# Predicting Illness for a Sustainable Dairy Agriculture:
Predicting and Explaining the Onset of Mastitis in Dairy Cows

Cathal Ryan,1,2 Christophe Guéret3, Donagh Berry,4,2
Medb Corcoran3, Mark T. Keane,1,2 Brian Mac Namee1,2
1School of Computer Science, University College Dublin, Belfield, Dublin 4,
Ireland
2VistaMilk, Science Foundation Ireland Research Centre
3Accenture Labs, Dublin
4Teagasc, Animal & Grassland Research and Innovation Centre, Moorepark, Fermoy
P61 P302, Co. Cork, Ireland

###### Abstract

> Mastitis is a billion dollar health problem for the modern dairy industry,
> with implications for antibiotic resistance. The use of AI techniques to
> identify the early onset of this disease, thus has significant implications
> for the sustainability of this agricultural sector. Current approaches to
> treating mastitis involve antibiotics and this practice is coming under ever
> increasing scrutiny. Using machine learning models to identify cows at risk
> of developing mastitis and applying targeted treatment regimes to only those
> animals promotes a more sustainable approach. Incorrect predictions from
> such models, however, can lead to monetary losses, unnecessary use of
> antibiotics, and even the premature death of animals, so it is important to
> generate compelling explanations for predictions to build trust with users
> and to better support their decision making. In this paper we demonstrate a
> system developed to predict mastitis infections in cows and provide
> explanations of these predictions using counterfactuals. We demonstrate the
> system and describe the engagement with farmers undertaken to build it.

## Introduction

If agriculture is to have a future it needs to be sustainable; for the dairy
sector this requires zero-carbon farming (using pasture rather than indoor
systems) and minimising the use of antibiotics (to avoid resistances arising).
Artificial intelligence (AI) has a major role to play in these efforts (see,
for example, the UN AI for Good initiative), and our work has focused on a
major animal health problem, mastitis, for dairy farms (which costs US$19-32B
per annum (?)). Current approaches to handling mastitis involve the
administration of large quantities of antibiotics, which is not sustainable
(?). The proliferation of sensors on modern dairy farms, and the data that
they generate, offers an opportunity to more sustainably address the mastitis
issue. Data collected from farms can be used to build machine learning models
that proactively identify cows likely to develop mastitis (?), enabling
farmers to treat those cows in targeted ways.

Predicting a cow’s likelihood to develop mastitis, however, has significant
consequences. If infections are missed then farmers will lose significant
income through discarded milk (?) and this may even lead to increased chances
of early culling of a cow (?). If healthy cows are flagged as mastitis risks,
farmers are likely to forego milking income and administer antibiotics
unnecessarily. Farmers can also use the output of a prediction model to gain
insights about how to improve the health status of their herd by identifying
contributors to mastitis infection. This opportunity, and the significant
consequences of actions taken by farmers based on model output, mean it is
very important that the output of prediction models used is explainable.

In this paper we describe a machine learning model built to predict the
propensity of a cow to develop mastitis, and an approach based on
counterfactuals to generate explanations of the predictions made by the model.
Moreover, we describe how engagement with stakeholders in the farming
community guided the design of this system, particularly in identifying
features that could actually be impacted and appropriate levels of change for
use in the counterfactual system. We demonstrate the effectiveness of this
approach with worked examples and suggest directions for future work.

## Related Work

Machine learning has been applied to the agricultural domain for tasks ranging
from grass growth prediction (?), to grass phenotype identification (?), to
disease detection within cows (?). In the latter case predicting mastitis has
received a significant amount of attention from researchers with approaches
based on simple statistical models (eg. (?)) and more sophisticated approaches
using machine learning (e.g. (?)). Approaches can be distinguished by the type
of input data they use, with some using simple features describing cows and
aggregate milking behaviour (?) and others using much more precise information
such as measurements of milk yield at the level of an individual quarter of a
cow’s udder (?). In this problem domain, there are considerable costs to miss-
prediction: potentially infected cows may need to be quarantined, their milk
cannot be used and expensive antibiotic treatment may be required. So, model
predictions need to be coupled with compelling explanations to allow farmers
to make actionable decisions.

Recent reviews of explainable AI have argued for the use of counterfactuals to
explain the causal importance of different features involved in a model’s
predictions (?; ?). For example, if the model makes a prediction that “ _Cow
#42 will remain healthy_ ” and the vet/farmer asks “ _why?_ ”, then the
explanation module can provide a counterfactual explanation saying

> _If cow #42 had a somatic cell count of 150 and a protein percentage of 5%
> she would be likely to succumb to mastitis_.

This information enables the end-user to assess whether they think the
critical differences between the proposed prediction and the counterfactual
explanation warrant acceptance of the model’s directions.

The growth in research interest in counterfactuals is largely driven by
computational, psychological and legal reasons. Computationally, many methods
generate counterfactuals by perturbing features of the original instance (?),
or nearest unlike neighbor (?); which can be seen as a follow on from the
popular LIME methods (?; ?). Psychologically, it has been shown that people
can readily and naturally understand the causal roles of features using
counterfactuals (?; ?; ?). Legally, it has been argued that counterfactuals
are GDPR compliant meeting the requirements for explaining automated decision-
making (?).

Figure 1: Proportion of infections found by sub-clinical mastitis prediction
model at different time horizons.

## Predicting Mastitis

We have developed a propensity model that uses the record of milking on a
particular day, a history of recent milkings, and details of the cow herself
to predict the likelihood that a cow will develop sub-clinical mastitis in the
next 7 days. To build this model we use a dataset originating from 7 different
research farms located around Ireland. This dataset contains details of over
2,300 cows over a period of 10 years and includes aspects of each cow’s
history including milk recordings, infection history, and genetic details.
Milk recording data is the richest data source and includes the volume of milk
yielded when a cow was milked; characteristic of that milk including fat,
lactose and protein content; and somatic cell count (SCC) recorded from the
collected milk, which is a useful indicator of mastitis (?). Milk yield is
available twice per day; fat, lactose, and protein content of milk and SCC are
measured approximately every 7 days.

Based on the raw data available, we generate a set of features describing a
cow on a particular day that form the basis of prediction. The milk yield and
milk characteristic data form multi-variate time series describing a cow’s
history from which multiple features could be generated—for example minimum,
maximum, mean, and skewness values of each variable over histories of 15 and
30 days. Any feature included in our prediction model, however, has the
potential to be included in the counterfactual explanations generated to
describe predictions. Including a model based on a large number of features
derived from time series, however, could lead to contradictory and
inactionable counterfactual explanations, such as the following, being
generated:

> If Cow #42 had a lower mean yield over the last 15 days and a higher minimum
> yield over the last 15 days she would be likely to succumb to mastitis

Therefore, in our model for each milk characteristic (e.g. yield and lactose
content), we use only the values from the current day (or the most recent
values) plus a single historical aggregate: skewness over the last 30 days.
The features used in the model are shown in the first column of Table 1. The
model trained is a gradient boosting model and we use data from 2010–2018 to
train it, and data from 2018–2019 to evaluate it. Figure 1 illustrates the
ability of the model to correctly predict sub-clinical mastitis cases at
different time horizons from 1 day in advance to 7 days in advance (measured
using the hold-out test set described above). Domain experts informed us that
capturing $70\%$ of infections at least 1 day in advance of their onset would
lead to a model more useful than current approaches. The model trained
achieves this from 5 days in advance.

Feature Name | Actionable | Actionable Time | Confidence | Size of Change
---|---|---|---|---
Somatic cell count (SCC) | No |  | Very high |
Yield | Yes |  | Low | 2 kg
Fat percentage | Yes | 1-2 days | Low | 0.05 percentage units
Protein percentage | Yes | 1-2 days | Low | 0.05 percentage units
Lactose percentage | No |  | Low |
Urea | Yes |  | Low |
Body condition score (BCS) | Yes | 2 weeks | Very high | 0.25 units
Weight | Yes | 1 week | Medium | 10 kg
Genetic merit | Yes | 5 years | Very high |
Parity | No |  | Very high |
Days since calving | No |  | Very high |
Calendar month | No |  | Low |
Infections for this lactation stage (farm) | No |  | Very high |
Infections for this lactation stage (cow) | No |  | Very high |
Infections for this cow | No |  | Very high |
| Proportion of cows infected per farm
---
and calving year
No |  | Very high |
Table 1: The features used in the model and an indication of whether or not
they are actionable and can be used to build confidence. For those that are
actionable, the time within which changes can be made, and the level of change
that is appropriate to suggest, also are provided.

