The Negative Energy in Generalized Vaidya Spacetime

Vitalii Vertogradov

Herzen State Pedagogical University, St. Petersburg, Russia ; The SAO RAS,
Pulkovskoe shosse 65, 196140, Saint-Petersburg, Russia.

Correspondence<EMAIL_ADDRESS>Tel.: +7(930)006-15-85

Abstract In this paper we consider the negative energy problem in generalized
Vaidya spacetime. We consider several models when we have the naked
singularity as a result of the gravitational collapse. In these models we
investigate the geodesics for particles with negative energy when the II type
of the matter field satisfies the equation of the state $P=\alpha\rho$
($\alpha\in[0\,,1]$).

Keywords: Generalized Vaidya spacetime; Naked singularity; Black hole;
Negative energy

## Introduction

In 1969 R. Penrose theoretically predicted the effect of the negative energy
formation in Kerr metric during the process of the decay or the collision.
Further the nature of the geodesics for particles with negative energy was
investigated [1, 2]. It was shown that in the ergosphere of a rotating black
hole closed orbits for such particles are absent. This geodesics must appear
from the region inside the gravitational radius. Also there was research
devoted to the particles with negative energy in Schwarzschild spacetime which
was conducted by A.A.Grib and Yu.V.Pavlov [3]. They showed that the particles
with negative energy can exist only in the region inside of the event horizon.
However, the Schwarzschild black hole is the eternal one and we must consider
the gravitational collapse to speak about the past of the geodesics for
particles with negative energy.

Black hole was considered to be the only result of critical gravitational
collapse. P. Joshi [4] showed that the result of the gravitational collapse
might be the naked singularity (For detailed information see [5, 20]). It
means that during the gravitational collapse process the time of the
singularity formation is less than the time of the apparent horizon formation
and there exists a family of non-spacelike, future-directed geodesics which
terminate at the central singularity in the past. M. Mkenyley et al
investigated the question about the gravitational collapse of generalized
Vaidya spacetime [6] and showed that the result of this collapse might be the
naked singularity. Further the condition for the mass function were obtained
[7, 8]. Vaidya spacetime is the one of the earliest examples of cosmic
censorship conjecture violation [9]. Usual Vaidya spacetime has the following
form:

$ds^{2}=-\left(1-\frac{2M(v)}{r}\right)dv^{2}+2\varepsilon
dvdr+r^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2})\,,$ (1.1)

where $v$ represents time and $\varepsilon=\pm 1$ depending on ingoing or
outgoing radiation. Vaidya spacetime is known as radiating Schwarzschild
spacetime. Here $M$ is the function of time and if the mass doesn’t depend on
time we have the Schwarzschild spacetime in which the time axis represents the
radial null geodesic. If $M$ depends not only on $v$ but also on $r$ then we
have generalized Vaidya spacetime.

In usual Vaidya spacetime we have non-zero right hand-side of the Einstein
equation. The matter is the type I of the matter field and represents so-
called null dust. The energy momentum tensor has the following form:

$T_{ik}=\mu L_{i}L_{k}\,,$ (1.2)

where $\mu$ is the energy density of the null dust and $L_{i}$ is a null
vector:

$L_{i}=\delta^{0}_{i}\,.$ (1.3)

The properties of generalized Vaidya spacetime will be described below.

So if we have the naked singularity formation then we have the question about
the negative energy in this case. There is no negative energy in our universe
so in the case of naked singularity formation this particles must be
forbidden. We consider one explicit model of Vaidya-Anti-de Sitter spacetime
when we have the eternal naked singularity and then we consider the negative
energy problem in generalized Vaidya spacetime when the type-II of the matter
field satisfies the equation of the state $P=\alpha\rho$ where
$\alpha\in[0\,,1]$.

This paper is organized as follows: In sec. 2 we briefly discuss methods which
we use preparing this paper, in sec. 3 we describe the generalized Vaidya
spacetime. In sec. 4 we consider the negative energy problem in Vaidya-Anti-de
Sitter spacetime. In sec. 5 we consider this problem in the generalized Vaidya
spacetime. Sec. 6 is the Discussion.

The system of units $G=c=1$ is used throughout the paper. Also latin indices
take values $0\,,1\,,2\,,3$ and greek indices - $1\,,2\,,3\,.$

## Geodesic Equation,
Energy and Angular Momentum.

In this section we provide necessary method to conduct this research.

This paper is devoted to the geodesics for particles with negative energy. So
first of all we should provide the necessary methods to find the geodesic
equations and the energy expressions.

The geodesic equation in metric $g_{ik}$ is given by:

$\frac{d^{2}x^{i}}{d\lambda^{2}}+\Gamma^{i}_{jk}\frac{dx^{j}}{d\lambda}\frac{dx^{k}}{d\lambda}=0\,,$
(2.4)

where $\Gamma^{i}_{jk}$ are Cristoffel symbols:

$\Gamma^{i}_{jk}=\frac{1}{2}g^{im}\left(g_{jm,k}+g_{mk,j}-g_{jk,m}\right)\,,$
(2.5)

here sign comma denotes partial derivative:

$g_{ik,j}=\frac{\partial g_{ik}}{\partial x^{j}}\,,$ (2.6)

and $g^{ik}$ is contravariant components of metric tensor:

$g^{ik}=\frac{g_{IK}}{|g|}\,,$ (2.7)

where $g_{IK}$ is the cofactor of the matrix $g_{ik}$ and $|g|$ is the
determinant of the matrix $g_{ik}$.

The equations (2.4) is the second order partial differential equations. But
for our purpose it is better to use the first order equations. For this aim we
use the lagrangian which is given by:

$\mathcal{L}=\frac{1}{2}g_{ik}\frac{dx^{i}}{d\lambda}\frac{dx^{k}}{d\lambda}\,.$
(2.8)

We will consider the spherical symmetry spacetime. So in our case we can put
$\theta=\frac{\pi}{2}$ and use only three coordinates $\\{v\,,r\,,\varphi\\}$.
Here $v$ is the time. We can obtain the energy and angular momentum
expressions using (2.8) i.e.:

$\begin{split}-E=\frac{\partial\mathcal{L}}{\partial\dot{v}}=g_{00}\dot{v}+g_{0\alpha}\dot{x}^{\alpha}\,,\\\
L=\frac{\partial\mathcal{L}}{\partial\dot{\varphi}}=g_{22}\dot{\varphi}+g_{2\alpha}\dot{x}^{\alpha}\,,\end{split}$
(2.9)

here dot sign denotes partial derivative with respect to affine parameter
$\lambda$. The expressions (2.9) gives only two equations. To solve this
system we need one more equation. We can use the following equation:

$g_{ik}\frac{dx^{i}}{d\lambda}\frac{dx^{k}}{d\lambda}=\delta\,,$ (2.10)

where $\delta=+1\,,-1\,,0$ denotes timelike, spacelike or null geodesic
respectively.

The proof that (2.9) and (2.10) are the geodesic equations and correspond to
(2.4) the reader can see in [21].

We have one important condition - the particles must move in future in time
i.e.:

$\dot{v}=\frac{dv}{d\lambda}>0\,.$ (2.11)

If we satisfy this condition then we can note that if $g_{0\alpha}=0$ in (2.9)
then we don’t have the negative energy because $-g_{00}$ is always positive.
If we consider $g_{00}>0$ then this situation corresponds to the black hole
and the particles with negative energy can exist only in the region inside the
event horizon.

Thus to have the particles with negative energy outside the event horizon the
metric must have off-diagonal term $g_{0\alpha}\neq 0$. In this paper we
consider the generalized Vaidya spacetime which contains off-diagonal term
$2dvdr$. So in this case we might expect the particles with negative energy to
exist outside the apparent horizon.

As we said above we want to prove the absence of particles with negative
energy outside the apparent horizon or in the case of the naked singularity
formation in the case of generalized Vaidya spacetime.

We have the naked singularity formation as a result of the continuous
gravitational collapse if the time of the apparent horizon formation is more
than the time of the singularity formation. Moreover there must exist a family
of non-spacelike, future-directed geodesics which terminate at the central
singularity in the past.

## The Generalized Vaidya Spacetime

The generalized Vaidya spacetime is widely used to describe the model of
gravitational collapse [6, 12, 7, 8, 13, 14] and the exterior metric of the
radiating stars [15, 16, 17, 18].

The generalized Vaidya spacetime has the following form [10]:

$\begin{split}ds^{2}=-\left(1-\frac{2M(r,v)}{r}\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}\,,\\\
d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}\,,\end{split}$ (3.12)

here $M(v,r)$ \- the mass function depending on coordinates $r$ and $v$ which
corresponds to advanced/retarded time.

The generalized Vaidya spacetime differs from usual Vaidya spacetime only by
dependence of the mass function on both $r$ and $v$. The right hand-side of
the Einstein equations is the mixture of two matter fields type -I and -II.
Like in Vaidya spacetime case the type-I of matter field is purely the null
dust but the type-II is cosmic strings.

We can write down the energy momentum tensor in the following form:

$T_{ik}=T^{(n)}_{ik}+T^{(m)}_{ik}\,,$ (3.13)

where the first term corresponds to the matter field type-I and the other one
corresponds to the matter field type-II [19].

Now let us write down the expression of the energy momentum tensor: [10]

$\begin{split}T^{(n)}_{ik}=\mu L_{i}L_{k}\,,\\\
T^{(m)}_{ik}=(\rho+P)(L_{i}N_{k}+L_{k}N_{i})+Pg_{ik}\,,\\\
\mu=\frac{2\dot{M}}{r^{2}}\,,\\\ \rho=\frac{2M^{\prime}}{r^{2}}\,,\\\
P=-\frac{M^{\prime\prime}}{r}\,,\\\ L_{i}=\delta^{0}_{i}\,,\\\
N_{i}=\frac{1}{2}\left(1-\frac{2M}{r}\right)\delta^{0}_{i}-\delta^{1}_{i}\,,\\\
L_{i}L^{i}=N_{i}N^{i}=0\,,\\\ L_{i}N^{i}=-1\,.\end{split}$ (3.14)

here $P$ \- pressure, $\rho$ \- density of the II matter field, $\mu$ \- the
density of the null dust and $L,N$ \- two null vectors.

