Preprint Nos. JLAB-PHY-24-4003; SLAC-PUB-17762; NJU-INP 088/24

Poincaré invariance, the Unruh effect, and black hole evaporation

Alexandre Deur∗

Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606,
USA

Stanley J. Brodsky

SLAC National Accelerator Laboratory, Stanford University, Stanford,
California 94309, USA

Craig D. Roberts

School of Physics, Nanjing University, Nanjing, Jiangsu 210093, China

Institute for Nonperturbative Physics, Nanjing University, Nanjing, Jiangsu
210093, China

Balša Terzić

Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USA

2024 April 28

In quantum field theory, the vacuum is widely considered to be a complex
medium populated with virtual particle + antiparticle pairs. To an observer
experiencing uniform acceleration, it is generally held that these virtual
particles become real, appearing as a gas at a temperature which grows with
the acceleration. This is the Unruh effect. However, it can be shown that
vacuum complexity is an artifact, produced by treating quantum field theory in
a manner that does not manifestly enforce causality. Choosing a quantization
approach that patently enforces causality, the quantum field theory vacuum is
barren, bereft even of virtual particles. We show that acceleration has no
effect on a trivial vacuum; hence, there is no Unruh effect in such a
treatment of quantum field theory. Since the standard calculations suggesting
an Unruh effect are formally consistent, insofar as they have been completed,
there must be a cancelling contribution that is omitted in the usual analyses.
We argue that it is the dynamical action of conventional Lorentz
transformations on the structure of an Unruh detector. Given the equivalence
principle, an Unruh effect would correspond to black hole radiation. Thus, our
perspective has significant consequences for quantum gravity and black hole
physics: no Unruh effect entails the absence of black hole radiation
evaporation.

##  1 Introduction

The Unruh effect is the name given to a theoretical prediction that an
accelerated observer can detect particles which are unobservable to an
inertial observer [1, 2, 3, 4]. The effect is often interpreted as a
consequence of the intrinsic complexity of the inertial frame vacuum, which is
commonly imagined to be populated by pairs of virtual particles that become
detectable in an accelerated frame. However, vacuum complexity can be viewed
as an artifact of formalisms that do not manifestly enforce causality [5],
like instant-form (IF) dynamics, a commonly used approach to quantization of
field theories. In contrast, the plainly causality-enforcing front-form (FF)
framework [6], has an essentially trivial vacuum, i.e., a structureless Fock
space ground state [7].

A question thus arises: Does acceleration complexify the FF vacuum, so that
the Unruh effect is also a feature of the FF? If not, then the Unruh effect is
itself an artifact, arising only in approaches that violate causality at
intermediate stages of a calculation. This conclusion would have profound
implications: by Einstein’s equivalence principle between gravity and
acceleration [8], there would then be no Hawking radiation [9] and no black
hole information paradox [10], a grave problem in astrophysics and quantum
gravity [11]. Alternatively, if the Unruh effect is a feature of FF dynamics,
then the usual interpretation of Unruh and Hawking phenomena, based on
particles created from of a supposedly complex quantum vacuum, is mistaken
because the effects would also emerge in the presence of a trivial vacuum.

Herein, we discuss the following three distinct cases: (1) IF Minkowski space
(i.e., inertial) vacuum, which is complex; (2) FF Minkowski space vacuum,
which is trivial; (3) “Rindler vacuum”, associated with an accelerating frame
for which “Rindler coordinates” are defined – see Supplemental Material (SupM)
Section B. The Rindler vacuum is complex because Rindler fields are quantized
at equal Rindler time. Importantly, the three vacua are inequivalent because
IF, FF and Rindler frames are not related by any Poincaré transformations.
Notably, although the Rindler vacuum is complex, it is, nevertheless, the
vacuum for a Rindler observer; hence, imperceptible by definition. In this
connection, the Unruh effect is a statement that a Rindler observer perceives
pairs that constitute the IF Minkowski space vacuum.

In Section 2, we summarize important features of Dirac’s three forms of
dynamics [6]. Section 3 explains why an Unruh effect is predicted when fields
are quantized using the common IF approach, whereas that is not so when fields
viewed from the inertial frame are quantized using the FF. The connection
between a complex vacuum and causality violation in IF dynamics is elucidated
in Section 4. Whilst these two sections provide intuitive perspectives, they
do not trace usual paths to a prediction of the Unruh effect. Since such paths
may be more easily followed by some readers, they are described in SupM
Sections B and C. SupM Section D presents a different perspective on the
problem by presenting a dynamical analysis. Section 5 discusses how the IF and
FF findings can be reconciled. This must be possible because Nature does not
depend upon the approach to its analysis. Conclusions are gathered in Section
6. (We use Natural Units throughout, viz. $\hbar\equiv 1\equiv c$. Thus,
energy and frequency are equivalent, as are momentum and wave number.)

##  2 Forms of Dynamics

There is no unique choice of time parameter in relativistic kinematics.
Parametrizing the worldline amounts to a specific convention for foliating the
four-dimensional (4D) spacetime into 3D space + 1D time. There are three
pertinent conventions (forms) fulfilling relativistic invariance, basic
causality, and fundamental spacetime symmetry requirements [6]: IF, where time
is the usual Galilean time (Fig. 1, left panel); FF, where time is aligned
tangentially to the light-cone (Fig. 1, right panel); and Point-Form (PF),
where the equal time hypersurfaces are hyperboloids. For future reference, the
FF coordinates are

$\displaystyle\tau\equiv x^{+}$ $\displaystyle:=$ $\displaystyle
t+z~{}~{}~{}{\rm(time)}\,;\qquad\quad\,x^{-}:=t-z~{}~{}~{}{\rm(space)};$ (1)
$\displaystyle p^{-}$ $\displaystyle:=$
$\displaystyle\omega-p~{}~{}~{}{\rm(energy)}\,;\qquad
p^{+}:=\omega+p~{}~{}~{}{\rm(momentum)}.$ (2)

The labels here highlight the FF associations. Importantly, distinct forms of
dynamics are not related by Poincaré transformations. Furthermore, they are
not equivalent: each form of dynamics possesses a different stability group,
viz. a distinct set of Poincaré generators that are purely kinematical. Such
generators operate in constant form-time hypersurfaces, whereas dynamical
operators evolve the system from one hypersurface to the next.

Figure 1:  Effect of boosts (blue worldline) and constant accelerations (red
worldline) on attached IF (left) and FF (right) frames. Thick arrows show the
frame axes of an observer at rest. The 45∘ dashed line is the trajectory for a
particle moving at light speed ($c\equiv 1$ in natural units). (The
$(-,+,+,+)$ metric is used in this figure, but $(+,-,-,-)$ is used for FF in
the main text, following usage in particle physics where FF dynamics is
employed. In that case, $x^{-}$ has the opposite direction, pointing left and
upward.)

The Poincaré group has 10 generators: 4 linear momenta for spacetime
translations; 3 angular momenta for space rotations; and 3 Lorentz boosts. The
FF stability group is the largest, with 7 generators: the linear momenta;
boosts; and one angular momentum. The IF and PF stability groups have only 6
generators: IF – linear and angular momenta; and PF – the boosts and angular
momenta.

Besides FF’s larger stability group, crucial advantages are offered by the
kinematical character of its boosts [12]. For instance, contrary to IF
dynamics, popular in textbooks, and to other forms of dynamics, such as PF and
oblique FF [13], FF dynamics manifestly preserves causality and the number of
particles in systems related by boosts. These things are key to its success in
proving exact results in high-energy physics – see, e.g., Ref. [14].

In stark contrast, IF does not straightforwardly respect causality because it
allows for vacuum-induced particle pairs, whose creation events are
uncorrelated with the worldline of the system; hence, acausal. Furthermore,
the dynamics induced when IF-boosting a system often complicates its
description with fictitious effects [12]. As elaborated in Section 4, since
commutation relations are imposed at equal form-time in canonical
quantization, viz. along the light-cone in the FF case, the FF vacuum is
trivial. (The possible existence of zero-modes [5] is irrelevant to the Unruh
effect – see SupM Section A.)

##  3 The Unruh effect

Most discussions of an Unruh effect focus on a scalar field in (1+1) spacetime
dimensions. Identification of the mechanism behind the effect in IF dynamics,
which holds for generic field and dimensions, shows that the effect is
predicted in these cases too. Consistently, the absence of an Unruh effect,
owing to triviality of the FF vacuum, is also valid for all cases.

The Unruh effect is predicted in IF dynamics because in non-inertial IF frames
a field $\phi$ cannot unambiguously be factored into distinct time-dependent
and space-dependent parts, namely, separated in a Poincaré-invariant manner
[15]. Consequently, in the decomposition of $\phi$ into modes with what may,
for ease of discussion, be called positive and negative frequency, $f_{p}$,
$f_{p}^{*}$:

$\displaystyle\phi=\int
dp\big{(}\hat{a}_{p}f_{p}+\hat{a}_{p}^{\dagger}f_{p}^{*}\big{)},$ (3)

the assignment of positive or negative frequency is frame dependent.
($\hat{a}_{p}$, $\hat{a}_{p}^{\dagger}$ are, respectively, annihilation,
creation operators.)