## Generating Useful Counterfactuals

Given a trained binary classification model $f(\dots)$, counterfactuals
represent a hypothetical input vector $\mathbf{x}^{\prime}$ giving the output
$f(\mathbf{x}^{\prime})=y^{\prime}$ where $y^{\prime}$ is the _positive_
class. This vector $\mathbf{x}^{\prime}$ is derived from an actual vector
$\mathbf{x}$ as a set of small changes $\Delta{}\mathbf{x}$ to apply to
$\mathbf{x}$ to change a _negative_ class outcome $f(\mathbf{x})=y$ into
$y^{\prime}$. That is, we seek to optimize $\Delta{}\mathbf{x}$ under the
constraint $f(\mathbf{x}+\Delta{}\mathbf{x})=y^{\prime}$. In our
implementation, $\Delta{}\mathbf{x}$ is found using the COBYLA (?) constrained
optimization approach. We use a Manhattan distance weighted feature-wise with
the inverse median absolute deviation, as it promotes sparsity and is more
robust to outliers.

For our use-case the _positive_ class is a prediction that a cow will succumb
to sub-clinical mastitis so we seek to provide support for explaining why a
given cow is predicted not to succumb to sub-clinical mastitis. Considering
that users find explanations more useful when only a small number of changes
appear (?) we set a constraint to change no more than 3 features of the model.
We further take a number of additional and domain-specific constraints into
account as outlined in Table 1. These were established through consultation
with a group of domain experts (farmers and animal health experts). For some
of the features a minimum magnitude of change is imposed as, for instance, it
is not meaningful to ask for milk yield to be changed by 0.1 litre per
milking. All of the features come with an indication of _actionability_ and
_confidence_. The latter is, to the best of our knowledge, new to applied
research on counterfactuals. Some features that are not actionable for the
farmers could be still relevant to report on in the counterfactual in order to
build confidence and trust in the model. The best example is somatic cell
count (SCC): there is little the farmer can do about this metric in the short
term but it is part of their own mental model for mastitis. Surfacing this
feature as part of the explanation for the predicted outcome is expected to
contribute to building a trust relationship between the farmer and the
prediction system.

## Worked Example

We use the predictions made on our hold-out test set to demonstrate the
ability of the system developed to predict mastitis and generate useful
counterfactual explanations. We randomly sampled 10,000 test instances
predicted to be healthy with a high prediction score (greater than 0.8) and
generated counterfactual explanations for each of these predictions with the
system described in the previous section.

Figure 2 summarises the prediction scores for the original model predictions
and the prediction scores of counterfactual instances generated. We can see
that the new counterfactual instances consistently result in low prediction
scores. Table 2 shows a sample of the counterfactuals generated (values in
parentheses are those associated with the counterfactuals). The following text
would be used to present the first counterfactual to a farmer:

> If cow #42 had an increase of one and a half units with respect to Yield she
> would be likely to succumb to mastitis.

We can see that only features marked as actionable or confidence building are
changed, and that the changes are in the units specified as actionable.

Figure 2: Probability Changes | Feature Changed
---
$\Delta$ Feature | Prediction Score
| Yield
---
1.5 | 0.979 (0.368)
| Fat
---
1 | 0.979 (0.192)
Table 2: Original versus counterfactual example

## Conclusions & Future Work

In this paper we demonstrate a system built to predict mastitis in cows and
generate explanations of its predictions using a counterfactual approach. We
took input from stakeholders to understand which features could be actionable
and which would help build confidence in the system’s predictions. We
demonstrate the ability of this system to create meaningful explanations while
keeping the number of changes to the original data as small as possible. We
are currently presenting this system to farmer focus groups for feedback.

##  Acknowledgments

This publication has emanated from research conducted with the financial
support of Science Foundation Ireland (SFI) and the Department of Agriculture,
Food and Marine on behalf of the Government of Ireland under Grant Number
[16/RC/3835].

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# Möbius Mirrors

Michael R.R. Good1,2 Eric V. Linder2,3 1Physics Department, Nazarbayev
University, Nur-Sultan, 010000 Kazakhstan
2Energetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan, 010000
Kazakhstan
3Berkeley Center for Cosmological Physics & Berkeley Lab, University of
California, Berkeley, CA 94720, USA

###### Abstract

An accelerating boundary (mirror) acts as a horizon and black hole analog,
radiating energy with some particle spectrum. We demonstrate that a Möbius
transformation on the null coordinate advanced time mirror trajectory uniquely
keeps invariant not only the energy flux but the particle spectrum. We clarify
how the geometric entanglement entropy is also invariant. The transform allows
generation of families of dynamically distinct trajectories, including
$\mathcal{PT}$-symmetric ones, mapping from the eternally thermal mirror to
the de Sitter horizon, and different boundary motions corresponding to Kerr or
Schwarzschild black holes.

##  I Introduction

Black holes represent a fascinating intersection of spacetime and information,
probing the nature of each, and their relation. The relation is intimately
tied with the horizon, and one can study other systems with horizons to
elucidate the connections. Using the moving mirror model DeWitt (1975);
Fulling and Davies (1976); Davies and Fulling (1977), the accelerating
boundary correspondence (e.g. Schwarzschild Good  _et al._ (2016), Reissner-
Nordström Good and Ong (2020), Kerr Good  _et al._ (2021)) maps hot black
holes to particular moving mirror trajectories with horizons. While a close
connection between the particle and energy creation by moving mirrors Walker
(1985); Unruh and Wald (1982) and black holes has been known for sometime,
complete investigation of these exact analogs are ongoing. Quasi-thermal
solutions typify geometric end-states like black hole remnants Chen  _et al._
(2015) (asymptotic constant-velocity mirrors Good  _et al._ (2017, 2019); Good
(2018); Good and Ong (2015); Good (2017)). Complete black hole evaporation
models are characterized by asymptotic zero-velocity mirrors Walker and Davies
(1982); Good  _et al._ (2020a, b); Good and Linder (2017); Good  _et al._
(2013); Good and Linder (2018, 2019).

While entanglement entropy Holzhey  _et al._ (1994), and hence information, is
tied directly to the mirror trajectory, the distant observer only detects
energy flux and particle production. We investigate the connection between
these by considering special transformations of the mirror trajectory such
that the energy flux remains invariant. These are Möbius transformations
arising from the Schwarzian operator in the quantum stress energy tensor, and
correspond to the $SL(2,\mathbb{R})$ group symmetry.

Using this, we can explore the relation between mirrors, and hence spacetimes,
with identical flux, such as thermal emission from de Sitter space Good  _et
al._ (2020c) and the Carlitz-Willey Carlitz and Willey (1987) mirror,
investigate cases with merely asymptotically identical flux, and probe the
zero energy, but nonzero particle, production of uniformly accelerating
motion, whose asymptotic dynamics corresponds to extremal black holes Liberati
_et al._ (2000); Good (2020); Good  _et al._ (2021); Rothman (2000); Foo and
Good (2021).

In Sec. II we describe the Möbius transform and its effects, applying it to
eternally thermal flux and Planckian particle production in Sec. III. We
consider the de Sitter spacetime in particular in Sec. IV, Schwarzschild and
Kerr black holes in Sec. V, and the uniform acceleration case in Sec. VI,
concluding in Sec. VIII.

##  II Möbius Transformations

The quantum stress tensor for an accelerating boundary correspondence (ABC)
indicates an energy flux $\mathcal{F}$ produced from the boundary (horizon) to
an observer at infinity,

$-24\pi\mathcal{F}(u)=\\{p,u\\}\equiv\frac{p^{\prime\prime\prime}}{p^{\prime}}-\frac{3}{2}\left(\frac{p^{\prime\prime}}{p^{\prime}}\right)^{2}\,,$
(1)

where $p(u)$ is the trajectory, i.e. mirror position $v$ as a function of $u$,
where $u=t-x$ and $v=t+x$ are the null coordinates, also called the retarded
and advanced times. It can be derived via point-splitting Fulling and Davies
(1976), or via the Schwinger term in the Virasoro algebra Reuter (1989). We
use natural units, e.g. $\hbar=k_{B}=1$, throughout.