This model must be physically reasonable so the energy momentum tensor should
satisfy weak, strong and dominant energy conditions [11]. It means that $\rho$
must be positive and for any non-spacelike vector $v^{i}$:

$T_{ik}v^{i}v^{k}>0\,,$ (3.15)

and the vector $T_{ik}v^{i}$ must be timelike. Weak energy conditions also
demands that the energy density must be non-negative I.e.

$\rho>0\,,\mu>0\,.$ (3.16)

The dominant energy condition means that the energy density should not be less
than the pressure. If we violate the dominant energy condition then our matter
moves along spacelike geodesics which is unphysical.

Strong and weak energy conditions demand:

$\begin{split}\mu\geq 0\,,\\\ \rho\geq 0\,,\\\ P\geq 0\,.\end{split}$ (3.17)

The dominant energy condition imposes following conditions on the energy
momentum tensor:

$\begin{split}\mu\geq 0\,,\\\ \rho\geq P\geq 0\,.\end{split}$ (3.18)

The properties of generalized Vaidya spacetime has been studied for the
equation of the state $P=\alpha\rho$ where $\alpha$ belongs to the interval
$[0\,,1]$ in articles [7, 8]. If we satisfy this equation of the state then
the mass function $M(r,v)$ has the form

$M(r,v)=C(v)+D(v)r^{1-2\alpha}\,,$ (3.19)

where $C(v)$ and $D(v)$ are an arbitrary functions of time $v$. It is also
worth noting that the case $\alpha=\frac{1}{2}$ is the special one and we
don’t consider it here.

When we speak about the naked singularity formation then it means that the
time of the singularity formation is less than the time of the apparent
horizon formation. And there is a family of non-spacelike future-directed
geodesics which terminate at the central singularity in the past. The
singularity forms at the time $v=0$ at $r=0$. So now we should know the
equation of the apparent horizon.

To derive the apparent horizon equation we need a tangent vector $u^{i}$:

$u^{i}=-\left(1-\frac{2M(v,r)}{r}\right)\frac{d}{dv}+2\frac{d}{dr}\,.$ (3.20)

Vector $u^{i}$ tangent to a congruence of outgoing null geodesies. However it
doesn’t satisfy the geodesic equations

$u_{i;k}u^{k}=\xi u_{i}\neq 0\,,$ (3.21)

where sign ’;’ denotes the covariant derivative and

$\xi=2\frac{M(v,r)-rM^{\prime}(v,r)}{r^{2}}\,.$ (3.22)

We can’t calculate the expansion of the outgoing null geodesies. First of all
we need to use an affine parameter $\dot{\lambda}$ and rescaled tangent vector
$k^{i}$:

$k^{i}=e^{-\gamma}u^{i}\,,$ (3.23)

where

$\frac{\partial\gamma}{\partial\lambda}=\xi(\lambda)\,,$ (3.24)

where $\lambda$ is an original affine parameter. To calculate the expansion
$\Theta$ which is

$\Theta=k^{i}_{;i}\,,$ (3.25)

now using (3.22), (3.23) and (3.25) we can write down the expansion
expression:

$\Theta=2\frac{e^{-\gamma}}{r^{2}}\left(1-\frac{2M(v,r)}{r}\right)\,.$ (3.26)

Note that the term $2\frac{e^{-\gamma}}{r^{2}}$ doesn’t have impact on the
sign of $\Theta$. Finally we have the apparent horizon equation

$1-\frac{2M(v,r)}{r}=0\,.$ (3.27)

## The Negative Energy in Vaidya-Anti-de Sitter spacetime

First of all, we decided to consider the energy problem in Vaidya-Anti-de
Sitter spacetime because in this case we have:

  * •

The eternal naked singularity i.e. the singularity will be never covered with
the apparent horizon,

  * •

Under some conditions we have two apparent horizons,

  * •

at the limit $\lim\limits_{r\to\infty}$ our spacetime becomes Minkowski
spacetime and like in Kerr black hole the energy which we consider is the
energy with regard to infinity.

In the case of Vaidya-Anti-de Sitter spacetime the type-II of the matter field
satisfies the equation of the state:

$P=\rho\,.$ (4.28)

it means $\alpha=1$ and using (3.19) we obtain:

$M(v,r)=C(v)+D(v)r^{-1}\,.$ (4.29)

Here we must consider the function $D(v)$ properly. It can’t be positive
because the density expression is given by:

$\rho=-\frac{D(v)}{r^{4}}<0\,,$ (4.30)

and if $D(v)>0$ then we violate energy conditions (3.17) and (3.18). So to
satisfy these conditions we must assume that $D(v)<0$ or we can write:

$D(v)=-D^{\prime}(v)\,,$ (4.31)

where now function $D^{\prime}(v)$ is positive.

We want to investigate the question about the singularities in this spacetime
which are formed at $v=0$ and $r=0$. But in this case we must impose some
extra conditions on functions $C(V)$ and $D^{\prime}(v)$ not to violate energy
conditions (3.17) and (3.18) because if
$\dot{C}(v)<\frac{\dot{D^{\prime}}(v)}{r}$ at some point then $\mu$ becomes
negative and we violate our energy conditions. So we must consider two cases:

  1. 1.

$D^{\prime}(v)\equiv\mu$, where $\mu$ is positive real constant. In this case
$\dot{D^{\prime}}(v)\equiv 0$ and if we impose the following condition
$C(V)\geq 0$ then we satisfy all energy condition.

  2. 2.

$D^{\prime}(0)=0$ but in this case even if $\lim\limits_{v\to 0,r\to
0}\frac{D^{\prime}(v)}{r}=X_{1}$ where $X_{1}$ finite positive constant and if
$C(v)>X_{1}$ then anyway we would violate energy conditions at the later
stage.

So the Vaidya-Anti-de Sitter spacetime is given by:

$ds^{2}=-\left(1-\frac{2C(v)-2\mu
r^{-1}}{r}\right)+2dvdr+r^{2}(d\theta+\sin^{2}\theta d\varphi^{2})\,.$ (4.32)

The apparent horizon equation in the case of Vaidya-anti-de Sitter spacetime
is given by:

$\frac{2C(v)}{r}-\frac{2\mu}{r^{2}}-1=0\,.$ (4.33)

The time of the singularity formation is $v=0$. So solving the above equation
we obtain:

$\begin{split}r^{2}-2C(v)r+2\mu=0\,,\\\
\frac{D}{4}=C^{2}(v)-2\mu\,.\end{split}$ (4.34)

From this equation we can conclude that if

$C^{2}(v)<2\mu\,,$ (4.35)

then the apparent horizon will be never formed.

In this paper we will not prove that the radial null geodesics can terminate
at the central singularity in the past.

To obtain the energy expression we use the lagrangian:

$\mathcal{L}=\frac{1}{2}g{ik}\dot{x}^{i}\dot{x}^{k}\,.$ (4.36)

In Vaidya-Anti-de Sitter spacetime the Lagrangian is given by:

$2\mathcal{L}=-\left(1-\frac{2M(v,r)}{r}\right)\dot{v}+2\dot{v}\dot{r}+r^{2}(\dot{\theta}+\sin^{2}\theta\dot{\varphi}^{2})\,.$
(4.37)

Due to spherical symmetry we can put $\theta=\frac{\pi}{2}$ and
$\dot{\theta}=0$.

Using (4.37) we can obtain the energy and angular momentum expressions:

$\begin{split}-E(v)=\frac{\partial\mathcal{L}}{\partial\dot{v}}=-\left(1-\frac{2C(v)-2\mu
r^{-1}}{r}\right)\dot{v}+\dot{r}\,,\\\
L=\frac{\partial\mathcal{L}}{\partial\dot{\varphi}}=r^{2}\dot{\varphi}\,,\end{split}$
(4.38)

here $E(v)\,,L$ are energy and angular momentum respectively and dot
represents the partial derivative with respect to affine parameter $\lambda$.

The radial geodesic is given by

$\dot{r}=\pm\sqrt{E^{2}(v)-\left(1-\frac{2C(v)-2\mu
r^{-1}}{r}\right)\left(\frac{L^{2}}{r^{2}}-\delta\right)}\,,$ (4.39)

where $\delta=-1$ in the case of timelike geodesics and $0$ in the case of
null ones.

The Vaidya-Anti-de Sitter spacetime contains the eternal naked singularity if
we can satisfy the condition (4.35) in this case $\Theta$ is always positive.
We also have the condition $\dot{v}>0$ because we must go to future in time.
So the first term in the energy expression

$E(v)=\left(1-\frac{2C(v)-2\mu r^{-1}}{r}\right)\dot{v}-\dot{r}\,,$ (4.40)

is always positive. Thus to obtain the negative energy $\dot{r}$ must be
positive. If the negative energy exist it must move from the center to
infinity. Due to these conditions and the equation (4.39) we obtain:

$\left(1-\frac{2C(v)-2\mu
r^{-1}}{r}\right)\dot{v}=E-\sqrt{E^{2}(v)-\left(1-\frac{2C(v)-2\mu
r^{-1}}{r}\right)\left(\frac{L^{2}}{r^{2}}-\delta\right)}\,.$ (4.41)

Let’s look at the expression which is under the root. We can rewrite it in the
form

$E^{2}+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)\,.$ (4.42)

The expression in round bracket is positive i.e. $\delta=0$ or $\delta=-1$,
$g_{00}<0$ is the condition for the positivity of the expansion $\Theta$. So
$g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)<0$. Due to this, the whole
expression (4.42) is less than $E^{2}$. It means that the right hand-side of
the equation (4.41)

$E^{2}-|\sqrt{E^{2}+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)}|_{<|E|}<0\,,$
(4.43)

is negative. So the negative energy in the case of the naked singularity
formation is possible only if we violate the condition $\dot{v}>0$. So we can
conclude that there is no non-spacelike geodesics for particles with negative
energy in the case of the naked singularity. In the next part we will show
that in generalized Vaidya spacetime if we have the naked singularity
formation that the particles with negative energy are forbidden.

