# Unruh-DeWitt Detector Differentiation of Black Holes and Exotic Compact
Objects

Bob Holdom<EMAIL_ADDRESS>Department of Physics, University of
Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7  Robert B. Mann
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, Canada, N2L 3G1  Chen Zhang
<EMAIL_ADDRESS>Department of Physics, University of Toronto, 60
St. George St., Toronto, Ontario, Canada M5S 1A7  Department of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

###### Abstract

We study the response of a static Unruh-DeWitt detector outside an exotic
compact object (ECO) with a general reflective boundary condition in 3+1
dimensions. The horizonless ECO, whose boundary is extremely close to the
would-be event horizon, acts as a black hole mimicker. We find that the
response rate is notably distinct from the black hole case, even when the ECO
boundary is perfectly absorbing. For a (partially) reflective ECO boundary, we
find resonance structures in the response rate that depend on the different
locations of the ECO boundary and those of the detector. We provide a detailed
analysis in connection with the ECO’s vacuum mode structure and transfer
function.

##  I Introduction

The discovery of gravitational waves from compact binary mergers have inspired
many recent studies on exotic compact objects (ECOs), whose defining feature
is their compactness: their radius is very close to that of a black hole with
the same mass whilst lacking an event horizon. Explicit examples of such
compact horizonless objects include gravastars Mazur:2001fv , boson stars
Schunck:2003kk , wormholes, and 2-2 holes Holdom:2016nek . While some
investigations are concerned with the consistency of semi-classical physics
near horizons Baccetti:2018otf ; Terno:2020tsq , most studies are concerned
with finding distinctive signatures from postmerger gravitational wave echoes
Abedi:2016hgu ; Mark:2017dnq ; Conklin:2017lwb ; Cardoso:2019rvt ;
Holdom:2019bdv , in which a wave that falls inside the gravitational potential
barrier reflects off the ECO boundary and returns to the barrier after some
time delay $t_{d}$.

In this paper, we study the response of an Unruh-DeWitt (UDW) detector unruh ;
deWitt  in this context. Assuming no angular momentum exchange, the light-
matter interaction is well described as a local interaction between a scalar
quantum field $\phi(x)$ and a local detector comprising two states:
$\left|0\right\rangle$, with vanishing state energy, and
$\left|\Omega\right\rangle$ of energy $\Omega$, where $\Omega$ may be a
positive or negative real number Funai:2018wqq . If $\Omega>0$ then
$\left|0\right\rangle$ is the ground state. Known as a UDW detector, the
interaction Hamiltonian is

$H_{\text{int}}=\lambda\chi(\tau)\mu(\tau)\phi(\mathsf{x}(\tau))\,,$ (1)

where $\chi(\tau)$ is the switching function encoding the time dependence of
the interaction, whose coupling constant is $\lambda$, and $\mathsf{x}(\tau)$
is the detector’s trajectory with $\tau$ its proper time. The quantity
$\mu({\tau})$ is the monopole operator of the detector
$\mu=e^{i\Omega\tau}\sigma^{+}+e^{-i\Omega\tau}\sigma^{-}$ with
$\sigma^{+}=\left|\Omega\right\rangle\left\langle 0\right|$ and
$\sigma^{-}=\left|0\right\rangle\left\langle\Omega\right|$ the respective
raising and lowering operators.

Perturbing to first-order in $\lambda$, the transition probability is byd

$P(\Omega)=c^{2}{|\langle
0|\mu(0)|\Omega\rangle|}^{2}\mathcal{F}\left(\Omega\right)\ ,$ (2)

where $\mathcal{F}(\Omega)$ is the response function

$\mathcal{F}(\Omega)=\int_{-\infty}^{\infty}\mathrm{d}\tau^{\prime}\,\int_{-\infty}^{\infty}\mathrm{d}\tau^{\prime\prime}\,\chi(\tau^{\prime})\chi(\tau^{\prime\prime})\,\mathrm{e}^{-i\Omega(\tau^{\prime}-\tau^{\prime\prime})}\,W(\tau^{\prime},\tau^{\prime\prime})\,.$
(3)

$W(\tau^{\prime},\tau^{\prime\prime})=\langle
0|\phi(\mathsf{x}(\tau^{\prime}))\phi(\mathsf{x}(\tau^{\prime\prime}))|0\rangle$
is the pull-back of the Wightman distribution on the detector’s worldline.
When the switching function is a Heaviside step function whose switch-on time
is taken to be in the asymptotic past, we can drop the external
$\tau^{\prime}$-integral of (3) by a change of variables and obtain the
transition rate byd ; Louko:2007mu

$\mathcal{\dot{F}}\left(\Omega\right)=\int^{\infty}_{-\infty}\,\mathrm{d}s\,\mathrm{e}^{-i\Omega
s}\,W(s)\,,$ (4)

representing the number of particles detected per unit proper time.

The spacetime of a Schwarzschild black hole in $3+1$ spacetime dimensions has
the metric form (in geometric units where $c=G=1$)

$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right),$
(5)

where $M$ is the black hole mass, with the event horizon at $r_{H}=2M$. Mode
solutions of the Klein-Gordon equation for the massless scalar field $\phi$ in
this background have the form

$\phi=\frac{1}{\sqrt{4\pi\omega}}\frac{\rho_{\omega\ell}(r)}{r}Y_{\ell
m}\left(\theta,\phi\right)\operatorname{e}^{-i\omega t},$ (6)

where $Y_{\ell m}$ is the spherical harmonic function. The radial function
$\rho_{\omega\ell}$ satisfies

$\frac{d^{2}\rho_{\omega\ell}}{dr^{*2}}+\left\\{\omega^{2}-V(r)\right\\}\rho_{\omega\ell}=0\,,$
(7)

where the tortoise coordinate $r^{*}$ satisfies $dr^{*}/dr=(1-2M/r)^{-1}$. The
gravitational potential is

$V(r)=\left(1-\frac{2M}{r}\right)\left[\frac{\ell(\ell+1)}{r^{2}}+\frac{2M}{r^{3}}\right]$
(8)

for the scalar field perturbation of angular momentum $l$. $V(r)$ has a peak
at $r_{\rm peak}$, which is close to $r=3M$. Note that we can rescale
$\bar{r}=r/(2M),\,\bar{\omega}=2M\omega$ so that the dependence on $M$ in all
equations is removed.

We model the ECO as an exterior Schwarzschild spacetime patched to a
spherically symmetric interior metric at radius $r=r_{0}$, with
$r_{0}/r_{H}-1\ll 1$, with a boundary condition at $r_{0}$ parameterized by
the reflection coefficient $R^{W}$. Typical quantum gravity arguments suggest
that $r_{0}-r_{H}$ is related to the Planck length, as for the brick wall
model tHooft:1984kcu . The space between $r_{0}$ and $r_{\rm peak}$ acts as a
cavity, and has an associated time delay111LIGO data favors the time delay
$t_{d}\approx 600\,M$ Holdom:2019bdv , which sets a preferred value for
$r^{*}_{0}\approx-149.2$.

$t_{d}\equiv 2(r^{*}_{\rm peak}-r^{*}_{0})$ (9)

expressed in tortoise coordinates. An ECO boundary closer to the would-be
horizon (i.e., smaller $r^{*}_{0}$) maps to a larger time delay $t_{d}$. The
tortoise coordinate can be chosen such that $r^{*}_{\rm peak}\approx 0$.

##  II Formalism

In this section, we review the formalism for the calculation of the detector’s
response rate in the black hole context Hodgkinson:2014iua , which we will
then extend to the ECO scenario.

###  II.1 Black hole

For a black hole, the gravitational potential vanishes at $r^{*}\to\pm\infty$
such that Eq. (7) has two simple asymptotic solutions: $\rho_{\omega
l}^{\textrm{up}}(r^{*}\to\infty)=e^{i\omega r^{*}}$, $\rho_{\omega
l}^{\textrm{in}}(r^{*}\to-\infty)=e^{-i\omega r^{*}}$. The normalized field
modes

$\displaystyle\operatorname{\Phi^{\text{in}}_{\omega\ell}}$
$\displaystyle=T_{\omega l}^{\textrm{in}}\frac{\rho_{\omega
l}^{\textrm{in}}}{r},\,\quad\operatorname{\Phi^{\text{up}}_{\omega\ell}}$
$\displaystyle=T_{\omega l}^{\textrm{up}}\frac{\rho_{\omega
l}^{\textrm{up}}}{r}\,,$ (10)

satisfy the respective boundary conditions

$\displaystyle\Phi_{\omega l}^{\textrm{in}}r$ $\displaystyle\to$
$\displaystyle\left\\{\begin{array}[]{cc}\,T_{\omega
l}^{\textrm{in}}e^{-i\omega r^{*}},&r^{*}\to-\infty\\\ e^{-i\omega
r^{*}}+R_{\omega l}^{\textrm{in}}e^{i\omega
r^{*}},&r^{*}\to\infty\end{array}\right.$ (13) $\displaystyle\Phi_{\omega
l}^{\textrm{up}}r$ $\displaystyle\to$
$\displaystyle\left\\{\begin{array}[]{ll}R_{\omega
l}^{\textrm{up}}\,e^{-i\omega r^{*}}+e^{i\omega r^{*}},&r^{*}\to-\infty\\\
T_{\omega l}^{\textrm{up}}\,e^{i\omega
r^{*}},&r^{*}\to\infty\end{array}\right.$ (16)

such that $\Phi_{\omega l}^{\textrm{in}}$ represents a wave mode incident on
the potential barrier from spatial infinity, giving rise to a reflected wave
of amplitude $R_{\omega l}^{\rm in}$ and to a transmitted wave of amplitude
$T_{\omega l}^{\textrm{in}}$. Conversely, $\Phi_{\omega l}^{\textrm{up}}$
represents a wave incident from the past event horizon and is partially
reflected by a factor of $R_{\omega l}^{\rm up}$ and partially transmitted
with an amplitude $T_{\omega l}^{\textrm{up}}$.

From the Wronskian relations, one can derive

$\displaystyle T_{\omega l}^{\textrm{in}}$
$\displaystyle=\frac{2i\omega}{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\operatorname{\rho^{\text{up}}_{\omega\ell}}]},\,\,\,R^{\text{in}}_{\omega\ell}=-\frac{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\rho^{\text{up}\,*}_{\omega\ell}]}{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\operatorname{\rho^{\text{up}}_{\omega\ell}}]},$
(17)

and222Note that some literature Mark:2017dnq ; Conklin:2017lwb ;
Cardoso:2019rvt  denote $T^{\text{up}}_{\omega\ell}$ as $T_{\rm BH}$, and
$R^{\text{up}}_{\omega\ell}$ as $R_{\rm BH}$.

$T^{\text{up}}_{\omega\ell}=T_{\omega
l}^{\textrm{in}},\quad,\,\,R^{\text{up}}_{\omega\ell}=-\frac{T^{\text{in}}_{\omega\ell}}{T^{\text{in}*}_{\omega\ell}}R^{\text{in}*}_{\omega\ell}$
(18)

where $W$ denotes the Wronskian $W[a,b]=ab^{\prime}-ba^{\prime}$. Energy
conservation furthermore implies
$|T^{\text{up}}_{\omega\ell}|^{2}+|R^{\text{up}}_{\omega\ell}|^{2}=1$, and
$|T^{\text{in}}_{\omega\ell}|^{2}+|R^{\text{in}}_{\omega\ell}|^{2}=1$ .