In another frame, $\phi=\int
dp^{\prime}\big{(}\hat{b}_{p^{\prime}}f_{p^{\prime}}+\hat{b}_{p^{\prime}}^{\dagger}f_{p^{\prime}}^{*}\big{)}$,
with $\hat{b}_{p^{\prime}}=\alpha\hat{a}_{p}+\beta\hat{a}_{p}^{\dagger}$:
$\beta\neq 0$ if one of the frames is non-inertial. A Bogolyubov
transformation [16] expresses this mixing:

$\displaystyle\begin{bmatrix}\hat{b}_{p^{\prime}}\\\
\hat{b}_{p^{\prime}}^{\dagger}\end{bmatrix}=\begin{bmatrix}\alpha_{11}&\beta_{12}\\\
\beta_{21}&\alpha_{22}\end{bmatrix}\begin{bmatrix}\hat{a}_{p}\\\
\hat{a}_{p}^{\dagger}\end{bmatrix}.$ (4)

Since $\hat{a}_{p}^{\dagger}$ and $\hat{b}_{p^{\prime}}^{\dagger}$ are
creation operators, the vacuum of frame 1, satisfying
$\hat{a}_{p}\ket{0}_{1}\equiv 0$ by definition, is perceived from frame 2 as
containing particles:
$\hat{b}_{p^{\prime}}\ket{0}_{1}=(\alpha_{11}\hat{a}_{p}+\beta_{12}\hat{a}_{p}^{\dagger})\ket{0}_{1}=\beta_{12}\ket{p}\neq
0$.

Typical formal analyses leading to a prediction of the Unruh effect, involving
Rindler frames, are provided in SupM Sections B and C. Here, we approach the
problem by beginning with a boost and observing the effect of making the
rapidity, $\theta$, time dependent, so that the boosted frame accelerates. The
analysis is performed for IF and FF dynamics in parallel, using normal fonts
for IF-specific formulae and bold fonts for their FF equivalents. Since
directions transverse to the boost are irrelevant, we use a (1+1)D spacetime,
as is typical. The positive frequency modes are then (recall $\tau=x^{+}$):

$\displaystyle{f_{p}\propto e^{i(\omega
t-pz)}~{}~{}~{}{\rm(IF)}}~{}~{}{\rm;}\qquad\bm{f_{p^{+}}\propto
e^{-\nicefrac{{i}}{{2}}(p^{-}\tau+p^{+}x^{-})}~{}~{}~{}{\rm(FF)}}.$ (5)

The IF analysis employs the conventional relativistic metric ${(-,+,+,+)}$,
with which the Unruh effect is usually discussed, whereas the FF derivation
follows the particle physics convention $\bm{(+,-,-,-)}$. The latter implies
the FF product $\bm{a^{\mu}b_{\mu}}$ $\bm{=}$
$\bm{(1/2)(a^{+}b^{-}+a^{-}b^{+})}$ in (1+1)D spacetime.

###  3.1 Lorentz Boost

A Lorentz boost relates the coordinates of two inertial frames:

$\begin{array}[]{llll}t^{\prime}=t\cosh\theta-z\sinh\theta&({\rm
IF})\,;&\quad\boldsymbol{\tau^{\prime}={\rm e}^{\theta}\tau}&({\rm FF})\,,\\\
z^{\prime}=z\cosh\theta-t\sinh\theta&({\rm
IF})\,;&\quad\boldsymbol{x^{-\prime}={\rm e}^{-\theta}x^{-}}&({\rm
FF})\,,\end{array}$ (6)

or, for energy-momentum:

$\begin{array}[]{llll}\omega^{\prime}=\omega\cosh\theta-p\sinh\theta&({\rm
IF})\,;&\quad\boldsymbol{p^{+\prime}={\rm e}^{\theta}p^{+}}&({\rm FF})\,,\\\
p^{\prime}=p\cosh\theta-\omega\sinh\theta&({\rm
IF})\,;&\quad\boldsymbol{p^{-\prime}={\rm e}^{-\theta}p^{-}}&({\rm
FF})\,.\end{array}$ (7)

The inverse transforms are straightforwardly obtained by changing
$\theta\to-\theta$ and switching $t^{\prime}\leftrightarrow t$, etc.:

$\begin{array}[]{llll}t=t^{\prime}\cosh\theta+z^{\prime}\sinh\theta&({\rm
IF})\,;&\quad\boldsymbol{\tau={\rm e}^{-\theta}\tau^{\prime}}&({\rm FF})\,,\\\
z=z^{\prime}\cosh\theta+t^{\prime}\sinh\theta&({\rm
IF})\,;&\quad\boldsymbol{x^{-}={\rm e}^{\theta}x^{-\prime}}&({\rm
FF})\,.\end{array}$ (8)

Figure 1 shows the effect of the boost (blue straight worldline with green
coordinate axes): in IF dynamics, it generates a rotation of the coordinate
axes by an angle determined by the rapidity, $\theta$, whereas axes
orientation is unaffected in FF.

In the boosted frame, the frequencies in Eq. (5) can be identified via a time
derivative of the field modes:

$\begin{array}[]{llll}\displaystyle\frac{\partial f_{p}}{\partial
t^{\prime}}=\frac{\partial x^{\mu}}{\partial
t^{\prime}}\partial_{\mu}f_{p}=\left[\frac{\partial t}{\partial
t^{\prime}}\partial_{t}+\frac{\partial z}{\partial
t^{\prime}}\partial_{z}\right]f_{p}&({\rm
IF})\,;&\displaystyle\boldsymbol{\frac{\partial
f_{p^{+}}}{\partial\tau^{\prime}}=\frac{\partial
x^{\mu}}{\partial\tau^{\prime}}\partial_{\mu}f_{p^{+}}=\left[\frac{\partial\tau}{\partial\tau^{\prime}}\partial_{\tau}+\frac{\partial
x^{-}}{\partial\tau^{\prime}}\partial_{x^{-}}\right]f_{p^{+}}}&({\rm
FF})\,.\end{array}$ (9)

Using Eqs. (5), (8), (7), this becomes:

$\begin{array}[]{lllll}&\displaystyle\frac{\partial f_{p}}{\partial
t^{\prime}}=i\left[\omega\cosh\theta-p\sinh\theta\right]f_{p}&({\rm
IF})\,;&\quad\displaystyle\boldsymbol{\frac{\partial
f_{p^{+}}}{\partial\tau^{\prime}}=-e^{-\theta}\frac{i}{2}p^{-}f_{p^{+}}}&({\rm
FF})\,,\\\\[7.74997pt] \Rightarrow&\displaystyle\frac{\partial f_{p}}{\partial
t^{\prime}}=i\omega^{\prime}f_{p}&({\rm
IF})\,;&\quad\displaystyle\boldsymbol{\frac{\partial
f_{p^{+}}}{\partial\tau^{\prime}}=-\frac{i}{2}p^{-\prime}f_{p^{+}}}&({\rm
FF})\,,\end{array}$ (10)

i.e., the frequency in a boosted frame is the boosted frequency,
$\omega^{\prime}$ or $\boldsymbol{p^{-\prime}}$. The same holds for negative
frequencies, $f_{p}^{*}$ and $f_{p^{+}}^{*}$. So, for a time-independent
boost, there is no mixing of positive and negative frequencies; the Bogolyubov
transformation is diagonal; and the vacua of the original and boosted inertial
frames coincide.

We next consider a time-dependent rapidity, $\theta(t)$, viz. transformation
to a non-inertial frame.

###  3.2 Time-dependent boost: acceleration

Accelerated frames can be formalized by making $\theta$ vary with time – see
Fig. 1. Consider, therefore, $\theta=ut$ or
$\boldsymbol{\theta=\upsilon\tau}$, with $u\neq 0\neq\boldsymbol{\upsilon}$
and both small, so that acceleration is constant to a good approximation.
Importantly, $\theta$ does not explicitly depend on space:
$\nicefrac{{\partial\theta}}{{\partial x}}=0$ and
$\boldsymbol{\nicefrac{{\partial\theta}}{{\partial x^{-}}}=0}$. Using IF
dynamics and the appropriate entries in Eq. (6) – Eq. (8), then Eq. (10)
yields:

$\displaystyle\frac{\partial f_{p}}{\partial t^{\prime}}$
$\displaystyle=\left[\left([1+z^{\prime}\partial_{t^{\prime}}\theta]\cosh\theta+t^{\prime}\partial_{t^{\prime}}\theta\sinh\theta\right)\partial_{t}+\left([1+z^{\prime}\partial_{t^{\prime}}\theta]\sinh\theta+t^{\prime}\partial_{t^{\prime}}\theta\cosh\theta\right)\partial_{z}\right]f_{p}$
$\displaystyle=i\left[\omega^{\prime}+\partial_{t^{\prime}}\theta(\omega
z-tp)\right]f_{p}\,;$ (11)

whereas, using the FF,

$\displaystyle\boldsymbol{\frac{\partial f_{p^{+}}}{\partial\tau^{\prime}}}$
$\displaystyle\boldsymbol{=\left[{\rm
e}^{-\theta}(1-\tau^{\prime}\partial_{\tau^{\prime}}\theta)\partial_{\tau}+{\rm
e}^{\theta}\partial_{\tau^{\prime}}\theta
x^{-\prime}\partial_{x^{-}}\right]f_{p^{+}}}$
$\displaystyle=\boldsymbol{-\frac{i}{2}\left[p^{-\prime}+\partial_{\tau^{\prime}}\theta(x^{-}p^{+}-\tau
p^{-})\right]f_{p^{+}}\,.}$ (12)

