# Quantum Energy Inequalities along stationary worldlines

Christopher J. Fewster *  — 1, and Jacob Thompson *  —1,2,
1  _Department of Mathematics, University of York, Heslington, York YO10 5DD,
United Kingdom._
2  _School of Mathematics and Statistics, The University of Sheffield, Hicks
Building, Hounsfield Road,_
_Sheffield S3 7RH, United Kingdom._
chris.fewster<EMAIL_ADDRESS>
###### Abstract

Quantum energy inequalities (QEIs) are lower bounds on the averaged energy
density of a quantum field. They have been proved for various field theories
in general curved spacetimes but the explicit lower bound is not easily
calculated in closed form. In this paper we study QEIs for the massless
minimally coupled scalar field in four-dimensional Minkowski spacetime along
stationary worldlines – curves whose velocity evolves under a $1$-parameter
Lorentz subgroup – and find closed expressions for the QEI bound, in terms of
curvature invariants of the worldline. Our general results are illustrated by
specific computations for the six prototypical stationary worldlines. When the
averaging period is taken to infinity, the QEI bound is consistent with a
constant energy density along the worldline. For inertial and uniformly
linearly accelerated worldlines, this constant value is attained by the
Minkowski and Rindler vacuums respectively. It is an open question as to
whether the bounds for other stationary worldlines are attained by other
states of interest.

##  1 Introduction

Even if a classical field theory obeys local energy conditions, such as
positivity of energy density, the corresponding quantum field theory (QFT)
will fail to do so, as a result of a general theorem [10]. In fact, it is
typical that the expectation value of energy density at any given point can be
made arbitrarily negative by a suitable choice of the quantum state [12]. This
surprising fact is deeply related to the uncertainty principle. As
unrestricted negative energy densities, or negative energy fluxes, could
produce a range of effects, ranging from violations of the second law of
thermodynamics [22] and evasion of the classical singularity theorems of
Penrose [47] and Hawking [31], to the ability to construct warp drives [1] or
wormholes [46], it is important to understand what restrictions might be
imposed by QFT itself. In this paper we will be concerned with _Quantum Energy
Inequalities_ (QEIs), which provide lower bounds on local averages of the
expected energy density, independent of the quantum state.

Starting with results of Ford and Roman [22, 23, 24] QEIs have been derived
for a variety of quantum fields in flat and curved spacetimes. References and
discussion may be found in the recent reviews [14, 39]. At the simplest level,
the QEIs constrain the magnitude and duration of violations of classical
energy conditions, placing stringent constraints on attempts to use quantum
fields to provide the exotic matter required to construct exotic spacetimes
[25, 48, 18]. It has also been shown that QEI restrictions are sufficient for
modified versions of singularity theorems to hold [16].

For example, consider the real scalar field of mass $m\geq 0$ in any globally
hyperbolic spacetime $(M,g)$, recalling that global hyperbolicity demands only
that the spacetime possesses a global Cauchy surface. Let $\gamma(s)$ be any
smooth timelike curve, parameterised by proper time. It was shown in [11] that
the energy density of the quantum field along $\gamma$ obeys the QEI

$\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\int_{0}^{\infty}\frac{\mathop{}\\!\mathrm{d}{\alpha}}{\pi}\widehat{g\otimes
gT}(-\alpha,\alpha)>-\infty,$ (1.1)

which holds for all real-valued compactly supported smooth test functions $g$,
and all Hadamard states $\omega$ of the field. Here, the hat denotes a Fourier
transform, defined according to the convention
$\hat{g}(\alpha)=\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}\,e^{i\alpha
s}g(s)$, and we employ units where $\hbar=c=1$, which will be in force
throughout this paper. On the left-hand side, the normal ordering is conducted
with respect to an arbitrary Hadamard reference state $\omega_{0}$, whose two-
point function is used to construct the distribution $T(s,s^{\prime})$ that
appears on the right-hand side. Recall also that the Hadamard states form a
large class of physically reasonable states, determined by their short-
distance structure [36, 44]. The two most important features of the QEI (1.1)
are that the right-hand side is completely independent of the state $\omega$,
and that the bound is finite – which is proved using the microlocal properties
of Hadamard states uncovered by Radzikowski [49]. Discussion of QEIs for other
QFTs, including non-free models, may be found in [14, 39]; see [27] for a very
recent development.

Although the lower bound in (1.1) is explicit and rigorous, it is not easy to
compute in closed form except in special cases. Those examples where explicit
calculation is possible are therefore particularly valuable because they can
provide insight into the nature of QEI bounds in general. To the best of our
knowledge this has only been achieved when $T$ exhibits translational
invariance $T(s+r,s^{\prime}+r)=T(s,s^{\prime})$ which occurs, for instance,
when $(M,g)$ is a stationary spacetime, $\gamma$ is a timelike Killing orbit
and $\omega_{0}$ is stationary. Translational invariance allows us to write,
with an abuse of notation, $T(s,s^{\prime})=T(s-s^{\prime})$, from which one
easily finds that the QEI (1.1) simplifies to