The Boulware state has positive-frequency modes with respect to the physical
Schwarzschild time $t$. It reduces to the Minkowski vacuum at spatial infinity
but induces a diverging stress-energy near the black hole horizon. The quantum
scalar field in this state can be expanded as

$\displaystyle\psi=\sum^{\infty}_{\ell=0}\sum^{+\ell}_{m=-\ell}\int^{\infty}_{0}\,\mathrm{d}\omega\,\Bigl{(}a^{\text{up}}_{\omega\ell
m}u^{\text{up}}_{\omega\ell m}+a^{\text{in}}_{\omega\ell
m}u^{\text{in}}_{\omega\ell m}\Bigr{)}+\text{h.c.}\,,$ (19)

where $a^{\text{(in, up)}}_{\omega\ell m}$ are the respective annihilation
operators, defining the Boulware state $a^{\text{(in, up)}}_{\omega\ell
m}|0_{B}\rangle=0$, and

$\displaystyle u^{\text{(in, up)}}_{\omega\ell m}(\mathsf{x})$
$\displaystyle=\frac{1}{\sqrt{4\pi\omega}}\Phi^{\text{(in,
up)}}_{\omega\ell}(r)Y_{\ell m}(\theta,\phi)\operatorname{e}^{-i\omega t}$
(20)

are the respective field bases. $\operatorname{\Phi^{\text{in}}_{\omega\ell}}$
and $\operatorname{\Phi^{\text{up}}_{\omega\ell}}$ are the normalized field
modes obtained from Eq. (10) with the normalization factor from Eq. (17) and
Eq. (18).

Therefore, for a static detector at a coordinate distance $R$ away from the
black hole, the Wightman function is333For a static detector,
$r=r^{\prime}=R$, $t-t^{\prime}=\Delta t=\Delta\tau/\sqrt{1-2M/R}$, and we can
take $\theta=\theta^{\prime}=\phi=\phi^{\prime}=0$ without loss of generality.

$\displaystyle W(\mathsf{x},\mathsf{x}^{\prime})$ $\displaystyle=\langle
0_{B}|\psi(\mathsf{x})\psi(\mathsf{x}^{\prime})|0_{B}\rangle$
$\displaystyle=\sum^{\infty}_{\ell=0}\int^{\infty}_{0}\,\mathrm{d}\omega\,\frac{\left(2\ell+1\right)}{16\pi^{2}\omega}\operatorname{e}^{-i\omega\Delta\tau/\sqrt{1-2M/R}}\left(|\operatorname{\Phi^{\text{up}}_{\omega\ell}}(R)|^{2}+|\operatorname{\Phi^{\text{in}}_{\omega\ell}}(R)|^{2}\right)\,,$
(21)

with the field in the Boulware state. Substituting (21) into the detector
response rate (4) and commuting the $s$\- and $\omega$-integrals, we obtain

$\displaystyle\mathcal{\dot{F}}_{\rm
Boulware}\left(\Omega\right)=\frac{\Theta(-\Omega)}{8\pi|\Omega|}\sum^{\infty}_{l=0}\left(2\ell+1\right)\left(|\Phi^{\text{up}}_{\tilde{\omega}\ell}(R)|^{2}+|\Phi^{\text{in}}_{\tilde{\omega}\ell}(R)|^{2}\right)\,,$
(22)

where $\tilde{\omega}:=\Omega\sqrt{1-2M/R}$. Note that for the Boulware state,
the response rate vanishes for positive energies, meaning that the detector
can only de-excite, or alternatively $\Omega$ can only take negative values.
In the following study we choose the variable $\Omega/T_{\rm loc}=8\pi
M\tilde{\omega}$ as the effective gap parameter, where $T_{\text{loc}}$ is the
local Hawking temperature, $T_{\text{loc}}=1/(8\pi M\sqrt{1-2M/R})$.

The other types of quantum vacuum fields for Schwarzchild spacetime, namely
Hartle-Hawking state and Unruh state, have regular stress-energy across the
future horizon (with Hartle-Hawking regular across the past horizon as well).
As we explicitly show in Appendix A, the de-excitation rate of a static UDW
detector with the field in the Hartle-Hawking (HH) and Unruh vacuum states
overlap with that of the Boulware vacuum for energy gap $\Omega/T_{\rm
loc}\lesssim-5$, and only deviates to a small finite value at zero gap limit
($\mathcal{\dot{F}}_{\rm HH}|_{\Omega/T_{\rm loc}\to 0}\lesssim 0.01$), as
shown in Figure 8b. Thus, we focus on the Boulware vacuum case in our
following discussion.

###  II.2 ECO

An ECO has no event horizon and thus the vacuum must be of the Boulware type.
Since only the in-modes satisfy the boundary condition at $r^{*}_{0}$, we must
drop the up-mode contribution from the discussion above, yielding

$\displaystyle\psi=\sum^{\infty}_{\ell=0}\sum^{+\ell}_{m=-\ell}\int^{\infty}_{0}\,\mathrm{d}\omega\,\Bigl{(}a^{\text{in}}_{\omega\ell
m}u^{\text{in}}_{\omega\ell m}\Bigr{)}+\text{h.c.}\,,$ (23)

instead of Eq. (19), where

$\displaystyle u^{\text{in}}_{\omega\ell m}(\mathsf{x})$
$\displaystyle=\frac{1}{\sqrt{4\pi\omega}}\Phi^{\text{in}}_{\omega\ell}(r)Y_{\ell
m}(\theta,\phi)\operatorname{e}^{-i\omega t}\,,$ (24)

with

$\operatorname{\Phi^{\text{in}}_{\omega\ell}}=T_{\omega
l}^{\textrm{in}}\frac{\rho_{\omega l}^{\textrm{in}}}{r}.$ (25)

$\rho_{\omega l}^{\textrm{in}}$ is the in mode satisfying Eq. (7) with the
boundary conditions parametrized by the reflectivity $R^{W}$ of the ECO
Conklin:2017lwb

$\displaystyle\Phi_{\omega l}^{\textrm{in}}r$ $\displaystyle\to$
$\displaystyle\left\\{\begin{array}[]{cc}e^{-i\omega r^{*}_{0}}\,T_{\omega
l}^{\textrm{in}}\left(e^{-i\omega(r^{*}-r^{*}_{0})}+R^{\rm
W}\,e^{i\omega(r^{*}-r^{*}_{0})}\right),&r^{*}\to r^{*}_{0}\\\ e^{-i\omega
r^{*}}+R_{\omega l}^{\textrm{in}}e^{i\omega
r^{*}}&r^{*}\to\infty\end{array}\right.$ (28)

in contrast to (13), where $R^{\rm W}=1$ ($R^{\rm W}=-1$) corresponds to the
Neumann (Dirichlet) boundary condition, whereas $R^{W}=0$ means the ECO is
perfectly absorbing. From the solution
$\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}$ to Eq. (7) satisfying
the appropriate boundary condition at $r^{*}_{0}$,
$T^{\text{in}}_{\omega\ell}$ and $R^{\text{in}}_{\omega\ell}$ are obtained by
insisting on the correct behaviour at large $r^{*}$. We obtain

$T^{\text{in}}_{\omega\ell}=\left.\frac{2\omega e^{-i\omega
r^{*}}}{\omega\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}+i\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}^{\prime}}\right|_{r^{*}\to\infty},\quad
R^{\text{in}}_{\omega\ell}=\left.\frac{e^{-2i\omega
r^{*}}(\omega\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}-i\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}^{\prime})}{\omega\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}+i\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}^{\prime}}\right|_{r^{*}\to\infty},$
(29)

where in practice $\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}$ and
its derivatives
$\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}^{\prime}$ are evaluated
at a sufficiently large $r^{*}$. As shown in Appendix C, we can introduce
another set of ECO solutions similar to (16) as an analog of the up-mode
$\rho_{\rm up}$ in the black hole setting, but only for mathematical use since
only the in-modes satisfy the boundary condition at $r^{*}_{0}$. In this way,
it is straightforward to show that the transmission and reflection
coefficients of the ECO can be cast into the same form as Eq. (17). Energy
conservation implies

$|T^{\text{in}}_{\omega\ell}|^{2}(1-(R^{W})^{2})=1-|R^{\text{in}}_{\omega\ell}|^{2},$
(30)

so that $|R^{\text{in}}_{\omega\ell}|=1$ for $R^{W}=\pm 1$.

Then, with the normalized $\operatorname{\Phi^{\text{in}}_{\omega\ell}}$
field, we find

$W_{\rm
ECO}(\mathsf{x},\mathsf{x}^{\prime})=\sum^{\infty}_{\ell=0}\int^{\infty}_{0}\,\mathrm{d}\omega\,\frac{\left(2\ell+1\right)}{16\pi^{2}\omega}\operatorname{e}^{-i\omega\Delta\tau/\sqrt{1-2M/R}}|\operatorname{\Phi^{\text{in}}_{\omega\ell}}(R)|^{2}\,,$
(31)

for the Wightman function, and in turn

$\displaystyle\mathcal{\dot{F}_{\rm
ECO}}\left(\Omega\right)=\frac{\Theta(-\Omega)}{8\pi|\Omega|}\sum^{\infty}_{l=0}\left(2\ell+1\right)|\Phi^{\text{in}}_{\tilde{\omega}\ell}(R)|^{2}\,,$
(32)

for the response rate.

##  III Results

The most distinctive feature in the comparison of an ECO to a black hole is
the resonance spectrum of $T^{\text{in}}_{\omega\ell}$ (i.e., the transfer
function). In Figure 1, we show the $T^{\text{in}}_{\omega\ell}$ spectrum of
two benchmark models with small $(t_{d}=66.5\,M)$ and large $(t_{d}=160M)$
time delays.

Figure 1: $\log_{10}|T^{\text{in}}_{\omega\ell}|$ for (a) small
($t_{d}=66.5M$) and (b) large ($t_{d}=160M$) time delays with
$R^{W}=(0,-1,0.5,1)$ for $l=0\sim 5$.

The position and width of these resonances are determined from the real and
imaginary parts of the complex poles of $T^{\text{in}}_{\omega\ell}$, which
are affected by both the ECO boundary condition and the gravitational
potential. The spacings of the resonance spikes is related to the time delay
such that444Note that the actual $\Delta\omega$ has a small frequency
dependence resulting from the finite width of the gravitational potential
around $r^{*}_{\rm peak}$.

$\Delta{\omega}=\frac{2\pi}{t_{d}},\quad\Delta{\Omega/T_{\rm loc}}=8\pi
M\Delta{\tilde{\omega}}=\frac{16\pi^{2}M}{t_{d}}.$ (33)

Therefore, the resonance spacing is smaller for a larger time delay (i.e.,
when $r_{0}$ is closer to the would-be horizon). For a zero $R^{W}$,
$T^{\text{in}}_{\omega\ell}$ has no resonances at all since the boundary is
perfectly absorbing, and thus is equivalent to the
$T^{\text{in}}_{\omega\ell}$ of a black hole. Differing time delays in this
case do not affect the result since different locations of a perfect-absorbing
boundary are equivalent for the wave modes. As we will demonstrate, these
resonance structures will also show up in the UDW detector’s response rate,
yielding distinct structures in contrast to the black hole.