Now let’s consider the violation of the condition (4.35). In this case the
equation (4.34) has two different roots:

$r_{\pm}=C(v)\pm\sqrt{C^{2}(v)-2\mu}\,.$ (4.44)

In this case we have two different apparent horizons at $r=r_{-}$ and
$r=r_{+}$. The expansion $\Theta$ is positive in two regions:

$\begin{split}0\leq r<r_{-}|\Theta>0\,,\\\ r_{-}<r<r_{+}|\Theta<0\,,\\\
r_{+}<r<\infty|\Theta>0\,.\end{split}$ (4.45)

Again if we consider the particles with negative energy in the regions where
$\Theta$ is positive then we are in the previous case and such particles are
forbidden. If we consider the region where $\Theta$ is negative then we should
put $g_{00}>0$. So let’s consider this case. The energy expression has the
same form as (4.38):

$E(v)=\left(1-\frac{2C(v)-2\mu r^{-1}}{r}\right)\dot{v}-\dot{r}\,.$ (4.46)

In this case we have two possibilities:

  1. 1.

$\dot{r}>0$ the movement from the singularity to infinity,

  2. 2.

$\dot{r}<0$ the movement to the singularity.

Let’s consider the first case when $\dot{r}>0$. As we showed in previous case
the expression (4.42) is less than $E^{2}$ but in this case $g_{00}>0$ and it
means that the expression (4.42) is bigger than $E^{2}$. Hence the right hand-
side of (4.41) is more than zero. And it means that again the non-spacelike
geodesics for particles with negative energy are forbidden due to the
violation of the condition $\dot{v}>0$.

Now let’s return to the second case $\dot{r}<0$. Here we have:

$\left(1-\frac{2C(v)-2\mu
r^{-1}}{r}\right)\dot{v}=E(v)-\sqrt{E^{2}+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)}\,.$
(4.47)

We can see that the right hand-side of the expression above is always
negative. In this case the non-spacelike geodesics for particles with negative
energy are not forbidden.

Now let’s consider the question about the existence of circular or closed non-
circular orbits. Both cases demand that the effective potential must be zero
at some point $r_{0}$. First of all let’s define the effective potential which
is given by:

$v_{eff}=-(\dot{r})^{2}=-\left[E^{2}+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)\right]\,.$
(4.48)

We know that $E<0$ and both $g_{00}$ and $\frac{L^{2}}{r^{2}}-\delta$ are
positive. So the whole expression in square brackets (4.48) are positive and
strictly more than zero. Thus there is no any $r_{0}$ that $v_{eff}(r_{0})=0$.
So in this case the closed non-circular and circular orbits are absent.

We showed that in the region $r_{-}<r<r_{+}$ the particles with negative
energy can exist. However in this particular case this is unphysical because
we demand $g_{00}>0$ and it means that all components of the metric tensor
i.e. $g_{00}\,,g_{01}\,,g_{22}\,,g_{33}$ are positive and the line element is
spacelike in this region. To have the non-spacelike line element we must
change the sign in front of the off-diagonal term and consider not falling
matter but radiation. We consider this case in the section below.

## The Negative Energy in Generalized Vaidya Spacetime

In the case of generalized Vaidya spacetime we will consider only the case
when the type-II of the matter field satisfies the equation of the state
$P=\alpha\rho$ where $\alpha\in[0\,,1]$. In this case we can satisfy all
energy conditions (3.17), (3.18).

The result which was obtained in the previous section is valid in generalized
Vaidya spacetime e.g. if we have the naked singularity formation then the
negative energy is impossible. Let’s prove it.

The line element we write down in the following form:

$ds^{2}=-\left(1-\frac{2M(v,r)}{r}\right)dv^{2}+2\varepsilon
dvdr+r^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2})\,,$ (5.49)

here $\varepsilon=\pm 1$ corresponding ingoing or outgoing matter
respectively.

The energy and the angular momentum has the following form:

$\begin{split}E(v)=\left(1-\frac{2M(v,r)}{r}\right)\dot{v}-\varepsilon\dot{r}\,,\\\
L=r^{2}\dot{\varphi}\,.\end{split}$ (5.50)

The radial geodesic is given by:

$\dot{r}^{2}=E(v)^{2}-\left(1-\frac{2M(v,r)}{r}\right)\left(\frac{L^{2}}{r^{2}}-\delta\right)\,.$
(5.51)

Note that the radial equation doesn’t depend on $\varepsilon$.

Now we consider the case of the naked singularity formation. It means that
$g_{00}<0$. Let’s start with the case $\varepsilon=+1$. In this case the first
expression (5.50) becomes:

$E(v)=\left(1-\frac{2M(v,r)}{r}\right)\dot{v}-\dot{r}\,.$ (5.52)

Here the negative energy can exist only in the case of $\dot{r}>0$. It gives:

$\left(1-\frac{2M(v,r)}{r}\right)\dot{v}=E(v)+\sqrt{E^{2}(v)+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)}\,.$
(5.53)

Like in previous section the expression under the root is less than $E^{2}(v)$
(due to the fact that $g_{00}<0$). Hence it means that the right hand-side of
the equation above is less than zero. However the left hand-side must be
always positive ($-g_{00}\dot{v}$) and it can be negative only in the case of
violating the main principle $\dot{v}>0$. So in the case $\varepsilon=+1$ and
the naked singularity formation the particles with negative energy are
forbidden.

Now let’s consider the second case $\varepsilon=-1$. In this case we have:

$E=-g_{00}\dot{v}+\dot{r}\,.$ (5.54)

In this case the only possibility for the negative energy to exist is the
condition $\dot{r}<0$. It gives:

$-g_{00}\dot{v}=E+\sqrt{E^{2}+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)}\,.$
(5.55)

This equation coincides with (5.53) above. It means that in general case if we
have the naked singularity formation the particles with negative energy are
forbidden.

Now let’s consider the case $g_{00}>0$ i.e. the black hole formation. Note,
that in this case $\varepsilon$ must be equal to $-1$ otherwise the line
element will be spacelike which is unphysical.

The first expression of (5.50) becomes:

$E=-g_{00}\dot{v}+\dot{r}\,.$ (5.56)

Here we have two possibilities $\dot{r}>0$ and $\dot{r}<0$. Let’s consider the
first case. When $\dot{r}>0$ we have the negative energy if:

$-g_{00}\dot{v}=E(v)-\sqrt{E^{2}(v)+g_{00}\left(\frac{L^{2}}{r}-\delta\right)}\,.$
(5.57)

Now let’s prove that there is no circular and elliptical orbits inside the
apparent horizon. Under the notion ’elliptical’ we mean non-circular closed
orbits. The circular orbit can exist if:

$\begin{split}v_{eff}(r_{0})=0\,,\\\
\frac{dv_{eff}(r)}{dr}|_{r=r_{0}}=0\,,\end{split}$ (5.58)

here $v_{eff}$ is the effective potential. In our case the effective potential
has the form

$v_{eff}=-\dot{r}^{2}=-\left[E^{2}+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)\right]\,.$
(5.59)

From (5.51) we can see that $E^{2}>0\,,-g_{00}<0$ and
$\frac{L^{2}}{r^{2}}-\delta>0$ and because $E(v)\neq 0$ the condition
$v_{eff}(r_{0})=0$ is impossible. The elliptical orbits also has the condition
$v_{eff}(r_{0})=0$. So in this case we can conclude that there is no circular
and elliptical orbits for particles with negative energy.

Now let’s consider the second case when $\dot{r}<0$. In this case we have:

$-g_{00}\dot{v}=E+\sqrt{E^{2}+g_{00}\left(\frac{L^{2}}{r^{2}}-\delta\right)}\,.$
(5.60)

Again the expression under the root is bigger than $E^{2}$ due to $E<0\,,E\neq
0$ and positivity of $g_{00}$ and $\frac{L^{2}}{r^{2}}-\delta$. And it means
we can’t satisfy the condition $\dot{v}>0$. So in this case the particles with
negative energy are forbidden.

## Discussion

The main goal of this article is to prove the absence of the particles with
negative energy in the case of the naked singularity formation in the
generalized Vaidya spacetime. We have considered one explicit example of
Vaidya-Anti-de Sitter spacetime and proved that in the case of naked
singularity formation the particles with negative energy are forbidden. Then
we have considered the general case and also proved the absence of the
particles with negative energy in the case of the naked singularity formation.

According to the Penrose process the particles with negative energy can exist
in the ergoregion of a rotating black hole. Geodesics for such particles have
been considered in [1]. It has been shown that this type of geodesic can’t
cross the static limit and these particles are forbidden in the region outside
of the static limit. These geodesics have been also studied in the case of
Schwarzschild spacetime [3]. In this metric the particles with negative energy
can exist only in the region inside the event horizon and are forbidden
outside it. But in the case of the naked singularity formation there is no the
apparent horizon and such particles can exist in our universe. Vaidya
spacetime is one of the first examples of the naked singularity formation. We
hasn’t considered the case of usual Vaidya spacetime in this paper because it
a particular case of the generalized Vaidya spacetime and the results were
obtained in this article are also valid in the case of usual Vaidya metric.

Due to off-diagonal term in generalized Vaidya spacetime there is a
possibility for the existence of the particles with negative energy but they
can exist only in the region inside the apparent horizon. If the horizon is
absent then we can have such particles only if they move back in time which is
unphysical.

There are many models of the gravitational collapse when the result is the
naked singularity (See for example [4]). The physics demands that the
particles with negative energy can not exist in the case of the naked
singularity formation. If they existed then the distant observer would be able
to detect but it is forbidden. In these models if the metric doesn’t have off-
diagonal term then due to the fact that $g_{00}<0$ the particles with negative
energy can not exist. If the metric has off-diagonal term $g_{0\alpha}\neq 0$
then these metrics should be investigated to prove the absence of such
particles in the case of the naked singularity formation. The generalized
Vaidya spacetime is one of such type of metrics and as we have proven the
negative energy for this metric is forbidden in the case of the naked
singularity formation.

acknowledgments The author says thanks to professor Pankaj Joshi for
scientific discussion. This work was supported by RFBR grant 15-02-06818-a.The
work was performed within the SAO RAS state assignment in the part ”Conducting
Fundamental Science Research”.