Evidently, the IF and FF expressions are similar: Eq. (11) cf. Eq. (12).
Moreover, reviewing Eqs. (6) – Eq. (8), one sees that $\omega z-tp$ =
$\omega^{\prime}z^{\prime}-t^{\prime}p^{\prime}$ and
$\boldsymbol{x^{-}p^{+}-\tau p^{-}}$ =
$\boldsymbol{{x^{-\prime}p^{+\prime}-\tau^{\prime}p^{-\prime}}}$, viz. these
combinations are Poincaré invariant scalar products, which we will write as
$\tilde{p}^{\mu}x_{\mu}=p^{\mu}\tilde{x}_{\mu}$; hence,

$\begin{array}[]{llll}\displaystyle\frac{\partial f_{p}}{\partial
t^{\prime}}=i\left[\omega^{\prime}+p^{\mu}\tilde{x}_{\mu}\partial_{t^{\prime}}\theta\right]f_{p}&({\rm
IF})\,;&\quad\displaystyle\boldsymbol{\frac{\partial
f_{p^{+}}}{\partial\tau^{\prime}}=-\tfrac{i}{2}\left[p^{-\prime}+p^{\mu}\tilde{x}_{\mu}\partial_{\tau^{\prime}}\theta\right]f_{p^{+}}}&({\rm
FF})\,.\end{array}$ (13)

In comparison with the inertial case, Eq. (10), each entry in Eq. (13) has
acquired an additional term, viz. a Poincaré-invariant factor multiplying the
rapidity’s time derivative. Of course, the appearance of that derivative means
the new term violates Poincaré invariance, as was to be expected since
inertial and accelerated frames are not related by genuine boosts
($\theta=\,$constant).

Considering the IF case, the time direction changes along the accelerated
worldline; so, time and space mix – see Fig. 1. This is expressed in the
result
$\partial_{t^{\prime}}\theta(t)=u\cosh(ut^{\prime})[1+uz^{\prime}+ut^{\prime}\tanh(ut^{\prime})]=u\,{\rm
sech}(ut)/[1-uz+ut\tanh(ut)]$. Therefore, it is impossible to develop a
Poincaré-invariant separation of the field $\phi$ into time and space
components. So, the definitions of positive and negative frequencies are
different in each non-inertial frame [15]: the Bogolyubov transformation is
not diagonal.

In the FF case, the time direction remains fixed along the accelerated
worldline – see Fig. 1. Thus, the time-derivative of $\theta$ is solely a
function of time:
$\boldsymbol{\partial_{\tau^{\prime}}\theta(\tau)=\upsilon{\rm
e}^{-\upsilon\tau^{\prime}}[1-\upsilon\tau^{\prime}]=\upsilon{\rm
e}^{-\upsilon\tau}/[1+\upsilon\tau]}$. Consequently, with time and space
directions remaining mutually perpendicular, $\phi$ can still be unambiguously
expanded over positive and negative frequency modes, even for accelerated
frames. The Bogolyubov transformation remains diagonal, and the vacua of
frames coincide and are trivial: _there is no Unruh effect in FF dynamics_.

###  3.3 Interpretation

There is a straightforward explanation for arriving at a prediction of the
Unruh effect when using IF dynamics but not with the FF. Namely, boost
operators are dynamical in the IF – see Section 2: they mix kinematics and
dynamics, leading to non-conservation of particle number. In an accelerating
frame, this mixing changes with time. Thus, non-stationary dynamics emerges
with energy being transferred between systems, viz. an observable effect,
unless cancelled by some other process. In marked contrast, FF boosts are
kinematical operators. Hence, no dynamical effects are introduced by
acceleration. So, an Unruh detector remains quiet.

##  4 Connection to commutation relations

Canonical quantization of fields is performed at constant proper time, i.e.,
with fields defined along the fixed $z$-direction for the IF inertial frame
(Fig. 1, left panel), along the varying $z$-direction for the IF accelerated
frame, and along the fixed $45^{\circ}$ direction for both inertial and
accelerated FF frames (Fig. 1, right panel).

In canonical quantization, the Heisenberg uncertainty principle originates
from commutation relations [17]. Since they are imposed at equal time, the
uncertainty principle also operates on equal time hypersurfaces. This results
in complex vacuum structure when time is defined as IF time because events
that should be causally linked, viz. timelike-separated, may instead become
spacelike-separated owing to position uncertainty.

Consider Fig. 2, which shows events $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ at
times $t_{1}$ and $t_{2}$ for the propagation of a massive field. If
$\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are spacelike-separated, the time-
ordering of $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ is frame-dependent with,
in some frames, $t_{2}<t_{1}$. Then, $Z$-graphs appear where event
$\mathcal{E}_{2}$ is the spontaneous appearance of a particle-antiparticle
pair. The antiparticle annihilates at later time $t_{1}$ with the initial
particle. This description is acausal since the pair appearance is spontaneous
and cannot be caused by the original particle propagation owing to the
spacelike interval. $Z$-graphs contribute negative probabilities to the
propagation, so disconnected creation-annihilation diagrams (vacuum loops),
balancing the negative probabilities, must be introduced to preserve unitarity
[18]. In accelerated frames, the virtual particles in the vacuum loops may
borrow 4-momentum from the acceleration process and, and unless this is
somehow cancelled, become real and observable, leading to the Unruh effect.

An acceleration of a FF frame continuously rescales the axes (Fig. 1, right
panel) without reorienting them; hence, the uncertainty principle continues to
operate along the light-cone direction $x^{-}$. The events $\mathcal{E}_{1}$
and $\mathcal{E}_{2}$ remain timelike-separated, causality is preserved, and
unitarity forbids vacuum loops [12]. Vacuum simplicity persists in an
accelerated FF frame and, without vacuum loops in any type of FF frame, there
is no Unruh effect. Consistently, momentum conservation also prohibits vacuum
loops: FF particles must have non-negative momenta $p^{+}$ [5, 19, 20]; and
since for the vacuum $p^{+}=0$, by definition,111From the FF momentum boost
formula, Eq. (7), $p^{+\prime}\propto p^{+}$; so, the vacuum momentum is zero
in all FF frames, i.e., $p^{+}=0$ for the vacuum, even in boosted and
accelerated frames. This contrasts with non-relativistic intuition and with
the IF framework, in which boosts mix energy and momentum. Thus, zero-mode
($\omega\to\infty$, $p=0$) vacuum contributions acquire non-zero momentum
under an IF boost. one of the particles of the putative vacuum loop would have
$p^{+}<0$, which is forbidden.

Figure 2:  Origin of IF vacuum complexity. A field $\varphi$ propagates
between events $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. Equal-time commutation
relations imply uncertainty relations along the fixed-time hypersurface
(horizontal dotted line). In IF dynamics, timelike events with respect to
$\mathcal{E}_{1}$ may become spacelike owing to the Heisenberg uncertainty
$\Delta z_{{}_{2}}$ (left panel). Thus, the time-ordering of $\mathcal{E}_{1}$
and $\mathcal{E}_{2}$ is frame-dependent. When $t_{2}<t_{1}$, causality-
violating $Z$-graphs arise. Their negative probabilities are compensated by
vacuum loops, rendering the vacuum complex. The loops’ virtual particles may
borrow 4-momentum from an accelerated frame and become observable in that
frame (Unruh effect). In FF dynamics, commutation relations are set at fixed
FF time $\tau$. Thus, the uncertainty $\Delta x^{-}$ never accesses the
spacelike region: $\mathcal{E}_{2}$ is always timelike with respect to
$\mathcal{E}_{1}$ (right panel). No $Z$-graphs arise, virtual loops are
forbidden by unitarity, and the vacuum is trivial. Without vacuum loops, no
Unruh effect emerges.

##  5 Reconciling Instant and Front Form Descriptions

Physics is independent of any choice of frame or framework. Therefore, IF and
FF conclusions must finally agree. Here, we describe how one may reconcile the
apparently conflicting pictures drawn above.

Following Ref. [21], which reconciled the IF and FF descriptions of deuteron
structure, we consider an Unruh detector [3, 22] made of deuterium atoms. Such
a detector would monitor the deuteron distribution between its ground state
$\ket{0}$ (the spin-0 singlet state) and its excited state $\ket{1}$ (the
spin-1 triplet state). The temperature $T$ is obtained from the population
distributions $N_{i}$: $N_{i}=N\exp{(-E_{i}/[k_{B}T])}/Z$, where $N$ is the
total number of atoms; $E_{i}$, the energy of state $\ket{i}$, $k_{B}$,
Boltzmann’s constant, and $Z$, the partition function. Irrespective of the
Unruh effect, the detector structure in the IF case continuously changes with
time because of a combination of three facts: (1) an IF boost is dynamical;
(2) the rapidity becomes time-dependent for accelerations,
$\theta\to\theta(t)$; and (3) a boost is equivalent to a Rindler time
translation – see SupM Section C.1. Specifically, the detector structure
evolves because the $\ket{0}$ and $\ket{1}$ populations mix owing to an IF-
induced spin-orbit force [12, 23]. State $\ket{1}$ depopulates in favor of
$\ket{0}$, i.e., the detector cools.

