segment that
are important. Firstly, the part of the slice given by $r=\bar{r}$ can be made
arbitrarily long, while still staying within the black hole radius $r_{h}$.
Secondly, excitations on this part can have either sign of the energy $E$, as
measured from infinity. Given the above two aspects of the $r=\bar{r}$ segment
of the slice, we can make states inside the black hole as follows. We place
$n_{quanta}$ quanta of wavelength $\lambda\sim r_{h}$ along this segment, each
having a spin that can be $\uparrow$ or $\downarrow$. Let the proper distance
between quanta be $\sim r_{h}$. Furthermore, let alternating quanta have
energies that are of opposite signs (i.e. $E,-E,E,-E,\dots$). By choosing
different values for the spins, the number of states ($N_{states}$) on this
segment is given by

$N_{states}=2^{n_{quanta}}\ .$ (3.4)

By choosing the $r=\bar{r}$ part of the slice to be sufficiently long, we can
place an arbitrarily large number of such quanta and thus get an arbitrarily
large value for $N_{states}$. Thus the entropy

$S\equiv\log N_{states}\ ,$ (3.5)

can be made arbitrarily large and in particular we can make

$S>S_{bek}\ ,$ (3.6)

where $S_{bek}={A\over 4G}$, where $A$ is the area of the black hole horizon.
This is called the ‘bags of gold’ problem or the problem of ‘unbounded
entropy’: we can store an entropy in the hole which is arbitrarily larger than
the Bekenstein entropy $S_{bek}$.

The bags of gold problem is closely related to the Hawking puzzle. The
evaporation of the hole generates negative energy quanta on the $r=\bar{r}$
part of the slice, of the kind used in the above construction of states. If we
keep feeding the black hole with quanta of wavelength $\lambda\sim r_{h}$,
then these quanta end up on our $r=\bar{r}$ slice as the positive energy
quanta used in the above construction. The entanglement entropy $S_{ent}$ of
the black hole with its radiation can be made arbitrarily large and in
particular we can have

$S_{ent}>S_{bek}\ .$ (3.7)

###  3.3 Some definitions

If we include the backreaction of the negative energy quanta that fall inside
the hole, then the mass $M$ of the hole slowly decreases, and we reach the
endpoint of evaporation as $M\rightarrow 0$. One possibility at this stage is
that the hole evaporates away completely as far as the usual part of spacetime
$r\geq 0$ is concerned; i.e. the location $r=0$ returns to being a normal part
of spacetime with the vacuum state in its vicinity. But before the endpoint of
evaporation, the interior of the hole contained the matter which made the hole
as well as the negative energy quanta which fell into the hole in the
evaporation process. We can imagine that this interior region pinches off into
a ‘baby universe’ that is disconnected from the usual $r\geq 0$ part of
spacetime. This situation is depicted in fig. 6.

Figure 6: The disconnected baby universe. Before the endpoint of evaporation,
the black hole interior contains the matter which made the hole, as well as
the negative energy Hawking quanta from the evaporation process. One may
imagine that this interior region pinches off into a ‘baby universe’ that is
disconnected from the rest of the spacetime.

In some situations we will call some part of a spacelike slice an ‘island’.
With the good slices we have chosen, this island will be, roughly speaking,
the $r=\bar{r}$ segment of the spacelike slice. The exact location of the
upper endpoint of this island will be determined by an optimization process,
but this exact location will not be of importance for the physical argument.
The relevant aspect of the island will be that it contains the negative energy
members of the Hawking pairs (except perhaps for the final few, depending on
where the exact upper endpoint of the island is).

###  3.4 The Euclidean hole

The analytic continuation $t\rightarrow-i\tau$ converts the Schwarzschild
metric (3.1) to the metric of the Euclidean hole

$ds^{2}=\Big{(}1-{r_{h}\over r}\Big{)}d\tau^{2}+{dr^{2}\over 1-{r_{h}\over
r}}+r^{2}\big{(}d\theta^{2}+\sin^{2}\\!\theta\,d\phi^{2}\big{)}\ .$ (3.8)

The radial coordinate ranges over $r_{h}\leq r<\infty$ and the ‘Euclidean
time’ direction $\tau$ is taken to be compact, with $0\leq\tau<4\pi r_{h}$;
this period corresponds to the inverse temperature of the hole

$T^{-1}\equiv\beta=4\pi r_{h}=8\pi GM\ .$ (3.9)

The $r,\,\tau$ directions form a cigar whose tip lies at $r=r_{h}$; the metric
is smooth at this tip with the chosen periodicity of $\tau$. Note that the
Euclidean hole has no horizon or any region interior to a horizon. There is no
place where quanta can have negative energy as seen from infinity, and there
is no analogue of pair creation. Thus the Euclidean metric (3.8) does not
exhibit the ‘bags of gold’ problem or the problem of growing entanglement
entropy ($S_{ent}$). The metric (3.8) can however be thought of as a saddle
point for some path integral in the gravity theory.

###  3.5 (1+1)-dimensional gravity

Arguments in the wormhole paradigm have often been made with the help of two
dimensional gravity theories. In two dimensions, the Einstein action is
topological. For Euclidean signature, we have

${1\over 2\pi}\left(\int
d^{2}x\sqrt{g\,}\,R+2\int_{\partial}dy\sqrt{h\,}\,K\right)=\chi\ ,$ (3.10)

where $R$ is the 2-d bulk curvature, $K$ is the extrinsic curvature at
boundaries and $\chi$ is the Euler number. Varying such an action does not
determine the metric. We can get an action which does have stationary points
by including a scalar field $X$; we can also write this field as

$X=e^{-2\Phi}\ ,$ (3.11)

where $\Phi$ is the dilaton. The most general form of the action is then

$S=C\int d^{2}x\sqrt{-g\,}\left({1\over 2}RX-{1\over 2}U(X)(\nabla
X)^{2}+V(X)\right)\ .$ (3.12)

In terms of $\Phi$ this has the form

$S={1\over 2}C\int d^{2}x\sqrt{-g\,}\
e^{-2\Phi}\left(R-\tilde{U}(\Phi)(\nabla\Phi)^{2}+2\tilde{V}(\Phi)\right)\ .$
(3.13)

Two dimensional theories of this form can be obtained by dimensionally
reducing a $D>2$ dimensional Einstein gravity theory on the angular sphere; in
that case $X\sim r^{D-2}$.

In the CGHS model [18], Hawking radiation was computed with a particular
choice of the two-dimensional theory. They considered the action suggested by
the worldsheet action in string theory, and coupled this to $N_{f}$ free
scalar fields $f_{i}$, giving

$S_{CGHS}={1\over 2\pi}\int
d^{2}x\sqrt{-g\,}\left(e^{-2\Phi}\big{(}R+4(\nabla\Phi)^{2}+4\lambda^{2}\big{)}-{1\over
2}\sum_{i}(\nabla f_{i})^{2}\right)\ .$ (3.14)

Hawking radiation was computed for the scalar fields $f_{i}$ and the Page
curve was found to be monotonically rising, just as in Hawking’s original
computation. The important difference from Hawking’s computation is the
following. In the CGHS model, the gravity theory is fully quantum; we do not
make the assumption that the matter fields travel on a fixed curved space. The
reason that we anyway find a definite gravity background $\\{g,\Phi\\}$ is
that in two dimensions the gravity theory has no propagating degrees of
freedom, so the path integral over gravity variables can be gauge fixed to a
particular configuration of $g$ and $\Phi$. Thus the CGHS model tells us that
treating gravity quantum mechanically in 1+1 dimensions does not change
Hawking’s conclusion that the Page curve will keep monotonically rising as the
evaporation proceeds. In the CGHS model we do not take into account the
backreaction created by the negative energy quanta falling into the hole.