$\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{\alpha}|\hat{g}(\alpha)|^{2}Q(\alpha),$
(1.2)

where

$Q(\alpha)=\dfrac{1}{2\pi^{2}}\int_{-\infty}^{\alpha}\mathop{}\\!\mathrm{d}{u}\
\hat{T}(u);$ (1.3)

the QEI (1.2) is also valid for complex-valued $g$. Taking the massless free
field as an example, averaging along an inertial worldline in Minkowski space
and using the Minkowski vacuum as the reference state $\omega_{0}$, this
results in $Q(\alpha)=\alpha^{4}\Theta(\alpha)/(16\pi^{3})$. Using the
evenness of $|\hat{g}|^{2}$ together with Parseval’s theorem then yields

$\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\frac{1}{16\pi^{2}}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g^{\prime\prime}(s)|^{2}.$
(1.4)

Similar expressions are known for massive fields and in Minkowski spacetime of
general dimension [15]; for some curved spacetime examples see [20]. Another
explicit example arises where $\gamma$ is a uniformly linearly accelerated
worldline in four-dimensional Minkowski spacetime with proper acceleration
$a$, in which case the QEI (1.2) becomes [17]

$\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{1}{16\pi^{2}}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}\left(\lvert
g^{\prime\prime}(s)\rvert^{2}+2a^{2}\lvert
g^{\prime}(s)\rvert^{2}+\dfrac{11}{30}a^{4}\lvert g(s)\rvert^{2}\right),$
(1.5)

and is again valid for all Hadamard states $\omega$ and complex-valued test
functions $g$.

Using such expressions the scaling behaviour of the bound is easily understood
and phenomena such as ‘quantum interest’ may be explored [26, 21, 14]. For
example, let $g_{\lambda}(s)=\lambda^{-1/2}g(s/\lambda)$, where $g$ is
normalised so that
$\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}=1$. Then (1.5)
implies

$\liminf_{\lambda\xrightarrow{}\infty}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g_{\lambda}(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{11a^{4}}{480\pi^{2}},$
(1.6)

reducing to the Averaged Weak Energy Condition (AWEC)

$\liminf_{\lambda\xrightarrow{}\infty}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{\tau}|g_{\lambda}(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq
0$ (1.7)

in the limit $a\to 0$, which can also be obtained directly from (1.4). An
interesting observation is that the lower bound in (1.6) is exactly the
constant energy density of the Rindler vacuum state along the accelerated
worldline, while the lower bound in (1.7) is the energy density of the
Minkowski vacuum.

As closed form expressions for QEI bounds are relatively few in number, it is
of interest to find others. The purpose of this paper is to present a
calculation of the QEI bound for a massless scalar field along any stationary
worldline in $4$-dimensional Minkowski spacetime. By a stationary worldline,
we mean any timelike curve $\gamma(s)$, parameterised by proper time $s$,
whose velocity vector evolves under a $1$-parameter subgroup of the Lorentz
group: $\dot{\gamma}(s)=\exp(sM)\dot{\gamma}(0)$ for some fixed
$M\in\mathfrak{so}(1,3)$ and future-pointing unit timelike $\dot{\gamma}(0)$.
The family of stationary worldlines includes physically important examples
including inertial, uniformly linearly accelerated and uniformly rotating
worldlines. We briefly recall their significance.

Stationary worldlines have a long history in relativity and quantum field
theory. Kottler [40], Synge [51] and Letaw [41] (see also [43]) all obtained
them as the solutions to four-dimensional Frenet-Serret equations subject to
constancy of the curvature invariants of the worldline; the name ‘stationary
worldlines’ is due to Letaw. The three curvature invariants are the curvature,
which measures the proper acceleration, and the torsion and hypertorsion,
which specify its proper angular velocity. More details are given in Section
2. Stationary worldlines are equivalently described as the orbits of timelike
Killing vector fields in Minkowski spacetime. There are also overlaps with the
theory of rigid motions in special relativity that goes back to Born [3] and
Herglotz [32]; in particular, any _rotational_ rigid motion is the flow of a
timelike Killing vector by the Herglotz–Noether theorem, although the same
theorem allows any $C^{2}$ timelike curve to be a flow line of an
_irrotational_ rigid motion. See [28] for discussion and references. By a
Poincaré transformation, any stationary worldline can be reduced to one of six
prototypes: the inertial, uniformly linearly accelerated, and uniformly
rotating worldlines have already been mentioned, while the three remaining
ones have spatial projections corresponding to a semicubical parabola, a
catenary or a helix. We will give more detail as we discuss each case
separately later on.

Turning to quantum field theory, the uniformly linearly accelerated trajectory
is of course at the heart of the extensive literature on the Unruh effect [52]
describing the thermal excitation of a detector travelling along the
accelerated trajectory [52, 8]. Coordinates adapted to this trajectory
naturally cover the Rindler wedge, to which the Minkowski vacuum state
restricts as a thermal state at the Unruh temperature relative to time
translation along the accelerated trajectory. This has attracted the attention
of other theorists to understand what excitations may be expected for
detectors following other stationary worldlines [41, 29] (and references
therein) and whether there are ground states associated to coordinates based
on these worldlines differing from restrictions of the Minkowski vacuum [42].
Proposals to observe the Unruh effect, on the other hand, focus on circular
trajectories because they are confined to a finite volume [2, 30]. In summary,
there is good motivation to deepen the understanding of physical effects seen
along stationary worldlines.