## References

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# Compressibility and the Equation of State of an Optical Quantum Gas in a Box

Erik Busley  Leon Espert Miranda  Andreas Redmann  Christian Kurtscheid
Kirankumar Karkihalli Umesh  Frank Vewinger  Martin Weitz  Julian Schmitt
<EMAIL_ADDRESS>Institut für Angewandte Physik, Universität Bonn,
Wegelerstraße 8, 53115 Bonn, Germany

###### Abstract

The compressibility of a medium, quantifying its response to mechanical
perturbations, is a fundamental property determined by the equation of state.
For gases of material particles, studies of the mechanical response are well
established, in fields from classical thermodynamics to cold atomic quantum
gases. Here we demonstrate a measurement of the compressibility of a two-
dimensional quantum gas of light in a box potential and obtain the equation of
state for the optical medium. The experiment is carried out in a
nanostructured dye-filled optical microcavity. We observe signatures of Bose-
Einstein condensation at high phase-space densities in the finite-size system.
Strikingly, upon entering the quantum degenerate regime, the measured density
response to an external force sharply increases, hinting at the peculiar
prediction of an infinite compressibility of the deeply degenerate Bose gas.

Quantum gases of atoms, exciton-polaritons, and photons provide a test bed for
many-body physics under both in- and out-of-equilibrium settings [1, 2, 3].
Experimental control over dimensionality, potential energy landscapes, or the
coupling to reservoirs offer wide possibilities to explore different phases of
matter. For cold atomic gases, thermodynamic susceptibilities and transport
properties have been extracted from density measurements [4, 5, 6, 7, 8, 9]
and have proven to be direct manifestations of the equation of state (EOS). In
general, the EOS of a material, e.g., its pressure–volume relation, describes
both the thermodynamic state of a system under a given set of physical
conditions as well as its response to perturbations, as mechanical
compression. Experimental investigations of the EOS in quantum gases
constitute a tool for the characterisation of phases and the identification of
phase transitions, enabling important tests of physical models in a wide range
of systems, from the ideal gas to superfluids and the interior of stars.

Quantum gases of light have so far been experimentally realized in low-
dimensional settings, mostly two-dimensional (2D) systems [3]. Thermalized
photon gases with non-vanishing chemical potential $\mu$, as well as Bose-
Einstein condensation (BEC) have been demonstrated in dye-filled optical
microcavities at harmonic confinement [10, 11, 12], including measurements of
density-insensitive thermodynamic quantities [13]. In contrast, the isothermal
compressibility $\kappa_{T}=n^{-2}(\partial n/\partial\mu)_{T}$ at temperature
$T$ depends on the (local) particle density $n$ in the gas; for a systematic
study, it thus is desirable to avoid spatially inhomogeneous density
distributions inherent to harmonically trapped gases, and instead prepare
uniform samples, where applying a spatially uniform force directly allows one
to compress the gas and probe $\kappa_{T}$.

Notably, BEC does not occur in the infinite 2D homogeneous Bose gas given that
thermal fluctuations at finite temperatures destroy long-range order [14].
While interactions nevertheless stabilize a superfluid through the
Berezinskii-Kosterlitz-Thouless (BKT) transition, the infinite 2D ideal gas is
doomed to remain quantum degenerate without forming a condensate. For a
finite-sized homogeneous gas in a box, however, condensation is expected to be
possible if the correlation length exceeds the system size at large phase-
space densities [15]. In ultracold atoms, the crossover between saturation-
driven BEC and interaction-driven BKT superfluidity has been investigated in
2D harmonically trapped Bose gases by tuning the interactions exploiting a
Feshbach resonance [16], while studies of homogeneous gases in box potentials
have focused on the interacting regime [17, 18, 19]. In uniform gases of
exciton-polaritons [20], on the other hand, the observation of BEC is hampered
by reservoir-induced interactions and non-equilibrium effects. Importantly, up
to now the compressibility and the EOS have not been determined for optical
quantum gases.

Here we examine a 2D quantum gas of photons in a box potential. In the finite-
size homogeneous system, we observe BEC, as evidenced from the measured
position and momentum distributions. In subsequent experiments, a mechanical
force is exerted onto the photon gas prepared in a regime around the phase
transition. By studying the density response to minute forces, we measure both
the bulk isothermal compressibility and the EOS of the optical quantum gas.

Fig. 1: Trapping a homogeneous 2D photon gas. (A) A spatially structured
microcavity filled with dye molecules confines and thermalizes the uniform
photon gas. The locally elevated surface of one of the cavity mirrors realizes
a repulsive box potential. (B) Example surface density of a uniform photon gas
in a square box with $L=$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. (C)
Spectral distributions of the cavity emission exhibit an exponential decay
consistent with $T\approx$300\text{\,}\mathrm{K}$$, indicating a thermal
equilibrium photon gas. The population in the low-frequency modes is enhanced
as $N_{\text{c}}/N$ approaches unity.

Our homogeneous 2D photon gases are prepared in a nanostructured optical
microcavity filled with a liquid dye solution [21], as illustrated in Fig.
1(A). The photons in the short cavity with mirror spacing on the order of the
optical wavelength form a 2D gas of particles with an effective mass $m$,
which are described by their transverse momentum
$k=\sqrt{k_{x}^{2}+k_{y}^{2}}$. To spatially confine the unbound plane-wave
states of a 2D homogeneous system, we implement a box potential of size $L$ as
a container for the photons. The box potential is realized using a
nanostructuring technique [21, 22], which imprints a position-dependent static
surface elevation onto one of the cavity mirrors. The locally reduced cavity
length results in a repulsive potential. Thermalization of the photon gas to
room temperature is achieved by absorption and re-emission processes on the
dye molecules [10]. Figure 1(B) shows an exemplary density distribution of a
trapped gas recorded by imaging the cavity emission, and Fig. 1(C) gives
optical frequency spectra over the trapped range $V_{0}/\hbar\simeq
2\pi\times$9\text{\,}\mathrm{THz}$$ above the cavity low-frequency cutoff
$\nu_{\mathrm{c}}=mc^{2}/(2\pi\hbar)$, where $\hbar$ is the reduced Planck’s
constant and $c$ the speed of light, for different total particle numbers $N$.
All spectra show an exponential decay of the population in the high-frequency
states consistent with
$k_{\text{B}}/(2\pi\hbar)\times$300\text{\,}\mathrm{K}$$, with Boltzmann’s
constant $k_{\text{B}}$, which we attribute as evidence for the gas to be
thermalized. Moreover, the population in the lowest-lying states is enhanced
as we increase $N$ beyond a critical value $N_{\text{c}}$, which signals the
emergence of a low-entropy phase as we discuss in detail in the following.

Fig. 2:  Bose-Einstein condensation of photons in a box. (A) Density
distributions (top), line profiles (middle) and momentum-space distributions
(bottom; $k_{0}$ denotes the trap limit, $k_{\mathrm{NA}}$ the imaging limit)
as $N$ is increased beyond $N_{\text{c}}=3.5(4)\times 10^{3}$. Below
$N_{\text{c}}$, the flat-top density and Gaussian $k$-space distribution
resemble a thermal gas in the normal phase. Above $N_{\text{c}}$, the ground
state becomes massively populated and deforms the cloud, more directly
observed in momentum space. (B) Normalized central spatial density versus $N$,
starting from $n_{0}L^{2}/N=1$ in the normal phase, and approaching the
expected value of $4$ when the ground state dominates (symbols as in (A)). (C)
$N_{\text{c}}$ and $\mathcal{D}_{\mathrm{c}}=N_{\text{c}}(\lambda/L)^{2}$
(inset), extracted from data as in (B), exhibit the predicted scaling (line)
with the box size.

We explore the quantum degenerate behavior in the finite-size homogeneous
system. Figure 2(A) shows surface densities in the box for different $N$.
Below the critical photon number $N_{\text{c}}$, the bulk density is uniform
as for a normal gas, while above, a macroscopic occupation of the ground state
$|\psi_{1,1}(x,y)|^{2}=4L^{-2}\cos^{2}(\pi x/L)\cos^{2}(\pi y/L)$ with
$|x|,|y|\leq L/2$ is observed; the qualitative change is evident from the line
profiles. The corresponding momentum distributions below $N_{\text{c}}$ in
Fig. 2(A) resemble a Maxwell-Boltzmann distribution
$f(k)\propto\exp(-k^{2}/\sigma_{k}^{2})$ with
$\sigma_{k}=\sqrt{2mk_{\text{B}}T}/\hbar$; for our data,
$\sigma_{k}=$2.4(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ gives
$T=$295(33)\text{\,}\mathrm{K}$$. Upon increasing the photon number, the
population at small $k$ is enhanced and ultimately dominated by a strongly
occupied ground state at $k=0$, the condensate. For large $k$, we observe
signatures of both microscope aperture and finite trap depth [21]. A closer
inspection in $x$ and $k$-space (Fig. 2(A), inset) confirms that the ground
state is Heisenberg-limited with an uncertainty product $\Delta x\Delta
k=0.7(1)$, in agreement with theory $\Delta x\Delta
k=\sqrt{\pi^{2}/12-1/2}\simeq 0.6$.