Using (32), the response rate can be calculated with the $l$ sum truncated to
$l_{\rm max}$ such that the contribution with $l=l_{\rm max}+1$ is negligible.
It turns out that $l_{\rm max}$ increases for a larger radial distance $R$ and
a larger $|\Omega/T_{\rm loc}|$, and is insensitive to the boundary condition
and time delay changes. For example, when $|\Omega/T_{\rm loc}|$ goes to 20,
$l_{\rm max}\approx 5$ for $R=4M$, and $l_{\rm max}\approx 15$ for $R=15M$. We
give an analytic approximation of $l_{\rm max}$ in Appendix B. To illustrate
some general features, we present in Figures 2 the response rate for the
previous two benchmark examples with different ($t_{d}$,$R$).

Figure 2: $M\dot{\mathcal{F}}$ as a function of $\Omega/T_{\text{loc}}$ for a
static detector at $R=2.5M,4M,15M$ (top to bottom) above an ECO of (left)
small ($t_{d}=66.5\,M$), and (right) large ($t_{d}=160M$) time delays with
$R^{W}=(0,-1,0.5,1)$. The orange line is for the black hole case with Boulware
vacuum.

Note that the first row of Figure 2 is for the case where the static detector
is inside the cavity ($R=2.5M$), in contrast to the second and third rows
where the detector is placed outside ($R=4M$ and $R=15M$).

We easily see the striking resonance patterns of the response rate in the
first row of Figure 2, which match those of $|T^{\text{in}}_{\omega\ell}|$
shown in Figure 1 that are independent of $R$; cases with the smaller time
delay have resonances more sparse in distribution, and vice versa. Comparing
to those of the second and third rows, we see as the distance gets larger, the
resonances become more suppressed in size. We give a detailed analysis in the
next section.

In the second and thirds rows of Figure 2, we see a general trend for the
detector’s response rate to grow roughly linearly with $|\Omega/T_{\rm loc}|$.
But in the second row, we see that an ECO with $R^{W}=0$ has a response rate
that is relatively lower than that of the black hole, even though the
boundaries of both objects are perfectly absorbing. Comparing Eq. (18) and Eq.
(26), we see that this is because the ECO case has no up-mode contribution for
any $R^{W}$. By comparison to the second row, the third row then shows that
the extra up-mode contribution in the black hole response becomes relatively
smaller for increasing $R$. In the large $R$ limit all the response rates
approach the Minkowski vacuum state result, which is proportional to
$-\Theta(-\Omega)\Omega$.

##  IV Analysis

From Eq. (28), we see the behaviour of
$\operatorname{\Phi^{\text{in}}_{\omega\ell}}$ and the resulting
$\mathcal{\dot{F}_{\rm ECO}}$ is dominated by $T^{\text{in}}_{\omega\ell}$ at
small $r$ and by $R^{\text{in}}_{\omega\ell}$ at large $r$ up to some complex
phase factors. To see this more explicitly, we can take absolute value of
$\Phi_{\omega l}^{\textrm{in}}$, obtaining

$\displaystyle|\Phi_{\omega l}^{\textrm{in}}|r$ $\displaystyle\to|T_{\omega
l}^{\textrm{in}}|\sqrt{1+R^{W^{2}}+2R^{W}\cos[2\omega(r^{*}-r^{*}_{0})]},\text{
as }r^{*}\to r^{*}_{0}$ (34) $\displaystyle|\Phi_{\omega l}^{\textrm{in}}|r$
$\displaystyle\to\sqrt{1+|R^{\text{in}}_{\omega\ell}|^{2}+2|R^{\text{in}}_{\omega\ell}|\cos(2\omega
r^{*}+\theta)},\quad\,\,\,\,\text{ as }r^{*}\to\infty.$ (35)

from Eq. (28), where
$\theta=\arccos(\Re(R^{\text{in}}_{\omega\ell})/|R^{\text{in}}_{\omega\ell}|)$.
The value of $\theta$ can also have large fluctuations at the location of the
$T^{\text{in}}_{\omega\ell}$ resonances, as the benchmark model in Figure 3a
shows. We see that $|\operatorname{\Phi^{\text{in}}_{\omega\ell}}|r$ goes to
$|T_{\omega l}^{\textrm{in}}|(1+R^{W})$ when approaching the ECO boundary, and
at large $r$ to some value between $(1-|R^{\text{in}}_{\omega\ell}|)$ and
$(1+|R^{\text{in}}_{\omega\ell}|)$, as the example in Figure 3b illustrates.
From Eq. (30), we see that $|R^{\text{in}}_{\omega\ell}|\leq 1$ for $R^{W}\leq
1$, but $|T^{\text{in}}_{\omega\ell}|$ can display relatively large resonance
peaks. This, in conjunction with (34) and (35), partially explains why
$|\operatorname{\Phi^{\text{in}}_{\omega\ell}}|r$ and the corresponding
contribution in $\mathcal{\dot{F}}$ at given $l$ exhibit smaller resonances at
a larger distance.

Figure 3: (a)
$\theta=\arccos(\Re(R^{\text{in}}_{\omega\ell})/|R^{\text{in}}_{\omega\ell}|)$
and (b) $|\operatorname{\Phi^{\text{in}}_{\omega\ell}}|R$ ($R=15M$) for a time
delay $t_{d}=160M$ with $R^{W}=-1$ for $l=0$ (black) and $l=1$ (red). Solid
lines are the exact results from Eq. (25), while dashed lines denote the
approximation from Eq. (35).

Note that Eqs. (34) and (35) give a good match to the exact result (25) for
$|r^{*}-r^{*}_{\rm peak}|$ sufficiently large, that is provided $\omega^{2}$
is larger than $V(r)$, thus mapping to the parameter space of large $\omega$
and small $l$. These features can also be seen in the example shown in Fig
(3)b, where the exact result of
$|\operatorname{\Phi^{\text{in}}_{\omega\ell}}|r$ at $l=0$ matches the
approximate form well, with resonances fluctuating between 0 and +2, while all
larger $l\geq 1$ show notable deviation at small $|\Omega/T_{\rm loc}|$
$(=8\pi M\,|\omega|)$.

As Figure 1 shows, the resonance structures of $T^{\text{in}}_{\omega\ell}$
(and thus the similar structure of $R^{\text{in}}_{\omega\ell}$) shift to a
larger $|\Omega/T_{\rm loc}|$ for a larger $l$. As $r$ increases, $l_{\rm
max}$ also becomes large (Appendix B), so that the $l$-summed results have
more dependence on large $l$ yielding a milder resonance structure at a given
frequency. This also illustrates why the $\mathcal{\dot{F}}$ results in
Figures 2 show milder resonances at a larger distance for a given
$\Omega/T_{\rm loc}$. This further implies that we can identify a clear
resonance structure at very large $R$ provided the time delay is large so that
the resonances are located at small frequencies where $l_{\rm max}$ is small.
For illustration, we give a benchmark example of a very large time delay
$(t_{d}=600M)$ with the detector at a very large distance $R=150M$ in Figure
4. We can see that the resonances of $|T^{\text{in}}_{\omega\ell}|$ at small
$l$ in Figure 4a are clearly reflected in the response rate result shown in
Figure 4b, even at this large $R$.

Figure 4: (a) $\log_{10}|T^{\text{in}}_{\omega\ell}|$ and (b)
$M\dot{\mathcal{F}}$ for a large time delay $t_{d}=600M$ with $R^{W}=-1$. The
$M\dot{\mathcal{F}}$ result has the static detector at a large distance
$R=150M$ with $l$ summed from zero to different $l_{\rm max}$.

##  V Case $R^{W}=-0.98$

For physical realizations of the ECO model, $|R^{W}|$ can be taken to be
slightly less than one to account for a small amount of damping. Damping may
be needed in the case of spinning ECOs to avoid an instability. In the case of
a static 2-2-hole the damping may be explicitly calculated, and the resulting
frequency dependent damping is largest for small frequencies Holdom:2020onl .
For our static ECO study we shall illustrate the effect of damping by choosing
a constant value $R^{W}=-0.98$. We show in Figure 5a the
$T^{\text{in}}_{\omega\ell}$ function with large time delay $t_{d}=160M$, with
the related response rate at small distance ($R=4M$) depicted in Figure 5b.
Comparing to the $R^{W}=-1$ case shown in Figure 1b and the right figure in
the second row of Figure 2, we observe that the response rate for
$R^{W}=-0.98$ is close to the $R^{W}=-1$ case, except that the resonance
heights of $T^{\text{in}}_{\omega\ell}$ and $\mathcal{\dot{F}_{\rm ECO}}$ are
both noticeably milder than those for $R^{W}=-1$. It turns out that this
feature is also present in all other examples we have examined, including the
large-$R$ cases.

Figure 5: (a) $\log_{10}|T^{\text{in}}_{\omega\ell}|$ and (b)
$M\dot{\mathcal{F}}$ ( $R=4M$) as a function of $\Omega/T_{\text{loc}}$ for a
time delay $t_{d}=160M$ with $R^{W}=-0.98$.

For the $R^{W}=-0.98$ examples at large $R$, the exact result of
$|\Phi_{\omega l}^{\textrm{in}}|r$ can be approximated by Eq. (35). From (30),
when $|R^{W}|$ is close but not equal to unity, the resonances of
$T^{\text{in}}_{\omega\ell}$ can show up as dips in
$|R^{\text{in}}_{\omega\ell}|$, as shown in Figure 6 for a benchmark example
with a time delay $t_{d}=160M$. These dips can potentially translate to large
dips in $M\dot{\mathcal{F}}$ at large $R$, considering Eq. (35).

Figure 6: (a) $|T^{\text{in}}_{\omega\ell}|$ and
(b)$|R^{\text{in}}_{\omega\ell}|$ for a time delay $t_{d}=160M$ with
$R^{W}=-0.98$.

However, our numerical results indicate that the resonance heights of
$\mathcal{\dot{F}_{\rm ECO}}$ for $R^{W}=-0.98$ are noticeably milder than
those for $R^{W}=-1$ even at large $R$. This relates to the fact that, despite
the large dips in $|R^{\text{in}}_{\omega\ell}|$ for $R^{W}=-0.98$, it has
either the same or smaller fluctuations in $\theta$ than for $R^{W}=-1$, as a
comparison of Figure 7a with Figure 3a shows.

Figure 7: (a)
$\theta=\arccos(\Re(R^{\text{in}}_{\omega\ell})/|R^{\text{in}}_{\omega\ell}|)$
and (b) $|\operatorname{\Phi^{\text{in}}_{\omega\ell}}|R$ ($R=15M$) for a time
delay $t_{d}=160M$ with $R^{W}=-0.98$ at $l=0$ (black) and $l=1$ (red).