The main result of this paper is that the QEI (1.2) along any stationary
worldline in Minkowski spacetime may be given explicitly as

$\int\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{1}{16\pi^{2}}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}\left(\lvert
g^{\prime\prime}(s)\rvert^{2}+2A\lvert g^{\prime}(s)\rvert^{2}+B\lvert
g(s)\rvert^{2}\right),$ (1.8)

where $A$ and $B$ are expressed in terms of the curvature $\kappa$, torsion
$\tau$, and hypertorsion $\upsilon$ as

$A=\kappa^{2}+\tau^{2}+\upsilon^{2}$ (1.9)

and

$B=\frac{1}{90}\left(3\kappa^{4}+62\kappa^{2}\tau^{2}+30(\kappa^{2}+\tau^{2}+\upsilon^{2})^{2}\right),$
(1.10)

and the inequality (1.8) holds for all Hadamard states $\omega$ and all smooth
compactly supported test functions $g$. Because general stationary worldlines
describe a richer range of behaviour than that of constant linear
acceleration, the above formulae provide a more refined understanding than
that given by (1.5) of the way QEI bounds along a general worldline are
sensitive to its curvature invariants.

To interpret the QEI (1.8), it is useful to consider its scaling behaviour. As
before, we take a test function $g_{\lambda}$ which is just a scaled version
of the test function $g$, namely $g_{\lambda}(s)=\lambda^{-1/2}g(s/\lambda)$,
so the support width of $g_{\lambda}$ is proportional to $\lambda$. Observing
that $g_{\lambda}^{(k)}(s)=\lambda^{-k-1/2}g^{(k)}(s/\lambda)$, we find

$\int\mathop{}\\!\mathrm{d}{s}|g_{\lambda}(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{\|g^{\prime\prime}\|^{2}}{16\pi^{2}\lambda^{4}}-\dfrac{A\|g^{\prime}\|^{2}}{8\pi^{2}\lambda^{2}}-T_{\text{reg}}(0)\|g\|^{2},$
(1.11)

where the norms are those of $L^{2}(\mathbb{R})$, i.e.,
$\|g\|^{2}=\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}$. Here
we have written $T_{\text{reg}}(0)=B/(16\pi^{2})$ for reasons that will become
clear later – see, for example, equation (1.13) and the arguments presented in
Section 3. For sampling times shorter than the curvature scales, i.e.,
$\lambda\ll\min\\{\kappa^{-1},\tau^{-1},\upsilon^{-1}\\}$, the leading term
dominates, reflecting the fact that any worldline is approximately inertial on
short enough timescales. At intermediate and long timescales relative to
curvature scales, the bound will receive corrections from, and eventually be
dominated by, the last two terms in (1.11), showing that the QEI is sensitive
to the curvature invariants of the worldline $\gamma$. In the limit
$\lambda\to+\infty$, and with $g$ normalised so that $\|g\|=1$, we obtain the
remarkably simple formula

$\liminf_{\lambda\xrightarrow{}\infty}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g_{\lambda}(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-
T_{\text{reg}}(0),$ (1.12)

which bounds the average energy density along the entire trajectory. In
particular, the QEI is consistent with the existence of a constant
renormalised energy density $-T_{\text{reg}}(0)$ along $\gamma$, and this is
the most negative value that any constant energy density could take. An
intriguing question is whether or not this value is attained by some Hadamard
state, or a sequence of Hadamard states in a limiting sense, which we will
address in Section 6.

The derivation of (1.8) requires a number of innovations. Although the point-
split energy density can be obtained easily enough for any given stationary
worldline, its Fourier transform does not have a closed form – as far as we
know – for three of the six prototypes. In Section 3, we develop a new method
for computing the QEI bound for massless fields in four-dimensions that avoids
the use of Fourier transforms. The result is that the QEI will take the form
(1.8) provided that the point-split energy density takes the form

$T(s,s^{\prime})=\lim_{\epsilon\to
0+}\left(\frac{3}{2\pi^{2}(s-s^{\prime}-i\epsilon)^{4}}-\frac{A}{4\pi^{2}(s-s^{\prime}-i\epsilon)^{2}}\right)+T_{\text{reg}}(s-s^{\prime}),$
(1.13)