The notation $\\{p,u\\}$ denotes the Schwarzian derivative, and this signals
that an ABC has an underlying symmetry in the form of the Möbius
transformations of $SL(2,\mathbb{R})$,

$p(u)\to\frac{ap(u)+b}{cp(u)+d},\quad ad-bc=1\,.$ (2)

(Since $p$ nominally has spacetime dimension one, we can make it dimensionless
by interpreting it as $\kappa p$, with $\kappa$ a normalizing factor.)

Thus, a trajectory

$P(u)=\frac{ap(u)+b}{cp(u)+d}\equiv Mp(u)\,$ (3)

with $ad-bc=1$ has the same energy flux as the original $p(u)$. Later, we will
show that it has invariant particle flux seen by an observer as well.

We can divide the transforms into two cases, when $c=0$ and when $c\neq 0$.
When $c=0$ then $P=(a/d)p+b/d$. The $b/d$ piece is a shift in $p$, the
translation part of the Möbius transform. This does not contribute to
$P^{\prime}$ and hence not to the energy flux, and since $P$ enters the
Bogolyubov beta coefficient $\beta_{\omega\omega^{\prime}}$ as
$e^{-i\omega^{\prime}P}$ then a constant addition in $P$ is a pure phase and
will cancel in $|\beta_{\omega\omega^{\prime}}|^{2}$. Nor does the translation
change the rapidity $\eta(u)=(1/2)\ln p^{\prime}(u)$ and so it is not of much
interest. It can, however, alter the position of the mirror at infinity, e.g.
whether the mirror starts at null coordinate $v=0$ or some finite value.

Thus the $c=0$ case (hence $d=1/a)$ is

$P=a^{2}p\qquad(c=0)\,.$ (4)

This represent the dilatation part of the Möbius transform. Note that since
$\eta(u)=(1/2)\ln p^{\prime}$ then a constant multiplicative factor for $P$ is
just an additive shift of $\eta\to\eta+\ln a$. This does have physical
consequences, as an additive factor to the entropy flux, and a multiplicative
factor to the mirror acceleration.

When $c\neq 0$ then we can multiply the numerator and denominator in $P$ by
$c$ to get

$P=\frac{acp+ad-ad+bc}{c(cp+d)}=\frac{a}{c}-\frac{1}{c(cp+d)}\,,$ (5)

where we have used $ad-bc=1$. Again ignoring the constant term, we can write
($s\equiv cd$)

$P=\frac{-1}{c^{2}p+s}\qquad(c\neq 0)\,.$ (6)

Now $\eta(P)$ is not simply related to $\eta(p)$, and the entropy and
acceleration are likewise different for the two trajectories, though the
energy flux is the same. When $d=0$ ($s=0$) then the Möbius transform is an
inversion of $p$, with a minus sign.

Equations (4) and (6) are the two transforms exhibiting the symmetry of the
ABC due to the Schwarzian, hence leaving the energy flux invariant. We are
also interested in whether this carries over to the particle creation and its
spectrum. Recall that the particle creation per mode per mode is
$N_{\omega\omega^{\prime}}=|\beta_{\omega\omega^{\prime}}|^{2}$, where
$\omega$ is the outgoing and $\omega^{\prime}$ the ingoing frequency, and the
particle spectrum seen by a distant observer is
$N(\omega)=\int_{0}^{\infty}d\omega^{\prime}\,|\beta_{\omega\omega^{\prime}}|^{2}$.
In particular, for the thermal case $N(\omega)$ has a Planckian spectrum of
particles. The Bogolyubov beta coefficient follows

$\beta_{\omega\omega^{\prime}}=\frac{-1}{2\pi}\sqrt{\frac{\omega}{\omega^{\prime}}}\int_{u_{\textrm{min}}}^{u_{\textrm{max}}}du\,e^{-i\omega
u-i\omega^{\prime}p(u)}\ .$ (7)

While we cannot evaluate this for $P(u)$ coming from arbitrary $p(u)$, we
examine several physically important cases in the following sections.

##  III Eternal Thermal Mirror

We begin with the classic case of constant thermal energy emission, having
particles distributed in a Planck distribution as discovered by Carlitz-Willey
Carlitz and Willey (1987) (see Good and Linder (2018); Good (2013); Good  _et
al._ (2013) for its trajectory in spacetime coordinates and further detail).
This corresponds to a particular nonuniformly accelerating mirror, given by

$p(u)=\frac{-1}{\kappa}\,e^{-\kappa u}\ .$ (8)

Figure 1 shows the trajectory as the solid black line, starting asymptotically
inertial (no acceleration) albeit at light speed, and evolving dynamically
toward asymptotic infinite acceleration with horizon at $v_{H}=0$ in advanced
time $v$. Since $p(u)$ is the function label for $v$, we see from Eq. (8) that
the mirror is limited to the bottom and left quadrants of the Penrose diamond.

Figure 1: Penrose diagram of constant energy flux mirrors with Planck
distributed particles emission, showing Carlitz-Willey (black) Eq. (8), and a
transformed case (blue), Eq. (9) with an equivalent spectrum. The de Sitter
mirror (green), Eq. (18), is also related by Möbius transform. Here $\kappa=2$
for illustration.

Now let us apply the Möbius transform, specifically the negative inversion of
Eq. (6) with $c=-b=1$, $a=d=0$. This gives

$P(u)=\frac{1}{\kappa}\,e^{\kappa u}\,,$ (9)

and is equivalent to the transform $x\to-x$, $t\to-t$, i.e. a $\mathcal{PT}$
symmetry. The trajectory is limited to the right and top quadrants of the
Penrose diagram, plotted in Figure 1 as the solid blue line. The early time
behavior illustrates the mirror climbing out of a horizon and approaching an
asymptotic inertial end state at time-like future infinity.

Explicit calculation indeed shows that the energy flux for both trajectories,
as observed by a witness at future-null infinity, are identical, here the
thermal constant,

$\mathcal{F}=\frac{\kappa^{2}}{48\pi}\,.$ (10)

One also finds that for the transformed case, like the original Carlitz-Willey
mirror, the particles radiated are in a Planck distributed spectrum,

$N_{\omega\omega^{\prime}}=\frac{1}{2\pi\kappa\omega^{\prime}}\,\frac{1}{e^{2\pi\omega/\kappa}-1}\,,$
(11)

with temperature $T=\kappa/2\pi$.

This result is not void of physical implications. An observer who discovers
constant energy flux emission and hot particles (with a temperature) will not
be able to distinguish between the accelerated boundary conditions, nor
determine the origin of such radiation. Moreover, whether the horizon happened
in the past, or in the future, cannot be determined.

To examine the particle creation in detail, we investigate the Bogolyubov beta
function under the Möbius transform. For this eternal thermal mirror, Eq. (8),
the transform of $p(u)=-(1/\kappa)e^{-\kappa u}$ in the $c=0$ case is given by
Eq. (4); evaluating Eq. (7) yields

$\beta_{\omega\omega^{\prime}}\sim(-i\omega^{\prime}a^{2})^{-i\omega/\kappa}\,\Gamma(i\omega/\kappa)\,.$
(12)

We now see how the extra constant $a$ from the (dilatation) transformation
does not propagate to $|\beta_{\omega\omega^{\prime}}|^{2}$:

$(ix)^{iy}=x^{iy}\,e^{-(\pi/2)y\,sgn(x)}\,$ (13)

and so when multiplying it by its complex conjugate we simply get $e^{\pi
y}=e^{-\pi\omega/\kappa}$, independent of $x=-\omega^{\prime}a^{2}$.
Alternately, one can view $x^{iy}=e^{iy\ln x}$ and so $a^{2}$ enters as a pure
phase.