Since no cooling occurs in the FF, and physics is independent of the
theoretical framework, an IF source of temperature is required to maintain the
initial balance of $\ket{0}$ and $\ket{1}$, despite the continuous
depopulation of $\ket{1}$. The ready candidate is the Unruh temperature. The
first IF effect – the cooling – has generally been overlooked. Unruh detectors
are typically described in terms of energy levels without considering the
dynamics of the internal detector structure responsible for those
levels.222This is highlighted by the fact that the masses of the particles
constituting the Unruh detector are not mentioned in descriptions. Were the
internal dynamical structure underlying the energy levels discussed, masses
would appear, as they do when dynamics is described. Without such
consideration, one cannot account for the IF boost dynamical effect.
Overlooking detector cooling, there is no compensation mechanism for the Unruh
effect; hence, it seems objectively observable.

In FF dynamics, none of this occurs because boosts are kinematical (no
cooling) and the FF vacuum is trivial (no Unruh temperature). Since physics is
independent of choice of frame and form, then the two IF effects must
compensate each other so that IF and FF agree. _It is now evident, however,
that since boost-induced dynamics are overlooked in the usual IF analyses that
lead to an Unruh effect, it has hitherto been erroneously identified as
observable._

##  6 Conclusion

It is popularly claimed that the Unruh effect needs no empirical verification
beyond that involved in validating Poincaré-invariant quantum field theory
(QFT) as a description of Nature. We have exposed the flaw in that position:
FF quantization is a powerful predictive tool in QFT; yet, this approach
reveals that prediction of an Unruh effect is an erroneous consequence of
causality-violating aspects of conventional IF dynamics. Properly, the Unruh
effect is cancelled in IF dynamics by another pseudo-effect, namely, cooling
of the Unruh detector owing to dynamical evolution under a succession of IF
boosts. According to the equivalence principle between gravity and
acceleration, the Unruh effect corresponds to Hawking radiation from black
holes. Thus, our analysis suggests that there should also be no black hole
evaporation – at least, not via the currently envisioned process [9]; and,
consequently, neither (currently elusive [24]) black hole explosion [25] nor
black hole information paradox. If so, then the origin of these grave
difficulties is similar to that of the cosmological constant problem [26],
viz. arising from using a computational formalism which violates causality at
intermediate steps [27, 12, 28, 29] and ascribing physical reality to mere
mathematical artifices. Hence, they would have a similar resolution, in this
case being eliminated by a manifestly causality preserving treatment of QFT.
Consequently, notwithstanding popular perspectives, one is left with a final
point that should not be lightly dismissed; namely, neither the Unruh effect
nor any of its corollaries can be considered real without empirical
verification.

Acknowledgments. We are grateful to G. D. de Téramond for useful discussions.
This material is based upon work supported by U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, contract DE-AC05-06OR23177 (AD);
U.S. Department of Energy contract DE-AC02-76SF00515 (SJB); National Natural
Science Foundation of China, grant no. 12135007 (CDR); and the U. S. National
Science Foundation award no. 1847771 (BT).

Author contributions. AD performed the calculations and drafted the
manuscript. SJB verified the calculations and the applicability of the front
form discussion, and reviewed and corrected the manuscript. CDR verified the
calculations and their context within quantum field theory, introduced
concepts to the discussion, and contributed materially to writing the
manuscript. BT verified the calculations and the applicability of the
discussion to cosmology, and reviewed and corrected the manuscript.

Competing interests. The authors declare no competing interests.

Additional information. Supplementary information is available online. It
includes a discussion of the front form vacuum and alternate demonstrations of
the Unruh effect in the instant and front forms. Correspondence and requests
for materials may be addressed to A. Deur. Reprint and permission information
is available at … (to be updated upon publication)

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Supplemental Material

##  Appendix A FF vacuum

That virtual particle loops are absent from the FF vacuum is firmly
established. Nevertheless, the role of zero-momentum modes in the FF vacuum is
a topic of intense discussion, e.g., such issues as whether, in flat
spacetime, the vacuum energy can be renormalized away [30, 31]. Though
important in themselves, these questions are irrelevant to the Unruh effect
because $p^{+}=0$ modes cannot transfer momentum (equally, kinetic energy) to
the particles forming the Unruh detector/thermometer: zero-modes cannot heat
thermometers.

We note that studies of the vacuum in quantum field theory often employ a
perturbative formalism; thus, one can consider whether the nonperturbative
vacuum could be nontrivial. This is also irrelevant to our subject because no
field coupling nor other expansion parameter is active in predictions of the
Unruh effect. If such were the case, then one could always choose a small
coupling for which a perturbative treatment is sufficient, since typical
analyses find the Unruh effect to occur universally, for strong and weak
fields alike. Furthermore, vacuum structure, whether trivial or complex, is
not invoked: the demonstrations only use coordinate definitions of the forms
of dynamics, the consequent quantization conditions, and generic properties of
quantum field theory. Only the interpretation of the result invokes the vacuum
(perturbative) structure as it provides an intuitive picture.

Additionally, discussions of the Unruh effect are often set for simplicity in
(1+1)D [4, 15], where the triviality of the non-perturbative FF vacuum is
clear: quantum chromodynamics has been solved exactly in (1+1)D, with the
meson spectrum recovered in the number of colors $N_{c}\to\infty$ limit using
FF [32] or IF [33]. Both studies involve obtaining the Hartree-Fock solution
for its vacuum, but with the FF calculation being considerably simpler owing
to vacuum simplicity.

Often, FF results are formally recovered by boosting IF results to the
infinite momentum frame (IMF) [34]. This is also the case here since in the
IMF, the Rindler, IF and FF times coincide – see Fig. 1. From Ref. [35], this
implies the same vacua and, in turn, no Unruh effect.

Yet, one also needs to allow for the possibility of zero-modes in the IF-IMF
vacuum. Some of these modes correspond to real particles – rather than the
null-momentum vacuum loops (“tadpole graphs”) responsible for zero-point
energy – and might contribute to the static properties of bound states [36].
However, consistent with the FF result, even these particular IF-IMF zero-
modes cannot lead to the dynamics underlying the Unruh effect for several
reasons. First, such modes manifest themselves only within bound states,
potentially contributing inside to long-distance field correlations (sometimes
called “condensates”) without support outside a bound state [27, 12, 28, 29].
Second, IF-IMF vacuum loops or condensates cannot produce pairs of physical
particles for the same reason as for the FF (see Section 4): all particles
must have nonnegative IF-IMF momenta. Since the vacuum or zero-modes have null
momenta by definition, and since momentum is conserved, this excludes loops as
one of the particles would have a negative momentum. Considering the IMF limit
offers a simple interpretation of the absence of the Unruh effect: in the IMF
the acceleration must fall to zero; thus, so must the Unruh temperature.

Thus, herein, FF vacuum triviality refers to the absence of loops of particles
with non-zero momenta, without considering zero-modes, which might play a role
in some physics but are irrelevant to the Unruh effect – see also SupM Sec. E.
Furthermore, FF vacuum triviality is not employed in the FF analyses of
Section 3 and SupM Sections B – D: an Unruh effect is precluded in FF
quantization for the same reason that the FF vacuum is trivial; so, the latter
need not be assumed. Discussing the triviality of the vacuum is useful chiefly
in interpreting the FF results.

##  Appendix B Standard derivation of the Unruh effect

###  B.1 Analyticity Argument Method

Here, we follow a standard IF approach to the Unruh effect [15] and
simultaneously present the analogue in FF dynamics. We use red fonts for IF-
specific formulae, which are taken directly from Ref. [15], and blue fonts for
their FF equivalents. Since the Unruh effect is usually discussed in the
“relativity convention” metric, viz. $(-,+,+,+)$, while the FF is generally
discussed in the “particle physics convention” $(+,-,-,-)$, then to preserve
the familiar formulae, we use $(-,+,+,+)$ in the IF case and $(+,-,-,-)$ in
FF.

Following Ref. [15], we consider massless scalar particles in (1+1)D spacetime
for simplicity. The spacelike line element is thus:

$\begin{array}[]{llll}{\color[rgb]{1,0,0}ds^{2}=-dt^{2}+dz^{2}}&\rm{(IF)};&\quad{\color[rgb]{0,0,1}ds^{2}=d\tau
dx^{-}}&\rm{(FF)}.\end{array}$ (B.1)

A uniformly accelerated observer follows a trajectory:

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}t(\rho)=\frac{1}{\alpha}\sinh(\alpha\rho)}&{\rm(IF)};&\displaystyle\quad{\color[rgb]{0,0,1}\tau(\rho)\equiv
t(\rho)+z(\rho)=\frac{1}{\alpha}e^{\alpha\rho}}&{\rm(FF)},\\\\[7.74997pt]
\displaystyle{\color[rgb]{1,0,0}z(\rho)=\frac{1}{\alpha}\cosh(\alpha\rho)}&{\rm(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}x^{-}(\rho)\equiv
t(\rho)-z(\rho)=\frac{1}{\alpha}e^{-\alpha\rho}}&{\rm(FF)},\end{array}$ (B.2)

where $\alpha$ is the acceleration and $\rho$ a parameter. The trajectory
describes a  hyperboloid in the IF and a  hyperbola in the FF:

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}t^{2}(\rho)=z^{2}(\rho)-\alpha^{2}}&{\rm(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}\tau(\rho)=\frac{1}{\alpha^{2}x^{-}(\rho)}}&{\rm(FF)}.\end{array}$
(B.3)

Working with these kinematics, Rindler coordinates $\eta$, $\xi$ can be
defined, along with the Rindler frame. Their relation to the Minkowski (i.e.,
inertial) frame is:

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}t=\frac{1}{a}e^{a\xi}\sinh(a\eta)}&{\rm(IF)};&\displaystyle\quad{\color[rgb]{0,0,1}\tau=\frac{1}{a}e^{a(\eta+\xi)}}&{\rm(FF)},\\\\[7.74997pt]
\displaystyle{\color[rgb]{1,0,0}z=\frac{1}{a}e^{a\xi}\cosh(a\eta)}&{\rm(IF)};&\displaystyle\quad{\color[rgb]{0,0,1}x^{-}=\frac{1}{a}e^{a(-\eta+\xi)}}&{\rm(FF)},\end{array}$
(B.4)

with the metric

$\displaystyle ds^{2}=e^{2a\xi}(-d\eta^{2}+d\xi^{2}).$ (B.5)

Thus, $\eta$ can be identified as the Rindler proper time and $\xi$ as the
Rindler space coordinate.