where the regular part $T_{\text{reg}}$ must satisfy various conditions,
whereupon the coefficient $B$ is given by $B=16\pi^{2}T_{\text{reg}}(0)$ as
before. In Section 4, we apply these ideas to stationary worldlines, resulting
in formulae for the point-split energy density in terms of functions easily
computed from the Lorentzian distance between two points on the curve and a
tetrad that is adapted to it, in a manner we describe. Most of the required
conditions on $T_{\text{reg}}$ follow directly from this analysis, and the
values $A$ and $B$ are identified in terms of Taylor coefficients of these
functions. Appendix A gives more detail on our methods, while in Appendix B
the relevant Taylor coefficients are evaluated in terms of curvature
invariants thus establishing (1.9) and (1.10). In Section 5, we work through
each prototype in turn, providing explicit formulae for the point-split energy
density that allow the remaining technical condition to be verified, and also
as a check on our Taylor series calculations. In three cases, (inertial
worldlines, linearly accelerated worldlines and the semicubical parabola), a
closed form may be found for $\hat{T}$, and we can also check our calculations
by using (1.2) and (1.3). Finally, in Section 6, we discuss the physical
significance of our results and some open problems. Two further appendices
contain additional computations: Appendix C computes a quantum inequality for
the Wick square along stationary worldlines following the same general method
of the main text, while Appendix D records the calculation of the minimally
coupled stress-energy tensor in the Rindler vacuum and Rindler thermal states,
which is needed for our discussion.

##  2 Stationary worldlines

Throughout this paper we work on $4$-dimensional Minkowski spacetime, with
metric
$\eta=\mathop{}\\!\mathrm{d}{t}^{2}-\mathop{}\\!\mathrm{d}{x}^{2}-\mathop{}\\!\mathrm{d}{y}^{2}-\mathop{}\\!\mathrm{d}{z}^{2}$,
and we employ the inertial coordinates $(t,x,y,z)$ except where otherwise
specified. A stationary worldline is any smooth curve
$\gamma:\mathbb{R}\to\mathbb{R}^{4}$, whose velocity vector $\dot{\gamma}$ is
a future-pointing unit timelike vector evolving under a $1$-parameter subgroup
of the Lorentz group $SO(1,3)$, i.e.,

$\dot{\gamma}^{\mu}(s)=\exp(sM)^{\mu}_{\phantom{\mu}\nu}\dot{\gamma}^{\nu}(0),$
(2.1)

where $M$ is any fixed element of $\mathfrak{so}(1,3)$ (which requires
precisely that $M_{\mu\nu}$ is antisymmetric). As every component of
$\exp(sM)$ is analytic in $s$, it follows that the Cartesian components of
$\dot{\gamma}(s)$ and, integrating, the Cartesian coordinates of $\gamma(s)$,
are also $s$-analytic. An equivalent definition of a stationary worldline is
that $\gamma$ is an orbit of a future-pointing timelike Killing vector field

$\xi^{\mu}(x)=M^{\mu}_{\phantom{\mu}\nu}(x^{\nu}-\gamma(0)^{\nu})+\dot{\gamma}^{\mu}(0),$
(2.2)

which is necessarily timelike in a neighbourhood of $\gamma$ and a future-
pointing unit vector on $\gamma$.

Finally, stationary worldlines can also be described as the solutions to the
Frenet-Serret equations with constant curvatures [40, 51, 41]. Here, the
curvature invariants of a general timelike curve $\gamma(s)$, parameterised by
proper time, are defined as follows. Suppose a right-handed tetrad
$e_{a}^{\mu}$ has been chosen along $\gamma$ so that

$\gamma^{(k+1)}(s)\in\operatorname{span}\\{e_{0}(s),\ldots,e_{k}(s)\\}\qquad(0\leq
k\leq 3),\qquad\text{and}\qquad\dot{\gamma}(s)=e_{0}(s),$ (2.3)

in which case we say that $e_{a}^{\mu}$ is _adapted_ to $\gamma$. If the
tetrad also satisfies

$e_{1}(s)^{\mu}\ddot{\gamma}(s)_{\mu}\leq 0,\qquad
e_{2}(s)^{\mu}\dddot{\gamma}(s)_{\mu}\leq 0,$ (2.4)

then it will be called a Frenet–Serret tetrad. If the tetrad is defined by
$e_{a}(s)=\exp(sM)e_{a}(0)$, then it is adapted (respectively, Frenet–Serret)
if and only if (2.3) holds at $s=0$ (resp., (2.3) and (2.4) hold at $s=0$).
Explicit formulae resulting from a Gram–Schmidt procedure are given in [41].
Expanding the derivatives of the tetrad vectors in terms of the tetrad, one
obtains the generalized Frenet–Serret equations

$\dot{e}_{a}^{\mu}=K_{a}^{\phantom{a}b}e_{b}^{\mu},$ (2.5)

where $K_{ab}$ is antisymmetric and tridiagonal (due to (2.3)). Thus it takes
the form

$K_{\bullet\bullet}(s)=\begin{pmatrix}0&-\kappa(s)&0&0\\\
\kappa(s)&0&-\tau(s)&0\\\ 0&\tau(s)&0&-\upsilon(s)\\\
0&0&\upsilon(s)&0\end{pmatrix},$ (2.6)

which defines the curvature $\kappa$, torsion $\tau$ and hypertorsion
$\upsilon$. Here, and elsewhere in this paper, bullets are used to indicate
tensorial type, when displaying tensorial components in vector or matrix form.
Explicitly, one has

$\kappa=e_{0\mu}\dot{e}_{1}^{\mu}=-e_{1\mu}\dot{e}_{0}^{\mu},\qquad\tau=e_{1\mu}\dot{e}_{2}^{\mu}=-e_{2\mu}\dot{e}_{1}^{\mu},\qquad\upsilon=e_{2\mu}\dot{e}_{3}^{\mu}=-e_{3\mu}\dot{e}_{2}^{\mu}.$
(2.7)

The choices made when specifying the Frenet–Serret tetrad ensure that $\kappa$
and $\tau$ are nonnegative, while $\upsilon$ can take any real value.