To quantify the transition point, we study the normalized spatial central
density $n_{0}L^{2}/N$ as a function of the particle number. Figure 2(B) shows
the transition at $N_{\text{c}}=3.5(4)\times 10^{3}$; the limiting cases
$n_{0}L^{2}/N=1$ ($=4$) for small (large) $N$ are well understood to arise
from the uniform normal gas (inhomogeneous condensate) density with $N/L^{2}$
($4N/L^{2}$) at the center. The critical particle number scales with the
predicted $N_{\text{c}}\sim(L/\lambda)^{2}\log(L/\lambda)$, see Fig. 2(C),
where
$\lambda=2\sqrt{\pi}/\sigma_{k}\simeq$1.47\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ denotes the thermal wavelength. The logarithmic scaling of the critical
phase-space density $\mathcal{D}_{\mathrm{c}}$ shown in the inset is
understood from the dependence of the coherence length reaching the system
size [15, 21]. At the largest investigated box sizes, we have
$\mathcal{D}_{\mathrm{c}}=N_{\text{c}}(\lambda/L)^{2}=6.3(8)$. For interacting
2D gases characterized by an interaction strength $\tilde{g}$ and realized in
ultracold atoms [6, 7, 15, 16, 17, 19], the BKT phase transition to a
superfluid usually occurs before BEC; for example, for homogeneous gases,
$\mathcal{D}_{\mathrm{c,BKT}}=\log(380/\tilde{g})=6.5$ for $\tilde{g}=0.6$
[19]. Quite distinctly, for photon gases in dye-microcavities, self-
interactions with $\tilde{g}\leq 10^{-6}$ [23] imply a significantly larger
$\mathcal{D}_{\mathrm{c,BKT}}\geq 20$ and, accordingly, both phases are
expected to be well separated.

We identify the BEC-like nature of the phase transition by extracting the
ground and excited state populations, $N_{0}$ and $N_{\mathrm{exc}}$,
respectively, from the momentum space distributions as a function of the total
particle number $N$. Figure 3(A) shows the visible saturation of the normal
part, which indicates that interaction effects are very small [24]; in
particular, it gives evidence that in our homogeneous finite-size system BEC
is prevalent, instead of BKT. This interpretation is supported by the deduced
caloric properties of the gas, see Fig. 3(B), which closely follow the ideal
Bose gas prediction. For the internal energy $U=\langle E\rangle
N_{\text{c}}/N^{2}k_{\text{B}}T$, where $\langle E\rangle$ denotes the average
transverse energy, we observe a crossover from a quadratic to a linear scaling
in the condensed and normal phase, respectively, as a function of
$N_{\text{c}}/N$. Correspondingly, its derivative $\partial
U/\partial(N_{\text{c}}/N)$ is a smooth function, highlighting that the heat
capacities in the normal and condensed phases are linked without any
discontinuities [8, 13]. In the normal-gas phase each particle can accommodate
only $\approx 0.5k_{\text{B}}T$ of thermal energy, as well understood from the
finite trap depth [21].

Fig. 3:  Saturation and caloric properties. (A) Population in the ground (red)
and excited states (blue), $N_{0}$ and $N_{\mathrm{exc}}$, respectively,
extracted from $k$-space distributions (as in Fig.2(A), bottom) versus photon
number $N$. At large $N$, the visible saturation of the normal part
$N_{\mathrm{exc}}$ indicates that the phase transition is BEC-like. (B)
Normalized internal energy (red), showing a crossover from quadratic
(condensed gas, $N_{\text{c}}/N<1$) to linear (normal, $N_{\text{c}}/N\gg 1$)
scaling, and the derivative (blue), which reflects the specific heat per
particle, versus $N_{\text{c}}/N$. In the normal phase, the latter reaches a
value near $0.5$, which is understood from the finite trap depth $V_{0}$, and
is below the value of $1$ expected for $V_{0}\rightarrow\infty$. Solid lines
are finite-size theory.

To determine the isothermal compressibility $\kappa_{T}=n^{-2}(\partial
n/\partial\mu)_{T}$ of the optical quantum gas, we exert a force onto the
photons by tilting one of the cavity mirrors, which superimposes a linear
potential $U(x)=U_{0}x/L$ to the box, and measure the density response. Figure
4(A) shows the displaced center-of-mass $\langle x\rangle$ as a function of
the tilt $U_{0}$. In local density approximation (LDA), with chemical
potential $\mu(x)=\mu_{0}-U(x)$, the center-of-mass to first order relates to
the compressibility, following $\langle x\rangle/L=-\kappa_{T}nU_{0}/12$ [5,
21, 25]. For small $U_{0}$, the data in Fig. 4(A) confirms the linear behavior
and shows an enhancement of the density response when going from the normal to
the condensed phase. The visible saturation for large $U_{0}$ is caused by the
finite box size, which limits the displacement.

Figure 4(B) shows the compressibility, extracted from a linear fit of the
region near $U_{0}=0$, see Fig. 4(A), along with theory for the infinite and
finite system. Below the critical density for condensation
$N_{\text{c}}/L^{2}\simeq$2.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-2}$,
the photon distribution is spatially homogeneous, as visible in Figs. 2(A,B).
Here LDA can be applied [25] to extract $\kappa_{T}$ using data with small
tilts ($U_{0}\lesssim\mu$), even in the case of very small interactions [21].
In the condensed phase, on the other hand, the LDA ceases to be valid, and the
corresponding region is shaded in gray. Within the region of validity, the
compressibility compares well with theory. Remarkably, at densities
$n\geq$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-2}$, we observe a sharp
increase of the compressibility; the onset is in good agreement with the
prediction for an infinite non-interacting system (dashed line) [21]. The
corresponding function
$\kappa_{T}=[\exp(n\lambda^{2})-1]/(k_{\text{B}}Tn^{2}\lambda^{2})$ exhibits a
minimum at $n\simeq
1.59/\lambda^{2}\simeq$0.74\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-2}$,
close to the measured value of $$0.79(5)\text{\,}\mathrm{\SIUnitSymbolMicro
m}$^{-2}$. It is well understood that as the thermal wave packets spatially
overlap the classically expected decrease in compressibility with density (it
is harder to compress a dense gas than a dilute one) is replaced by a
compressibility increase stemming from the quantum-statistical occupation of
low-lying energy levels, reducing the energy cost for compression as compared
to the classical gas case. In the extreme high-density limit of an infinite-
size deeply degenerate gas, bosons can be added to the system at essentially
vanishing energy cost, meaning that $(\partial\mu/\partial n)_{T}$ gradually
approaches zero as $\mu\rightarrow 0$, so that the compressibility takes
arbitrarily large values.

Fig. 4:  Compressibility and equation of state. (A) A linear potential
realizes a force $F_{0}=U_{0}/L$ that compresses the photon gas (inset).
Depending on its normal (blue) or quantum degenerate (green) state, the gas
shows a different center-of-mass response, fitted in the linear response
region (lines). (B) The compressibility $\kappa_{T}$ decreases with density
$n=N/L^{2}$, until above $n\simeq$0.8\text{\,}\mathrm{\SIUnitSymbolMicro
m}$^{-2}$ quantum degeneracy sets in and the Bose gas becomes highly
compressible as a consequence of the gradually vanishing energy cost to
increase the density. At $n\geq$2.6\text{\,}\mathrm{\SIUnitSymbolMicro
m}$^{-2}$ (shaded), the $\kappa_{T}$ extraction ceases to be valid; this range
of validity has been confirmed numerically [21]. For comparison, we show
finite-size theory (red line, shading indicates uncertainty in $V_{0}$, which
causes the vertical shift), where the kink at large densities is due to the
finite box size, and the prediction for the infinite 2D Bose gas (dashed). (C)
From the density profiles (insets: exemplary density deviation from $U_{0}=0$
and line profile) we extract the equation of state by combining data for
different local chemical potentials [21]. Lines show theory for finite-depth
box (solid) and infinite-system (dashed).

Finally, we study the equation of state $n=f(\mu,T)$ of the photon gas. Figure
4(C) shows the variation of the density $n$ as a function of the chemical
potential $\mu$, as determined from combining recorded density profiles of the
gas in a tilted box at different $N$; the insets give an exemplary density
deviation from the unperturbed case and the corresponding line profile $n(x)$.
As above, the conversion from position to chemical potential relies on the
potential gradient $U_{0}/L\approx k_{\text{B}}\
$0.6\text{\,}\mathrm{K}\mathrm{/}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
which acts as a calibrated differential energy scale
$\mathrm{d}\mu=U_{0}/L\thinspace\mathrm{d}x$. Note that our data exhibits a
larger slope than the EOS of the infinite 2D Bose gas,
$n(\mu)=-\lambda^{-2}\log[1-\exp(\mu/k_{\text{B}}T)]$ (dashed line) owing to
the finite trap depth, as we confirm by numerical calculations [21]. Except
for the condensed regime, where LDA is invalid, our method reliably extracts
the EOS of a quantum gas of light.

We have demonstrated a measurement of the compressibility of an optical
quantum gas and determined its EOS. The experiment is carried out using a 2D
photon gas inside a box potential, both below and above the phase transition
to a BEC. Compression of optical gases may have direct consequences for
thermodynamic machines with light as a work medium [26]. A further interesting
perspective is the exploration of sound [17, 18, 19, 20]. The required dynamic
manipulation of optical quantum gases is feasible by, e.g., electro-optic trap
modulation or spatiotemporally resolved pumping of the dye reservoir [27].
Beyond ideal gas theory, a nonvanishing healing length can be achieved by
adding either Kerr media or exploiting the weakly dissipative nature of photon
condensates [28]. The demonstrated homogeneous quantum gas of light in a box
opens new possibilities for studies of universal phenomena in 2D, including
critical behavior [29] and the non-equilibrium Kardar-Parisi-Zhang phase [30].

Acknowledgments. We thank F. König for experimental assistance; and D. Luitz,
P. Christodoulou, Z. Hadzibabic, and R. Lopes for fruitful discussions. This
work was supported by the DFG within SFB/TR 185 (277625399) and the Cluster of
Excellence ML4Q (EXC 2004/1–390534769), and by the EU within the Quantum
Flagship project PhoQuS (820392). J.S. acknowledges support from an ML4Q
Independence grant.