Next consider the eternal thermal $c\neq 0$ case. For the $\mathcal{PT}$
transformed mirror, i.e. Eq. (9), the integral is very similar, giving

$\beta_{\omega\omega^{\prime}}\sim(i\omega^{\prime}/\kappa)^{1-i\omega/\kappa}\,\Gamma(1-i\omega/\kappa)\,,$
(14)

and this provides an identical $|\beta_{\omega\omega^{\prime}}|^{2}$ to that
from Eq. (12) and hence the original Carlitz-Willey case: a thermal, Planckian
particle spectrum.

The more general case when $d\neq 0$, i.e. Eq. (6) with $s\neq 0$, is
considerably less tractable. The Bogolyubov beta function involves a Whittaker
function,

$\beta_{\omega\omega^{\prime}}\sim\Gamma\left(\frac{-i\omega}{\kappa}\right)\,W_{i\omega/\kappa,-1/2}(i\omega^{\prime}/s)\,,$
(15)

but the spectrum per mode squared, $N_{\omega\omega^{\prime}}$

$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{4\pi\kappa\omega^{\prime}\sinh(\pi\omega/\kappa)}\left|U\left(\frac{-i\omega}{\kappa},0,\frac{i\omega^{\prime}}{s}\right)\right|^{2}.$
(16)

is not the orthodox thermal spectrum, Eq. (11), and thus
$|\beta_{\omega\omega^{\prime}}|^{2}$ is not invariant for a general
trajectory transformation. Here $U$ is the confluent hypergeometric function.

When the argument of $W$ is large, e.g. the high frequency limit, then
$W_{a,b}(z)\to e^{-z/2}z^{a}$. Since $z=i\omega^{\prime}/s$ is imaginary, the
first factor is a pure phase and cancels in the modulus, while $z^{a}$ gives a
factor $e^{-\pi\omega/(2\kappa)}$ using Eq. (13). Thus the
$|\beta_{\omega\omega^{\prime}}|^{2}$ is the same as the original, Carlitz-
Willey mirror,

$|\beta_{\omega\omega^{\prime}}|^{2}\sim\left|\Gamma\left(\frac{-i\omega}{\kappa}\right)\right|^{2}\,e^{-\pi\omega/\kappa}\sim\frac{1}{e^{2\pi\omega/\kappa}-1}\,,$
(17)

a Planckian particle distribution. Outside the high frequency regime the
invariance of $|\beta_{\omega\omega^{\prime}}|^{2}$ is broken. However, at the
level of the actual (observable) particle spectrum, $N(\omega)=\int
d\omega^{\prime}\,|\beta_{\omega\omega^{\prime}}|^{2}$, the thermal spectrum
is preserved. See Section VII for the proof.

This indicates that in a Möbius transformation with $cd\neq 0$, the particle
production from the new mirror is not identical for the (unobservable)
particle per mode per mode distribution. The low-frequency situation is
reminiscent of the ‘soft’ (zero energy) particle per mode per mode
distribution of certain mirrors, such as in the uniformly accelerated case; we
treat that in more detail in Section VI. However, the particles reaching a
distant observer would be blueshifted, and therefore in the high frequency
limit where the per mode per mode distribution is invariant. And as mentioned
in the previous paragraph, the observable particle spectrum $N(\omega)$ is
thermal and invariant.

##  IV de Sitter

A de Sitter space has a horizon and eternally thermal emission; the
accelerated boundary correspondence for it is a mirror with trajectory Good
_et al._ (2020c)

$p(u)=\frac{2}{\kappa}\,\tanh\frac{\kappa u}{2}\,.\qquad({\rm de\ Sitter})$
(18)

Using the Möbius transformations we can convert this into the Carlitz-Willey
(CW) eternal thermal mirror of Eq. (8), or the reverse:

$\displaystyle P_{\rm dS}$ $\displaystyle=$
$\displaystyle\frac{-1}{c^{2}p_{\rm CW}+s}\qquad{\rm with}\
c^{2}=\frac{\kappa}{4}=-s,$ (19) $\displaystyle P_{\rm CW}$ $\displaystyle=$
$\displaystyle\frac{-1}{c^{2}p_{\rm dS}+s}\qquad{\rm with}\
c^{2}=\frac{\kappa}{4},\ s=\frac{1}{2}\,.$ (20)

As before we have dropped constant contributions to $P$ as they only give a
constant phase (though they do affect the $v$ location of the horizon). See
Figure 1 for a plot of the de Sitter mirror (green), Eq. (18) with $\kappa=2$.

We can use the same transformations for anti-de Sitter space, where
$p=(2/\kappa)\tan(\kappa u/2)$, as long as we take $\kappa_{\rm AdS}=\pm
i\kappa_{\rm dS}$, since AdS has negative eternal thermal flux
$F=-\kappa^{2}_{\rm dS}/(48\pi)$.

As for the eternal thermal mirror with $d\neq 0$ (recall $s=cd$), the
Bogolyubov beta coefficient resolves to Eq. (15), which by converting to the
confluent hypergeometric function $M$ is equivalent to Eq. (16) of Good  _et
al._ (2020c), and has the same ‘soft’ particles in
$|\beta_{\omega\omega^{\prime}}|^{2}$, while keeping an invariant thermal
particle spectrum $N(\omega)$.

##  V Schwarzschild and Kerr

The Schwarzschild ABC has mirror trajectory Good  _et al._ (2016)

$p(u)=v_{H}-\frac{1}{\kappa}\,W\left(e^{-\kappa(u-v_{H})}\right)\,,$ (21)

where $W$ is the Lambert $W$ function, also known as a product log.
Calculating the particle spectrum of this under Möbius transforms is not
analytically tractable, so we make some general observations.

Figure 2 plots the Schwarzschild mirror trajectory as the black curve. The
other curves in the figure show the application of Möbius transforms, and one
can verify that they have identical energy flux to the original Schwarzschild
mirror. (In fact, one can even create spacelike trajectories with the same
energy flux.)

Figure 2: Penrose diagram of some identical flux mirrors. The original
Schwarzschild mirror, Eq. (21), is black. The colors, (red, blue, purple),
correspond to $P(u)$ with schematic form: $1/W$, $-W/(W+1)$, $1/(W-1)$,
respectively. The purple line (both branches are shown) is a Möbius transform
of the Schwarzschild mirror with $u$-horizon at $u=-1$, where the mirror, Eq.
(22), runs into the observer at right null-infinity. The energy flux detects
nothing unusual.

Note that one of the Möbius transforms, with $a=0,d=c=1,b=-1$, of the
Schwarzschild mirror (with $\kappa=1$, $v_{H}=0$ for simplicity) gives the
purple curve,

$P(u)=\frac{1}{W\left(e^{-u}\right)-1}$ (22)

with identical energy flux and a horizon in $u$, but where the mirror smashes
into our intrepid observer at null-infinity. Despite this, the witness will
register nothing unusual happening in the energy flux! Interestingly, one can
imagine wrapping the space so the points $(u,v)=(-1,+\infty)$ and
$(-1,-\infty)$ join, making the purple mirror trajectory continuous
(speculatively, a wormhole?). We consider an implication of this sort of idea
in Sec. VI.2.

In the hopes of dealing with simpler expressions than product logs, and
exploring the Kerr metric for which $p(u)$ is unknown, one might use the label
function $f(v)$ describing the null coordinate $u$. For Kerr (and
Schwarzschild as a limiting case),

$f(v)=v-\frac{1}{2g}\frac{1+\delta}{\delta}\,\ln|gv|+\frac{1}{2g}\frac{1-\delta}{\delta}\,\ln|gv-\delta|\,,$
(23)

where $g=1/(4M)$ and $\delta=\sqrt{1-a^{2}/M^{2}}$, with $M$ and $a$ the mass
and spin parameter of the black hole. For the Schwarzschild case $a=0$,
$\delta=1$, $g=\kappa$, and the last term in $f(v)$ vanishes. However, the
Möbius transformation only applies to $p(u)$. See Appendix A for discussion of
transformations applied to $f(v)$.