As the curvature invariants are constant along stationary worldlines, it is
easy to compute higher derivatives of the tetrad,

$\frac{\mathop{}\\!\mathrm{d}^{k}}{\mathop{}\\!\mathrm{d}s^{k}}e_{a}^{\mu}=(K^{k})_{a}^{\phantom{a}b}e_{b}^{\mu},\qquad(K^{k})_{a}^{\phantom{a}b}=K_{a}^{\phantom{a}c_{1}}K_{c_{1}}^{\phantom{c_{1}}c_{2}}\cdots
K_{c_{k-1}}{}^{b}.$ (2.8)

For example, the first three derivatives of the velocity $u=\dot{\gamma}$ may
be computed as

$\dot{u}^{\mu}=\dot{e}_{0}^{\mu}=\kappa
e_{1}^{\mu},\qquad\ddot{u}^{\mu}=\kappa^{2}e_{0}^{\mu}+\kappa\tau
e_{2}^{\mu},\qquad\dddot{u}^{\mu}=\kappa(\kappa^{2}-\tau^{2})e_{1}^{\mu}+\kappa\tau\upsilon
e_{3}^{\mu}.$ (2.9)

It is also possible to give a general formula for $\gamma(s)$ in terms of $M$,
$\gamma(0)$ and $\dot{\gamma}(0)$. As $M^{\bullet}_{\phantom{\bullet}\bullet}$
is antisymmetric with respect to $\eta$, there is a unique decomposition

$\dot{\gamma}(0)^{\mu}=M^{\mu}_{\phantom{\mu}\nu}v^{\nu}+k^{\mu},$ (2.10)

where $M^{\mu}_{\phantom{\mu}\nu}k^{\nu}=0$. One then has

$\gamma(s)^{\mu}=\exp(sM)^{\mu}_{\phantom{\mu}\nu}v^{\nu}+sk^{\mu}+\gamma(0)^{\mu}-v^{\mu}.$
(2.11)

Any stationary worldline $\gamma$ may be related to one of six basic types by
a proper orthochronous Poincaré transformation. Note that $\gamma(s)$ is
determined by the initial position, $\gamma(0)\in\mathbb{R}^{4}$, the initial
four-velocity $\dot{\gamma}(0)$ and the element $M\in\mathfrak{so}(1,3)$ that
fixes the evolution $\dot{\gamma}(s)=\exp(sM)\dot{\gamma}(0)$. Under a
Poincaré transformation $x\mapsto\Lambda x+w$, $\gamma$ is mapped to
$\tilde{\gamma}(s)=\Lambda\gamma(s)+w$, whose velocity evolves according to
the $1$-parameter Lorentz subgroup $\exp(s\Lambda M\Lambda^{-1})$ and which is
therefore also a stationary worldline. As the Lorentz transformation maps a
Frenet–Serret tetrad for $\gamma$ to a Frenet–Serret tetrad for
$\tilde{\gamma}$, it follows from (2.7) that the curvature invariants of
$\tilde{\gamma}$ are identical to those of $\gamma$. Using the classification
of conjugacy classes in $\mathfrak{so}(1,3)$ [50], we may choose $\Lambda$ in
such a way that $\tilde{M}=\Lambda M\Lambda^{-1}$ is one of five possible
types: (a) the zero element, generating the trivial subgroup of $SO(1,3)$, (b)
a generator of boosts in the $tx$-plane, corresponding to a hyperbolic
subgroup of $SO(1,3)$, (c) a generator of rotations in the $yz$-plane,
corresponding to an elliptic subgroup of $SO(1,3)$, (d) a generator of a null
rotation that fixes the null vector $\partial_{t}+\partial_{x}$ but acts
nontrivially on all other null vectors, corresponding to a parabolic subgroup
of $SO(1,3)$; (e) the sum of a generator of boosts in the $tx$-plane and a
generator of rotations in the $yz$ plane, corresponding to a loxodromic
subgroup of $SO(1,3)$. In each case, Lorentz transformations that commute with
the $1$-parameter subgroup in question can be used to arrange that
$\dot{\tilde{\gamma}}(0)$ takes a convenient form.