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## Supplementary Information

###  .1 Experimental scheme

The photon gas is operated inside a dye-filled optical microcavity, see Fig.
S1, which consists of two plane Bragg mirrors each with a maximum reflectivity
of $99.998\%$ and spaced by
$D_{0}=q\lambda_{0}/2\tilde{n}\approx$1.4\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$, where $q=7$ is the longitudinal wave number and
$\lambda_{0}=$583\text{\,}\mathrm{nm}$$ the optical wavelength. As a dye
medium we use rhodamine 6G solved in ethylene glycol (refractive index
$\tilde{n}=1.44$). Owing to the small value of $D_{0}$, the wave vector
component $k_{z}=\pi q/D_{0}\tilde{n}$ introduces a large energy gap $\hbar
k_{z}c/q\gg k_{\text{B}}T$ between adjacent modes, with reduced Planck’s
constant $\hbar$, the speed of light $c$, Boltzmann’s constant $k_{\text{B}}$
and temperature $T=$300\text{\,}\mathrm{K}$$. The gap effectively freezes out
the motional degree of freedom of the photons along $z$, making the system 2D.
In the transverse direction, the quasi-continuous momenta $k_{x,y}$ describe
the in-plane motion of the particles in the homogeneous system. Our experiment
allows the single-pulse-resolved detection of the cavity emission, and the
acquired data is averaged over 5-10 shots.

One of the cavity mirrors exhibits a reflectivity maximum at
$550\text{\,}\mathrm{nm}$ and a bandwidth of $50\text{\,}\mathrm{nm}$, where
the reflectivity is above $99.99\%$; in addition, it is equipped with an
absorptive silicon layer below the high-reflectivity Bragg coating, which
enables nanostructuring of the mirror surface (see section “Potential
creation” and ref. [22]). The opposing mirror consists of a custom dielectric
coating, which contains two stacked reflection bands of
$50\text{\,}\mathrm{nm}$ spectral width, the first one centered at
$570\text{\,}\mathrm{nm}$ (on top; in contact with the dye), the second one
around $700\text{\,}\mathrm{nm}$ (below; sandwiched between top-layers and
glass substrate). While the top coating confines the photon gas, the lower
coating prevents residual isotropic fluorescence from reaching the analysis
part. The cavity emission transmitted through this mirror is guided into the
analysis part of the experiment.

Fig. S1: Sketch of the dye-filled microcavity experiment. Photons transmitted
through one of the cavity mirrors are collected by a microscope objective, and
their position $n(x,y)$, momentum $n(k_{x},k_{y})$ or energy $n(E)$ spectral
distributions are detected using intensified cameras. To precisely co-align
the cavity mirrors (or induce linear gradient potentials) calibrated piezo
mounts are used.

To maintain a steady-state photon number inside the cavity, continuous pumping
is required to compensate losses from mirror transmission. We optically pump
the dye with a laser at $532\text{\,}\mathrm{nm}$ under a $42^{\circ}$ angle
(mirror transmission $>50\%$); the pump light is temporally chopped into
$500\text{\,}\mathrm{ns}$-pulses at $30\text{\,}\mathrm{Hz}$ repetition rate
to minimize photobleaching. Operating in the optically dense regime
$\Gamma\tau\gg 1$, most fluorescence photons are reabsorbed by the dye film at
a rate $\Gamma$ during the cavity lifetime $\tau$. This leads to a detailed
balance between molecules and photons if the ratio of absorption and emission
rates scales with a Boltzmann-factor, which is the case provided the used dye
fulfills the Kennard-Stepanov relation [31]. The photon gas thermalizes with
the dye heat bath and acquires the rovibrational temperature of the molecules,
which is at room temperature [10]. At fixed $T$, the phase-space density of
the photon gas is determined by the pumping; controlling the excitation level
of the dye molecules allows us to vary the chemical potential of the photons
and prepare gases in the normal to quantum-degenerate regime [32, 33]. To
illuminate the system as homogeneously as possible, the pump beam diameter is
adjusted individually to roughly twice the system size for each box.

The cavity emission is collected by a long working distance microscope
objective (Mitutoyo $20\times$, $\mathrm{NA}=0.42$) and imaged onto an EMCCD
camera (Andor iXon Ultra 897) to record the spatial- or momentum-space
distributions, respectively, as sketched in Fig. S1. The latter is achieved by
two interchangeable sets of lenses in the optical path, enabling us to image
either the cavity or the Fourier plane onto the camera sensor. The obtained
momentum distributions shown in the bottom panels of Fig. 2(A) of the main
text are cropped at the maximum transverse momentum $k_{\mathrm{NA}}$
collected by our microscope and allow to identify the trap depth
$V_{0}=1.4k_{\text{B}}T$, which sets an upper limit for the energy of the
photons that can be contained in the box; for momenta beyond
$k_{0}=\sqrt{2mV_{0}}/\hbar\approx$2.9\text{\,}\mathrm{\SIUnitSymbolMicro
m}$^{-1}$, the signal indeed vanishes. Moreover, we obtain broadband energy-
resolved spectra of the cavity emission by dispersing the collimated light
with a diffraction grating (1200 ll/mm) and detecting its first order on an
ICCD camera; the spectral resolution is $\Delta\nu=$0.6\text{\,}\mathrm{THz}$$
(FWHM), see Fig. 1(C) of the main text. To identify the individual energy
levels of the photon gas in the box potential (Fig. S2(E)), an Echelle grating
with 316 ll/mm is at our disposal; operating in a high diffraction order gives
a spectral resolution $\Delta\nu=$19\text{\,}\mathrm{GHz}$$ (FWHM).

To prepare uniform photon gases in the box and apply potential gradients for
the compression of the optical gas by tilting the cavity mirrors, a high-
precision alignment of the cavity mirrors is required. For example, a cavity
length variation of $1\text{\,}\mathrm{\text{Å}}$ at $q=7$ results in a
potential energy $\hbar(2\pi\times 36)$ GHz, which exceeds the ground state
energy $E_{1,1}=\hbar(2\pi\times 13)$ GHz in a
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ sized box, perturbing the ground
state wave function. To achieve the desired sub-Å precision, both cavity
mirrors are fixed in 2” mirror mounts with complementary piezo actuators
technologies: the first mount (Thorlabs POLARIS-K2S2P) is equipped with two
continuous-voltage piezo actuators, which allow for step-free adjustment of
the mirror tilt angle in $x$ and $y$ direction; the second mount uses piezo
actuators (PI N-470, $20\text{\,}\mathrm{nm}$ step size), which allow for
discrete, but well-countable calibrated mirror tilts. One step induces a
height difference of 10pm over a $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$
sized box, which corresponds to a potential energy difference
$k_{\text{B}}\times$0.16\text{\,}\mathrm{K}$=\hbar(2\pi\times 3.3)$ GHz and is
sufficient for alignment. For the compressibility measurements, we have
verified that the discrete steps provide reproducible tilt angles.

###  .2 Potential creation

The box potentials are created by nanostructuring one of the plane mirrors
prior to its operation in the dye-filled cavity, see Fig. S2 and ref. [22] for
a detailed description of the method. We note that recent other work has
reported an alternative technique to create potentials for microcavity photon
gases, including box potentials, using focused ion beam milling [34]. In our
work, the mirrors used as writing samples contain a
$30\text{\,}\mathrm{nm}$-thin silicon layer below their dielectric Bragg
coating, which enables the structuring process: A $532\text{\,}\mathrm{nm}$
laser beam is focused through the quartz glass substrate onto the silicon
layer (beam diameter $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$), where part of
the light is absorbed and converted into heat. Above a threshold laser power
near $30\text{\,}\mathrm{mW}$, the heating induces a local elevation of the
dielectric layer stack with Gaussian transverse profile of
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ width (FWHM). The process is
attributed to a thermally-induced mechanical stress enhancement between the
alternating Bragg layer materials, which– above a threshold value– leads to a
detachment, often called delamination, of neighboring layers [35]. The 2D box
structures shown in Fig. S2 are created by steering the focused laser beam
across the silicon layer using a galvo scanner. We use unity-aspect-ratio box
potentials of sizes $L=\\{20,30,...,90\\}$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ with a potential wall increasing from 10% to 90% over
$3.7(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ distance, independent of $L$.
The achieved variable structure heights $\Delta D\leq$60\text{\,}\mathrm{nm}$$
are controlled by the laser power. For $\Delta D\geq$25\text{\,}\mathrm{nm}$$,
however, we observe that transverse wave vectors
$k_{0}\geq$3\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ are not confined by
our box; in other words, the spectra show a high-energy cutoff. The underlying
mechanism of the effect is presently not understood and requires further
studies. To mitigate this limitation, we restrict our studies in the present
work to structure heights (trap depths) below $25\text{\,}\mathrm{nm}$
($1.4k_{\text{B}}T$), where the potential depth exhibits a systematic behavior
in accordance with eq. (S1) (see section “Two-dimensional photon dispersion”).

Fig. S2: Nanostructure box potentials. (A) Height profiles of the
nanostructured mirror surface for box potentials with
$L\approx\\{20,40,90\\}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. (B)
Normalized cuts of the mirror surface height $\Delta D$ for all investigated
sizes $L\approx\\{20,30,...,90\\}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
(C) Zoomed-in view of the box slope after shifting the cuts from (B)
horizontally; independent of $L$, all curves exhibit a similar decay from 90
to 10% over $3.7(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (D) The spectral
width of the trapped photon gas emission versus $q$ for two nanostructures
along with parameter-free theory based on eq. (S1) (lines). (E) Lowest-lying
energies of a particle in a box of size
$L=$22(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (blue, top panel) and
$36(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (red, bottom), respectively,
relative to the cavity low-frequency cutoff. The high-resolution spectra
($\Delta\nu=$19\text{\,}\mathrm{GHz}$$) show resonances (circles and diamonds,
extracted from multiple spectra) close to the theory predictions (triangles).