Figure B.1: Coordinate frames: Rindler (black), Minkowski IF (red), and FF
(blue). As in Fig. 1, we use the $(-,+,+,+)$ metric for both IF and FF.

Equations (B.4) provide a coordinate system only on the right spacelike
quadrant of the Minkowski space delimited by $|t|=z,z\geq 0$ and denoted
quadrant I in Fig. B.1. Coordinates for the left spacelike quadrant
($|t|=z,z\leq 0$, quadrant IV), are obtained by flipping Minkowski coordinate
signs. This is because the Rindler space coordinate $\xi$ is a ray originating
at $z=0$ rather than a line spanning $[-\infty,+\infty]$. Therefore, in
quadrant IV, $\xi$ points left and the Rindler time is reversed: $\eta$ flows
downward. The reasoning applies to IF and equaly to FF since the FF Killing
field is

${\color[rgb]{0,0,1}\partial_{\eta}=\frac{\partial\tau}{\partial\eta}\partial_{\tau}+\frac{\partial
x^{-}}{\partial\eta}\partial_{x^{-}}=a(\tau\partial_{\tau}-x^{-}\partial_{x^{-}})}\,,$
(B.6)

with components $(2a\tau,-2ax^{-})$. Thus, $\partial_{\eta}$ is orthogonal to
the $\tau=0$ hypersurface, as in the IF case and, like the IF,
$\partial_{\eta}$ defines positive- and negative-modes for a Fock basis. In
all, in quadrant IV,

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}t=-\frac{1}{a}e^{a\xi}\sinh(a\eta)}&{\rm(IF)};&\displaystyle\quad{\color[rgb]{0,0,1}\tau=-\frac{1}{a}e^{a(\eta+\xi)}}&{\rm(FF)},\\\\[7.74997pt]
\displaystyle{\color[rgb]{1,0,0}z=-\frac{1}{a}e^{a\xi}\cosh(a\eta)}&{\rm(IF)};&\displaystyle\quad{\color[rgb]{0,0,1}x^{-}=-\frac{1}{a}e^{a(-\eta+\xi)}}&{\rm(FF)}.\end{array}$
(B.7)

In Rindler coordinates, the Klein-Gordon equation for massless free fields is:

$\displaystyle\Box\phi=\frac{1}{a}e^{-2a\xi}(-\partial^{2}_{\eta}+\partial^{2}_{\xi})\phi=0.$
(B.8)

(The extension to massive fields is straightforward and without qualitative
import.) Solutions are given by planes waves:
$g_{p}=(4\pi\omega)^{-\nicefrac{{1}}{{2}}}{\rm e}^{-i\omega\eta+ip\xi}$, with
$\omega=|p|$. In Minkowski frames, the Klein-Gordon equation is:

$\begin{array}[]{llll}{\color[rgb]{1,0,0}\Box\phi=(-\partial^{2}_{t}+\partial^{2}_{z})\phi=0}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\Box\phi=(\partial^{+}\partial^{-})\phi=0}&{\rm(FF)}.\end{array}$
(B.9)

Here,  $\displaystyle\partial^{+}\equiv 2\nicefrac{{\partial}}{{\partial
x^{-}}}$,  $\displaystyle\partial^{-}\equiv
2\nicefrac{{\partial}}{{\partial\tau}}$. They are the operators associated
with FF momentum, $p^{+}=\omega+p$, and FF energy, $p^{-}=\omega-p$,
respectively. The plane wave solutions of Eqs. (B.9) are:

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}f_{p}\propto{\rm
e}^{-i\omega
t+ipz}}\,,\;{\color[rgb]{1,0,0}f_{p}^{\ast}}&{\rm(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}f_{p^{+}}\propto{\rm
e}^{-\nicefrac{{i}}{{2}}(p^{+}x^{-})}}\,,\;{\color[rgb]{0,0,1}f_{p^{+}}^{\ast}}&{\rm(FF)}.\end{array}$
(B.10)

There is no  ${\rm e}^{-\nicefrac{{i}}{{2}}(p^{-}\tau)}$ component in the FF
wave function because massless particles are also zero-energy in FF dynamics:
$p^{-}\equiv\omega-p=0$.

Since in quadrant IV, future Minkowski time (IF or FF) flows in the “$-\eta$”
direction – see Fig. B.1, two sets of positive frequency Rindler modes must be
introduced: one for quadrant I and the other for quadrant IV,

$\displaystyle g_{p}^{({\rm
I})}=\begin{cases}\displaystyle\frac{1}{\sqrt{4\pi\omega}}{\rm
e}^{-i\omega\eta+ip\xi}&\rm I\\\ 0&\rm IV\\\ \end{cases}$ (B.11a)
$\displaystyle g_{p}^{({\rm IV})}=\begin{cases}0&\rm I\\\
\displaystyle\frac{1}{\sqrt{4\pi\omega}}{\rm e}^{+i\omega\eta+ip\xi}&\rm IV\\\
\end{cases}$ (B.11b)

so that $\omega\geq 0$ in both quadrants. The modes $g_{p}^{({\rm I})}$ and
$g_{p}^{({\rm IV})}$ can be analytically continued outside their respective
quadrant, which enables $\phi$ to be expressed in terms of four Rindler
creation/annihilation operators:

$\displaystyle\displaystyle\phi=\int dp\big{(}\hat{b}_{p}^{({\rm
I})}g_{p}^{({\rm I})}+\hat{b}_{p}^{({\rm I}){\dagger}}g_{p}^{({\rm
I})*}+\hat{b}_{p}^{({\rm IV})}g_{p}^{({\rm IV})}+\hat{b}_{p}^{({\rm
IV}){\dagger}}g_{p}^{({\rm IV})*}\big{)}.$ (B.12)

In contrast, Minkowski space not being partitioned, two creation/annihilation
operators are sufficient:

$\begin{array}[]{llll}{\color[rgb]{1,0,0}\phi=\int
dp\big{(}\hat{a}_{p}f_{p}+\hat{a}_{p}^{\dagger}f_{p}^{*}\big{)}}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\phi=\int
dp^{+}\big{(}\hat{a}_{p^{+}}f_{p^{+}}+\hat{a}_{p^{+}}^{\dagger}f_{p^{+}}^{*}\big{)}}&{\rm(FF)}\,.\end{array}$
(B.13)

The annihilation operators of Eqs. (B.12) and (B.13) define the Rindler and
Minkowski vacua, respectively:

$\begin{array}[]{llllll}\hat{b}^{({\rm I})}_{p}\ket{0_{R}}=\hat{b}^{({\rm
IV})}_{p}\ket{0_{R}}=0&{\rm(Rindler)};&\quad{\color[rgb]{1,0,0}\hat{a}_{p}\ket{0_{M}}_{\rm
IF}=0}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\hat{a}_{p^{+}}\ket{0_{M}}_{\rm
FF}=0}&{\rm(FF)}.\end{array}$ (B.14)

Exposing an Unruh effect hinges on showing that $\hat{b}^{({\rm I,IV})}_{p}$
cannot be expressed with $\hat{a}$ only but must also include
$\hat{a}^{\dagger}$. Then, a Rindler observer will detect particles from the
Minkowski vacuum, $\hat{b}^{({\rm I,IV})}_{p}\ket{0_{M}}\neq 0$.

The expressions of $\hat{b}^{({\rm I,IV})}_{p}$ in term of $\hat{a}$ and
$\hat{a}^{\dagger}$ can be obtained by finding Rindler modes $h^{({\rm
I,IV})}_{p}$ that have the same vacuum as the Minkowski modes $f^{({\rm
I,IV})}$ [37]. Using Eqs. (B.4), (B.7), the relationship between the phases of
the Rindler and Minkowski modes in the two spacetime quadrants is (recall the
different metric signatures for IF and FF):

$\begin{array}[]{ll}{\rm
e}^{-a(\eta-\xi)}&=\left\\{\begin{array}[]{lllll}{\color[rgb]{1,0,0}-a(t-z)}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\phantom{-}ax^{-}}&{\rm(FF)}&\quad{\rm
I}\\\
{\color[rgb]{1,0,0}\phantom{-}a(t-z)}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}-ax^{-}}&{\rm(FF)}&\quad{\rm
IV}\end{array}\right.\\\ {\rm
e}^{a(\eta+\xi)}&=\left\\{\begin{array}[]{lllll}{\color[rgb]{1,0,0}\phantom{-}a(t+z)}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\phantom{-}a\tau}\phantom{{}^{-}}&{\rm(FF)}&\quad{\rm
I}\\\
{\color[rgb]{1,0,0}-a(t+z)}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}-a\tau}&{\rm(FF)}&\quad{\rm
IV}\end{array}\right.\end{array}\,.$ (B.15)

Above, for the sake of generality, we wrote $a\tau$, but for a (1+1)D massless
scalar field, contributions involving $\tau$ vanish because their phase is
$p^{-}\tau$, with $p^{-}=0$. Recall that $a$ is the acceleration and should
not be confused with the operators $\hat{a}$, $\hat{a}^{\dagger}$.