Taking these possibilities in turn: in case (a), all Lorentz transformations
commute with the trivial subgroup, so we may without loss assume that
$\tilde{\gamma}(s)=(s,0,0,0)$. In case (b), the subgroup of boosts parallel to
the $x$-axis commutes with itself and the subgroup of rotations in the
$yz$-plane. Thus, we may arrange that
$\dot{\tilde{\gamma}}(0)=\cosh\chi\partial_{t}+\sinh\chi\partial_{y}$ for some
$\chi\in\mathbb{R}$,111We could even arrange that $\chi\geq 0$, but it is
convenient not to insist on this. leading to two subcases: $\chi=0$, in which
case (after possible translation)

$\tilde{\gamma}(s)=(a^{-1}\sinh as,a^{-1}\cosh as,0,0)$ (2.12)

is a uniformly linearly accelerated worldline with $a\neq 0$, or $\chi\neq 0$,
in which case (up to translations)

$\tilde{\gamma}(s)=(a^{-1}\cosh\chi\,\sinh as,a^{-1}\cosh\chi\,\cosh
as,-s\sinh\chi,0)$ (2.13)

is a catenary. The curvature invariants (in either subcase) are
$\kappa=|a|\cosh\chi$ and $\tau=|a\sinh\chi|$, while the hypertorsion is
$\upsilon=0$. For convenience, the curvature invariants for all six prototypes
are tabulated in Table 1, in agreement with [43].

| Inertial | Linear Acc. | Catenary | Parabolic | Elliptic | Loxodromic
---|---|---|---|---|---|---
| $\kappa=\tau=\upsilon=0$ | $\kappa>0$ | $\kappa>\tau>0$ | $\kappa=\tau>0$ | $\tau>\kappa>0$ | $\kappa,\tau>0$
|  | $\tau=\upsilon=0$ | $\upsilon=0$ | $\upsilon=0$ | $\upsilon=0$ | $\upsilon\neq 0$
$\kappa$ | $0$ | $\lvert a\rvert$ | $\lvert a\rvert\cosh\chi$ | $\lvert a\rvert$ | $r\omega^{2}$ | $\sqrt{C^{2}a^{2}+V^{2}\omega^{2}}$
$\tau$ | $0$ | $0$ | $\lvert a\sinh\chi\rvert$ | $\lvert a\rvert$ | $\lvert\omega\rvert\sqrt{1+(r\omega)^{2}}$ | $(a^{2}+\omega^{2})C|V|/\kappa$
$\upsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $a\omega/\kappa$
Table 1: Curvature invariants for the stationary worldlines.

In case (c), the $1$-parameter parabolic subgroup takes the form

$P^{\bullet}_{\phantom{\bullet}\bullet}(s)=\begin{pmatrix}1+(as)^{2}/2&-(as)^{2}/2&0&as\\\
(as)^{2}/2&1-(as)^{2}/2&0&as\\\ 0&0&1&0\\\
as&-as&0&1\end{pmatrix}=\exp\begin{pmatrix}0&0&0&as\\\ 0&0&0&as\\\ 0&0&0&0\\\
as&-as&0&0\end{pmatrix}$ (2.14)

for some constant nonzero $a\in\mathbb{R}$, and commutes with Lorentz
transformations of the form

$\Lambda^{\bullet}_{\phantom{\bullet}\bullet}=\begin{pmatrix}1+r^{2}/2&-r^{2}/2&r\cos\theta&r\sin\theta\\\
r^{2}/2&1-r^{2}/2&r\cos\theta&r\sin\theta\\\ r\cos\theta&-r\cos\theta&1&0\\\
r\sin\theta&-r\sin\theta&0&1\end{pmatrix}$ (2.15)

which can be used to bring the initial velocity into the form
$\dot{\tilde{\gamma}}(0)=\cosh\chi\partial_{t}+\sinh\chi\partial_{x}$ for some
$\chi\in\mathbb{R}$. Conjugating $P^{\bullet}_{\phantom{\bullet}\bullet}(s)$
with a boost in the $tx$-plane results in
$P^{\bullet}_{\phantom{\bullet}\bullet}(\lambda s)$ for some $\lambda>0$; in
other words effectively rescaling $a$. Therefore there is no loss of
generality in assuming that the initial $4$-velocity is
$\dot{\tilde{\gamma}}(0)=\partial_{t}$, in which case the worldline (up to
translation) is the semicubical parabola,

$\tilde{\gamma}(s)=\left(s+\dfrac{1}{6}a^{2}s^{3},\dfrac{1}{6}a^{2}s^{3},0,\dfrac{1}{2}as^{2}\right).$
(2.16)

Next, the elliptic subgroup in case (d) commutes with boosts in the $tx$-plane
and rotations in the $yz$-plane. Accordingly, we may arrange the initial
velocity to be
$\dot{\tilde{\gamma}}(0)=\cosh\chi\partial_{t}+\sinh\chi\partial_{z}$ for some
$\chi\in\mathbb{R}$; the special case $\chi=0$ corresponds to inertial motion
and may be discarded. Up to a translation, this results in the uniformly
rotating worldline

$\tilde{\gamma}^{\bullet}(s)=\left(s\cosh\chi,0,r\cos{\omega s},r\sin\omega
s\right),$ (2.17)

where the radius $r>0$ and proper angular velocity $\omega\neq 0$ are related
to the initial rapidity by $r\omega=\sinh\chi$. The proper acceleration is
$\kappa=r\omega^{2}$, while the torsion is
$\tau=\lvert\omega\rvert\sqrt{1+(r\omega)^{2}}$ and the hypertorsion vanishes.