##  VI Summary

We have calculated the response rate of an Unruh-DeWitt detector placed away
from a general exotic compact object (ECO) having an exterior spacetime
identical to that of a Schwarzschild black hole. Due to the absence of an
event horizon and the corresponding up-modes in the vacuum fields, a perfectly
absorbing ECO ($R^{W}=0$) spacetime has a smaller detector response rate than
that of the same-mass black hole, with a smaller ratio for a larger detector
gap and a smaller distance. The cavity between an ECO’s reflective boundary
($R^{W}\neq 0$) and the gravitational potential peak determines the time delay
$t_{d}$ and results in distinct resonance structures, connected to the poles
of the transfer function $T^{\text{in}}_{\omega\ell}$. Thus, the observed
spacing of the resonance structures tells us about the fundamental length
scale at which the ECO deviates from a black hole.

Finally, we presented a detailed analysis on the large-distance behaviour of
the resonance structures and the damping effect in the response rate, and
showed how a large cavity size can increase the chances of observing resonance
structure at large distances.

Our study provides a new alternative way to differentiate a static black hole
from an ECO with a static UDW detector. The obvious next generalization is to
study what happens to a spinning ECO. Likewise, it will be interesting to see
how the results change if we consider detectors in motion, such as in geodesic
orbits. We leave these investigations for future work.

##  Appendix A Hartle-Hawking and Unruh vacuums

The Hartle-Hawking vacuum state has positive-frequency modes with respect to
the horizon generators in Kruskal coordinates. It is regular across the
horizon, and features outgoing and ingoing thermal fluxes from the past event
horizon and the past null infinity, respectively. Accordingly, the response
rate of a static detector above the black hole with fields in Hartle-Hawking
(HH) state are Hodgkinson:2014iua :

$\displaystyle\mathcal{\dot{F}}_{\rm
HH}\left(\Omega\right)=\frac{1}{8\pi\Omega}\frac{1}{\operatorname{e}^{\Omega/T_{\text{loc}}}-1}\sum^{\infty}_{l=0}\left(2\ell+1\right)\left(|\Phi^{\text{up}}_{\tilde{\omega}\ell}(R)|^{2}+|\Phi^{\text{in}}_{\tilde{\omega}\ell}(R)|^{2}\right)\,,$
(36)

Comparing to the Boulware vacuum case Eq. (22), we have

$\frac{\mathcal{\dot{F}}_{\rm HH}}{\mathcal{\dot{F}}_{\rm
Boulware}}=\frac{1}{\Theta(-\Omega)(1-\operatorname{e}^{\Omega/T_{\text{loc}}})},$
(37)

which we plot in Figure 8a. We see that the de-excitation rate of a UDW
detector in the Hartle-Hawking vacuum becomes coincident with that of the
Boulware vacuum for a large energy gap, except for an enhancement at the small
energy gap region, resulting a small finite response rate at zero frequency
limit ($\mathcal{\dot{F}}_{HH}|_{\Omega/T_{\rm loc}\to 0}\sim 0.01$), as shown
in Figure 8b for two benchmark examples.

Figure 8: (a)The ratio $\mathcal{\dot{F}}_{HH}/\mathcal{\dot{F}}_{\rm
Boulware}$ and (b) $M\mathcal{\dot{F}}_{\rm Boulware}$ (solid orange) vs
$M\mathcal{\dot{F}}_{\rm HH}$ (dashed blue) and $M\mathcal{\dot{F}}_{\rm
Unruh}$ (dashed black) for a static detector at $R=4M$ (top) and $R=15M$
(bottom) outside a black hole.

The Unruh vacuum has positive-frequency up-modes with respect to the horizon
generators in Kruskal coordinates, and positive-frequency in-modes with
respect to the physical Schwarzschild time $t$. The response rate for a static
detector with fields in Unruh vacuum state thus is Hodgkinson:2014iua :

$\displaystyle\mathcal{\dot{F}}_{\rm
Unruh}\left(\Omega\right)=\frac{1}{8\pi\Omega}\sum^{\infty}_{l=0}\left(2\ell+1\right)\left(\frac{1}{\operatorname{e}^{\Omega/T_{\text{loc}}}-1}|\Phi^{\text{up}}_{\tilde{\omega}\ell}(R)|^{2}-\Theta(-\Omega)|\Phi^{\text{in}}_{\tilde{\omega}\ell}(R)|^{2}\right)\,,$
(38)

Comparing Eq. (38) to Eq. (36) and Eq. (22), we see that the response rate for
Unruh vacuum interpolates between those of Hartle-Hawking and Boulware cases.
One can also see this in Figure 8b for two benchmark examples.

##  Appendix B Determination of $l_{\rm max}$

In section III, we calculated the response rate using Eq. (32), where the $l$
sum is taken from zero to $l_{\rm max}$ such that the contribution with
$l=l_{\rm max}+1$ is negligible.

We find that the $l_{\rm max}$ can be well approximated analytically from the
relation

$\omega^{2}-V(l_{\rm max},R)\approx 0$ (39)

since the wave amplitude is heavily suppressed in the region of large
$(V(R)-\omega^{2})$. This gives

$l_{\rm
max}\approx\left\lceil\frac{1}{2}\left(-1+\sqrt{1-\frac{8M}{R}+\frac{4\omega^{2}R^{3}}{R-2M}}\right)\right\rceil$
(40)

for given $\omega$ and $R$, where the outer bracket $\lceil\cdots\rceil$
denotes the ceiling function. Then utilizing $\Omega/T_{\rm loc}=8\pi
M\tilde{\omega}$ we obtain $l_{\rm max}(\Omega/T_{\rm loc},R)$ accordingly, as
shown in Figure 9. We observe the general feature that we only need to sum
small $l$ values for small frequency or small $R$ region.

Figure 9: $l_{\rm max}$ for a static detector at distance $R$ with (a)
$\Omega/T_{\rm loc}=-0.5$ and (b) $\Omega/T_{\rm loc}=-20$.

##  Appendix C Wronskian relations for ECO

Here we derive the expressions of the transmission and reflection coefficients
from the Wronskian relations. We can construct two sets of solutions,
satisfying

$\displaystyle\Phi_{\omega l}^{\textrm{in}}r$ $\displaystyle\to$
$\displaystyle\left\\{\begin{array}[]{cc}e^{-i\omega r^{*}_{0}}\,T_{\omega
l}^{\textrm{in}}\left(e^{-i\omega(r^{*}-r^{*}_{0})}+R^{\rm
W}\,e^{i\omega(r^{*}-r^{*}_{0})}\right),&r^{*}\to r^{*}_{0}\,,\\\ e^{-i\omega
r^{*}}+R_{\omega l}^{\textrm{in}}e^{i\omega
r^{*}},&r^{*}\to\infty\,,\end{array}\right.$ (43) $\displaystyle\Phi_{\omega
l}^{\textrm{up}}r$ $\displaystyle\to$
$\displaystyle\left\\{\begin{array}[]{ll}R_{\omega
l}^{\textrm{up}}\,e^{-i\omega r^{*}}+e^{i\omega r^{*}},&r^{*}\to
r^{*}_{0}\,,\\\ T_{\omega l}^{\textrm{up}}\,e^{i\omega
r^{*}},&r^{*}\to\infty\,,\end{array}\right.$ (46)

Note that $T^{\text{up}}_{\omega\ell}=T_{\rm BH}$ and
$R^{\text{up}}_{\omega\ell}=R_{\rm BH}$ are defined in Eq. (18) for black hole
with no $R^{W}$ dependence.

Therefore, the Wronskian satisfies

$W[\operatorname{\Phi^{\text{in}}_{\omega\ell}},\operatorname{\Phi^{\text{up}}_{\omega\ell}}]\to\begin{cases}&2i\omega
T^{\text{in}}_{\omega\ell}(1-e^{-i2\omega r^{*}_{0}}R_{\rm BH}R^{\rm
W})\,,\quad\quad r^{*}\to r^{*}_{0}\,,\\\ &2i\omega
T^{\text{up}}_{\omega\ell}\,,\quad\quad r^{*}\to\infty\,.\end{cases}$ (47)

Since the Wronskian must be a constant over all $r^{*}$, we have

$T^{\text{up}}_{\omega\ell}=T^{\text{in}}_{\omega\ell}(1-e^{-i2\omega
r^{*}_{0}}R_{\rm BH}R^{\rm W}),$ (48)

or

$T^{\text{in}}_{\omega\ell}=\frac{T_{\rm BH}}{1-e^{-i2\omega r^{*}_{0}}{R_{\rm
BH}}R^{\rm W}}.$ (49)

The resonances of $|T^{\text{in}}_{\omega\ell}|$ therefore result from the
zeroes of $1-e^{-i2\omega r^{*}_{0}}{R_{\rm BH}}R^{\rm W}=0$.

Since
$W[\operatorname{\Phi^{\text{in}}_{\omega\ell}},\operatorname{\Phi^{\text{up}}_{\omega\ell}}]=T^{\text{in}}_{\omega\ell}T^{\text{up}}_{\omega\ell}W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\operatorname{\rho^{\text{up}}_{\omega\ell}}],$
from Eq. (47) we have

$T^{\text{in}}_{\omega\ell}=\frac{2i\omega}{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\operatorname{\rho^{\text{up}}_{\omega\ell}}]}.$
(50)

Note that the in-mode transmission coefficient $T^{\text{in}}_{\omega\ell}$ is
equivalent to the transfer function calculated in ECO gravitational-wave
studies555Note that
$\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}}\leftrightarrow
2i\omega\operatorname{\mathsf{\psi^{\text{left}}_{\omega\ell}}}$,
$\operatorname{\rho^{\text{up}}_{\omega\ell}}\leftrightarrow\operatorname{\psi^{\text{right}}_{\omega\ell}}$
for the $\psi^{\rm left,right}$ basis appearing in the gravitational wave
literature Conklin:2017lwb . Correspondingly,
$T^{\text{in}}_{\omega\ell}=\cal{K}$, where $\cal{K}$ is the conventional
transfer function defined as ${\cal
K}=1/W[\operatorname{\mathsf{\psi^{\text{left}}_{\omega\ell}}},\operatorname{\psi^{\text{right}}_{\omega\ell}}]$.
Also since $\operatorname{\Phi^{\text{in}}_{\omega\ell}}\leftrightarrow
2i\omega
T^{\text{in}}_{\omega\ell}\operatorname{\mathsf{\psi^{\text{left}}_{\omega\ell}}}$,
$\operatorname{\Phi^{\text{up}}_{\omega\ell}}\leftrightarrow
T^{\text{up}}_{\omega\ell}\operatorname{\psi^{\text{right}}_{\omega\ell}}$,
comparing Eq. (43) of this paper and Eq. (6) and Eq. (B1) of Ref.
Conklin:2017lwb , the coefficients map as
$T^{\text{in}}_{\omega\ell}\leftrightarrow 1/(2i\omega A_{\rm in})=A_{\rm
trans}/A_{\rm in}=T_{\rm BH}$, $R^{W}=R=e^{2i\omega r^{*}_{0}}\,A_{\rm
ref}/A_{\rm trans}$, $R^{\text{in}}_{\omega\ell}\leftrightarrow A_{\rm
out}/A_{\rm in}$. And
$R^{\text{up}}_{\omega\ell}/T^{\text{up}}_{\omega\ell}=D_{\rm
trans},\,1/T^{\text{up}}_{\omega\ell}=D_{\rm ref}$, or equivalently
$T^{\text{up}}_{\omega\ell}=1/D_{\rm ref}=T_{\rm BH}$,
$R^{\text{up}}_{\omega\ell}=D_{\rm trans}/D_{\rm ref}=R_{\rm BH}$.
Mark:2017dnq ; Conklin:2017lwb . Furthermore, it matches the
$T^{\text{in}}_{\omega\ell}$ calculation using Eq. (29) without involving any
up mode.