In the RST model [19] the action (3.14) was slightly modified to a form where
the backreaction could be easily computed. The hole can then be seen to
evaporate away as the radiation proceeds. The Page curve again keeps rising
till the endpoint of evaporation. The entanglement entropy of this radiation
was computed as a function of time in [21, 20].

From these computations in (1+1)-dimensional theories, one can see that the
monotonically rising nature of the Page curve should not depend on precisely
which action of the type (3.12) we take. One theory which has been considered
in recent computations is Jackiw-Teitelboim gravity (JT gravity). The action
of JT gravity plus a matter CFT is

${S_{0}\over 4\pi}\left(\int
d^{2}x\sqrt{-g\,}\,R+2\int_{\partial}dy\sqrt{h}\,K+{1\over 4\pi}\int
d^{2}x\sqrt{-g\,}\,X(R+2)+{1\over
2\pi}\int_{\partial}dy\sqrt{h}\,X_{b}K\right)+S_{CFT}\ .$ (3.15)

Here the first two terms are the topological terms (3.10); they have been
included since one may need to sum over different topologies in some Euclidean
computations. The term $S_{CFT}$ denotes matter fields that we add to the
gravity action; we take these matter fields to define a conformal field theory
(CFT). A particular example of this CFT could be one given by a set of free
scalar fields.

Note that all these two dimensional theories are ‘incomplete’ theories of
gravity in the following sense. Black holes in these theories have a
Bekenstein entropy $S_{bek}>0$ given in terms of the value of the field $X$
(or equivalently $\Phi$) at the horizon. However, the theories do not have the
necessary degrees of freedom to manifest $\exp[S_{bek}]$ orthogonal quantum
states to account for this entropy. String theory on the other hand is a
‘complete’ theory, since we expect that there are in fact $\exp[S_{bek}]$
states to account for the entropy. Consider, for example, the black hole in
4+1 noncompact dimensions studied in [25]. The Bekenstein entropy is
reproduced by counting the number of states of branes carrying the given mass
and charges. Note however that these states differ from each other in the
configurations of branes in the 5 _compact_ directions. If we dimensionally
reduce on these compact directions, then we cannot manifest the states
required to account for the entropy. Similarly, fuzzball microstates differ
from each other in the way the compact directions fiber over the noncompact
directions. If we consider the dimensionally reduced theory, then there will
be no fuzzballs; there will be a unique state for the black hole as indicated
by the no-hair theorems.

##  4 The Page curve - I: What topology change can and cannot do

It has been argued that using a simple theory of gravity like JT gravity, one
can deduce that the Page curve of a black hole must come down like the Page
curve of a normal body. But how can this be, when we have already noted in the
above section that the Page curve in simple (1+1)-dimensional theories of
gravity keeps rising montonically? As we will see below, the important step in
the recent computations will be that a new ‘prescription’ will be added to the
(1+1)-dimensional theory. This prescription will, in turn, be equivalent to
requiring a certain nonlocality in the _exact_ theory. Our task is to clarify
this nonlocality requirement.

Our discussion of the Page curve will span three sections, so we start by
summarize the points that we will make in these three sections:

  1. (1)

It has been argued that the recent Page curve computations are similar to the
Gibbons-Hawking computation of entropy in the following sense. In the Gibbons-
Hawking computation, a semiclassical computation is able to reproduce the
entropy, while the actual significance of this entropy as a count of states
will only emerge when we know the exact quantum gravitational structure of the
hole. But we will argue that the recent semiclassical Page curve computations
are _not_ similar to the Gibbons-Hawking computation in this way. In the
Gibbons-Hawking computation we start with a path integral of the exact quantum
gravity theory that should yield the entropy; this path integral is then
argued to have a semiclassical saddle point which we use. But in the Page
curve computations, we start with a quantity that is _not_ the entanglement
entropy that we wanted to compute. Instead, we modify the path integral for
the Rényi entropies by a ‘prescription’. It is this prescription that contains
the nonlocal effects that the wormhole paradigm must invoke in order to avoid
a monotonically rising Page curve.

  2. (2)

It has been argued that the prescription arises automatically when we take
into account the fact that in gravity one can have topology change. We will
argue that such is _not_ the case. We will see explicitly how topology change
affects the structure of the Hilbert space and the definition of the inner
product. We will then note that these effects of topology change do _not_
generate a link between different replica copies, either in the Euclidean
setting or in the Lorentzian setting.

  3. (3)

In the wormhole paradigm, one seeks to nevertheless introduce such links
between copies, arguing that they are a feature of the semiclassical Gibbons-
Hawking type of path integral that emerges as an approximation of the exact
theory. But here we must remember that the exact variables are related to the
effective semiclassical variables through the relation
$g_{ef\\!f}=F[g_{exact}]$ (eq.(1.11)), which then forces the behavior of all
other effective quantities through (1.12) and (1.13). Thus we cannot postulate
that the effective semiclassical theory will have links between copies if
there is no corresponding dynamics in the _exact_ theory that corresponds to
the prescription of introducing these links. We will not be able to identify
any clear postulate in the exact theory which can give the replica wormhole
prescription in the effective theory. In a later section we will look at some
models of nonlocality that have been proposed in the exact theory. We will see
that for models where the nonlocalities stay within the black hole interiors,
there is a loss of unitarity, while with models that have nonlocalities
involving also the region far from the hole, there is a violation of normal
low-energy physics far from the hole.

In the remainder of this section we will make a first pass at the role of
different topologies in Lorentzian and in Euclidean signature.

###  4.1 Topology change in (1+1)-dimensional gravity: the Lorentzian theory

It is sometimes said that the new ‘prescription’ in the (1+1)-dimensional
theory just takes into account the fact that we must allow topology change in
a gravity theory. We will see that such is not the case. It is true that in
the CGHS or RST computations the topology of the (1+1)-dimensional spacetime
was taken to be the trivial one, similar to the topology that Hawking assumed
in his (3+1)-dimensional computation. But we will also see that allowing
topology change in the black hole region will _not_ change Hawking’s
conclusion that the Page curve keeps monotonically rising. The reason for this
is that the effective small corrections theorem does not care about which
topologies contribute to the dynamics of the black hole interior.

One might argue that since we do not really know how quantum gravity behaves,
it is possible that there are new rules for amplitudes in quantum gravity,
which need not hold in non-gravitational quantum theories. But actually we do
know a lot about quantizing gravity, especially in the (1+1)-dimensional case.
The string worldsheet theory is a (1+1)-dimensional quantum gravity theory,
since we need to sum over both the quantum fields $X^{\mu}(\tau,\sigma)$ on
the worldsheet as well as the metric $g_{ab}(\tau,\sigma)$ on the worldsheet.
The worldsheet theories with central charge $c<1$ yield quantum gravity
theories that have been solved in multiple ways: through dynamical
triangulations [26, 27], in light cone gauge (KPZ) [28] and in conformal gauge
(DDK) [29, 30]. All these ways of studying 2-d quantum gravity include the
possibility of topology change. Let us therefore review from first principles
what topology change means in 1+1 dimensions and what constraints we have from
unitarity on such a theory. In this way we will understand what aspects of the
dynamics we can change and what we cannot.