Lastly, in case (e), the loxodromic subgroup is generated by a linear
combination of a $tx$-boost generator and a $yz$-rotation generator. As it
commutes with $tx$-boosts and $yz$-rotations, we may assume without loss that
the initial velocity is
$\dot{\tilde{\gamma}}(0)=\cosh\chi\partial_{t}+\sinh\chi\partial_{z}$ for
$\chi\in\mathbb{R}\setminus\\{0\\}$; the possibility $\chi=0$ corresponds to a
hyperbolic worldline and is rejected. Up to a translation, this results in the
worldline

$\gamma^{\bullet}(s)=(Ca^{-1}\sinh(as),Ca^{-1}\cosh(as),V\omega^{-1}\cos(\omega
s),V\omega^{-1}\sin(\omega s)),$ (2.18)

where $C=\cosh\chi$ and $V=\sinh\chi$, which undergoes both rotation in the
$yz$-plane at constant proper angular velocity $\omega\neq 0$ and constant
distance $|V/\omega|$ from the $x$-axis, while undergoing uniform acceleration
in the $x$-direction controlled by $a\neq 0$ (the cases where one or both of
$a$ or $\omega$ vanish are already covered under (a), (b) and (d)). The
curvature invariants for this worldline are

$\kappa=\sqrt{C^{2}a^{2}+V^{2}\omega^{2}},\qquad\tau=(a^{2}+\omega^{2})C|V|/\kappa,\qquad\upsilon=a\omega/\kappa.$
(2.19)

##  3 Reformulation of the QEI bound

We study the massless minimally coupled scalar field in $4$-dimensional
Minkowski spacetime, with field equation
$\Box\phi=\eta^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0$ and classical stress-
energy tensor

$T_{\mu\nu}=(\nabla_{\mu}\phi)\nabla_{\nu}\phi-\tfrac{1}{2}\eta_{\mu\nu}\eta^{\alpha\beta}(\nabla_{\alpha}\phi)\nabla_{\beta}\phi.$
(3.1)

Consider an observer following a timelike curve $\gamma$, parameterised by
proper time, with $4$-velocity $u^{\mu}=\dot{\gamma}^{\mu}$. This observer
sees energy density

$T_{\mu\nu}u^{\mu}u^{\nu}=\frac{1}{2}\sum_{a=0}^{3}(e_{a}^{\mu}\nabla_{\mu}\phi)^{2},$
(3.2)

where $e_{a}^{\mu}$ ($0\leq a\leq 3$) is a tetrad defined around $\gamma$ with
$e_{0}^{\mu}|_{\gamma}=u^{\mu}$.

In quantum field theory, the stress-energy tensor requires renormalisation.
Let

$G(x,x^{\prime})=\langle\phi(x)\phi(x^{\prime})\rangle_{\omega}$ (3.3)

be the Wightman function of the field in a state $\omega$. The Wick square has
expectation value

$\langle{:}\phi^{2}(x){:}\rangle_{\omega}=(G-G_{0})(x,x),$ (3.4)

where

$G_{0}(x,x^{\prime})=\lim_{\epsilon\to
0+}\frac{-1}{4\pi^{2}((t-t^{\prime}-i\epsilon)^{2}-\|\bm{x}-\bm{x}^{\prime}\|^{2})}$
(3.5)

is the Wightman function of the Poincaré invariant vacuum $\omega_{0}$. This
expression makes sense if (like $\omega_{0}$) $\omega$ is a _Hadamard state_
[36, 44], because the difference $G-G_{0}$ is then a smooth function.
Similarly, the renormalised stress-energy tensor has expectation value

$\langle{:}T_{\mu\nu}(x){:}\rangle_{\omega}=D_{\mu\nu}(x)-\tfrac{1}{2}\eta_{\mu\nu}\eta^{\alpha\beta}D_{\alpha\beta}(x),$
(3.6)

where

$D_{\mu\nu}(x)=\left[\\!\left[(\nabla\otimes\nabla)(G-G_{0})\right]\\!\right]_{\mu\nu}(x)$
(3.7)

and the double square brackets denote a coincidence limit.

Although the classical energy density (3.2) is everywhere nonnegative, the
quantised energy density may assume negative expectation values. The QEIs
provide lower bounds on averaged expectation values, for which a prototype is
a lower bound on the following expression

$\int\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}(\mathcal{Q}\phi)^{2}\mathbf{:}\rangle_{\omega}(\gamma(s)),$
(3.8)

where $\mathcal{Q}$ is a partial differential operator with smooth real
coefficients and $g\in\mathcal{C}_{0}^{\infty}(\mathbb{R})$ is a smooth real-
valued test function. In the case where $\mathcal{Q}$ is the identity, (3.8)
is an averaged Wick square, while by considering a sum of similar terms for
$\mathcal{Q}_{a}=2^{-1/2}e^{\mu}_{a}\nabla_{\mu}$ for $0\leq a\leq 3$, we can
bound averages of the energy density along $\gamma$.