The cavity mirrors (before nanostructuring) exhibit an r.m.s. surface
roughness of $\delta D\approx$0.4\text{\,}\mathrm{\text{Å}}$$ within a
$(20\times 20)$\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{2}$ region. For our
experimental parameters, this corresponds to a ’noise floor’ potential
$\hbar(2\pi\times 15)$\text{\,}\mathrm{GHz}$$ of the same order as the ground
state energy $E_{1,1}$ in a $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$-sized
box, and we observe its influence on the ground mode profiles in boxes with
$L\geq$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ due to the rapidly
decreasing ground state energy, $E_{1,1}\propto 1/L^{2}$. For the used
moderate-height nanostructures, we do not find evidence for a reduction of the
mirror reflectivity; correspondingly, typical photon lifetimes
$\tau\approx$300\text{\,}\mathrm{ps}$$ in the box-cavity are sufficiently long
to ensure thermalization of the photon gas, which is established within
roughly the (spectrally averaged) absorption time
$\Gamma^{-1}\approx$50\text{\,}\mathrm{ps}$$ at our dye concentration of $1$
mmol/l and cutoff wavelength $\lambda_{0}=$583\text{\,}\mathrm{nm}$$.

###  .3 Two-dimensional photon dispersion

Photons trapped in a microcavity environment exhibit a modified dispersion
relation as compared to free photons. The highly anisotropic dimensions of the
resonator ($D_{0}\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
longitudinally versus $L\approx 20-90$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
transversally) freeze out the motional degree of freedom in $z$ direction,
making the photons formally equivalent to massive particles, which are deeply
2D and experience in-plane potentials imposed by the mirror surface height
profiles. Starting from the free photon energy-momentum relation, $E=\hbar
kc/\tilde{n}$ with wave vector $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$, we
apply the paraxial approximation $k_{x,y}\ll k_{z}=\pi q/(D_{0}-\Delta D)$
with local cavity length variation $\Delta D\equiv\Delta D(x,y)$; note that we
define $\Delta D>0$ for delaminated, i.e., elevated mirror surfaces. To first-
order one obtains

$E\approx
m\left(\frac{c}{\tilde{n}}\right)^{2}+\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m}+\left(\frac{c}{\tilde{n}}\right)^{3}\frac{m^{2}}{\pi\hbar
q}\Delta D,$ (S1)

where $m=\pi\hbar q\tilde{n}/D_{0}c$ is the effective photon mass, and the
quadratic dependence on momentum underlines the phenomenology of a massive
particle in 2D. The first term gives the rest energy of the photon, the second
its kinetic energy, and the last term describes a potential energy, which is
repulsive in our experiments with $\Delta D>0$. For typical experimental
parameters, $q=7$ and $\lambda_{0}=$583\text{\,}\mathrm{nm}$$, we get
$m=7.8\times 10^{-36}$\text{\,}\mathrm{kg}$$ and a potential energy
$k_{\text{B}}T$ every $\Delta D=$17\text{\,}\mathrm{nm}$$ of mirror surface
elevation. To validate the trap depth $V_{0}$, we measure the energy width of
the cavity emission spectrum $\Delta E$ for different resonator lengths $q$
and structure heights $\Delta D$. Figure S2(D) shows the confirmed
$1/q$-scaling of the potential depth predicted by eq. (S1) without any free
parameters for two nanostructures. Note that experimentally $q\geq 7$ and
$\Delta D<$25\text{\,}\mathrm{nm}$$, such that the achieved maximum trap depth
is limited to about $1.4k_{\text{B}}T$.

###  .4 Density, critical point, and thermodynamics

We turn to the thermodynamics of the 2D photons in the box, derive their
surface density distribution and define a critical particle number for BEC in
the finite-size system. A particle in a square-box potential of infinite trap
depth and size $L$, is restricted to quantized eigenenergies
$E_{n_{x},n_{y}}=\pi^{2}\hbar^{2}(n_{x}^{2}+n_{y}^{2})/2mL^{2}$ and
eigenfunctions
$\psi_{n_{x},n_{y}}(x,y)=2L^{-1}\sin[n_{x}\pi(x/L+1/2)]\sin[n_{y}\pi(y/L+1/2)]$,
with $|x|,|y|\leq L/2$. A weighted sum yields the surface density of the gas

$n_{\mu,T}(x,y)=\sum_{n_{x},n_{y}}{\frac{|\psi_{n_{x},n_{y}}(x,y)|^{2}}{\exp[(E_{n_{x},n_{y}}-\mu)/k_{\text{B}}T]-1}}.$
(S2)

Here, the chemical potential $\mu\leq E_{1,1}$ determines the total particle
number, $N=\int{\mathrm{d}x\mathrm{d}y\ n_{\mu,T}(x,y)}$, independently of
temperature $T$. Experimentally, we control $\mu$ by varying the excitation
level of the dye molecules via the pump laser power, while
$T=$300\text{\,}\mathrm{K}$$ is fixed by the thermalization of the photons to
the dye heat bath, as observed in the broadband spectral photon distributions
shown in Fig. 1(C) of the main text. Moreover, by performing spectroscopy of
the lowest-lying single-particle states of the photon gas, one can validate
whether the trapping potential is indeed well-described by a box. Figure S2(E)
shows corresponding high-resolution spectra for two boxes of
$L=$22(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$36(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ containing several resonances.
The identified resonance frequencies are in good agreement with theory, and
for the two boxes $E_{1,1}/2\pi\hbar=$42(3)\text{\,}\mathrm{GHz}$$ and
$17(2)\text{\,}\mathrm{GHz}$, respectively, above the low-frequency cavity
cutoff $m(c/\tilde{n})^{2}$. A comparison with thermal energy
$k_{\text{B}}T/2\pi\hbar\approx$7.5\text{\,}\mathrm{THz}$$ indicates that the
eigenenergies– even for small system sizes– can be considered quasi-
continuous.

Equation (S2) allows for an estimate of the critical particle number
$N_{\text{c}}$ for condensation of the gas in the box. We expand the low-
momentum part of the Bose-Einstein distribution
$n_{k}=4\pi\lambda^{-2}/(k^{2}+k_{\mathrm{c}}^{2})$ with thermal wavelength
$\lambda=2\pi\hbar/\sqrt{2\pi mk_{\text{B}}T}$, and find an expression for the
correlation length $\xi=k_{c}^{-1}=\hbar/\sqrt{2m|\mu|}$. Upon increasing $N$,
also $\xi$ increases and eventually spans the system size,
$\xi(N_{\text{c}})=L$. This condition defines a critical particle number

$N_{\text{c}}=-\frac{L^{2}}{\lambda^{2}}\log\left[1-\exp\left(-\frac{\lambda^{2}}{L^{2}}\frac{1+2\pi^{2}}{4\pi}\right)\right],$
(S3)

which gives $N_{\text{c}}\approx 4500$ for
$L=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$\lambda\approx$1.47\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The log-term
gives a critical phase-space density $\mathcal{D}_{\mathrm{c}}\approx 6.1$,
which lies slightly above the measured value of
$\mathcal{D}_{\mathrm{c,exp}}=4.7(5)$, see the inset of Fig. 2(C); for
$L=$48\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, on the other hand, we find
$\mathcal{D}_{\mathrm{c,exp}}=6.2(6)$, in good agreement with the predicted
$\mathcal{D}_{\mathrm{c}}\approx 6.5$. In the limit $L\gg\lambda$, which is
fulfilled for our boxes, eq. (S3) predicts a scaling
$N_{\text{c}}\sim(L/\lambda)^{2}\log(L/\lambda)$, which we confirm
experimentally in Fig. 2(C) of the main text.

The experimentally achievable box potentials differ from the infinite-depth
textbook scenario and exhibit both a finite trap depth and slope width, see
Fig. S2. Additionally, the lowest-energy wave functions are altered in the
presence of a potential gradient $U_{0}/L$ if their energy
$E_{n_{x},n_{y}}<U_{0}$ (see section “Compressibility from center-of-mass
response”). For theoretical predictions, we employ numerical wave functions
obtained by solving the Schrödinger equation with potentials inferred from
cuts through the box along $x$ and $y$, respectively. The density distribution
is then evaluated using eq. (S2). Since the computation is sensitive to
residual detection noise on the measured height profiles of the box
nanostructures, we first fit a Gaussian-convolved step function to the surface
scan data and use it to model the box potential. To account for the
experimentally observed finite trap depth $V_{0}$, we discard all modes with
energy above $1.4k_{\text{B}}T$. A numerical estimate for the critical
particle number, $N_{\text{c}}\approx 3600$, is obtained in analogy to our
measurements in Fig. 2(B). To model the photon gas compressibility, we follow
the same procedure described above, but superimpose the box with a linear
potential $U(x)=U_{0}x/L$.

Fig. S3: Effect of trap depth on caloric properties. Derivative of the
internal energy $U$ with respect to $N_{\text{c}}/N$ for different trap depths
$V_{0}/k_{\text{B}}T=\\{0.6,1.6,4.0\\}$ (left). In the normal-gas phase,
$N_{\text{c}}/N\gg 1$, the curves saturate at values for the specific heat per
particle below the prediction for $V_{0}\rightarrow\infty$ at a value of 1.
The latter is restored as $V_{0}$ increases (right). For our experiments,
$V_{0}\approx 1.4k_{\text{B}}T$.