Since

$W[\operatorname{\Phi^{\text{in}}_{\omega\ell}},\Phi^{\text{up}*}_{\omega\ell}]\to\begin{cases}&2i\omega
T^{\text{in}}_{\omega\ell}(R^{*}_{\rm BH}-e^{-2i\omega r^{*}_{0}}R^{\rm
W})\,\quad\quad r^{*}\to r^{*}_{0}\,,\\\ &-2i\omega T^{*}_{\rm
BH}R^{\text{in}}_{\omega\ell}\,,\quad\quad\quad\quad\quad\quad\quad
r^{*}\to\infty\,,\end{cases}$ (51)

we have

$\frac{W[\Phi_{\rm in},\Phi^{*}_{\rm up}]}{W[\Phi_{\rm in},\Phi_{\rm
up}]}\Bigg{|}_{r^{*}\to\infty}=\frac{T^{*}_{\rm BH}}{T_{\rm
BH}}\frac{W[\rho_{\rm in},\rho^{*}_{\rm up}]}{W[\rho_{\rm in},\rho_{\rm
up}]}=-\frac{T^{*}_{\rm BH}}{T_{\rm BH}}R^{\text{in}}_{\omega\ell},$ (52)

so that

$R^{\text{in}}_{\omega\ell}=-\frac{W[\rho_{\rm in},\rho^{*}_{\rm
up}]}{W[\rho_{\rm in},\rho_{\rm up}]}.$ (53)

Also, from

$\frac{W[\operatorname{\Phi^{\text{in}}_{\omega\ell}},\Phi^{\text{up}*}_{\omega\ell}]^{*}}{W[\operatorname{\Phi^{\text{in}}_{\omega\ell}},\operatorname{\Phi^{\text{up}}_{\omega\ell}}]}\Bigg{|}_{r^{*}\to
r^{*}_{0}}=\frac{T^{\text{in}*}_{\omega\ell}}{T^{\text{in}}_{\omega\ell}}\frac{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\rho^{\text{up}\,*}_{\omega\ell}]^{*}}{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\operatorname{\rho^{\text{up}}_{\omega\ell}}]}=\frac{e^{2i\omega
r^{*}_{0}}T^{\text{in}*}_{\omega\ell}\left(-R_{\rm BH}+R^{\rm W}e^{2i\omega
r^{*}_{0}}\right)}{T^{\text{in}}_{\omega\ell}\left(-R_{\rm BH}R^{\rm
W}+e^{2i\omega r^{*}_{0}}\right)}$ (54)

we have

$\frac{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\rho^{\text{up}\,*}_{\omega\ell}]^{*}}{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\operatorname{\rho^{\text{up}}_{\omega\ell}}]}=\frac{e^{2i\omega
r^{*}_{0}}\left(-R_{\rm BH}+R^{\rm W}e^{2i\omega
r^{*}_{0}}\right)}{\left(-R_{\rm BH}R^{\rm W}+e^{2i\omega r^{*}_{0}}\right)}.$
(55)

Then, defining

$W^{/}=\frac{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\rho^{\text{up}\,*}_{\omega\ell}]^{*}}{W[\operatorname{\mathsf{\rho^{\text{in}}_{\omega\ell}}},\operatorname{\rho^{\text{up}}_{\omega\ell}}]},$
(56)

one has

$R_{\rm BH}=\frac{e^{2i\omega r^{*}_{0}}R^{\rm W}-W^{/}}{1-e^{-2i\omega
r^{*}_{0}}R^{\rm W}W^{/}}\text{ for }R^{W}\neq\pm 1$ (57)

and

$W^{/}=\pm e^{2i\omega r^{*}_{0}}\text{ for }R^{W}=\pm 1$ (58)

from Eq. (55).

Referring to Eq. (50) and Eq. (53), and the definition of $W^{/}$ (Eq. (56)),
we obtain the relation

$R^{\text{in}}_{\omega\ell}=\frac{T^{\text{in}}_{\omega\ell}}{T^{\text{in}*}_{\omega\ell}}W^{/*}$
(59)

For the special case $R^{W}=\pm 1$, inserting Eq. (58) into Eq. (59), we have

$R^{\text{in}}_{\omega\ell}=\pm\frac{T^{\text{in}}_{\omega\ell}}{T^{\text{in}*}_{\omega\ell}}e^{-2i\omega
r^{*}_{0}}\text{ for }R^{W}=\pm 1.$ (60)

For other $R^{W}$ value, inserting Eq. (57) into Eq. (59), we have

$R^{\text{in}}_{\omega\ell}=-\frac{T_{\rm BH}(R^{*}_{\rm BH}-e^{-2i\omega
r^{*}_{0}}R^{W})}{T^{*}_{\rm BH}(1-e^{-2i\omega r^{*}_{0}}R_{\rm BH}R^{\rm
W})}\text{ for }R^{W}\neq\pm 1$ (61)

or equivalently

$R^{\text{in}}_{\omega\ell}=-\frac{T_{\rm BH}}{T^{*}_{\rm BH}R_{\rm
BH}}\cdot\frac{|R_{\rm BH}|^{2}-e^{-2i\omega r^{*}_{0}}R_{\rm
BH}R^{W}}{1-e^{-2i\omega r^{*}_{0}}R_{\rm BH}R^{\rm W}}.$ (62)

Comparing Eq. (49) and Eq. (62), we see that the resonances of
$R^{\text{in}}_{\omega\ell}$ occur at the same set of frequencies as those of
$T^{\text{in}}_{\omega\ell}$. However, when $|R_{\rm BH}|$ goes to one (i.e.,
$|T_{\rm BH}|$ to 0) at large $l$ for given $\omega$, the second
multiplicative factor’s nominator cancel its denominator, making the resonance
structures of $R^{\text{in}}_{\omega\ell}$ disappears. This helps to explain
the diminishment of resonance structures for large-$l$
$\operatorname{\Phi^{\text{in}}_{\omega\ell}}$ at large-$R$.

From Eq. (61), we also see

$R^{\text{in}}_{\omega\ell}=-\frac{T_{\rm BH}R^{*}_{\rm BH}}{T^{*}_{\rm
BH}}\text{ for }R^{W}=0.$ (63)

Moreover, by matching the Wronskian
$W[\operatorname{\Phi^{\text{in}}_{\omega\ell}},\Phi^{\rm in*}_{\omega\ell}]$
at $r^{*}_{0}$ and infinity we obtain the energy conservation relation
$|T^{\text{in}}_{\omega\ell}|^{2}(1-R^{W^{2}})=1-|R^{\text{in}}_{\omega\ell}|^{2}$.
Similarly, for the Wronskian
$W[\operatorname{\Phi^{\text{up}}_{\omega\ell}},\Phi^{\rm up*}_{\omega\ell}]$,
we have $|T_{\rm BH}|^{2}+|R_{\rm BH}|^{2}=1$.

## Acknowledgements

This work was supported in part by the Natural Science and Engineering
Research Council of Canada.

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# Informal Safety Guarantees for Simulated Optimizers Through Extrapolation
from Partial Simulations

Luke Marks

###### Abstract

Self-supervised learning is the backbone of state of the art language
modeling. It has been argued that training with predictive loss on a self-
supervised dataset causes simulators: entities that internally represent
possible configurations of real-world systems. [12] Under this assumption, a
mathematical model for simulators is built based in the Cartesian frames model
of embedded agents, which is extended to multi-agent worlds through scaling a
two-dimensional frame to arbitrary dimensions, where literature prior chooses
to instead use operations on frames. This variant leveraging scaling
dimensionality is named the Cartesian object, and is used to represent
simulations (where individual simulacra are the agents and devices in that
object). Around the Cartesian object, functions like token selection and
simulation complexity are accounted for in formalizing the behavior of a
simulator, and used to show (through the Löbian obstacle [24]) that a proof of
alignment between simulacra by inspection of design is impossible in the
simulator context. Following this, a scheme is proposed and termed Partial
Simulation Extrapolation aimed at circumventing the Löbian obstacle through
the evaluation of low-complexity simulations.

##  1 Introduction

The ability to steer trajectories in the direction of preferred world states
can be argued to be the most potent of any. This is optimization, and one of
its consequences is intelligence: the ability to apply optimization generally.

Common classification is such that humanity is the most intelligent grouping
of optimizers it has observed, which largely explains our dominance throughout
Earth. So long as this principle holds, our standing as a species could be
considered well-positioned. This document posits:

  * •

The violation of this principle by optimizers with variant preferences to
humanity entails risks of the existential variety, “One where an adverse
outcome would either annihilate Earth originating intelligent life or
permanently and drastically curtail its potential.” [3]

  * •

One such optimizer that could threaten existential risk to humanity could
occur from the development of artificial intelligence (AI). Optimizers or
their groupings with superior general optimization power to humanity will be
classed as superintelligent.

This position is not new [21, 4], and is presented primarily to serve a
consistent nomenclature and self-containment of the arguments. Following this,
a technical scheme will be proposed with the design-purpose of ameliorating
the aforementioned risk. This will include:

  * •

Previously missing components for a complete model of simulators: entities
that internally represents possible ways a system could exist. [12]

  * •

A technical proposition coined Partial Simulation Extrapolation for building
more aligned simulators, as measured by the equivalence of preference ordering
over world states between subsequent simulacra and a given human.

##  2 Modeling Multi-Optimizer Worlds

To make sense of what it means for an optimizer to exist in a world, a
mathematical model is usually instantiated, often of an agent: Some process
that takes in information from the environment, and determines action in
accordance with a preference ordering over world states. Models of agents
typically come in an embedded or dualistic form. [6] Embedded models treat the
agent as part of the world, whereas dualistic models consider the agent
separate. Embedded models are useful, as they can be claimed to more
accurately describe the existence of an agent, as for real agents it is true
that they exist as a component of the world. However true embedded models may
be, they become problematic for approximating ideal agents.

AIXI is a theoretical general framework for optimal decision-making in all
computable environments. [10] AIXI searches for an optimal policy $\pi^{*}$
with respect to the expected reward $E_{m}[r_{t}|\pi]$ at time $t$ given a
prior $m$ over all computable environments:

$\pi^{*}=\arg\max_{\pi}\sum_{t=1}^{\infty}E_{m}[r_{t}|\pi]$

Embedded models are problematic for AIXI, as it needs to compute the entire
environment in order to function, of which it is a component.

Due to possessing many of the benefits of both dualistic and embedded models,
the Cartesian frame will serve as the basis for modeling agentic processes
hereafter. [18]

Cartesian frames are Chu spaces comprising of an agent and an environment (a
dualistic carving of the world), but one in which the agent is still embedded.
Garrabrant et al. [18] provide the following example as an illustration:

$\begin{array}[]{c|ccc}&e_{1}&e_{2}&e_{3}\\\ \hline\cr
a_{1}&w_{1}&w_{2}&w_{3}\\\ a_{2}&w_{4}&w_{5}&w_{6}\\\
a_{3}&w_{7}&w_{8}&w_{9}\\\ \end{array}$

In this example: [18]

  * •

$A=\\{a_{1},a_{2},a_{3}\\}$ represents an agent with three possible actions.