We can now identify which Rindler modes match the spacetime dependence of the
Minkowski modes  $f_{p}$ or  $f_{p^{+}}$. Using Eqs. (B.11) and (B.15) and the
Rindler dispersion relation, $\omega=p$, the Rindler mode in quadrant I that
has the same phase as  $f_{p}$ or  $f_{p^{+}}$ is:

$\begin{array}[]{llll}{\color[rgb]{1,0,0}\sqrt{4\pi\omega}g_{p}^{({\rm
I})}=a^{i\omega/a}(-t+z)^{i\omega/a}}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\sqrt{4\pi\omega}g_{p}^{({\rm
I})}=(ax^{-})^{ip^{+}/2a}}&{\rm(FF)}.\end{array}$ (B.16)

The available Rindler mode in quadrant IV, Eq. (B.11b), does not have the same
phase as in Eq. (B.16). To match the phase, we must consider the complex
conjugate of $g_{p}^{({\rm IV})}$ with negative momentum $k:=-p<0$:

$\begin{array}[]{llll}{\color[rgb]{1,0,0}\sqrt{4\pi\omega}g_{-p}^{({\rm
IV})*}=a^{i\omega/a}e^{\pi\omega/a}(-t+z)^{i\omega/a}}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\sqrt{4\pi\omega}g_{-p}^{({\rm
IV})*}=e^{\pi p^{+}/2a}(ax^{-})^{ip^{+}/2a}}&{\rm(FF)}.\end{array}$ (B.17)

The Rindler dispersion relation, $\omega=|k|$, and the definition,
$p^{+}\equiv\omega+k$, impose $p^{+}=0$ on the domain $k=-p\leq 0$. Thus, in
the FF case, there is no available Rindler mode in quadrant IV. Thus, Eq.
(B.17) becomes:

$\begin{array}[]{llll}{\color[rgb]{1,0,0}\sqrt{4\pi\omega}g_{-p}^{({\rm
IV})*}=a^{i\omega/a}e^{\pi\omega/a}(-t+z)^{i\omega/a}}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\sqrt{4\pi\omega}g_{-p}^{({\rm
IV})*}={\rm constant}=0}&{\rm(FF)}.\end{array}$ (B.18)

The rightmost result follows because $g_{-p}$ must be a plane wave, by
definition of the Fourier decomposition of $\phi$, Eq. (B.12), and a nonzero
value for the constant would violate this constraint.

With Eqs. (B.16) and (B.18), the Rindler modes can now be expressed in terms
of the Minkowski modes:

$\begin{array}[]{llll}{\color[rgb]{1,0,0}\sqrt{4\pi\omega}\big{(}g_{p}^{({\rm
I})}+e^{-\pi\omega/a}g_{-p}^{({\rm
IV})*}\big{)}=a^{i\omega/a}\big{(}-t+z\big{)}^{i\omega/a}}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\sqrt{4\pi\omega}g_{p}^{({\rm
I})}=(ax^{-})^{ip^{+}/2a}}&{\rm(FF)}.\end{array}$ (B.19)

Plainly, only the Rindler mode $g^{({\rm I})}_{p}$ matches the phase of the
Minkowski FF modes $f_{p^{+}}$ – see Eq. (B.10), and not  $g^{({\rm
IV})*}_{-p}$; the Bogolyubov transformation is diagonal. Hence, there is no
Unruh effect in the FF framework. Notwithstanding this, concluding details are
included below.

The IF combination in Eq. (B.19) is not yet properly normalized. The
normalized IF mode is (the FF mode remains unchanged):

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}h_{p}^{({\rm
I})}=\frac{1}{\sqrt{2\sinh(\nicefrac{{\pi\omega}}{{a}})}}\big{(}e^{\pi\omega/2a}g_{p}^{({\rm
I})}+e^{-\pi\omega/2a}g_{-p}^{({\rm
IV})*}\big{)}}&{\rm(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}~{}~{}~{}g_{p}^{({\rm
I})}=\frac{1}{\sqrt{4\pi\omega}}\big{(}ax^{-}\big{)}^{ip^{+}/2a}}&{\rm(FF)}.\end{array}$
(B.20)

The normalized Rindler modes, $h_{p}$, have corresponding
creation/annihilation operators $\hat{c}_{p}$, which, by the construction
above, act on the Minkowski vacuum as $\hat{c}_{p}\ket{0_{M}}=0$ and
$\hat{c}^{\dagger}_{p}\ket{0_{M}}=\ket{p}$. Using Eq. (B.20) and analogous
reasoning for $\hat{b}_{p}^{({\rm IV})}$, one obtains the relationship between
$\hat{c}_{p}$, $\hat{c}^{\dagger}_{p}$ and the original Rindler operators:

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}\hat{b}^{({\rm
I})}_{p}=\frac{1}{\sqrt{2\sinh(\nicefrac{{\pi\omega}}{{a}})}}\big{(}{\rm
e}^{\pi\omega/2a}\hat{c}_{p}^{({\rm I})}+{\rm
e}^{-\pi\omega/2a}\hat{c}_{-p}^{({\rm
IV})\dagger}\big{)}}&{\rm(IF)};&\quad{\color[rgb]{0,0,1}\hat{b}^{({\rm
I})}_{p}=\hat{a}_{p^{+}}}&{\rm(FF)},\\\
\displaystyle{\color[rgb]{1,0,0}\hat{b}^{({\rm
IV})}_{p}=\frac{1}{\sqrt{2\sinh(\nicefrac{{\pi\omega}}{{a}})}}\big{(}{\rm
e}^{\pi\omega/2a}\hat{c}_{p}^{({\rm IV})}+{\rm
e}^{-\pi\omega/2a}\hat{c}_{-p}^{({\rm
I})\dagger}\big{)}}&{\rm(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}\hat{b}^{({\rm
IV})}_{p}=\hat{a}_{p^{-}}}&{\rm(FF)}.\end{array}$ (B.21)

Acting now with, e.g., $\hat{b}^{({\rm I})}_{p}$ on the Minkowski vacuum, one
finds:

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}\hat{b}^{({\rm
I})}_{p}\ket{0_{M}}_{\rm
IF}=\frac{1}{\sqrt{2\sinh(\nicefrac{{\pi\omega}}{{a}})}}\big{(}{\rm
e}^{-\pi\omega/2a}\ket{-p}\big{)}}&{\rm(IF)};&\quad\displaystyle\hat{b}^{({\rm
I})}_{p}\ket{0_{M}}_{\rm FF}=\hat{a}_{p^{+}}\ket{0_{M}}_{\rm
FF}=0&{\rm(FF)}.\end{array}$ (B.22)

Evidently, in a Rindler frame, following this reasoning, one can observe
particles from the IF Minkowski vacuum but not from the (trivial) FF vacuum.

The crucial point is the transition between Eqs. (B.16) and (B.19): modes with
negative momentum, $k=-p$, are introduced. This is allowed in the IF: it
simply means waves moving toward “$-z$” (for quadrant I); and they can be
detected by an observer in that quadrant. In the FF, on the other hand,
$p^{+}>0$ always; waves with $p^{+}\leq 0$ would propagate backward in IF
time; thus violating causality. Therefore, they do not exist; so, neither do
Unruh particles. In other words, because, formally,
$f_{-p^{+}}=f^{\ast}_{p^{+}}$, one must set $f_{-p^{+}}=0$ or one would double
count in the Fourier expansion, Eq. (B.13). (Note that $f^{\ast}_{p^{+}}$
cannot be invoked for Eq. (B.17) since the procedure is to match Rindler modes
to $f_{p^{+}}$, not to $f^{\ast}_{p^{+}}$.)

###  B.2 Method directly deriving the Bogolyubov transformation

In Section B.1, the Bogolyubov transformation was derived following the
original analyticity argument [37]. Here we discuss the direct derivation –
see e.g., Ref. [4]. The steps need not be fully detailed because the argument
precluding the Unruh effect in FF is the same as in Section B.1.

The Bogolyubov transformation is

$\begin{array}[]{llllll}\displaystyle g_{p}^{({\rm
I})}&=&\displaystyle\int_{0}^{\infty}\frac{d\omega}{\sqrt{4\pi\omega}}\big{(}\hat{\alpha}_{p}^{({\rm
I})}f_{p}+\hat{\beta}_{p}^{({\rm I})}f_{p}^{*}\big{)}\,,&\quad g_{p}^{({\rm
VI})}&=&\displaystyle\int_{0}^{\infty}\frac{d\omega}{\sqrt{4\pi\omega}}\big{(}\hat{\alpha}_{p}^{({\rm
VI})}f_{p}+\hat{\beta}_{p}^{({\rm VI})}f_{p}^{*}\big{)}\,,\end{array}$ (B.23)

with $\hat{\beta}_{p}^{({\rm VI})}=-{\rm
e}^{-\pi\omega/a}\hat{\alpha}_{p}^{({\rm I})\ast}$, $\hat{\beta}_{p}^{({\rm
I})}=-{\rm e}^{-\pi\omega/a}\hat{\alpha}_{p}^{({\rm IV})\ast}$.