Let us start with a very simple model; this model will have all the features
that we wish to highlight. In 1+1 dimensions, the spatial sections are
1-dimensional, thus we first consider a single line segment. Let this segment
have $N$ lattice points along it and on each lattice point we have a single
bit whose value can be $0$ or $1$. Thus there are $2^{N}$ states on this
segment; we label these states as

$|\psi_{i}^{(N)}\rangle\ ,\quad i=1,\dots,2^{N}\ .$ (4.1)

We can assume that the spacing between lattice points is the Planck length.

Now suppose this line segment described a (1+1)-dimensional cosmology at some
fixed time. The cosmology can expand, so that at some later time we have a
longer line segment. Since we have kept the spacing between lattice sites as
the Planck length, we will have $N^{\prime}>N$ lattice sites on this new
segment and thus a larger number of states allowed for the quantum bits on the
segment. At first this looks confusing, since the dimension of the Hilbert
space should not change during time evolution. But the answer is simple. Since
this is a theory of quantum gravity, the spacelike slice is described not only
by the quantum fields on it, but also by the metric on it. Our Hilbert space
consists of the union of the Hilbert spaces for segments of all different
lengths $N\geq 0$, with $2^{N}$ states $|\psi_{i}^{(N)}\rangle$ on each such
segment. The evolution can then take us from one quantum state on a segment of
one length $N$ to some quantum state of a segment of a different length
$N^{\prime}$. The transition Hamiltonian $H$ must satisfy

$\langle\psi_{j}^{(N^{\prime})}|H|\psi_{i}^{(N)}\rangle=\left(\langle\psi_{i}^{N}|H|\psi_{j}^{(N^{\prime})}\rangle\right)^{*}\
.$ (4.2)

That is, the amplitude for any given transition must be the complex conjugate
of the amplitude of the reverse transition. So, even though the metric on the
segment can change, we still have a well defined notion of unitarity.

This setup is sufficient to understand the quantum gravitational setting of
(1+1)-dimensional models like the CGHS model or the RST model, where spacetime
could stretch but not change topology. Now suppose we do wish to allow
topology change. What should we do? In our (1+1)-dimensional situation the
answer is simple: all we can do is to allow the possibility that a line
segment can break into two segments or two such segments can join to form one.
A basis of our Hilbert space now consists of the following configurations.
Each configuration has some number $k\geq 0$ of line segments with the $i$th
segment having length $N_{i}$ and $2^{N_{i}}$ possible states of the quantum
bits on the segment. Note that if two segments in a configuration have the
same length and the same configuration of quantum bits, then they are
indistinguishable, i.e. they are like two bosons of the same species. This
aspect will be important when we talk about baby universes later. The
Hamiltonian then gives transition amplitudes between these basis states, which
can be nonzero between states having a different number of line segments. The
amplitudes must still satisfy the analogue of (4.2)

$\langle\\{\psi_{j}^{(N^{\prime})}\\}|H|\\{\psi_{i}^{(N)}\\}\rangle=\left(\langle\\{\psi_{i}^{(N)}\\}|H|\\{\psi_{j}^{(N^{\prime})}\\}\rangle\right)^{*}\
,$ (4.3)

where we have used the symbol $\\{\psi_{i}^{(N)}\\}$ to denote a collection of
line segments. We have not specified what the transition amplitudes are and
choosing different values for these amplitudes will define different
(1+1)-dimensional quantum gravity theories. With our choice of matter field (a
single bit per lattice site) there are no other freedoms in defining the
overall _structure_ of the theory. In particular, we must satisfy (4.3) if we
wish to preserve unitarity. This is important to note since we will find that
in some of the recent models of black hole evaporation using baby universes,
the evolution chosen turns out to be _not_ unitary.

We have gone through these simple points in detail since there has been much
confusion about what postulates we can or cannot add to a quantum gravity
theory like JT gravity. The above discussion was in the Lorentzian theory,
where the black hole problem is actually defined. We will now turn to the
Euclidean theory, which has also been used to make indirect computations of
entanglement entropy. Again, our goal will be to understand the significance
of different postulates that we may or may not add to a theory like JT
gravity.

###  4.2 The small corrections theorem and topology change

Having seen the nature of topology change in gravity, we remark here on the
fact that the possibility of topology change in the black hole interior has no
effect on the derivation of the effective small corrections theorem. To see
this, consider the (1+1)-dimensional case in which we studied topology change
in section 4.1 above. Topology change could lead to a 1-dimensional spatial
segment to break into two segments, or two such segments could join to form
one. There are two possibilities to consider:

  1. (a)

Suppose this breaking of segments happens at the horizon and in such a way
that it invalidates the effective semiclassical dynamics at the horizon.
Furthermore, this breaking happens often enough that this effective
semiclassical description is invalidated for a significant fraction of emitted
quanta (i.e. for a fraction that is order unity). In that case the central
assumption of the wormhole paradigm is invalidated, since we do not have
semiclassical dynamics at the horizon in some effective variables. So we will
not consider this possibility further.

  2. (b)

Suppose the line segment breaks in two very occasionally, so that such a break
typically happens when the segment becomes very long. As a example, we may say
that such a break happens when the segment holds $\sim S_{bek}$ negative
energy quanta $\\{c\\}$. But we see that such a process of breaking segments
has no effect on the effective small corrections theorem, since its derivation
does not need to know what happens to the $c$ quanta that fall into the hole,
only that the evolution of the black hole region is unitary.

###  4.3 Using Euclidean path integrals for (1+1)-dimensional gravity

The physical gravity theory is the Lorentzian one, however, its Euclidean
continuation can be a useful tool to compute certain quantities; for instance,
the Gibbons-Hawking computation of $S_{bek}$. We will now consider another use
of the Euclidean path integral: as a means of generating entangled states
between two disconnected regions. We will then make the observation that this
kind of Euclidean path integral does _not_ imply that there is an interaction
between the two disconnected spaces. If we do argue for such an interaction,
then this interaction would be a _new_ postulate; i.e. not something that
follows from a Euclidean path integral in the gravity theory. This issue will
be relevant in our understanding of how JT gravity has been used in the recent
computations of the Page curve.

In the previous subsection we took our line segments to be open. We can
equally well consider closed loops; let us do that here since then the
corresponding Euclidean manifolds will be simpler in that they will not have a
boundary. Consider the space made of two disconnected circles; let the spatial
coordinate on these circles be $\sigma_{1}$ and $\sigma_{2}$ respectively. On
each circle let there be a free scalar field $X$. There is no interaction
Hamiltonian connecting these two circles, but the overall state $|\Psi\rangle$
of the system can have entanglement between the two circles, as in the case of
the generically nonfactorizable state

$|\Psi\rangle=\sum_{n}c_{n}|\psi_{1,n}\rangle|\psi_{2,n}\rangle\ ,$ (4.4)

where $|\psi_{1,n}\rangle$ and $|\psi_{2,n}\rangle$ are states on the circles
$1$ and $2$ respectively. Despite there being no interaction between the two
circles, we may introduce an interaction as an artificial technique to
generate the entangled state. Thus consider the scalar field on the Euclidean
cylinder of length $\tau$; the two ends of this cylinder are the two circles
$1$ and $2$ that we had started with. The path integral over $X$ on this
cylinder generates the entangled state on the two circles

$|\Psi\rangle_{thermal}=\sum_{n}e^{-\tau E_{n}}|E_{1,n}\rangle|E_{2,n}\rangle\
,$ (4.5)

where $|E_{1,n}\rangle$ and $|E_{2,n}\rangle$ are states of energy $E_{n}$ on
circles $1$ and $2$. While this generates a particular entangled state, more
general entangled states of the form (4.4) can be made by including higher
genus surfaces in place of the cylinder and/or adding operator insertions
$\hat{O}$ on this Euclidean cylinder; such possibilities are depicted in fig.
7(b).