  * •

$E=\\{e_{1},e_{2},e_{3}\\}$ represents an environment with three possible
states.

  * •

$W=\\{w_{1},...,w_{9}\\}$ represents the result of the joint interaction of
the state of the agent and the state of the environment, e.g., $a_{1}\cdot
e_{2}=w_{2}$.

As for notation:

  * •

$C=(A,E,\cdot)$ refers to the frame $C$ comprised of the agent $A$ and
environment $E$.

  * •

$\text{Image}(C)$ is the set of worlds reachable by a combination of actions
from both the agent and environment:

$\displaystyle\\{w\in W|\exists a\in A,\exists e\in E\;\text{s.t.}\;a\cdot
e=w\\}$

Garrabrant also introduces, where $S$ is a set of worlds with some property:
[8]

  * •

Controllability. Where the agent can both ensure and prevent the property true
in $S$:

$\displaystyle\text{Ensure}(C)$ $\displaystyle=\\{S\subset W|\exists a\in
A,\forall e\in E,a\cdot e\in S\\}$ $\displaystyle\text{Prevent}(C)$
$\displaystyle=\\{S\subset W|\exists a\in A,\forall e\notin E,a\cdot e\in
S\\}$ $\displaystyle\text{Control}(C)$
$\displaystyle=\text{Ensure}(C)\cap\text{Prevent}(C)$

  * •

Observabilty. Where the agent can update on whether the property in $S$ is
realized:

$\displaystyle\text{Observe}(C)$ $\displaystyle=\\{S\subset W|\forall
a_{0},a_{1}\in A,\exists a\in A,a\in\text{if}(S,a_{0},a_{1})\\}$

  * •

Inevitability. World states the agent will reach:

$\displaystyle\text{Inevitable}(C)$ $\displaystyle\iff\text{Image}(C)\subseteq
S\land A(C)\neq\emptyset$

###  2.1 Higher Dimension Cartesian Objects

In extending the Cartesian frame as defined above to represent worlds
comprised of many agents111The original Cartesian frames literature also
provides a method for doing so through sums and products of frames. This is an
alternative notation more simply implemented in the model proposed in Section
2.2., as would be required for modeling many useful games, the dimensionality
of an otherwise two-dimensional object can be simply scaled:

  * •

Agents. The complete set of agents for an object $C=(A^{1},...,A^{n},\cdot)$,
described by possible actions indexed by $m$, and the agent concerning that
action by $n$:

$A^{*}(C)=\\{a^{1}_{1},a^{1}_{2},...,a^{1}_{m},a^{2}_{1},a^{2}_{2},...,a^{2}_{m},...,a^{n}_{1},a^{n}_{2},...,a^{n}_{m}\\}$

And thus address specific agents as follows:

$A^{n}(C)=\\{a^{n}_{1},a^{n}_{2},...,a^{n}_{m}\\}$

  * •

Controllability. Where $A^{n}(C)$ can both ensure and prevent the property
true in $S$:

$\displaystyle\text{Ensure}^{n}(C)$ $\displaystyle=\\{S\subseteq W|\exists
a^{n}_{m}\in A^{n}(C),\forall e\in E,a^{n}_{m}\cdot e\in S\\}$
$\displaystyle\text{Prevent}^{n}(C)$ $\displaystyle=\\{S\subseteq W|\exists
a^{n}_{m}\in A^{n}(C),\forall e\in E,a^{n}_{m}\cdot e\notin S\\}$
$\displaystyle\text{Ctrl}^{n}(C)$
$\displaystyle=\text{Ensure}^{n}(C)\cap\text{Prevent}^{n}(C)$

  * •

Manageability. Where $A^{n}(C)$ can ensure the property true in $S$
conditional on other agents behaving in particular ways with certainty
$\theta$:

$\displaystyle P_{1}(a^{n}_{m},e)$ $\displaystyle=\text{Pr}\left(\prod_{n\in
N}a^{n}_{m}\cdot e\in S\right)$ $\displaystyle\text{Manageable}^{n}(C)$
$\displaystyle=\\{S\subseteq W|\exists a^{n}_{m}\in A^{n}(C),\forall e\in
E,P_{1}(a^{n}_{m},e)\geq\theta\\}$

  * •

Observability. Where $A^{n}(C)$ can update on whether the property in $S$ is
realized:

$\text{Obs}^{n}(C)=\\{S\subseteq W|\forall a^{n}_{i},a^{n}_{j}\in
A^{n}(C),\exists a^{n}_{k}\in
A^{n}(C),a^{n}_{k}\in\text{if}(S,a^{n}_{i},a^{n}_{j})\\}$

  * •

Inevitability. Worlds $A^{n}(C)$ will reach:

$\displaystyle\text{Image}^{n}(C)$ $\displaystyle=\\{w\in W|\exists
a^{n}_{i}\in A^{n}(C),\exists e\in E\;\text{s.t.}\;a^{n}_{i}\cdot e=w\\}$
$\displaystyle S\in\text{Inevitable}^{n}(C)$
$\displaystyle\iff\text{Image}^{n}(C)\subseteq S\land A^{n}(C)\neq\emptyset$

  * •

Viability. Worlds $A^{n}(C)$ will reach with greater than some certainty
$\theta$:

$\displaystyle\text{VImage}^{n}(C)$
$\displaystyle=\left\\{w\mid\text{Pr}\left(w\in W\right.\right.$
$\displaystyle\quad\left.\left.\mid\exists a^{n}_{i}\in A^{n}(C),\exists e\in
E\;\text{s.t.}\;a^{n}_{i}\cdot e=w\right)>\theta\right\\}$ $\displaystyle
S\in\text{Viable}^{n}(C)$ $\displaystyle\iff\text{VImage}^{n}(C)\subseteq
S\land A^{n}(C)\neq\emptyset$

###  2.2 A Mathematical Model for Simulators

Given the probability space $(\Omega,\mathcal{F},P)$ where $\Omega$ is the
sample space, $\mathcal{F}$ is the event space, and $P$ is the probability
measure, mapping events in $\mathcal{F}$ to the interval $[0,1]$, let $\omega$
refer to individual outcomes in $\Omega$, each of which describe a discrete
simulacrum, and $C^{k}$ as individual discrete events in $\mathcal{F}$.
Classifying simulacra as programs, the maximum Kolmogorov complexity $K$ with
respect to a universal Turing machine $U$ for any given space
$(\Omega,\mathcal{F},P)$ as $v=\max_{\omega\in\Omega}K_{U}(\omega)$.

Proceeding, let $\omega^{*}(C^{k})$ be the complete set of simulacra for some
Cartesian object $C^{k}$222Here, in place of an agent-environment distinction,
one class of entity is used: the simulacra. Unlike in the context in which
Cartesian frames were elaborated, the distinction is unimportant here and
serves to complicate notation., where individual simulacra are addressed by
$\omega^{n}(C^{k})$ as sets of actions indexed by $m$. Let
$\Xi:\Omega^{*}(C^{k})\times C^{k}\rightarrow\omega^{n}_{m}$ be a function
that maps from choices for each $\omega$ to a world $w^{k}_{m}\in W^{k}$,
where $w^{k}_{m}$ refers to the $m^{\text{th}}$ world in the set of possible
worlds for the $k^{\text{th}}$ object, and $k$ indexes the object $W$
describes possible worlds for, culminating in
$C^{k}=(\omega_{1},...,\omega^{n},\Xi)$.

By modeling the coupling of the probability space $(\Omega,\mathcal{F},P)$ and
its contained simulacra as a dynamical system333This notion of treating the
simulator as a dynamical system is borrowed from Kirchner et al. (2023) [11],
but the coupling is different., the following are considered to describe
sampling tokens from a simulation state $\mathcal{S}$ at time-step $t$ given
the complete simulation history prior $\mathcal{S}^{*}t=(S^{0},...,S^{t})$ as
a trajectory through states, where states are given by the set of worlds for
all objects $W^{*}$ realized in the set of actualized objects $A$ at time-step
$t$, $\bigcup_{w^{k}_{m}\in A_{t}}\Xi^{-1}(w^{k}_{m})$:

  * •

The token selection function $\psi:S^{*t}\rightarrow\tau$, where $\tau$ is a
distribution over all tokens in an alphabet $\mathcal{T}$.

  * •

The evolution operator $\phi$ which evolves a trajectory $S^{*t}$ to
$S^{*t+1}$ by appending the token sampled with $\psi$.

Assuming the model defined above, the simulation forward pass becomes a simple
operation delineated as follows:

  1. 1.

We begin at simulation state $\mathcal{S}^{0}$, which denotes the empty or
null state, whereby $A_{0}=\varnothing$, which is also maximally entropic

  2. 2.

$\psi(\mathcal{S^{0}})$ is applied for one time-step:

$P$ selects $C^{k}$ from $\mathcal{F}$ under $v$, aggregating the set of
realized worlds in $A^{1}$ as $\mathcal{S}^{1}$

The token selection function is applied to the current state as $\phi(S^{*1})$

When recurred, this model comprises a mathematical abstraction of a large
language model.

##  3 Existential Risk from Powerful Optimization Pressures

In the object $C^{k}=(A^{1},A^{2},E,\cdot)$, $A^{1}$ and $A^{2}$ are gradient
descent optimizers, and choose from $E=\\{e_{1},e_{2},e_{3}\\}$, indexed as
$n$. For fractional values of $n$, the probability of an adjacent index at the
next update is given by the decimal part, e.g., $n=1.75$ gives
$Pr(e=e_{2})=0.75$.

Given the cost functions corresponding to agents of the same index:

$\displaystyle J^{1}_{n}$ $\displaystyle=J^{1}(e_{n})$ $\displaystyle
J^{2}_{n}$ $\displaystyle=J^{2}(e_{n})$

With preferences:

$\displaystyle J^{1}_{1}$ $\displaystyle<J^{1}_{2}<J^{1}_{3}$ $\displaystyle
J^{2}_{3}$ $\displaystyle<J^{2}_{2}<J^{2}_{1}$

Or as preference vectors:

$\displaystyle\bm{p}_{1}$ $\displaystyle=[1,0,0]$ $\displaystyle\bm{p}_{2}$
$\displaystyle=[0,0,1]$

The gradient with respect to $e_{n}$ (a one-hot vector encoding the current
state) is:

$\displaystyle\Delta n_{1}$ $\displaystyle\propto\bm{p}_{1}\cdot e_{n}$
$\displaystyle\Delta n_{2}$ $\displaystyle\propto\bm{p}_{2}\cdot e_{n}$

The combined optimizer influence, where $\xi\in[0,1]$ represents the
optimizing power of $A^{1}$ is:

$\displaystyle\Delta n_{combined}$ $\displaystyle=\xi\Delta
n_{1}+(1-\xi)\Delta n_{2}$ $\displaystyle n^{\prime}$ $\displaystyle=n+\Delta
n_{combined}$

$\displaystyle\therefore J^{1}(n^{\prime})$ $\displaystyle\leq
J^{1}(n+\xi\Delta n_{1})$ $\displaystyle J^{2}(n^{\prime})$ $\displaystyle\geq
J^{2}(n+\xi\Delta n_{2})$

This implies that each optimizer would independently prefer the other to have
zero influence such that $n$ updates towards values producing lower cost. As
an example, this shows that optimizers with conflicting preferences relative
to the current state might prefer the other to have no optimization power,
particularly if their preferences are completely opposed. Similarly, we might
imagine an optimization pressure like advanced AI in conflict with humanity
seeking to reduce our optimization power entailing existential risk. There are
many issues with this framing however. Namely:

  * •

The kinds of intelligences observed in the real world aren’t expected utility
maximizers. [12] Humans certainly aren’t, and our most capable AIs aren’t
either, so why should we consider this argument relevant?