To conclude our discussion of the thermodynamics of the 2D photon gas in the
box, we focus on the inverse particle number dependence ($N_{\text{c}}/N$) of
the normalized internal energy $U=\langle E\rangle
N_{\text{c}}/(N^{2}k_{\text{B}}T)$ (units of $k_{\text{B}}T$) and its
derivative, both shown in Fig. 3(B) of the main text. With respect to the
cavity low-frequency cutoff $m(c/\tilde{n})^{2}$, the average transverse
energy reads

$\langle
E\rangle=\sum_{n_{x},n_{y}}{\frac{E_{n_{x},n_{y}}-E_{1,1}}{\exp[(E_{n_{x},n_{y}}-\mu)/k_{\text{B}}T]-1}}.$
(S4)

The representation $U(N_{\text{c}}/N)$ gives insight into the effective
temperature behavior of the internal energy; the derivative $\partial
U/\partial(N_{\text{c}}/N)$ then provides a qualitative measure of the
specific heat. While in a uniform gas we have $N_{\text{c}}/N=T/T_{\text{c}}$
with a critical temperature $T_{\text{c}}$, in the box logarithmic finite-size
corrections modify this relation, but $N_{\text{c}}/N$ remains a purposeful
measure for the reduced temperature at the used
$L=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. In the normal-gas limit,
$N_{\text{c}}/N\gg 1$, one usually expects an internal energy (or specific
heat) per particle $U/N=k_{\text{B}}T$ (or $c_{V}/N=k_{\text{B}}$), owing to
the two motional degrees of freedom in the 2D homogeneous system. However, due
to our finite $V_{0}$ the sum in eq. (S4) is truncated and both quantities
take a value of roughly $0.5$ in their respective units. Numerically, this
result is directly obtained from eq. (S4) and shown in Fig. S3 for increasing
$V_{0}$, for which the infinite-depth prediction is gradually restored. For
the analysis of our experimental data, on the other hand, we replace the Bose-
Einstein distribution function in eq. (S4) with the recorded spectra shown in
Fig. 1(C) of the main text. Additionally, the sum iterates over energy
occupations $n(E)$ (obtained from spectrometer pixels) instead of eigenstates.
The photon numbers are calibrated by identifying the spectrum at the critical
point, setting it to $N_{\text{c}}$, and normalizing the total photon number
of all other spectra based on the relative integrated signals. The specific
heat data is obtained by numerical derivation.

Fig. S4: Density distributions of the photon gas in a tilted box. Exemplary
line profiles of the experimental (top row) and theoretical (bottom) density
distributions in a box of size $L=$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
at photon numbers $N\approx\\{3000,4500,6800,8500\\}$ (from left to right).
The tilt potential $U_{0}/k_{\text{B}}\approx$43\text{\,}\mathrm{K}$$ induces
a visible density gradient along the $x$ direction and displaces the cloud’s
center-of-mass, which is more pronounced for larger $N$.

###  .5 Compressibility from center-of-mass response

The isothermal compressibility $\kappa_{T}=-V^{-1}(\partial V/\partial
P)_{T}=n^{-2}(\partial
n/\partial\mu)_{T}=[\exp(n\lambda^{2})-1]/(k_{\text{B}}Tn^{2}\lambda^{2})$ of
the infinite 2D Bose gas, as calculated from the equation of state
$n(\mu)=-\lambda^{-2}\log[1-\exp(\mu/k_{\text{B}}T)]$, diverges in the quantum
degenerate regime towards large densities $n$, as indicated in Fig. 4(B)
(dashed line) of the main text. Experimentally, this pathological feature of
the infinite 2D homogeneous gas cannot be investigated. A finite-size system,
however, enables the study of the highly-compressible ideal gas, where the
energy cost to increase the particle density gradually vanishes,
$\partial\mu/\partial n\rightarrow 0$ and the volume of the gas (here: surface
area) can be reduced by applying infinitesimal pressures. Experimentally, we
probe the compressibility of our photon gases by squashing the photons in the
box. A linear gradient potential $U(x)=U_{0}x/L$ is applied along $x$ by
tilting one of the cavity mirrors (see section “Experimental scheme”), which
results in a spatially uniform force $F_{0}=U_{0}/L$ acting on the photons and
displacing the cloud’s center-of-mass

$\langle x\rangle=L\int_{-L/2}^{L/2}{\mathrm{d}x\ x\frac{n_{\mu,T}(x)}{N}}$
(S5)

with surface density $n_{\mu,T}(x)$ along $x$ at chemical potential $\mu$ and
temperature $T$ and photon number $N=L\int{\mathrm{d}x\ n_{\mu,T}(x)}$. Figure
S4 gives experimental and theoretical density line profiles in a tilted box.
Within certain limits, $\langle x\rangle$ gives a direct measure of
$\kappa_{T}$, as we derive below.

Our approach is based on the local density approximation, which connects $\mu$
of a spatially inhomogeneous system with the chemical potential of the
homogeneous system $\mu_{0}$ via the position-dependent potential energy; for
the case of a linear gradient potential, $\mu(x)=\mu_{0}-U_{0}x/L$. We perform
a variable transformation with $x=(\mu_{0}-\mu)L/U_{0}$ and
$\mathrm{d}x=-L/U_{0}\mathrm{d}\mu$ to eliminate the position dependence and
express the density as $n(\mu)$. From eq. (S5),

$\langle
x\rangle=\frac{L^{3}}{NU_{0}^{2}}\int_{\mu_{0}+U_{0}/2}^{\mu_{0}-U_{0}/2}{\mathrm{d}\mu(\mu-\mu_{0})n(\mu)}\simeq\frac{L^{3}}{NU_{0}^{2}}\int_{\mu_{0}+U_{0}/2}^{\mu_{0}-U_{0}/2}{\mathrm{d}\mu(\mu-\mu_{0})\left[n(\mu_{0})+(\mu-\mu_{0})\left.\frac{\partial
n}{\partial\mu}\right|_{\mu_{0}}\right]},$ (S6)

where we have expanded $n(\mu)$ to first order, indicating how the
compressibility $\kappa_{T}=n^{-2}(\partial n/\partial\mu)_{T}$ enters the
center-of-mass. Integration yields the relation used to extract $\kappa_{T}$:

$\frac{\langle
x\rangle}{L}=-\frac{U_{0}}{12}\frac{L^{2}}{N}\left.\frac{\partial
n}{\partial\mu}\right|_{\mu_{0}}=-\frac{\kappa_{T}n}{12}U_{0}$ (S7)

Note that the linearization in eq. (S6) is valid only if the potential
amplitude $U_{0}$ remains within certain limits, which differ for the normal
($U_{0}\ll\sqrt{40}k_{\text{B}}T$) and the condensed ($U_{0}\ll\sqrt{20}\mu$)
region; for the latter case where $\mu\rightarrow 0$, this requires very small
tilts $U_{0}$. While the general applicability of the method remains valid, it
requires the detection of small displacements of the photon gas at weak tilts.
Figure S5 shows a numerical calculation of the $\kappa_{T}$ extraction using
the center-of-mass method for three values of $U_{0}$ (solid lines),
indicating that small values of $U_{0}$ indeed improve the agreement with the
theory prediction for the 2D Bose gas in an untilted box. Our experimental
data in Fig. 4(A) highlights the change in sensitivity, on the one hand
confirming linearity $\langle x\rangle(U_{0})$ over a wide range of values of
$U_{0}$ in the normal phase, on the other hand revealing a nonlinear response
for the degenerate gas. For the $\kappa_{T}$ extraction, in all cases we
analyze the (narrow) linear response region only.

Fig. S5: Validity of the compressibility extraction. Numerical study of the
compressibility $\kappa_{T}$ extracted from the center-of-mass in tilted boxes
with potentials $U_{0}/k_{\text{B}}=\\{1,3,10\\}$\text{\,}\mathrm{K}$$ (blue,
red, green lines) versus density $n$, along with the theory prediction for a
2D ideal Bose gas in a non-tilted box (solid black) and for the infinite
system (dashed). The extraction in the degenerate region at densities
$n\gtrsim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-2}$ becomes more accurate
for smaller tilts $U_{0}$.  Fig. S6: Equation of state reconstruction by
density profile stitching. (A) The EOS of the infinite 2D homogeneous ideal
Bose gas (solid line) can be reconstructed from the central density of a Bose
gas in a finite-size box (circles) in the non-degenerate regime. The method
works also when extracting the maximum density of the gas in a box
superimposed with a gradient potential $U_{0}x/L$ (diamonds); here, both
infinite-system EOS (dashed) and data were shifted down. Inset: In the quantum
degenerate regime, the reconstruction loses its validity. (B) Illustration of
the numeric stitching process, which is exact for infinitely deep boxes
(numerically confirmed for $V_{0}>4k_{\text{B}}T$, deviations at $\mu=0$ due
to the spatial extent of the ground mode) but deviates for smaller $V_{0}$.
Two examples of calculated density profiles (solid and dashed) used for the
stitching are shown for the $0.5k_{\text{B}}T$ curve.

###  .6 Reconstruction of the equation of state

The equation of state (EOS) in the form $n=f(\mu,T)$ is a monotonous function,
see Fig. S6(A); hence the linearly increasing photon density profiles $n(x)$,
see Fig. S4, observed in the presence of the linear potential allow for a
reconstruction of the EOS. In the experiment, $\mu(x,N)$ is controlled both by
varying the total photon number $N$ or by applying a potential such that
$\mu(x)\rightarrow\mu_{0}-U_{0}x/L$ (local density approximation). A single
density profile partially yields the EOS over an energy range of width
$\mu=U_{0}$. To reconstruct $n(\mu)$ over larger ranges of $\mu$, multiple
profiles with partially overlapping densities, say $n_{1}(x)$ and $n_{2}(x)$,
but different $N$ are recorded at fixed $U_{0}$ and stitched together. A
relative shift by $\Delta\mu$, required to match
$n_{1}(\mu)=n_{2}(\mu+\Delta\mu)$, merges the profiles into a continuous
trace, see Fig. S6(B). Since experimental density profiles are subject to
noise, we use linear fits (excluding boundary of the box) as a reference for
the stitching procedure, which is carried out for $\sim$300 traces. For
$\mu/k_{\text{B}}>-$50\text{\,}\mathrm{K}$$, where the density profiles are
observed to become nonlinear, a corresponding set of profiles containing the
finite-size condensate peak is stitched manually.

We numerically find that due to finite trap depth $V_{0}$, the slope of
$n(\mu)$ increases, see Fig. S6(B). Note that this implies a larger
compressibility $\kappa_{T}=n^{-2}(\partial n/\partial\mu)_{T}$, as observed
in Fig. 4(B) of the main text. By calculating the surface density of the
photon gas in a finite-depth box and comparing it to its infinite-depth
counterpart, we analytically find from Boltzmann statistics that rescaling
$\mu\rightarrow\mu/[1-\exp(-V_{0}/k_{\text{B}}T)]$ in the infinite-system EOS
$n(\mu)$ reproduces the finite-depth result (solid line in Fig. 4(C) of the
main text).