The important point to note is that the Euclidean manifold generating the
entangled state in the above way is an artificial construction whose sole
purpose is to obtain the entangled state; this manifold joining the two
circles does not imply that there is an interaction Hamiltonian between the
two circles. Nevertheless, we will now modify the dynamics by adding a
‘prescription’ using the above kinds of Euclidean surfaces linking the two
circles to generate an interaction between the two circles.

Suppose we construct the state (4.5) using a path integral on the Euclidean
cylinder as described above. We can now evolve the state on each circle in
Lorentzian time, using the Hamiltonian of the free fields on the respective
circles. There is of course no interaction between the two circles in this
evolution.

Figure 7: The bottom and top of figure (a) depicts the generation of the
states $|\Psi\rangle$ and $\langle\Psi|$ respectively on the union of two
circles by Euclidean evolution (E). Under Lorentzian time evolution (L), these
states of their respective circles evolve independently; there is no
interaction. An interaction, shown by a horizontal Euclidean evolution, can
equally be written as operator insertions; one per circle, as in (b).

Now suppose we want to introduce an interaction between the circles. We wish
to introduce a cylinder that stretches from one circle to the other; one may
call this a ‘wormhole’. We cannot do this while staying in Lorentzian
signature, since it is not possible to put a continuous light cone structure
on a geometry of the kind in fig. 7(a) where the wormhole is a horizontal
cylinder connecting the two circles $1,2$. Thus we assume that the
prescription for the new interaction is as follows: (i) the Lorentzian
evolution changes to Euclidean on each circle, (ii) the Euclidean wormhole
joins these two Euclidean sections as in fig. 7(a) and, (iii) we return to
Lorentzian evolution on the two circles. If we wish to compute a complete
amplitude, we can take the inner product with the state (4.5) again; the
geometry giving this full amplitude is depicted in fig. 7(a).

We have now added a prescription that gives an interaction between the field
theories on the two circles $1,2$. Let us see what the nature of this
interaction is. In the physical problem of the black hole, one circle (say
circle $1$) will correspond to the space inside a black hole, while the other
circle (circle $2$) will describe the degrees of freedom far from the hole.
Since these two regions are far from each other, we should think of the
wormhole connecting them as ‘long’.

Figure 8: Two equivalent pictures where a wormhole linking two points
$z_{1},\,z_{2}$ on the plane can be thought of as operators inserted at these
two points in the computation of, for instance, correlation functions.

The wormhole interaction can instead be written as an operator that acts on
two Hilbert spaces: one on the worldsheet of the theory on circle $1$ and one
that acts of the worldsheet of the theory on circle $2$, as in fig. 7. We take
local coordinate patches $z_{1},z_{2}$ on the two worldsheets and let
$\hat{O}_{h_{i}}$ be a basis of operators in each patch. Then we can write the
effect of the wormhole interaction as an effective operator

$\hat{W}=|0\rangle_{1}\leavevmode\nobreak\ {}_{2}\langle
0|+\sum_{i}e^{-\beta^{\prime}h_{i}}|\hat{O}_{h_{i}}\rangle_{1}\leavevmode\nobreak\
{}_{2}\langle\hat{O}_{h_{i}}|\leavevmode\nobreak\ +\leavevmode\nobreak\ {\rm
Hermitian\leavevmode\nobreak\ conjugate}\ ,$ (4.6)

where $\beta^{\prime}$ governs the length of the wormhole and we have
separated the identity term and the contributions of higher dimension
operators (see fig. 8). The identity term gives no interaction between the two
circles. In the limit of a long wormhole, only operators $\hat{O}_{h_{i}}$
with low dimensions $h_{i}$ contribute significantly. An example of such a low
dimension operator is $\hat{O}=\partial X$, for which the effect of the
wormhole becomes

$\hat{W}\rightarrow e^{-\beta^{\prime}}\partial X(z_{1})\partial X(z_{2})\ .$
(4.7)

Let us look at the effect of this interaction on any of the components making
up the state (4.5). The state $|E_{1,n}\rangle$ contains excitations of the
form

$\hat{a}^{\dagger}_{1,k_{1}}\\!\cdots\hat{a}^{\dagger}_{1,k_{p}}|0\rangle_{1}\
,$ (4.8)

where the $\hat{a}^{\dagger}_{1,k_{i}}$ are creation operators on circle $1$
and $|0\rangle_{1}$ is the vacuum state on this circle. We have a similar
structure for the state $|E_{2,n}\rangle$ on circle $2$. In the operator (4.7)
we can expand each $\partial X$ in creation and annihilation operators on the
respective circles and by doing so we see that its action on (4.5) yields
terms of the following kind. One of the oscillator excitations
$\hat{a}^{\dagger}_{1,k}$ in (4.8) is annihilated by $\partial X(z_{1})$, and
an oscillator excitation $\hat{a}^{\dagger}_{2,k^{\prime}}$ is created on
circle $2$. Thus we can say that a particle vanishes from circle $1$ and
another particle appears on circle $2$. This is the nonlocal transport of
quanta that results from the prescription that we have added to the theory of
free scalar fields on circles $1$ and $2$.

###  4.4 Summary

Let us summarize what we have seen in this section. Firstly, considering the
Lorentzian theory. Whilst it is true that there can be topology change in
gravity, if we take a (1+1)-dimensional theory for instance then the role of
this topology change is well understood: all that can happen is that the
spacelike slice – which is a 1-dimensional manifold – can split into two
segments, or two such segments can join to form one. The rules for this
splitting and joining should ensure that the the evolution is unitary. We
understand these rules in many formalisms where (1+1)-dimensional gravity has
been studied and do not have the freedom to add arbitrary rules to the quantum
gravity theory on the grounds that we do not know what role topology change
should play.

Now consider Euclidean computations. Here we have to be careful about a new
issue: there is a ‘technical tool’ that we can use to generate an entangled
state between two noninteracting regions. This tool is a path integral over a
cylinder that connects two noninteracting circles. We must be careful to not
confuse this technical tool with a real interaction between the two
noninteracting circles. If we nevertheless postulate such Euclidean cylinders
between two circles imply actual interactions in the theory, then roughly
speaking such interactions are of the form where we take a quantum from one
circle and place it on the other circle (eq.(4.8)); one could call this a
wormhole-type interaction between otherwise noninteracting regions.

##  5 The Page curve - II: The prescription of replacing Rényi entropies by
new quantities

In section 5.1 we review the definition of entanglement entropy and then how
Rényi entropies are given by appropriate traces when the entanglement is
between one part of a spacelike slice and its complement in section 5.2. In
section 5.3 we see how a ‘prescription’ is used in the wormhole paradigm to
replace the Rényi entropy by a new quantity.888For other critiques on path
integral justifications of the island formula, see [31, 32]. In section 5.4 we
recall our discussion of section 4 about what topology change can do, and we
note that the above prescription does _not_ follow from considerations of
topology change.