For IF, these coefficients are non-zero; hence, the Unruh effect is predicted.
In contrast, for FF, $\hat{\beta}_{p}^{({\rm VI})}$$=\hat{\alpha}_{p}^{({\rm
IV})*}=0$ because no FF modes are permitted in quadrant IV. Hence,
$\hat{\beta}_{p}^{({\rm VI})}=\hat{\beta}_{p}^{({\rm I})}=0$ and no Unruh
effect is perceived.

##  Appendix C The Unruh effect in thermal quantum field theory

Before examining the Unruh effect using thermal quantum field theory (QFT),
one must first recall the correspondence between Rindler time translations and
Minkowski boosts, and the implication for Rindler Hamiltonian definition in FF
dynamics.

###  C.1 Relation between Rindler time translations and boosts: implications
for the Hamiltonian

For IF and FF, a Rindler time translation corresponds to a Minkowski boost.
This is readily seen using Eqs. (B.4) and making a Rindler time translation
$\eta\to\eta+\Delta$, which leads to:

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}t^{\prime}=\frac{1}{a}{\rm
e}^{a\xi}\sinh(a\eta+a\Delta)=\frac{1}{a}{\rm
e}^{a\xi}\big{(}\sinh(a\eta)\cosh(a\Delta)+\cosh(a\eta)\sinh(a\Delta)\big{)}}&\rm{(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}{\tau}^{\prime}=\frac{1}{a}e^{a(\eta+\Delta+\xi)}}&\rm{(FF)},\\\\[7.74997pt]
\displaystyle{\color[rgb]{1,0,0}z^{\prime}=\frac{1}{a}{\rm
e}^{a\xi}\cosh(a\eta+a\Delta)=\frac{1}{a}{\rm
e}^{a\xi}\big{(}\cosh(a\eta)\cosh(a\Delta)+\sinh(a\eta)\sinh(a\Delta)\big{)}}&\rm{(IF)};&\displaystyle\quad{\color[rgb]{0,0,1}{x^{-}}^{\prime}=\frac{1}{a}{\rm
e}^{a(-\eta-\Delta+\xi)}}&\rm{(FF)}.\end{array}$ (C.1)

Hence,

$\begin{array}[]{llll}\displaystyle{\color[rgb]{1,0,0}t^{\prime}=t\cosh(a\Delta)+x\sinh(a\Delta)}&\rm{(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}{\tau}^{\prime}={\rm
e}^{a\Delta}\tau}&\rm{(FF)},\\\
\displaystyle{\color[rgb]{1,0,0}z^{\prime}=z\cosh(a\Delta)+t\sinh(a\Delta)}&\rm{(IF)};&\quad\displaystyle{\color[rgb]{0,0,1}{x^{-}}^{\prime}=e^{-a\Delta}x^{-}}&\rm{(FF)},\end{array}$
(C.2)

which are the usual boost formulae with rapidity $a\Delta$.

Now, since FF boosts are kinematical, the above correspondence makes, in FF
dynamics, the Rindler time translation operator a kinematical operator.
Therefore, it cannot serve as the Hamiltonian. In Euclidean space, where
thermal QFT is used to study the Unruh effect, the dilation operator defines
the FF Hamiltonian, as explained in the next section.

###  C.2 Thermal QFT

For a scalar field, the thermal QFT derivation of the IF Unruh effect [38]
utilizes the periodicity in Euclidean time (viz. imaginary time
$\tilde{t}:=it$) of a QFT at finite temperature $T=1/\beta$,
$0\leq\tilde{t}\leq\beta$.

Consider the two-point Schwinger function of the field $\phi(\tilde{t},z)$:

$\displaystyle G_{\beta}(\tilde{t},z)=\frac{1}{Z}{\rm Tr}~{}{\rm e}^{-\beta
H}\mathcal{T}[\phi(\tilde{t},z)\phi(0,0)],$ (C.3)

with $Z$ the partition function and $\mathcal{T}$ the imaginary-time ordering
operator. In the Boltzmann factor ${\rm e}^{-\beta H}$, $H$ is the operator
providing energy eigenvalues, not necessarily the time-evolution operator,
although it is both in IF dynamics. This implies ${\rm e}^{-\beta
H}\phi(\tilde{t},z){\rm e}^{\beta H}=\phi(\tilde{t}-\beta,z)$, which, with the
cyclicity property of a trace and Eq. (C.3), yields – see, e.g., Ref. [39, Eq.
(2.86)]):

$\displaystyle{\color[rgb]{1,0,0}G_{\beta}(\tilde{t},z)=G_{\beta}(\tilde{t}-\beta,z).}$
(C.4)

Using Eqs. (C.2), a shift in imaginary time $\tilde{t}\to\tilde{t}+i\Delta$,
$i\Delta=2\pi/a$, is seen to yield  $t^{\prime}=t$ and $z^{\prime}=z$ or
${\tau}^{\prime}=\tau$ and ${x^{-}}^{\prime}=x^{-}$. Namely, after a Wick
rotation, the Rindler metric makes a succession of boosts that correspond to a
single rotation in imaginary time; and the shift is just a change in
temperature. On the other hand, in FF dynamics, as shown in Section C.1, the
Rindler Hamiltonian is not the time translation operator; so, ${\rm e}^{-\beta
H}\phi(0,0){\rm e}^{\beta H}\neq\phi(\beta,0)$ and there is no cyclic relation
in FF.

Evidently, only in IF does the formal equivalence between $\beta$ and
periodicity yield an Unruh temperature  $\beta^{-1}=T=a/2\pi$.

The Rindler frame identifies with Dirac’s Point-Form (PF, see Section 2) after
switching time and space, which is nugatory in Euclidean space because the
axes are indistinguishable. The PF Hamiltonian being the dilation operator
[40], as evident from the continuous dilation of the axes attached to the red
wordline in Fig. 1, right panel, one may choose it as the FF Rindler
Hamiltonian. Then, one recovers Eq. (C.4), but the Rindler frame is no longer
equivalent to a cyclic frame under dilation factors $d_{1,2}$:
${\tau}^{\prime}=\frac{1}{a}e^{a(d_{1}\eta+d_{2}\xi)}\neq\tau$ except for the
trivial case $d_{1}=1=d_{2}$. Consequently, the thermal QFT approach reveals
no Unruh effect in FF dynamics.

##  Appendix D Dynamical derivation

Consider the Lagrangian of a free massive scalar field $\varphi$ in (1+1)D:

$\displaystyle\mathcal{L}=\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{1}{2}m^{2}\varphi^{2},$
(D.1)

with $m$ the field mass. Then change the perspective to a frame in which the
field appears shifted,

$\displaystyle\varphi\equiv\phi+a/m^{2}.$ (D.2)

In this frame, after subtracting a (physically immaterial) constant,
$\nicefrac{{a^{2}}}{{2m^{2}}}$, $\mathcal{L}$ becomes:

$\displaystyle\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}+a\phi\,,$
(D.3)

where $a$ acts as an external source term and provides a linear potential,
viz. a constant acceleration. This model was studied in Refs. [41, 42], albeit
not in the context of the Unruh effect. To see that $a\phi$ in Eq. (D.3)
produces a constant force, consider the classical mechanics equivalent,
$\mathcal{L}=\frac{1}{2}m\dot{z}^{2}-\frac{1}{2}kz^{2}-az$, where $k$ is the
harmonic oscillator constant. The potential term $az$ generates a constant
force $a$ in the (1+1)D case. In (1+2)D or (1+3)D spacetimes, $a\propto t$ and
$a\propto t^{2}$, respectively, are needed to obtain a constant force. This
may be understood as the flux from source $a$ diluting in 2D or 3D space. To
keep a constant force on the accelerating body moving away from the source,
the latter must intensify with $t$.

Suppose a detector sensitive to $\varphi$ is attached to the new frame. It may
be pictured as being composed of fermions that couple to $\phi$. The field
waves $\partial_{\mu}\phi\partial^{\mu}\phi$ induce transitions in the
detector, making it sensitive to $\phi$, and the $a\phi$ term accelerates the
detector.