##  VI Constant Acceleration Mirrors

Mirrors with constant acceleration have zero energy flux. However, the
particle production is a more subtle matter. Let us explore what Möbius
transformations of uniform acceleration look like. (Of course they will
preserve the zero energy flux.) The classic uniform acceleration mirror
trajectory is given by $p(u)=-1/(\kappa^{2}u)$, with acceleration
$\alpha(u)=p^{\prime\prime}(u)/[2(p^{\prime})^{3/2}]=-\kappa$.

###  VI.1 Branches Matter

If we Möbius transform with $c=0$, then $P(u)=a^{2}p=-a^{2}/(\kappa^{2}u)$.
This dilatation is merely a redefinition of $\kappa$, i.e. a different
constant acceleration. As mentioned, the energy flux remains zero, but the
particle production involves a subtlety: there are two branches, $u<0$ and
$u>0$ on either side of the horizon at $u=0$. These can be viewed as two
mirrors. A Penrose diagram of the dual-mirror system situation is given in
Figure 3 depicting the two mirrors (each shown for various $\kappa$
parameters) that are the two roots of the single spacetime coordinate
trajectory function. In spacetime coordinates, $x$ and $t$, hyperbolic motion
is

$x^{2}-t^{2}=\alpha^{-2}\,,$ (24)

so that there are actually two mirrors:

$x(t)=\pm\sqrt{\alpha^{-2}+t^{2}}\,,$ (25)

where $\alpha=-\kappa$ is the constant proper acceleration for each. The
mirror in the right quadrant of the diagram traverses $(u,v)=(-\infty,0)$ to
$(0,\infty)$ while the one in the left quadrant traverses $(0,-\infty)$ to
$(\infty,0)$.

Figure 3: Penrose diagram of the uniformly accelerated mirror, with
$\kappa=1/2,1,2$, for red, blue, green, respectively. Note that $\kappa=1$
(blue) is a straight line on the conformal diagram but is not stationary in
space. Horizons exist at $u=0$ and at $v=0$. The particle production and
energy emission is zero.

###  VI.2 Zero Particle Creation with Two Branches

For a $c\neq 0$ Möbius transform,

$P(u)=\frac{-1}{-c^{2}/(\kappa^{2}u)+s}=-\frac{1}{s}-\frac{c^{2}}{s(s\kappa^{2}u-c^{2})}\,,$
(26)

and we can ignore the first term as contributing a constant phase. Switching
to $U=s\kappa^{2}u-c^{2}$, we have $P(U)=-c^{2}/(sU)$. The integration in
$\beta_{\omega\omega^{\prime}}$, split into $[-\infty,0]$ and $[0,\infty]$,
but counting both branches gives

$\beta_{\omega\omega^{\prime}}\sim\sqrt{\frac{4q}{\omega}}\,e^{i\pi/2}\,\left[K_{-1}\left(-2\sqrt{q\omega}\right)+K_{-1}\left(2\sqrt{q\omega}\right)\right]\,,$
(27)

where $q=\omega^{\prime}/\kappa^{2}$ for $p(u)$ and
$q=(c^{2}/s^{2})\omega^{\prime}/\kappa^{2}$ for the transformed $P(u)$. Since
for the modified Bessel function of the second kind $K_{-1}(-z)=-K_{-1}(z)$,
we find $\beta_{\omega\omega^{\prime}}=0$ regardless of $q$ and hence $c$,
$s$.

Thus, when taking into account both branches, the transformation leads to both
invariant (zero) flux and (zero) particle production for the constant
acceleration case.

###  VI.3 Non-zero Particle Creation with One Branch

In the previous subsection we worked with a dual mirror system (two roots)
(see also Fabbri and Navarro-Salas (2005)). Now we will look at just the
single mirror system (one root) with uniform acceleration, which has long been
known to have the surprising result of non-zero particle production with zero
energy flux Birrell and Davies (1984). To compute the soft particle production
from a single uniformly accelerating mirror, we can shift it over to the
origin to avoid confusion with the previous system, expressing the single
uniformly accelerated mirror, again with $\alpha=-\kappa$, as

$p(u)=\frac{u}{1+\kappa u}\,,$ (28)

with early-time horizon positioned at $u_{H}=-\kappa^{-1}$. It has particle
creation that is solved by integrating from the early time horizon onward
(this branch only includes one root, and therefore only one mirror) via

$\beta_{\omega\omega^{\prime}}=\frac{-1}{2\pi}\sqrt{\frac{\omega}{\omega^{\prime}}}\int_{-\kappa^{-1}}^{\infty}du\,e^{-i\omega
u-i\omega^{\prime}u/(1+\kappa u)}\,.$ (29)

Using a simple substitution $X=u+1/\kappa$ gives a range of integration from
$[0,\infty]$ and

$\beta_{\omega\omega^{\prime}}=\frac{-i}{\pi\kappa}\,e^{i(\omega-\omega^{\prime})/\kappa}\,K_{1}\left(\frac{2}{\kappa}\sqrt{\omega\omega^{\prime}}\right)\,.$
(30)

The phase pre-factor is unimportant (as well as the sign of the beta) upon
complex conjugation, so

$|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{\pi^{2}\kappa^{2}}\left|K_{1}\left(\frac{2}{\kappa}\sqrt{\omega\omega^{\prime}}\right)\right|^{2}\,.$
(31)

Eq. (31) is the ‘soft’-spectrum per mode per mode of a single uniformly
accelerated mirror, distinctly non-thermal (non-Planckian) Fulling and Davies
(1976); Davies and Fulling (1977); Birrell and Davies (1984); in the high
frequency limit $K_{1}\to 0$ (exponentially), however, and as before the
distribution is independent of transformation. Thus, while the union of the
two branches preserves the particle flux per mode per mode (zero, for a
uniform acceleration) under a Möbius transform, a single branch does not,
except in the high frequency limit. However, as we shall see in Section VII,
the particle spectrum $N(\omega)$ is preserved.

###  VI.4 Thermal Uniform Acceleration

While a single uniformly accelerated mirror produces soft particles, despite
zero energy flux, and we have seen that the spectrum is decidedly non-
Planckian Birrell and Davies (1984), the story of the radiation is not told by
just the acceleration. Is there a context where a uniformly accelerated moving
mirror can have a temperature? Here we find that yes, it almost certainly can,
under very specific circumstances. This is similar to the Unruh effect Unruh
(1976) where an eternally uniformly accelerated observer (not mirror) sees a
Planck distributed particle radiation with temperature proportional to the
acceleration. The situation we will investigate will be in stark contrast to
the original situation of the Davies-Fulling effect for a single moving mirror
that eternally uniformly accelerates creating a Bessel distributed particle
radiation distribution per mode per mode, Eq. (31), with undefined temperature
and zero energy flux (soft particles). If we relax the eternal uniformity, and
look at what happens as a mirror asymptotically approaches uniform
acceleration, we find a particular mirror that can emit constant energy flux
indicative of thermal emission.

From the start of this section, recall that $p(u)=-1/(\kappa^{2}u)$ has
eternal uniform acceleration. This can be written in terms of
$f(v)=-1/(\kappa^{2}v)$. Now consider the trajectory

$f(v)=-\frac{1}{\kappa^{2}v}-\frac{1}{12\kappa^{4}v^{3}}\,.$ (32)

(One could obtain $p(u)$ by solving the cubic equation, but dealing with
$f(v)$ is simpler.) This asymptotically approaches the uniform acceleration
trajectory as $v\to-\infty$. The proper acceleration
$\alpha(v)=-f^{\prime\prime}(v)/[2f^{\prime}(v)^{3/2}]$ has an early time
limit

$\lim_{v\to-\infty}\alpha(v)=-\kappa\,\left[1+\mathcal{O}\left((\kappa
v)^{-2}\right)\right]\,,$ (33)

explicitly approaching uniform acceleration (accelerating leftward using the
Davies-Fulling convention Fulling and Davies (1976); Davies and Fulling
(1977)). The trajectory (green) is shown in Figure 4, along with the eternally
uniform acceleration case (red).

Figure 4: Penrose diagram of a uniformly accelerated mirror (red),
$f(v)=-1/(\kappa^{2}v$), (branch $v<0$ only), and an early-time asymptotically
uniformly accelerated mirror (green), Eq. (32). Here $\kappa=2$ for
illustration.