###  5.1 Entanglement entropies: review of notation

Consider a quantum system in a pure state $|\Psi\rangle$. Suppose there is
some way to separate the degrees of freedom of this system into two sets,
which we call subsystem $A$ and subsystem $B$. The Hilbert space
${\mathcal{H}}$ is assumed then to decompose as

${\mathcal{H}}={\mathcal{H}}_{A}\otimes{\mathcal{H}}_{B}\ .$ (5.1)

If $|\psi_{i}\rangle$ and $|\chi_{j}\rangle$ are orthonormal bases of states
on subsystems $A$ and $B$ respectively, then we can write the full pure state
on $\mathcal{H}$ as

$|\Psi\rangle=\sum_{i,j}C_{ij}|\psi_{i}\rangle|\chi_{j}\rangle\
,\quad\sum_{i,j}|C_{ij}|^{2}=1\ .$ (5.2)

The inner product of the full system $A\cup B$ factorizes into inner products
on the subsystems. For instance, using the product states
$|\Psi_{1}\rangle=|\psi_{1}\rangle|\chi_{1}\rangle$ and
$|\Psi_{2}\rangle=|\psi_{2}\rangle|\chi_{2}\rangle$ we get

$\langle\Psi_{1}|\Psi_{2}\rangle=\langle\psi_{1}|\psi_{2}\rangle\langle\chi_{1}|\chi_{2}\rangle\
.$ (5.3)

For more general $|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ the inner product
is obtained from the above relation using linearity.

Consider the state $|\Psi\rangle$ in (5.2) and suppose we wish to trace out
subsystem $B$ to get a density matrix $\rho_{A}$ describing system $A$ (a
reduced density matrix). We first write the bra corresponding to the ket
$|\Psi\rangle$

$\langle\Psi|=\sum_{i^{\prime},j^{\prime}}C^{*}_{i^{\prime},j^{\prime}}\langle\psi_{i^{\prime}}|\langle\chi_{j^{\prime}}|\
,$ (5.4)

and then obtain the density matrix of the full system as

$|\Psi\rangle\langle\Psi|=\Bigg{(}\sum_{i,j}C_{ij}|\psi_{i}\rangle|\chi_{j}\rangle\Bigg{)}\Bigg{(}\sum_{i^{\prime},j^{\prime}}C^{*}_{i^{\prime},j^{\prime}}\langle\psi_{i^{\prime}}|\langle\chi_{j^{\prime}}|\Bigg{)}\
.$ (5.5)

Finally we trace out subsystem $B$, getting the reduced density matrix for
subsystem $A$ to be

$\rho_{A}=\Bigg{(}\sum_{i,j}C_{ij}|\psi_{i}\rangle\Bigg{)}\Bigg{(}\sum_{i^{\prime},j^{\prime}}C^{*}_{i^{\prime},j^{\prime}}\langle\psi_{i^{\prime}}|\Bigg{)}\delta_{jj^{\prime}}\leavevmode\nobreak\
=\leavevmode\nobreak\
\sum_{i,i^{\prime}}\Bigg{(}\sum_{j}C_{ij}C^{*}_{i^{\prime}j}\Bigg{)}|\psi_{i}\rangle\langle\psi_{i^{\prime}}|\
.$ (5.6)

Note that the partial trace in (5.6) is done using the inner product on the
Hilbert space ${\mathcal{H}_{B}}$

$\langle\chi_{j^{\prime}}|\chi_{j}\rangle=\delta_{j^{\prime}j}\ .$ (5.7)

If we use some other matrix in place of the identity matrix in (5.7) to
perform the partial trace, then we are _not_ computing the desired reduced
density matrix but some other quantity. It is important to note this fact,
since we will see that in the wormhole paradigm the prescriptions that are
added are equivalent to replacing (5.7) by a different matrix.

The entanglement entropy $S_{ent}(A)$ (also called the von Neumann entropy or
the fine grained entropy of subsystem $A$) is given by

$S_{ent}(A)=-\operatorname{Tr}\big{[}\rho_{A}\log\rho_{A}\big{]}=-\sum\lambda_{i}\log\lambda_{i}\
,$ (5.8)

where $\lambda_{i}$ are the eigenvalues of $\rho_{A}$. The Rényi entropies
$S_{n}(A)$ are defined by

$S_{n}(A)=-{1\over
n-1}\log\big{[}\operatorname{Tr}\rho_{\\!A}^{\,n}\big{]}=-{1\over
n-1}\log\Big{[}\sum_{i}\lambda_{i}^{n}\Big{]}\ .$ (5.9)

This family of quantities provides a useful way of obtaining the (generally
difficult to calculate) $S_{ent}$ via the analytic continuation to
$n\in\mathbb{R}$, followed by taking the limit $n\rightarrow 1^{+}$. If this
can be well defined, then we get

$\lim_{n\rightarrow 1^{+}}S_{n}(A)=S_{ent}(A)\ .$ (5.10)

To get a rough sense of what these measures of entanglement describe, consider
the maximally entangled state between subsystems $A$ and $B$

$|\Psi\rangle={1\over\sqrt{N}}\sum_{i=1}^{N}|\psi_{i}\rangle|\chi_{i}\rangle\
,$ (5.11)

from which the eigenvalues of the reduced density matrix $\rho_{A}$ are
$\lambda_{i}={1\over{N}}$ and we get

$S_{ent}(A)=-\sum_{i=1}^{N}{1\over N}\log{1\over N}=\log N\ .$ (5.12)

Therefore, $S_{ent}(A)$ measures the number of terms in the sum in (5.11). For
the Rényi entropies, we have

$\operatorname{Tr}[\rho_{\\!A}^{\,n}]=\sum_{i=1}^{N}{1\over N^{n}}={1\over
N^{n-1}}\ ,$ (5.13)

and so

$S_{n}(A)=-{1\over n-1}\log\Big{[}{1\over N^{n-1}}\Big{]}=\log N\ .$ (5.14)

Hence, in this maximally entangled case the Rényi entropies are the same as
$S_{ent}(A)$. The Rényi entropies are less useful as a description of
entanglement when the $\lambda_{i}$ are not all comparable to each other. If
one eigenvalue $\lambda_{1}$ is large while the other $N-1$ are all equal,
$S_{ent}$ reflects the largeness of $N$ while the $S_{n}$ saturate to a value
set by $\lambda_{1}$ and do not reflect the large entanglement encoded by the
other $\lambda_{i\neq 1}$.

###  5.2 Computing the Rényi entropies

We now set up the computation of the second Rényi entropy $S_{2}(A)$ for a
1-dimensional system and then see how this computation is modified by a
prescription in the wormhole paradigm.

Figure 9: A 1-d discrete system where the first $M$ segments make up the
subset $B$ and the last $N$ segments make up subset $A$. At the center of each
segment shown is a scalar field degree of freedom $X$; this is the matter
field on our 1-dimensional spacelike slice.

In fig. 9 we depict a 1-d system consisting of $M+N$ line segments. The first
$M$ segments are labeled by an index $j=1,\dots,M$ and the last $N$ segments
are labeled by $i=1,\dots,N$. At the center of each segment we place a scalar
field degree of freedom $X$; this is the matter field on our 1-dimensional
spacelike slice. When we will discuss the gravitational theory later, we will
think of the first $M$ segments as the gravitational region containing the
black hole, while the last $N$ segments will describe the spacetime away from
the hole, including the region near infinity.