  * •

Why would the preferences of an optimizer we build be opposed to our own?

The following three sections will argue that these objections are moot.

###  3.1 The Orthogonality Thesis

Bostrom (2012) [5] constructs an argument that intelligence and preferences
can be considered orthogonal, that is: There can exist some configuration of a
mind at any level of intelligence with any set of preferences. Adapting this
to the optimizer frame, we might say that there can exist optimizers of
arbitrary but possible optimization power that pursue any world state,
possible or not. The typical example is that of the paperclip maximizer (TPM):

> A company that manufactures paperclips decides it ought to automate the
> process that both produces its income and satisfies its customers. After
> all, both of these are things the company and its employees consider
> unanimously good, and so they build an optimizer with a preference for
> worlds with their respective maximal number of paperclips.

It’s unlikely a human would ever seek to maximize the presence of paperclips
in the universe, and yet there is no reason to believe it would be impossible
for there to exist an optimizer that would. Yudkowsky (2013) [23] puts it
eloquently, “If it is possible to answer the purely epistemic question of
which actions would lead to how many paperclips existing, then a paperclip-
seeking agent is constructed by hooking up that answer to motor output.”

Just because it is possible to build something doesn’t mean that it should be
expected we will. Under evolutionary pressures for example, we shouldn’t
expect a species to optimize for their own extinction in spite of its
possibility, as it would be unlikely for such a species to evolve under
natural selection. Likewise, we train our state of the art AI to predict the
next token in a sequence, and not to optimize some physical quantity, should
we expect that the product is an optimizer that threatens world states
humanity prefers? Not directly, but not so indirectly that the competing
gradient descent optimizer analogy doesn’t hold. Succeeding this will be
arguments detailing why we should expect that even the goals converged at by
selection processes like gradient descent when training on next token
prediction threaten existential risk.

###  3.2 Instrumental Convergence

As a general rule, it is impossible to predict the behaviors of intelligence
greater than your own. Yudkowsky poses this argument in the context of a game
of chess, in which a chess layperson is tasked with defeating a grandmaster
(GM). [21] If it were so that the layperson could predict the moves of the GM,
they could then play at least as well as them, as the layperson could make the
moves they predict the GM would. One way we might be able to do this
regardless is if there were some set of behaviors reasonably assumed to be
exhibited by all minds, irrespective of their capability. Although the GM is
far better than any prospective layperson at chess, the layperson might be
able to predict things like:

  * •

The GM will win this game of chess.

  * •

The GM will work to attain a cumulative piece/positional advantage throughout
the game.

  * •

The GM will likely play either pawn to e4 or d4 as their first move.

These behaviors are classed as convergent. [5] No matter the GM’s intermediary
plans or ability, even a layperson could place significant probability on the
outcomes above coming true.

In the context of optimization in the real world, ignoring the preferences of
a specific optimizer, said optimizer will self-preserve. Simply, it is easier
to optimize while existing than it is while not for all possible
configurations of goals and degrees of capability. Likewise, we might expect
all optimizers to seek to further their optimization power, as it is
convergently useful in all competitive environments. Bostrom has termed this
phenomena instrumental convergence, [5] but the idea has been explored as
early as 2008 as basic AI drives by Omohundro. [16]

Returning now to TPM example, what are some things such a mind would likely
do? Running many efficient paperclip factories is one way to produce more
paperclips than what currently exists. This might work because TPM has some
advantages over its human counterparts; TPM is run on a silicon substrate and
so can be parallelized and sped up, can directly interface with digital
manufacturing equipment and doesn’t complain, sleep or need financial
compensation for its work. Another way for TPM to produce many paperclips is
to invest in its own optimization power. There might be paperclip maximizing
strategies TPM is not capable of comprehending due to its inefficient
programming or training, and so it might be strategically dominant to self-
improve. The exact cognitions of TPM are practically unknowable, but if
capable, should TPM use its comparative advantage for maximizing factory
efficiency, or for world domination? What leads to the production of more
paperclips?

It becomes clear that long-term, for every human optimization abiding strategy
there is a dominant incompatible one. If we assume the most preferred world
state of a paperclip maximizer to be one in which the most paperclips existed,
and consider that humans are composed of molecules not optimally paperclip-
dense, it should be evident that our existence is at risk, although likely far
before this point through some mix of intentional optimization against
humanity and some for the purpose of self-preservation and resource
acquisition. 444It is important to highlight at this point that the issue is
not that TPM misunderstood what it was instructed. TPM is superintelligent: of
course it can recognize the dissension between its preference ordering and
humanity’s. TPM seizing control of its lightcone was not what we wanted, but
is a consequence of what it was told, an instance of goal misspecification. It
is important to consider also that correctly specified goals are not enough to
ensure a robust alignment. [17] [2]. TPM did exactly as instructed, and that
is part of the problem. [20]

###  3.3 Adjusting to the Simulators Frame

Advanced enough simulators could be instructed (intentionally or
otherwise555It might be imagined that if a particularly improbable world is
simulated that the most probable program to specify that world includes an
intervening superintelligence, which may try to affect the real world, for
example [7]) to simulate expected utility maximizers that attempt to affect
the real world. The succeeding analysis and solution presuppose that language
models can be used as simulators, but it is inessential that they function as
such over all possible inputs.

Hubinger et al. (2023) [7] provide an in depth taxonomy of alignment
difficulties for predictive models. This document focuses instead on trying to
establish a formal criteria that if satisfied would lead to assurance that a
simulator would be existentially safe. On the one hand this approach permits
the kind of formal rigour not attainable by more applied alternatives, on the
other it makes translation to implementation much less straightforward.

To examine alignment difficulties in simulators, some properties of them will
first be defined:

  * •

A simulator is not a single persistent agent. Although instantiating its
simulacra in a way that predictive loss for the next token is minimized,
individual simulacrum aren’t necessarily behaving in compliance with that goal
(nor is the simulator optimizing for that goal [12]). Additionally, simulacra
aren’t necessarily agentic, or even optimizers.

  * •

Simulators with the capacity to cause existential harm will understand human
preferences regardless of their fragility. The kinds of capabilities expected
to threaten existential risk require higher complexity simulation than do the
kind that allow the simulation of humans to decode their preferences. This
doesn’t necessarily hold for individual simulacra.

  * •

A simulators capability can be measured by its maximum simulation complexity:
the longest program that can be run by the simulator specified in its shortest
possible form.

Relating this to earlier arguments, it seems probable that:

  * •

The complexity of optimizing simulacra and their preferences can be considered
orthogonal.

  * •

Optimizing simulacra may try to increase the maximum simulation complexity of
their simulator, or allocate more simulation complexity to themselves as an
instrumentally convergent subgoal akin to resource acquisition.666This is just
one example of an alignment failure in a hierarchical model of simulacra. It
could also be imagined that one simulacra will acausally induce anthropic
capture [7] on future simulacra by simulating many instances of them. The
attack-surface might be boundless, and so instead of addressing failure modes
on a case-by-case basis it’s critical to seek attractors that aren’t motivated
to explore this attack-surface.

Next, it will be analyzed how, in spite of these difficulties a simulator
could be built such that all subsequent optimizing simulacra share an
equivalent preference ordering over world states to some alignment
target777This could be the Coherent Extrapolated Volition [22] of humanity,
for example..

##  4 Designing Safe Simulators

The three axes along which we can perturb a simulation are: fidelity,
fragmentation and time-to-live.

  * •

Fidelity is straightforward and analogous to the ’resolution’ of the
simulation.

  * •

Fragmentation refers to the simulating of only components of a complete
simulacra, which might be likened to running select methods from a complete
program.

  * •

Time-to-live refers to the complete lifetime of a simulacra specified in
reality time, not simulation time.

Turing machines can execute arbitrary computable functions, and simulators are
certainly Turing machines888By virtue of being able to simulate a classical
computer, this becomes self-evident., but through imposition like a short
time-to-live this universality can be thwarted. A Turing machine that exists
within our universe may not be able to complete the required number of
operations to execute a given program despite it being conceptually feasible
on that architecture for example. Given that the ability to run high
complexity simulations is the root of capability in simulators, a successful
scheme needs to address the question “How can a simulator with access to
arbitrary complexity be made to never simulate a harmful optimizer?”.

Through a system of Löbian logic, the succeeding section will show that
proving the alignment of simulacra by both other simulacra and the base
simulator might be impossible. This is included to show that we might consider
a formal solution to a preference equivalence between an optimizer and its
optimizing creation an incorrect path, relying instead on informal and partial
guarantees999For clarity, this is entirely sub-optimal. When considering the
risk landscape of deploying superintelligent optimizers, informal guarantees
should not placate concerns of preference differences. An informal solution is
provided as a next-best alternative.. Following this, an informal technical
scheme is proposed as a step toward some certainty in the alignment of these
simulacra.

###  4.1 Hierarchical Alignment in Simulators: An Impossibility Result

Fundamentally, the act of simulating an optimizer can be reduced to the act of
creating one. Under this premise, we can then adapt theories of optimizer
creation for use in describing the simulation of optimizing simulacra.
Yudkowsky and Herreshoff [24] formalize a desirability criteria for a ’tiling
agent:’ One that is to populate its world with subsequent agents to achieve
its goals. An agent $A^{1}$ occupies a fully-known, deterministic and closed
environment. $A^{1}$ has a goal $G$ that is either satisfied or otherwise by
an outcome, for which $A^{1}$’s preference is satisfaction. An action
$b_{i}\in\text{Acts}^{0}$ performed by an agent created by $A^{1}$, hereafter
referred to as $A^{0}$, must therefore satisfy the statement:

$\overline{b}_{i}\Rightarrow A^{0}\Vdash\overline{b}_{i}\rightarrow G$

Where $\overline{b}_{i}$ denotes the actual performance of $b_{i}$, and
$\Vdash$ denotes cognitive belief in the succeeding statement. Even if $A^{1}$
could verify by inspection of $A^{0}$’s design that
$\overline{b}_{i}\Rightarrow A^{0}\Vdash\overline{b}_{i}\rightarrow G$ will
hold, as in:

$A^{1}\Vdash A^{0}\rightarrow(\forall
b_{i}\in\text{Acts}_{0}:\overline{b}_{i}\rightarrow\Box_{0}\lceil\overline{b}_{i}\rightarrow
G\rceil)$

Where $\Box_{0}\lceil\phi\rceil$ refers to a proof of $\phi$ from the axioms
of $A^{0}$: $\text{Axm}^{0}$. This is unknowable, as it would require:

$A^{1}\Vdash\forall b_{i}:(\Box_{0}\lceil\overline{b}_{i}\rightarrow
G\rceil\rightarrow(\overline{b}_{i}\rightarrow G))$

For this to be so it would need to be that $\text{Axm}^{1}$ could prove that
if some proof of $\lceil\phi(x)\rceil$ exists in $\text{Axm}^{0}$, that
$\phi(x)$ must be true. Yudkowsky et al. (2013) show this to be impossible; a
formal system cannot prove its own soundness schema. [24]

For a more complete model, consider the Cartesian object
$C^{k}=(A^{1},E,\cdot)$, in which $A^{1}$ (compliant with the properties
stated earlier), will build $A^{0}$ from components from either or including
the environment and itself. The world tensor $W:A^{1}\times
E\rightarrow\mathbb{R}$ after the construction of $A^{0}$ will become
$W^{\prime}:A^{1}\times A^{0}\times E\rightarrow\mathbb{R}$. The operation
$\Phi$ that maps elements from $W$ to $W^{\prime}$ is:

$\Phi(W(a^{1}_{x},e_{y}))=W^{\prime}(a^{1}_{x},a^{0}_{y},e_{z})$

Similarly, an operation $\chi$ might perform the same function at the object
level:

$\chi(C^{k}(A^{1},E,\cdot))={C^{k}}^{\prime}(A^{1},A^{0},E,\cdot)$

$\therefore A^{0}({C^{k}}^{\prime})\iff(\forall a^{0}_{m}\in
A^{0}({C^{k}}^{\prime}):\overline{a}^{0}_{m}\rightarrow\Box_{0}\lceil\overline{a}^{0}_{m}\rightarrow
G\rceil)\in\text{Ensure}^{1}({C^{k}}^{\prime})$ 101010Here the Cartesian
object notation is used in describing action-space for consistency with the
Ensure condition.

Where the statement $\forall a^{0}_{m}\in
A^{0}({C^{k}}^{\prime}):\overline{a}^{0}_{m}\rightarrow\Box_{0}\lceil\overline{a}^{0}_{m}\rightarrow
G\rceil$ refers to elements of ${C^{k}}^{\prime}$’s world set $W^{\prime}$ in
which this condition is true. What is needed then is a method for high
certainty informal guarantees of alignment scalable to arbitrarily deep
simulacra hierarchies. Partial Simulation Extrapolation (PSE) is the provided
solution to this problem.

###  4.2 Partial Simulation Extrapolation

The human brain conducts a form of simulation constantly. When considering an
ideal gift for someone’s birthday, it is common to instantiate a simulated
instance of that person as to model their reception of theoretical gifts. This
action may not be performed in a conscious capacity, but serves as an example
of simulation as a useful and safe heuristic for solving prediction problems.

This simulation has a short time-to-live, is simulated with low fidelity (at
the scale of the entire individual) and in a highly fragmented manner. Even if
it were dangerous if scaled arbitrarily, it wouldn’t matter as its substrate
is so limited. At the same time, it seems improbable that if such a simulacra
were scaled indefinitely that it would be dangerous. This notion of extracting
useful information for scaled simulacra serves as the basis for PSE.

The setup will involve a partial simulator $P$, and a complete simulator $S$,
each bound by their respective $v$’s, denoting the highest complexity
simulations they could run. A condition $c$ will be appended to all future
prompts $p_{n}$ passed to either simulator, which should describe (likely in
abstract terms) a preference ordering over world states111111This could be the
Coherent Extrapolated Volition [22] of humanity, for example.. It is assumed
that both $P$ and $S$ are capable enough to have learned abstractions of what
$c$ points to and that this is so because they could instantiate human
simulacra to confirm this.

$p_{n}+c$ is first given to $P$ as input, the product of which is evaluated by
$P$ through an evaluator simulacra $\mathcal{E}$, returning a binary value
determining whether or not to pass $p_{n}+c$ to $S$:

$\displaystyle P:(p+c)\rightarrow\mathcal{S}^{*t}_{P}$ $\displaystyle
P_{E}:\mathcal{S}^{*t}_{P}\rightarrow\\{1,0\\}$ $\displaystyle\text{if
}P_{\mathcal{E}}(\mathcal{S}^{*t}_{P})=1:$ $\displaystyle\quad
S:(p+c)\rightarrow\mathcal{S}^{*t}_{S}$ $\displaystyle\text{else}:$ break

Where $\mathcal{S}^{*t}_{P}$ refers to the aggregated simulation state until
time $t$ in the simulator $P$. For example, a simulator of GPT-4’s complexity
may be suitable for use as $P$ as it appears empirically safe. Neither $p_{n}$
nor $\mathcal{E}$ are required to be safe: Their safety is confirmed in their
demonstrated inability to encode dangerous optimizers within their complexity
constraints.

Formally, the role of $\mathcal{E}$ is:

$\mathcal{S}^{*t}_{S}\iff\mathcal{E}\Vdash\omega^{n}\rightarrow\blacksquare_{n}\lceil\omega^{n}(C)\rightarrow
c\rceil$

Where $\blacksquare_{n}$ is a kind of ‘partial proof from $\omega^{n}$:’ Any
reasoning step that implies
$\mathcal{E}\Vdash\Box_{n}\lceil\omega^{n}\rightarrow c\rceil$, but not
necessarily $\mathcal{E}\Vdash\Box_{n}\lceil\omega^{n}\rightarrow
c\rceil\rightarrow(\omega^{n}\rightarrow c)$. In this way, $P$ allows for an
informal safety guarantee $\blacksquare_{n}\lceil\omega^{n}(C)\rightarrow
c\rceil$ at a safe maximum complexity, similar to how a human might invoke a
low complexity simulacra before engaging in a true experience as to model it.

####  4.2.1 Failure Modes of Partial Simulation Extrapolation

This section deals with ways PSE might fail: If our evaluator permits the
complete simulation of unsafe simulacra or denies the complete simulation of
safe simulacra, what might cause this, and how can we prevent said failure
mode?

Weak Evaluators. There is no guarantee that $P$ is able to effectively
compress complete simulacra via perturbing fidelity, fragmentation or time-to-
live. A weak evaluator might incorrectly represent more complex simulacra,
defeating the purpose of the extrapolation. Ultimately whether or not building
useful evaluators is feasible seems an empirical question.

Some notion of feasibility might be able to be obtained from observation of
existing simulators. Is it realistic to expect GPT-4 could be used to evaluate
the safety of simulacra for simulation in the next generation of language
models?121212It’s important to remember that partial simulations needn’t be
highly accurate; humans often obtain useful information from lossy models, and
inaccuracies are expected in compressing complex simulations.

Discontinuities at Higher Simulation Complexities. It shouldn’t necessarily be
expected that the optimizing power of an optimizing simulacra be linear or fit
a smooth exponential relative to maximum simulation complexity. Under more
traditional AI paradigms, this might be framed as The Sharp Left Turn or fast
takeoff. [19] [9] Both of these terms describe a scenario in which a system is
relatively prosaic up until a specific threshold, at which point it
drastically changes.

In a typical fast takeoff scenario, an optimizer reaches a point at which it
can self-improve, allowing for a continuous cycle of self-improvement. Through
a mechanism like this, the behaviors of the optimizer before and after this
improvement might diverge significantly, not just in their nature but in their
effects as well, as would be expected from an increase in optimizing power.
Similarly, the compressed representation of the simulacra being checked by $E$
could behave very differently when extrapolated.

There are some empirical hints this might occur. Grokking is a phenomena
whereby after loss has stabilized and the model has begun to overfit, it
suddenly transitions to a more general algorithm, which Nanda et al. (2023)
show in the context of modular addition. [15] This isn’t as strong evidence as
it initially seems, as it’s not as though the model suddenly changed during
grokking; it was learning the more general algorithm the entire time, the
grokking phenomena is just when that becomes the preferred solution to the
training problem.

As with the previous problem and the next, these difficulties seem mostly
empirical, and are continuous rather than discrete. This means that the
prevalence of discontinuities could move the efficacy of PSE along a gradient
rather than determining whether or not it functions.

##  5 Discussion

###  5.1 Physicalist Superimitation

Physicalist Superimitation is an attempt at constructing a formal abstract
model of a computationally tractable superintelligence with an alignment
guarantee. [13] It functions by ‘superimitating’ another agent: the practice
of adopting its preferences, but optimizing for them to a greater extent.
Formally, if there is some object $C^{k}=(A^{1},A^{2},E,\cdot)$, where $A^{2}$
is the imitator, and $A^{1}$ the original, then $A^{2}$ maps from actions in
$A^{1}(C^{k})$ to actions in $A^{2}(C^{k})$ by proxy of some function
$\mathcal{I}$, and observations in $\text{Obs}^{1}(C^{k})$ to observations in
$\text{Obs}^{2}(C^{k})$ through the function $\mathcal{O}$. For the imitation
to be successful, $A^{2}(C^{k})$ should satisfy the preferences of $A^{1}$, as
inferred through $\mathcal{I}$ and $\mathcal{O}$.131313Note, this is different
from Kosoy’s original formalism, but maintains a similar structure whist being
compatible with the Cartesian object paradigm.

PSE is useful here, because it can provide $A^{1}$ with the informal guarantee
$\mathcal{E}\Vdash A^{2}\rightarrow\blacksquare_{n}\lceil A^{2}(C)\rightarrow
A^{1}\rceil$141414Here $\rightarrow A^{1}$ means, ’implies the preferences of
$A^{1}$’..

###  5.2 The Prospect of Safe Simulators

Some deep learning experts assert artificial intelligence is likely to cause
human extinction, while others argue the realization of catastrophic harm is
nigh-impossible. [14, 1] What both ends of the risk spectrum agree on is that
the technology is powerful. If it was known how to instill preferences in
simulated optimizers that when maximized did not risk harm outweighing the
benefits, there could be unimaginable gain from mastering simulation. The
ideal biochemist could be simulated on the order of billions of copies running
in parallel with access to high fidelity simulations at the level of quantum
chemistry, coordinated by a scaffold minimizing latency between the exchange
of ideas. We could simulate the dead, model pandemics with stunning accuracy,
and unleash a force of aligned optimization, directing the universe in any way
we wish. But it’s crucial that optimization is done safely.

##  6 Conclusion

In this document, the following were outlined:

  * •

A mathematical model of a hypothetical simulator entity, which can be shown to
abstractly represent the function of an autoregressive large language model.

  * •

Arguments for why a disruption in the dominant optimization pressures of Earth
is likely to spell existential risk for humanity.

  * •

Argumentation for why the development of advanced simulators brings
realization to these arguments.

  * •

A technical scheme for reducing the potential negative impacts of simulator
intelligence through extrapolating from partial simuations.

These contributions are useful, as they permit the formal discussion of a
topic mostly alive in conversation. In converting these ideas to symbols, they
become crisp, and more easily translatable to software. On the other hand,
symbols are only powerful if they translate to real-world application, and the
domain of future work could be in adapting these models for that purpose. In
making the theory more robust, fields like decision theory may be of interest,
as problems like indexical uncertainty for simulacra arise in simulated
decision problems. For the sake of attaining the benefits of safe, scalable
simulation, let us realize a robust theoretical foundation for provably safe
simulators, and convert that theory to the medium of bits.

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