The energy flux as derived by the quantum stress tensor for the moving mirror
model is also easily found in advanced time using the Schwarzian,
$24\pi\mathcal{F}(v)=\\{f(v),v\\}f^{\prime}(v)^{-2}$, which has an early time
limit,

$\lim_{v\rightarrow-\infty}\mathcal{F}(v)=\frac{\kappa^{2}}{48\pi}\,.$ (34)

This demonstrates conclusively that despite (asymptotically) uniformly
accelerating, the energy flux emission is a non-zero constant, unlike the
eternally uniformly accelerating mirror Fulling and Davies (1976). The key
here is that $\alpha^{\prime}(v)$ is not exactly zero, although it is
asymptotically zero. The constant energy flux of Eq. (34) is identical to that
of the de Sitter mirror Good  _et al._ (2020c) or the eternal thermal mirror
of Carlitz-Willey Carlitz and Willey (1987), which have explicitly been shown
to emit particles in a Planck distribution with temperature

$T=\frac{\kappa}{2\pi}\,.$ (35)

We study approaches to asymptotes further in Appendix A.

##  VII Particle Spectrum Invariance

In the previous sections we have concentrated on the energy flux and the
particle spectrum per mode per mode, i.e.
$N_{\omega\omega^{\prime}}=|\beta_{\omega\omega^{\prime}}|^{2}$. Concentrating
on the observable particle spectrum itself,

$N(\omega)=\int_{0}^{\infty}d\omega^{\prime}\,|\beta_{\omega\omega^{\prime}}|^{2}\,,$
(36)

its invariance under Möbius transformations can be shown quite generally111In
a late stage of writing we found an argument by Reuter (1989), of which ours
is an inverse; we are indebted to it for improving the derivation to that
given here..

To do so, it is most convenient to work with $f(v)$, i.e. the function label
for the retarded time $u$. Here the Bogolyubov beta coefficient takes the form

$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int_{v_{\textrm{min}}}^{v_{\textrm{max}}}dv\,e^{-i\omega^{\prime}v-i\omega
f(v)}\,.$ (37)

In computing the particle spectrum, the proof proceeds by carrying out the
integration over $\omega^{\prime}$ first, so that

$\displaystyle N(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}\omega}\int dv_{1}\int
dv_{2}\,e^{-i\omega[f(v_{1})-f(v_{2})]}$ (38) $\displaystyle\times\int
d\omega^{\prime}\,\omega^{\prime}\,e^{-i\omega^{\prime}(v_{1}-v_{2})}$
$\displaystyle=$
$\displaystyle\frac{-1}{4\pi^{2}\omega}\int\int\frac{dv_{1}dv_{2}}{(v_{1}-v_{2}-i\epsilon)^{2}}\,e^{-i\omega[f(v_{1})-f(v_{2})]}\,,$

where we used a real regulator $\epsilon>0$ that we then take the limit
$\epsilon\to 0$.

Now consider another mirror trajectory that incorporates the Möbius transform
of $p(u)$, i.e. the function label for $v$. We call $V_{i}=Mv_{i}$. Its
particle spectrum will simply be

$\tilde{N}(\omega)=\frac{-1}{4\pi^{2}\omega}\int\int\frac{dV_{1}dV_{2}}{(V_{1}-V_{2}-i\epsilon)^{2}}\,e^{-i\omega[\tilde{f}(V_{1})-\tilde{f}(V_{2})]}\,,$
(39)

where we indicate this mirror’s quantities with tildes.

However, a property of the Möbius transform is that the quantity

$\frac{P^{\prime}(u_{1})P^{\prime}(u_{2})}{[P(u_{1})-P(u_{2})]^{2}}=\frac{p^{\prime}(u_{1})p^{\prime}(u_{2})}{[p(u_{1})-p(u_{2})]^{2}}\,$
(40)

is invariant. Since $p(u)$ is the function label for $v$ (and the same
relation of $P(u)$ with $V$), this is precisely the integration “Jacobian”
$dV_{1}dV_{2}/(V_{1}-V_{2})^{2}$. Furthermore, since $f(v)$ is the label
function for $u$, and $u$ is kept fixed during the transform $p(u)\to P(u)$,
then $\tilde{f}(V=Mv)=f(v)$. This shows that Eq. (38) is identical to Eq.
(39), and hence that the particle spectrum

$\tilde{N}(\omega)=N(\omega)\,,$ (41)

is invariant between Möbius transformed mirrors.

In summary, the Möbius transformation preserves the energy flux and the
particle spectrum, though the Bogolyubov beta coefficients can differ. The
particle distribution per mode per mode can also deviate due to soft particles
when $cd\neq 0$, but is preserved in the observable (high frequency) limit.

##  VIII Conclusions

The anomalous quantum effect that breaks conformal symmetry of classical
relativity is particularly explicit and manifest in the symmetries of the
moving mirror model and its resulting radiation from vacuum. Although
gravitation does not extend to the (1+1)-dimensional spacetime of the model
(where $G_{\mu\nu}$ is identically zero), and the full physical significance
of the gravitational analog in the ABC is limited to the strength of the
analogy between (1+1) and (3+1)-dimensions, the tractability of fully
analytical results due to the additional simplicity gained by avoiding
spacetime curvature is an asset. As far as quantum fields and relativistic
thermodynamics are concerned this approach surrenders some valuable albeit
phenomenological insights.

Our work underscores the symmetry governing the conformal anomaly of the
moving mirror model. The most fundamental observables of the model are
invariant under Möbius transformations. This includes the energy flux
$\mathcal{F}$, the particle spectrum $N(\omega)$, and the von Neumann
entanglement (geometric) entropy $S_{\textrm{ren}}$. In select important cases
we have found that this invariance extends to the spectral measure
$N_{\omega\omega^{\prime}}=|\beta_{\omega\omega^{\prime}}|^{2}$. We summarize
our results:

  * •

Explicit demonstration of the dramatic dynamical change that a Möbius
transform inflicts on the mirror trajectory, while not impacting what the
observer sees. Humorously phrased, this gives a ‘Mirror of Dorian Gray’, or as
seen in Figure 1, ‘objects in moving mirror are closer (or further) than they
appear’; see start of Section III.

  * •

Full particle invariance $|\beta_{\omega\omega^{\prime}}|^{2}$ for the eternal
thermal mirror under dilatations or inversions ($cd=0$ case). However,
$|\beta_{\omega\omega^{\prime}}|^{2}$ can differ for transforms where $cd\neq
0$, though invariance is restored in the high frequency regime, and upon
integration over the ingoing modes to form the observable particle spectrum
$N(\omega)$; see end of Section III and Section VII.

  * •

Explicit Möbius transforms between the Carlitz-Willey mirror and de Sitter
mirror. Although both are eternally thermal, the Bogolyubov beta coefficients
and $|\beta_{\omega\omega^{\prime}}|^{2}$ are not invariant under the
transforms, due to changing horizon structure; see Section IV.

  * •

A Möbius transform on the Schwarzschild mirror can give a horizon unseen by
the stress tensor, Eq. (22) and Figure 2, demonstrating that the energy flux
may contain hidden horizons, or a wrapping of the spacetime itself; see
Section V.

  * •

Consistency of zero energy and zero particles for dual uniform accelerating
mirrors; see Section VI.2.

  * •

Shown how asymptotic uniform acceleration can radiate thermal energy flux; see
Section VI.4.

  * •

Shown that zero total energy time-dependent accelerating mirrors are
forbidden; see Appendix B.

  * •

Elucidation of invariance from Möbius transformations for entanglement
entropy; see Appendix C.