Let us consider the last $N$ segments as defining a subsystem $A$ and the
first $M$ segments as describing a subsystem $B$. Our goal is to trace out
system $B$ to get a density matrix $\rho_{A}$ for system $A$. We start with a
pure state $|\Psi\rangle$ for the full system $A\cup B$. Similarly to (5.2),
we write this state as

$|\Psi\rangle=\sum_{i_{1},j_{1}}C_{i_{1}j_{1}}|\psi_{i_{1}}\rangle|\chi_{j_{1}}\rangle\
,\quad\sum_{i_{1},j_{1}}|C_{i_{1}j_{1}}|^{2}=1\ ,$ (5.15)

where $|\psi_{i_{1}}\rangle$ and $|\chi_{j_{1}}\rangle$ are orthonormal bases
for the states of subsystems $A$ and $B$ respectively. We now take a second
set of $M+N$ line segments, on which we place the state dual to $|\Psi\rangle$

$\langle\Psi|=\sum_{i^{\prime}_{1},j^{\prime}_{1}}C^{*}_{i^{\prime}_{1}j^{\prime}_{1}}\langle\psi_{i^{\prime}_{1}}|\langle\chi_{j^{\prime}_{1}}|\
.$ (5.16)

To trace over the subset $B$ we take the outer product
$|\Psi\rangle\langle\Psi|$ using (5.15) and (5.16), and act with the delta
function $\delta_{j_{1},j^{\prime}_{1}}$.

Figure 10: A schematic representation of a method for computing the quantity
$\operatorname{Tr}(\rho^{\,2}_{\\!A})$, displaying only the spatial slice on
which a state is defined. This is in place of the full Euclidean manifold over
which a path integral generates the given state on that slice. In (a), a
partial trace (represented by the blue arrow) over the $B$ subsets, results in
the reduced density matrix $(\rho_{A})_{i_{1}i^{\prime}_{1}}$. In (b) we
multiply two copies of $(\rho_{A})_{i_{1}i^{\prime}_{1}}$ and perform a trace
over the remaining indices to get $\operatorname{Tr}(\rho^{\,2}_{\\!A})$.

In the 1-dimensional slices depicted in fig. 10, this operation corresponds to
identifying the values of the field variable $X$ in the first $M$ segments of
the bra and the ket states and summing over all possible values, yielding the
reduced density matrix $\rho_{A}$ on the $N$ segments describing set $A$.
Expanding $\rho_{A}$ in a basis as
$\sum_{i_{1},i^{\prime}_{1}}(\rho_{A})_{i_{1}i^{\prime}_{1}}|\psi_{i_{1}}\rangle\langle\psi_{i^{\prime}_{1}}|$
we see that the state on the last $N$ segments of (5.15) gives the ket of the
density matrix while the state on the last $N$ segments of (5.16) gives the
bra of the density matrix.

Since our goal is to compute $S_{2}(A)$, we need a second copy of the density
matrix $|\Psi\rangle\langle\Psi|$ on $A\cup B$. To get this second copy of
$\rho_{A}$, we take one more set of $M+N$ line segments, with the states

$|\Psi\rangle=\sum_{i_{2},j_{2}}C_{i_{2}j_{2}}|\psi_{i_{2}}\rangle|\chi_{j_{2}}\rangle\
,\quad\langle\Psi|=\sum_{i^{\prime}_{2},j^{\prime}_{2}}C^{*}_{i^{\prime}_{2}j^{\prime}_{2}}\langle\psi_{i^{\prime}_{2}}|\langle\chi_{j^{\prime}_{2}}|\
,$ (5.17)

such that $\sum_{i_{2},j_{2}}|C_{i_{2}j_{2}}|^{2}=1$. To trace over the set
$B$ and get the second copy of $\rho_{A}$ we take $|\Psi\rangle\langle\Psi|$
and act with the delta function $\delta_{j_{2},j^{\prime}_{2}}$. Now we have
two copies of $\rho_{A}$, one with components
$(\rho_{A})_{i_{1}i^{\prime}_{1}}$ and the other with
$(\rho_{A})_{i_{2}i^{\prime}_{2}}$. To compute $\rho_{\\!A}^{\,2}$ we must act
with $\delta_{i^{\prime}_{1},i_{2}}$ and then to get
$\operatorname{Tr}[(\rho_{A})^{2}]$ we must further act with
$\delta_{i_{1},i^{\prime}_{2}}$. Collecting together all these steps
gives999We write this as explicitly as possible in order to make very clear
the difference once a prescription is introduced in the following subsection.

$\displaystyle\operatorname{Tr}\big{[}(\rho_{A})^{2}\big{]}=\Bigg{(}\sum_{i_{1},j_{1}}C_{i_{1}j_{1}}|\psi_{i_{1}}\rangle|\chi_{j_{1}}\rangle\sum_{i^{\prime}_{1},j^{\prime}_{1}}C^{*}_{i^{\prime}_{1}j^{\prime}_{1}}\langle\psi_{i^{\prime}_{1}}|\langle\chi_{j^{\prime}_{1}}|\Bigg{)}$
$\displaystyle\Bigg{(}\sum_{i_{2},j_{2}}C_{i_{2}j_{2}}|\psi_{i_{2}}\rangle|\chi_{j_{2}}\rangle\sum_{i^{\prime}_{2},j^{\prime}_{2}}C^{*}_{i^{\prime}_{2}j^{\prime}_{2}}\langle\psi_{i^{\prime}_{2}}|\langle\chi_{j^{\prime}_{2}}|\Bigg{)}$
$\displaystyle\qquad\times\Big{(}\delta_{j_{1},j^{\prime}_{1}}\delta_{j_{2},j^{\prime}_{2}}\Big{)}\Big{(}\delta_{i_{1},i^{\prime}_{2}}\delta_{i^{\prime}_{1}i_{2}}\Big{)}\
.$ (5.18)

###  5.3 The ‘prescription’

We have gone through these elementary steps in detail so that we can now state
the ‘prescription’ that will be used in the wormhole paradigm to modify the
above computation. This prescription makes the following replacement of the
first bracket in the second line of (5.2)

$\delta_{j_{1},j^{\prime}_{1}}\delta_{j_{2},j^{\prime}_{2}}\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\
\delta_{j_{1},j^{\prime}_{1}}\delta_{j_{2},j^{\prime}_{2}}+C\delta_{j_{1},j^{\prime}_{2}}\delta_{j_{2},j^{\prime}_{1}}\
,$ (5.19)

where $C$ is a constant that will be specified below101010See for example an
overview in [33]..

Figure 11: In the wormhole prescription, the computation of the second Rényi
entropy depicted in fig. 10 is modified by a term generated by the partial
traces show in blue on two copies of the total density matrix. This
corresponds to $(\operatorname{Tr}[\rho_{A}])^{2}$. A simple redefinition of
indices gives the right-hand side.