To avoid infrared issues, suppose space is of finite extent, with length $L$
and circular boundary conditions. Then decompose the two fields as follows
(the $p^{\mu}_{n}$ spectrum is discrete because $L$ is finite):

$\displaystyle\phi=\frac{1}{\sqrt{2L}}\sum_{n}\big{(}\hat{A}_{p_{n}}e^{ip^{\mu}_{n}x_{\mu}}+\hat{A}_{p_{n}}^{\dagger}e^{-ip^{\mu}_{n}x_{\mu}}\big{)},$
(D.4a)
$\displaystyle\varphi=\frac{1}{\sqrt{2L}}\sum_{n}\big{(}\hat{a}_{p_{n}}e^{ip^{\mu}_{n}x_{\mu}}+\hat{a}_{p_{n}}^{\dagger}e^{-ip^{\mu}_{n}x_{\mu}}\big{)}.$
(D.4b)

The relation between $(\hat{A}_{p_{n}},\hat{A}^{\dagger}_{p_{n}})$ and
$(\hat{a}_{p_{n}},\hat{a}^{\dagger}_{p_{n}})$ can be obtained by expressing
Eq. (D.2) in terms of a translation operator, $U$:

$\displaystyle\varphi$ $\displaystyle=U^{\dagger}\phi U,\,\quad U={\rm
e}^{-ia{\mathsf{P}}/m^{2}}\,,$ (D.5)

where $\mathsf{P}$ is the 3-momentum integrated over a time $\Delta T$
characterizing the (Unruh) detector response, which is obtained from the
3-momentum density, viz. the canonical conjugate of $\varphi$:

$\displaystyle{\mathsf{p}}=\frac{-i}{\sqrt{2L}}\sum_{n}\sqrt{\frac{\omega_{n}}{2}}\big{(}\hat{a}_{p_{n}}{\rm
e}^{ip^{\mu}_{n}x_{\mu}}-\hat{a}_{p_{n}}^{\dagger}{\rm
e}^{-ip^{\mu}_{n}x_{\mu}}\big{)}.$ (D.6)

Focusing now on IF dynamics (red fonts):

$\displaystyle{\mathsf{P}}=\int_{\Delta T,\pm
L}d^{2}x^{\mu}\,{\mathsf{p}}={\color[rgb]{1,0,0}-i\sum_{n}\sqrt{\omega_{n}L}\Delta
T\big{(}\hat{a}_{p_{n}}-\hat{a}_{p_{n}}^{\dagger}\big{)}}.$ (D.7)

(Factors of $\pi$ are omitted here as they are immaterial to the discussion.)
Eqs. (D.5) – (D.7) give

$\displaystyle{\color[rgb]{1,0,0}U=\exp{\displaystyle\frac{a\Delta
T}{m^{2}}\sum_{n}\sqrt{\omega_{n}L}\big{(}\hat{a}_{p_{n}}-\hat{a}_{p_{n}}^{\dagger}\big{)}}},$
(D.8)

which becomes, upon using the Baker-Campbell-Hausdorff formula and the
commutation properties of $\hat{a}_{p_{n}}$ and $\hat{a}^{\dagger}_{p_{n}}$:

$\displaystyle{\color[rgb]{1,0,0}U=\prod_{n}\exp\left[\frac{(a\Delta
T)^{2}}{2m^{4}}\omega_{n}L\right]~{}\exp\left[-\frac{a\Delta
T\sqrt{\omega_{n}L}}{m^{2}}\hat{a}_{p_{n}}\right]}.$ (D.9)

The Unruh effect is a statement that although the vacuum of an inertial field
(here $\varphi$) is, by definition, without real particles, so that
$\hat{a}_{p_{n}}\ket{0_{M}}\equiv 0$, an accelerated field (here $\phi$) is
not:
${\color[rgb]{1,0,0}\hat{A}_{p_{n}}\ket{0_{M}}=U\hat{a}_{p_{n}}U^{\dagger}\ket{0_{M}}\neq
0}$. This again suggests that the IF predicts an observable Minkowski vacuum.
However, the detector involved in the observation process should also be
included in the analysis. Such would require incorporation of fermion fields
into the Lagrangian, Eq. (D.1), and subsequent study of the bound states
constituting the detector. Section 5 then indicates that the interaction of
$a\phi$ with the detector, i.e. the dynamical effect of the time-dependent
boost, will cancel the Unruh effect.

For the FF, however, the momentum density is ${\color[rgb]{0,0,1}{\sf
p}=\partial^{+}\varphi\equiv\nicefrac{{\partial\varphi}}{{\partial x^{-}}}}$,
viz. a simple space derivative. Therefore:

$\displaystyle{\mathsf{P}}=\int_{\Delta T,\pm L}{\color[rgb]{0,0,1}d\tau
dx^{-}\frac{\partial\varphi}{\partial x-}}={\color[rgb]{0,0,1}\int_{\Delta
T}d\tau\big{[}\varphi\big{]}_{-L}^{+L}}={\color[rgb]{0,0,1}0},$ (D.10)

owing to the periodic boundary conditions. Thus,
${\color[rgb]{0,0,1}U=\mathds{1}}$ and the trivial vacuum of $\varphi$ in
Minkowski space $\hat{a}_{p_{n}}\ket{0_{M}}=0$ remains such for the
accelerated field:
${\color[rgb]{0,0,1}\hat{A}_{p_{n}}\ket{0_{M}}=\hat{a}_{p_{n}}\ket{0_{M}}=0}$.
There is no Unruh effect in FF.

One may interpret the IF and FF results as follows.
(i) IF – ${\mathsf{p}}$ is the time derivative of the field,
${\mathsf{p}}=\nicefrac{{\partial\varphi}}{{\partial t}}$. Thus, it gives the
difference between $\varphi$ on two infinitesimally separated hypersurfaces,
i.e., its dynamical evolution.
(ii) FF – ${\mathsf{p}}$ is a space derivative of the field,
${\mathsf{p}}=\nicefrac{{\partial\varphi}}{{\partial x^{-}}}$, i.e.,
${\mathsf{p}}$ operates on a single hypersurface without dynamical evolution.
Consequently, the conclusions of the previous sections are recovered: (i) the
IF treatment _dynamically_ affects a seemingly nontrivial vacuum, whereas (ii)
the FF treatment does not impact upon a trival vacuum. These qualities trace
back to the dynamical vs. kinematical character of IF cf. FF boosts.

##  Appendix E Black hole evaporation

The equivalence principle entails that without an Unruh effect, there should
also be no Hawking effect, viz. black holes have zero temperature and do not
evaporate. Here, we briefly elaborate on this point.

The light-cone surface separating Minkowski spacetime into four quadrants,
Fig. B.1, delineates equivalent event horizons: the $x^{-}=0$ line is the
future (in Rindler time $\eta$) horizon for observers in quadrant I, and the
$x^{+}=0$ line is the past horizon of the Killing vector $\partial_{\eta}$
[15]. For both FF and IF, $\partial_{\eta}$ is the killing field associated
with boosts – see SupM Section C.1.

The equivalent redshift factor is provided by the magnitude of the norm of
$\partial_{\eta}$. In FF, $\partial_{\eta}=(2a\tau,-2ax^{-})$, cf. Eq. (B.6).
Given the FF coordinates, Eq. (B.4), the redshift factor is:

$\mathcal{Z}\equiv\sqrt{-(\partial_{\eta})^{\mu}(\partial_{\eta})_{\mu}}=e^{a\xi};$
(E.1)

and the equivalent redshift-normalized surface gravity is:

$\kappa=\sqrt{\nabla_{\mu}\mathcal{Z}\nabla^{\mu}\mathcal{Z}}=a.$ (E.2)

These expressions are the same as the IF ones, consistent with $\mathcal{Z}$
and $\kappa$ being Lorentz scalars.

A black hole temperature observed at infinity is $T=\mathcal{Z}T_{H}$ with
$T_{H}$ the Unruh temperature near the horizon. In the FF formalism $T_{H}=0$,
therefore $T=0$ and there is neither Hawking radiation nor black hole
evaporation. We note that $T_{H}$ being null, it is not proportional to the
acceleration, $a$, which in turn prevents $T_{H}$ or $T$ from being related to
the black hole surface gravity, in contrast to the traditional expression
$T=\kappa/(2\pi)$[9].

Considering curved spacetime provides a different and complementary
perspective on the Hawking and Unruh effects. It is known that the definition
of positive and negative frequency modes is ambiguous in curved spacetime.
Yet, they determine the vacuum state; and since in a curved spacetime there is
no preferred choice of the positive-frequency modes, there is also no unique
vacuum state, in contrast to Minkowski spacetime. This is the underlying
formal reason for the Unruh and Hawking effects, see Section 3. However, it
apparently contradicts the basic principle of General Relativity that in the
limit of vanishing distances, spacetime must tend to a Minkowski spacetime. In
other words, when distance scales become negligible in comparison with the
curvature radius, space appears flat, with a unique vacuum. Indeed, a naive or
classical vacuum has no special scales and one may, unhindered, reach the
small distance limit. However, in the IF framework, the vacuum is complex,
with a characteristic distance scale set by the size of the vacuum loops. This
scale forbids one from achieving the small distance limit, keeping the
definition of positive and negative modes ambiguous, and allowing for an Unruh
effect. This perspective makes the crucial role of a non-trivial vacuum clear.
It also provides an intuitive understanding of the thermal distribution. The
greater distance scale set by larger loops allows for a larger curvature,
therefore, by the equivalence principle, a larger acceleration and, in turn, a
larger Unruh temperature. Since large loops are less probable than the smaller
loops, so also are larger temperatures.

The FF vacuum is trivial; hence, without distance scales preventing approach
to the Minkowski limit. Positive and negative modes are well-defined,
precluding Hawking or Unruh effects. Furthermore, this perspective renders
evident why the zero-point energy (SupM Section A) is irrelevant for the Unruh
effect. By definition, zero-point energy occurs at a single point in space
with their only possible physical contribution being from the infinite
momentum loop, viz. with conjugate distance $\to 0$ [31].