The moving mirror model has come a long way over the five decades since DeWitt
DeWitt (1975), Davies and Fulling Fulling and Davies (1976); Davies and
Fulling (1977) demonstrated accelerated boundaries produced particles from the
quantum vacuum. It has flourished, with e.g. Schwinger Schwinger (1992), Unruh
Davies  _et al._ (1976), Wilczek Wilczek (1993) helping to establish
theoretical and experimental Chen and Mourou (2020, 2017) research directions
into the nature of particle creation, spacetime, and information. In 2021
alone, the dynamical Casimir effect field has been enriched by diverse, novel
ideas ranging over holography, qubits, relativity, and harvesting entropy Liu
_et al._ (2021); Reyes (2021); Akal  _et al._ (2021a, b); Lin and Chen (2021);
Kawabata  _et al._ (2021); Ageev (2021); Agustí  _et al._ (2021); Alves  _et
al._ (2020); Sato (2021). The connection of the Möbius transform to the group
$SL(2,\mathbb{R})$, Schwarzian derivatives, and conformal field theory offers
another avenue to a deeper understanding of the nature of information,
horizons, and vacuum particle creation.

###### Acknowledgements.

MG acknowledges funding from state-targeted program “Center of Excellence for
Fundamental and Applied Physics” (BR05236454) by the Ministry of Education and
Science of the Republic of Kazakhstan, and the FY2021-SGP-1-STMM Faculty
Development Competitive Research Grant No. 021220FD3951 at Nazarbayev
University. This work is supported in part by the Energetic Cosmos Laboratory.
EL is supported in part by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under contract no. DE-AC02-05CH11231.

##  Appendix A Transforming $f(v)$

As seen in the Schwarzschild case, the trajectory $p(u)$ in terms of the null
coordinate $u$ is not always tractable to general manipulation. It is
worthwhile exploring what the equivalent of the Möbius transformation is for
$f(v)$. That is, what mapping $f(v)\to\tilde{f}(V=Mv)\equiv g(v)$ keeps the
energy flux invariant. Recall that

$24\pi\mathcal{F}(v)=\frac{1}{f^{\prime}(v)^{2}}\left[\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3}{2}\left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{2}\right]\,.$
(42)

We write $g^{\prime}(v)=h(f)f^{\prime}(v)$ to obtain a differential equation
for $h(f)$. Denoting $\dot{h}=dh/df$, we have

$\ddot{h}-\frac{3}{2}\frac{\dot{h}^{2}}{h}+(h-h^{3})\,24\pi\mathcal{F}(f)=0\,,$
(43)

where $\mathcal{F}(f)$ is the energy flux we seek to keep invariant.

Consider the case of zero flux, i.e. the constant acceleration mirror of Sec.
VI. Here the mapping resolves to

$\displaystyle f(v)$ $\displaystyle\to$ $\displaystyle\tilde{f}(V=Mv)\equiv
g(v)$ (44)
$\displaystyle=k_{1}+k_{2}^{-2}f(v)\,\delta(k_{3})-\frac{\cancel{\delta}(k_{3})}{k_{3}^{2}f+k_{2}k_{3}}\,,$

where $k_{1},k_{2},k_{3}$ are constants. The $\delta$ notation indicates that
the second term only exists if $k_{3}=0$ (and $k_{2}\neq 0$) and the third
term only exists if $k_{3}\neq 0$ (so one never has both terms at once). We
indicate that when we apply to a Möbius transform to $v$, written as $V=Mv$,
then the function label for $u$ goes from $f(v)$ to $\tilde{f}(V)$. The
trajectory $f(v)=-1/(\kappa^{2}v)$ gives the constant acceleration, zero
energy flux case, so our results indicate that any mirror with trajectory

$g(v)=k_{1}-\frac{\delta(k_{3})}{k_{2}^{2}\kappa^{2}v}-\frac{\cancel{\delta}(k_{3})}{k_{2}k_{3}-k_{3}^{2}/(\kappa^{2}v)}\,,$
(45)

will also have zero energy flux. The case with $k_{3}=0$ is a dilatation, a
redefinition of $\kappa$. When $k_{2}=0$, $k_{3}\neq 0$, then
$g=\kappa^{2}v/k_{3}$ and this is an inversion, to the trivial constant
velocity, zero acceleration case. The case with $k_{3}\neq 0$, $k_{2}\neq 0$
has $g=-\kappa^{2}v/(k_{2}k_{3}\kappa^{2}v-k_{3}^{2})$, and the mirror has
$g(v=\pm\infty)=-1/(k_{2}k_{3})\neq 0$, $g(0^{\pm})=0^{\pm}$, unlike
$f(v=\pm\infty)=0^{\mp}$, $f(0^{\pm})=\mp\infty$.

When the flux is instead thermal, and hence constant, we can solve Eq. (43) to
find

$f(v)\to g(v)=k_{1}-\frac{1}{\kappa}\,\ln\left(\frac{e^{\kappa f}-1}{e^{\kappa
f}+1}\right)\,.$ (46)

One can verify that Carlitz-Willey $f(v)$ can be transformed into de Sitter,
and vice versa, with this formula.

It is also interesting to consider when two ABC have only asymptotically the
same energy flux, whether thermal or zero. The transformation teaches us that
the approach to thermality is nearly identical for the transformed trajectory
relative to the original thermal mirror, even if it is only thermal
asymptotically. Consider $g^{\prime}(v)=[1+\epsilon(v)]\,f^{\prime}(v)$. Then
Eq. (43) becomes $\ddot{\epsilon}=48\pi\mathcal{F}\epsilon$ to first order,
and

$\epsilon(f)\sim
e^{-\int^{f}_{f_{0}}df_{\star}\,\sqrt{48\pi\mathcal{F}(f_{\star})}}\,,$ (47)

as long as the $\dot{h}^{2}/h\sim\dot{\epsilon}^{2}$ term is small compared to
the leading order. Approaching thermality, $\mathcal{F}\to\mathcal{F}_{\rm
th}=\kappa^{2}/(48\pi)$, and the distance moved $\Delta f\to\infty$ as seen
from Eq. (23), or equally the eternal thermal or de Sitter cases, so indeed
$g^{\prime}(v)\to f^{\prime}(v)$ exponentially. Since the energy flux depends
only on $f^{\prime}$ (and its derivatives), then to leading order the
transformed and original mirror trajectories approach thermality in the same
way.

Similarly, when the flux approaches zero, then on that asymptote we can map
the uniform acceleration mirror to, for example, Schwarzschild, which has
$\mathcal{F}(v\to-\infty)\sim v^{-3}\to 0$. This can be done through the
transformation of Eq. (45) with $k_{2}=0$, $k_{3}=\kappa$, giving the leading
order in $f_{\rm Schw}(v)=v-\kappa^{-1}\ln(-\kappa v)\approx v$ as
$v\to-\infty$. Unfortunately, using the full Schwarzschild flux

$\mathcal{F}_{\rm Schw}(v)=\frac{\kappa^{2}}{48\pi}\frac{1-\kappa v}{(1-\kappa
v)^{4}}\,,$ (48)

in Eq. (43) is not tractable due to $\mathcal{F}(f)$ leading us back to
product logs.

##  Appendix B No Go Zero Energy

We have explored transformations that leave invariant the energy flux. If the
energy flux is unchanged then of course the total energy emitted is as well.
Suppose we now consider mirrors with the same total energy, without requiring
the energy flux $\mathcal{F}(u)$ be the same. For mirrors where the total
energy is infinite (e.g. the eternal thermal mirrors), this question is not so
interesting: there are many ways to add up to infinity. The other total energy
of particular interest is zero net energy, such as seen in the constant
acceleration case. There, the zero energy arises from uniform zero energy
flux. However, we know that negative energy flux can exist as well as positive
energy flux, and indeed is required under certain widespread circumstances,
such as unitarity Good  _et al._ (2020a); Bianchi and Smerlak (2014a, b).

Can we arrange mirror motion such that negative energy flux at some time
exactly cancels positive energy flux at some other time, leaving zero total
energy? We present here a no-go conjecture against this possibility. What we
find is that such a trajectory balancing the negative and positive energy
fluxes is always interrupted by a horizon, leaving a net positive or negative
total energy.

It is useful to write the energy flux in terms of the rapidity
$\eta(u)=(1/2)\ln p^{\prime}(u)$. Then the total energy is

$E=\int_{u_{-}}^{u_{+}}du\,\left[\eta^{\prime\prime}-\left(\eta^{\prime}\right)^{2}\right]\,,$
(49)

where $u_{-}$, $u_{+}$ are the limits of mirror motion, either finite or
infinite horizons.