The indices of type $j$ run over the subsystem $B$ that we trace over, which
in the black hole context describes the gravitational region containing the
black hole. With a little relabelling of indices we can see that the effect of
the prescription (5.19) in the computation (5.2) can be written in terms of
the reduced density matrix on $A$ as

$\operatorname{Tr}\big{[}(\rho_{A})^{2}\big{]}\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\
\Big{(}\operatorname{Tr}\big{[}(\rho_{A})^{2}\big{]}+C\big{(}\operatorname{Tr}[\rho_{A}]\big{)}^{2}\Big{)}\
.$ (5.20)

Let us note the effect of this prescription on entanglement entropies. Suppose
the state $|\Psi\rangle$ has the form

$|\Psi\rangle={1\over\sqrt{k}}\sum_{i=1}^{k}|\psi_{i}\rangle|\chi_{i}\rangle\
,$ (5.21)

then one finds

$\operatorname{Tr}[\rho_{A}]=1\
,\quad\operatorname{Tr}\big{[}(\rho_{A})^{2}\big{]}={1\over k}\ .$ (5.22)

The second Rényi entropy $S_{2}(A)$ for this state is given by (cf. (5.9))

$S_{2}(A)=-\log\Big{[}\operatorname{Tr}\big{[}\rho_{\\!A}^{\,2}\big{]}\Big{]}=\log
k\ .$ (5.23)

The entanglement between $A$ and $B$ rises with increasing $k$ and so here
$S_{2}(A)$ correctly reflects this. However, with the above prescription we
get

$S_{2}^{prescription}(A)=-\log\Big{[}\operatorname{Tr}\big{[}\rho_{\\!A}^{\,2}\big{]}+C\,\big{(}\operatorname{Tr}[\rho_{A}]\big{)}^{2}\Big{]}=-\log\bigg{[}\left({1\over
k}+C\right)\bigg{]}\ .$ (5.24)

Suppose we take the constant $C$ to be

$C=e^{-S_{bek}}\ ,$ (5.25)

then we see from (5.24) that for $k\ll S_{bek}$ we get

$S_{2}^{prescription}(A)\approx\log k\ ,$ (5.26)

while for $k\gtrsim S_{bek}$ we get

$S_{2}^{prescription}(A)\approx S_{bek}\ .$ (5.27)

Thus $S_{2}^{prescription}(A)$ is a quantity that behaves like the usual Rényi
entropy for low amounts of entanglement, but saturates to the value $S_{bek}$
for large values of the entanglement. It is important to note that
$S_{2}^{prescription}$ is _not_ the original quantity $S_{2}$ that we were
supposed to compute. So what is the reason that we are computing
$S_{2}^{prescription}$? It has sometimes been said that the modification
$S_{2}\rightarrow S_{2}^{prescription}$ arises because we must take topology
change into account in a theory of gravity. We now argue that this is not the
case; topology change can indeed occur in gravity, but it does not imply the
replacement $S_{2}\rightarrow S_{2}^{prescription}$.

###  5.4 Topology change

Topology change may appear to be something mysterious, however, we have
already seen in section 4.1 how to handle topology change in (1+1)-dimensions.
The fundamental structure of the quantum theory is _not_ altered, as far as
the notion of Hilbert space, inner products and unitarity are concerned. In
the (1+1)-dimensional case we considered, the spatial sections simply did not
have to consist of only a single line segment but instead could be made of
multiple line segments, with the full Hilbert space being the union of these
different possibilities. There has to be an inner product on this Hilbert
space and the evolution has to be unitary with respect to this inner product.

Figure 12: Introducing topology change in the 1-d system, the first $M$
segments which up the subset $B$ can be broken into sub-segments. The last $N$
segments still make up the subset $A$. Again, $X$ represents the matter field
on our 1-dimensional spacelike slice.

In fact this structure of having multiple line segments is not peculiar to
gravity, We could consider the quantum dynamics of a 1-dimensional polymer,
which can break into multiple line segments. A similar Hilbert space and
evolution will emerge, though the details of the Hamiltonian will depend on
the physics of the system.

With this in mind, let us turn to the computation of the Rényi entropies. In
the discussion of section 5.2, the subregion $B$ represents the region with
gravity. Allowing topology change then only changes the fact that the states
$|\chi_{j}\rangle$ describing this subset $B$ consist not just of one segment
but a linear combination of states with different number of segments. Recall
that we had taken a line segment to be made up of a certain number of links
and at the center of each link we had placed a scalar degree of freedom $X$.
In a connected segment, the Hamiltonian will typically have an interaction
linking the scalar field values at neighboring links $\alpha,\alpha+1$ of the
form

$H^{int}_{\alpha,\alpha+1}={1\over
2}{(X_{\alpha}-X_{\alpha+1})^{2}\over\delta^{2}}\ .$ (5.28)

There will be no such term between links that are on different line segments;
this fact will tell us when we have topologically disconnected segments.
Crucially, any entanglement entropy is simply a function of the state
$|\Psi\rangle$ on a spacelike slice with the inner product being needed to
trace out the subsystem $B$. The Hamiltonian is, however, _not_ involved in
the computation of entanglement. In a constrained system the allowed states
may be subject to a Hamiltonian constraint and we will discuss this issue
below, but once we have a state $|\Psi\rangle$ that is a physical state for
the system $A\cup B$, then to compute a Rényi entropies $S_{n}(A)$ all we need
is the inner product on the system $B$.

But now we see immediately that topology change does not lead to any
prescription like (5.19). All that happens is that the states
$|\chi_{j}\rangle$ now span both single segment and multi-segment
possibilities. The states labeled by indices
$i_{1},j_{1},i^{\prime}_{1},j^{\prime}_{1}$ in (5.2) pertain to the first copy
of $\rho_{A}$ and the states labeled by
$i_{1},j_{2},i^{\prime}_{2},j^{\prime}_{2}$ pertain to the second copy of
$\rho_{A}$. We do not end up mixing these two copies of $\rho_{A}$ in a new
way a suggested by the prescription (5.19). It is true that gravity can allow
for topology change, but this just changes the structure of the Hilbert space,
without changing how a quantity like $S_{2}(A)$ is to be computed.

So let us ask: why was a prescription like (5.19) suggested? To understand the
answer to this question, we will now cast the above computation of $S_{n}(A)$
in the language of path integrals.

##  6 The Page curve - III: Path integrals and the difference between Rényi
and Gibbons-Hawking type computations

In section 6.1 we see how to recast states in terms of path integrals and
observe that allowing topology change in the gravity theory does _not_ give a
wormhole that should connect different replica copies. In section 6.2 we
recall the Gibbons-Hawking computation of entropy and observe that the Rényi
entropy-inspired computations using added prescriptions are fundamentally
different from the Gibbons-Hawking computation: with the Rényi entropy
computations we start with a path integral prescription that is not the
correct one for the Rényi entropy for a general quantum system, while in the
Gibbons-Hawking computation the starting point is the correct path integral
that should count all microstates.

Section 6.3 discusses further, in relation to the sewing procedure in 2-d
CFTs, how having ‘wormholes’ in a Euclidean picture has no relation to
interactions in the real, Lorentzian theory (and thus cannot answer a
Lorentzian problem such as that of black hole information loss).

In section 6.4 we note that there are computations which show that the Page
curve comes down, but these computations hold for _normal_ systems where the
Page curve would automatically come down by the standard computation of Page
[34]. In these systems there is no analogue of the effective pair production
(1.2), so these computations do not address the goal of the wormhole paradigm.

###  6.1 Expressing states through path integrals

As we have already noted, the computation of entanglement entropies is related
to the state $|\Psi\rangle$ on a spacelike slice, it does not involve the
dynamics of the system. Thus there is no natural connection between the
computation of a quantity like $S_{n}(A)$ and a path integral in the theory.
So why should we try to use path integrals?