The first term is a total derivative, so

$E=\eta^{\prime}(u_{+})-\eta^{\prime}(u_{-})-\int
du\,\left(\eta^{\prime}\right)^{2}\,.$ (50)

To achieve zero energy, since the remaining integrand is positive we require
$\eta^{\prime}_{+}>\eta^{\prime}_{-}$.

Now if $\eta^{\prime}(u_{+})\to\pm\infty$ or
$\eta^{\prime}(u_{-})\to\pm\infty$ then the second term, involving
$\eta^{\prime}$, must win, giving negative total energy. If $\eta^{\prime}\to$
const ($\neq 0$), then if $u$ extends to $\infty$ again the integral will
dominate the boundary term, since the sum of a near constant over an infinite
interval is infinite. Thus we need either $\eta^{\prime}\to 0$ (asymptotic
inertia) or to cut off $u$ by a horizon at finite coordinate.

A horizon has $\eta=(1/2)\ln p^{\prime}\to\infty$, so there is a pole in
$p^{\prime}(u)$. Noting that $\eta^{\prime}=p^{\prime\prime}/(2p^{\prime})$,
we would normally expect $\eta^{\prime}\to\infty$, i.e. if the pole is of
order $m$ so $p^{\prime}\sim 1/(u_{+}-u)^{m}$ then
$\eta^{\prime}=m/[2(u_{+}-u)]\to\infty$ as $u\to u_{+}$ (the same holds for
obtaining the horizon with $\eta\to\infty$ by
$p^{\prime}\sim(u-u_{\star})^{m}\to 0$, i.e. the previous $m$ is negative).
Hence this reduces to the previously considered case, which did not give zero
flux. However, there are special cases where $p^{\prime\prime}\sim p^{\prime}$
($\eta^{\prime}\to{\rm const}$) and $p^{\prime\prime}/p^{\prime}\to 0$
($\eta^{\prime}\to 0$). The first of these has $p^{\prime}\sim e^{bu}$, which
blows up at $u_{+}=\infty$; this is not at finite $u$ and this is the
previously considered, failed case of $\eta^{\prime}\to{\rm const}$. The
second of these takes $\eta^{\prime}\to 0$ and is our remaining case to
assess.

For this last case, where $\eta^{\prime}\to 0$, then unless $\eta^{\prime}=0$
for all $u$, the range where it is nonzero will contribute to the integral and
again unbalance the terms, preventing zero energy. The only zero total energy
case is the zero energy flux case, when
$\eta^{\prime\prime}=(\eta^{\prime})^{2}$, and hence $\eta^{\prime}=-1/u$.
This is precisely the constant acceleration case with $p(u)=-1/(\kappa^{2}u)$.
This case does have a pole, at $u=0$, and $\eta^{\prime}\to\infty$ there, but
the terms balance in this one case. For poles of order $m$, the integral gives
$\int du\,u^{-2m}=(1-2m)^{-1}u^{-2m+1}$ while the boundary term gives
$u^{-m}$. Only when $-2m+1=-m$, i.e. the $m=1$ case, do the terms cancel each
other.

##  Appendix C Entanglement Invariance

Here we derive the (1+1)-dimensional entanglement entropy or ‘geometric
entropy’ in conformal field theory (CFT) and show its fundamental invariance
under Möbius transformations of the trajectory. We then break the invariance
by referencing a static mirror on one side of the system, which amounts to a
choice of a reference frame for the observer, deriving the relationship to the
rapidity of the mirror (see e.g. Good (2020); Fitkevich  _et al._ (2020);
Myrzakul  _et al._ (2021); Bianchi and Smerlak (2014b)).

We start with the entropy of a system in (1+1)-D CFT using $\epsilon$ as a UV
cut-off Holzhey  _et al._ (1994),

$S=\frac{1}{6}\ln\frac{L}{\epsilon}\,,$ (51)

where $L$ is the size of the arbitrary system in general. For us, the model
will be the mirror trajectory which measures the size of the system by keeping
track of accessible (1+1) dimensional spacetime in which the quantum field is
free to propagate. That is, for a general and arbitrary moving mirror
trajectory $p(u)$,

$L\equiv p(u)-p(u_{0})\,,$ (52)

where $u$ and $u_{0}$ are null coordinates that form the region in the system
which we are considering, and geometrically $\epsilon$ is asymmetrically
smeared, i.e. $\epsilon^{2}\equiv\epsilon_{p}\epsilon_{p_{0}}$. Here $p(u)$ is
the trajectory of the mirror in null coordinates (it is the usual advanced
time function of retarded time $u$). The smearing and dynamics of the mirror
are related as

$\epsilon_{p}=p^{\prime}(u)\epsilon_{u}\,,\qquad\epsilon_{p_{0}}=p^{\prime}(u_{0})\epsilon_{u_{0}}\,.$
(53)

These $\epsilon_{p}$ smearings are coarse grained parameters at the end of the
sub-systems, that parametrize how well the observer distinguishes the
subsystem from the rest of the universe Holzhey  _et al._ (1994).

Substituting Eqs. (53) into Eq. (51) yields the bare entropy of the system,

$S_{\textrm{bare}}=\frac{1}{12}\ln\frac{\left[p(u)-p(u_{0})\right]^{2}}{p^{\prime}(u)p^{\prime}(u_{0})\epsilon_{u}\epsilon_{u_{0}}}\
.$ (54)

The vacuum entropy of the system can be found by considering a static mirror
where $L=u-u_{0}$ and $\epsilon^{2}=\epsilon_{u}\epsilon_{u_{0}}$. Thus,

$S_{\textrm{vac}}=\frac{1}{12}\ln\frac{(u-u_{0})^{2}}{\epsilon_{u}\epsilon_{u_{0}}}\
.$ (55)

Even though the entropies above are defined in terms of smearing, this
dependence can be removed by an intuitive renormalization (see also Bianchi
and Smerlak (2014b)) via

$S_{\textrm{ren}}\equiv
S_{\textrm{bare}}-S_{\textrm{vac}}=\frac{1}{12}\ln\frac{[p(u)-p(u_{0})]^{2}}{p^{\prime}(u)p^{\prime}(u_{0})(u-u_{0})^{2}}\
.$ (56)

The geometric entropy Eq. (56) is invariant under Möbius transformations, as
seen from Eq. (40). Thus a Möbius transformation of mirror trajectories
preserves energy flux, particle spectrum, and geometric entropy.

Going further, if we define the derivative $p^{\prime}(u_{0})$ as

$p^{\prime}(u_{0})=\frac{p(u)-p(u_{0})}{u-u_{0}}\,,$ (57)

then the dynamic meaning of the entropy becomes clear as we deviate $\delta u$
away from $u_{0}$. We obtain the Möbius invariant,

$S_{\textrm{ren}}=\frac{1}{12}\ln\frac{p^{\prime}(u_{0})}{p^{\prime}(u)}=-\frac{1}{6}(\eta-\eta_{0})\
.$ (58)

Choosing a static reference frame for time $u_{0}$ sets $p^{\prime}(u_{0})=1$
or $\eta_{0}=0$, giving

$S(u)=-\frac{1}{12}\ln p^{\prime}(u)=-\frac{\eta}{6}\ ,$ (59)

which is the common definition of entanglement entropy in terms of rapidity,
e.g. Bianchi and Smerlak (2014b); Chen and Yeom (2017). Interestingly Eq. (59)
is not preserved by a Möbius transform. This is clear from $\eta(u)=(1/2)\ln
p^{\prime}(u)$ and $P(u)=(ap+b)/(cp+d)$, hence
$P^{\prime}=p^{\prime}/(cp+d)^{2}$. The reason for this is that rapidity
$\eta=\tanh^{-1}\dot{z}$ is not by itself invariant, but depends on the
coordinate frame in which the mirror velocity $\dot{z}$ is measured. Both the
invariant and variant von Neumann entropies, Eq. (58) and Eq. (59), measure
the degree of quantum entanglement between the two subsystems, past and
future. The more fundamental measure of entanglement of the system is Eq.
(58), which has invariance, like the particles and energy. The operation of
choosing a static mirror reference choice breaks the symmetry of the model:
rapidity is always measured with respect to some frame.

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