In 2-d CFTs, the path integral has been useful in the computation of
entanglement in the following way. We often wish to compute the entanglement
of a subregion $A$ when the overall state $|\Psi\rangle$ on our spatial slice
is the _vacuum_ $|0\rangle$. In this case we can generate the state
$|\Psi\rangle=|0\rangle$ on our spacelike slice as follows. Working in
Euclidean signature a path integral over the lower half plane generates the
state $|0\rangle$ on the upper boundary of this half-plane. Let us call the
2-d manifold spanning the lower half-plane ${\mathcal{M}}_{1}$. We generate
the dual state $\langle\Psi|=\langle 0|$ by a similar path integral on an
upper half-plane, calling this manifold $\widetilde{\mathcal{M}}_{1}$. We then
perform the trace over the subset $B$ by joining ${\mathcal{M}}_{1}$ and
$\widetilde{\mathcal{M}}_{1}$ along the region representing $B$ (the
complement of $A$). The states on the edges of ${\mathcal{M}}_{1}$ and
$\widetilde{\mathcal{M}}_{1}$ that are in the segment $A$ then give the
density matrix $\rho_{A}$. If we wish to compute the entanglement for some
state other than $|\Psi\rangle=|0\rangle$, we can insert appropriate operators
in the manifolds ${\mathcal{M}}_{1}$ and $\widetilde{\mathcal{M}}_{1}$ to
alter the evolution.

But here we note that getting $\rho_{A}$ from a path integral in this way is
just a trick that makes the computation easier; the original definition of the
density matrix as a partial trace over the subset $B$ actually involves only
the state $|\Psi\rangle$ on the 1-dimensional slice. Thus we should be careful
to not modify the path integral computation in an arbitrary way through a
‘prescription’, since then we will not be actually computing the density
matrix $\rho_{A}$ or the entropies $S_{n}(A)$.

Figure 13: In (a) states $\psi_{i}$ and $\psi_{j}$ are shown on the circles
that they are defined on. In (b) the states $\psi^{\prime}_{i}$ and
$\psi^{\prime}_{j}$ evolve in Euclidean time to the states $\psi_{i}$ and
$\psi_{j}$ respectively in a 2-dimensional gravity theory, depicted as
cylinders. In (c) we allow the 2d gravity theory to have topology change,
which is shown as the addition of handles to the cylinders.

As an example, let us start with an ordinary CFT on a circle. In fig. 13(a) we
depict the bra and ket states $|\Psi_{i}\rangle$ and $\langle\Psi_{j}|$ for
such a CFT with the inner product between these states denoted by

$\langle\Psi_{j}|\Psi_{i}\rangle\ .$ (6.1)

How should we get this inner product from a path integral? In fig. 13(b), we
depict an _amplitude_ between states $|\Psi^{\prime}_{i}\rangle$ and
$\langle\Psi^{\prime}_{j}|$ which we write this

$\langle\Psi^{\prime}_{j}|\Psi^{\prime}_{i}\rangle_{amplitude}\ .$ (6.2)

But why should this amplitude (6.2) have anything to do with the inner product
(6.1)? If we expand the state $|\Psi^{\prime}_{i}\rangle$ into energy
eigenstates, then these different eigenstates evolve with different factors

$\sum_{n}C_{n}|E_{n}\rangle\rightarrow\sum_{n}C_{n}e^{-E_{n}\tau}|E_{n}\rangle\
.$ (6.3)

So the states $|\Psi^{\prime}_{i}\rangle$ and $\langle\Psi^{\prime}_{j}|$ in
the amplitude (6.2) will have to be different from the states
$|\Psi_{i}\rangle$ and $\langle\Psi_{j}|$ in the inner product (6.1). If we
evolve the states in the amplitude through a large time
$\tau\rightarrow\infty$, then we will end up with just the vacuum states at
the middle of the cylinder, as the coefficients of all higher energy states
are subleading.

But now consider not a simple CFT but a theory of gravity on our 2-d cylinder;
the string world sheet theory is an example of such a gravity theory. The
physical states then all have the same energy since they satisfy the momentum
and Hamiltonian constraints (in terms of the stress tensor modes
$L_{0},\bar{L}_{0}$)

$(L_{0}-1)|\Psi\rangle=0\ ,\quad(\bar{L}_{0}-1)|\Psi\rangle=0\ .$ (6.4)

Thus any state from the physical Hilbert space does not suffer a change in the
relative weights of its parts as it evolves down the cylinder and after
absorbing a suitable power of $e^{\tau}$, we can identify
$|\Psi^{\prime}_{i}\rangle$, $\langle\Psi^{\prime}_{j}|$ with
$|\Psi_{i}\rangle$, $\langle\Psi_{j}|$. Thus the inner product (6.1) in this
case can be written as an amplitude of the type (6.2).

Now suppose the 2-d gravity theory describing our cylinder is allowed to have
topology change. This means that the 1-dimensional circle describing the
spatial sections of the cylinder can break up into two such circles and vice
versa. The evolution of the state on the cylinder can now have handles, as
depicted in fig. 13(c). Note that the physical quantity that we need for the
definition of entanglement entropy is an inner product like (6.1); recasting
this as an amplitude (6.2) is just something that we have done for our present
purposes.

Now we come to the crucial step. Suppose we wish to compute the second Rényi
entropy $S_{2}(A)$. For this purpose we create two copies of the bra/ket pair,
as in fig. 14(a), and identify the segments along the subset $B$ of each
bra/ket pair. The further traces in the second line of (5.2) then give
$\operatorname{Tr}[(\rho_{A})^{2}]$, which we then use in (5.9) to compute
$S_{2}(A)$. But now suppose someone wishes to rewrite this computation using
path integrals. The bra and ket states for each copy are created by a time
evolution, similar to the evolution we did for the string world sheet to write
(6.1) as an amplitude (6.2).

Figure 14: In (a) states are shown on their respective circles. In (b) the
states evolve in Euclidean time in a 2-dimensional gravity theory, with all
topology changes allowed. This is shown as cylinders with handles. In (c) we
show handles which surpass the cylinders and join two different cylinders.
This is forbidden, as we argue in text.

This time evolution can have handles on the 2-d worldsheet if the Hamiltonian
allows for topology change, _but there will be no handles that connect the 2-d
manifolds of one bra/ket pair to the manifolds for the other bra/ket pair._
This is because the time evolution over these 2-d manifolds was just a trick
to reproduce the inner products needed for the computation of
$\operatorname{Tr}[(\rho_{A})^{2}]$ and in this trace the only delta functions
are the ones given in (5.2). If we use a path integral to reproduce this
result for $\operatorname{Tr}[(\rho_{A})^{2}]$, then we can only have those
handles that arise in the evolution that gives rise to the states in fig. 14;
we cannot include any other kinds of handles at will.

To summarize, we have seen that entanglement entropies are a property of a
given state; the dynamics of the theory is not involved. There is no natural
appearance of a path integral in the computation of entanglement entropies,
since the path integral describes the dynamical evolution of states. We can
however use the path integral as a trick to recast the computation of a
quantity such as Rényi entropy. But while this path integral may manifest
handles within the computation of each copy of the density matrix, there are
no handles between different copies; i.e. we find no analogue of a ‘replica
wormhole’.

###  6.2 The Gibbons-Hawking computation

Let us consider the Gibbons-Hawking computation of entropy, starting from
first principles; this will allow us to see the differences between the
Gibbons-Hawking computation and the Euclidean arguments for the Page curve.
The Gibbons-Hawking argument proceeds in the following steps:

  1. (A)

First consider _any_ quantum system, not necessarily one with gravity. We