A lower bound on (3.8) was established in [11] – in fact the bound applies to
general timelike curves in arbitrary globally hyperbolic spacetimes for
massive as well as massless fields. In our case it asserts that

$\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}(\mathcal{Q}\phi)^{2}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\int_{0}^{\infty}\frac{\mathop{}\\!\mathrm{d}{\alpha}}{\pi}\yhwidehat{g\otimes
gT}(-\alpha,\alpha)>-\infty$ (3.9)

holds for all real-valued compactly supported smooth test functions $g$, and
all Hadamard states $\omega$, where

$T(s,s^{\prime})=\langle\mathcal{Q}\phi(\gamma(s))\mathcal{Q}\phi(\gamma(s^{\prime}))\rangle_{\omega_{0}}=((\mathcal{Q}\otimes\mathcal{Q})G_{0})(\gamma(s),\gamma(s^{\prime})).$
(3.10)

Here, the vacuum two-point function enters because normal ordering is
performed relative to the vacuum; the general results of [11] also allow for
any Hadamard state to be used as the reference state for this purpose. At a
more formal level, $T$ is the pull-back of the distribution
$(\mathcal{Q}\otimes\mathcal{Q})G_{0}$ by the map
$(s,s^{\prime})\mapsto(\gamma(s),\gamma(s^{\prime}))$, and its existence is
owed to the special properties of the Hadamard condition and the fact that
$\gamma$ is timelike – see [11] for full details and rigorous proofs.

As already mentioned, a QEI for the energy density involves a sum of such
bounds, leading to (1.1) with

$T(s,s^{\prime})=\frac{1}{2}\sum_{a=0}^{3}((\nabla_{e_{a}}\otimes\nabla_{e_{a}})G_{0})(\gamma(s),\gamma(s^{\prime})).$
(3.11)

While it is usually not hard to obtain the distribution $T$ for a given
timelike curve in Minkowski spacetime, assuming that $G_{0}$ is given, it is
not usually possible to find the Fourier transform required to compute the QEI
bound (3.9) in closed form.

The situation is somewhat simplified if $T(s,s^{\prime})$ is translationally
invariant, in which case one has the bound given by (1.2) and (1.3). This can
be taken a little further, on observing that $|\hat{g}(\alpha)|^{2}$ is even,
so only the even part
$Q_{\text{even}}(\alpha)=\tfrac{1}{2}(Q(\alpha)+Q(-\alpha))$ of $Q$
contributes to (1.2), resulting in the bound

$\int\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}(\mathcal{Q}\phi)^{2}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{\alpha}\lvert\hat{g}(\alpha)\rvert^{2}Q_{\text{even}}(\alpha),$
(3.12)

which is the final form of our prototypical quantum inequality.

A convenient expression for $Q_{\text{even}}$ may be found by manipulating
equation (1.3) in the following way:

$\displaystyle Q_{\text{even}}(\alpha)$
$\displaystyle=\dfrac{1}{4\pi^{2}}\left[\int_{-\infty}^{\alpha}\hat{T}(u)\mathop{}\\!\mathrm{d}{u}+\int_{-\infty}^{-\alpha}\hat{T}(u)\mathop{}\\!\mathrm{d}{u}\right]$
$\displaystyle=\dfrac{1}{4\pi^{2}}\left[2\int_{-\infty}^{0}\hat{T}(u)\mathop{}\\!\mathrm{d}{u}+\int_{0}^{\alpha}\hat{T}(u)\mathop{}\\!\mathrm{d}{u}-\int_{0}^{\alpha}\hat{T}(-u)\mathop{}\\!\mathrm{d}{u}\right]$
$\displaystyle=\dfrac{1}{2\pi^{2}}\left[\int_{-\infty}^{0}\hat{T}(u)\mathop{}\\!\mathrm{d}{u}+\int_{0}^{\alpha}\hat{T}_{\text{odd}}(u)\mathop{}\\!\mathrm{d}{u}\right],$
(3.13)

where $\hat{T}_{\text{odd}}(u)=\tfrac{1}{2}(\hat{T}(u)-\hat{T}(-u))$. In the
above calculation, $\hat{T}$ is assumed to be continuous, as is the case for
the examples we will study.

Evaluating $Q_{\text{even}}$ from (3) requires several steps. Computing $T$ is
a tedious but straightforward calculation best handled using computer algebra.
In the simplest cases, the transform may be evaluated in closed form, which
(as will be seen later) is the case for the inertial, uniformly accelerated
and semicubical parabola worldlines, but is not possible (to our knowledge) in
the case of the other stationary worldlines. However, this obstacle can be
circumvented, as we now describe.

Using the Minkowski vacuum as the reference state, we will show in Section 4
that the point-split energy density along a stationary worldline may be
written in the form

$T(s,s^{\prime})=T_{\text{sing}}(s-s^{\prime})+T_{\text{reg}}(s-s^{\prime}),$
(3.14)

where $T_{\text{sing}}$ is given by the distributional limit

$T_{\text{sing}}(s)=\lim_{\epsilon\to
0+}\left(\frac{3}{2\pi^{2}(s-i\epsilon)^{4}}-\frac{A}{4\pi^{2}(s-i\epsilon)^{2}}\right)$
(3.15)

for some constant $A$ (the sign is chosen for later convenience) and
$T_{\text{reg}}$ is smooth, real and even, and decaying as
$\mathcal{O}(s^{-2})$ as $|s|\to\infty$. In particular, $T_{\text{reg}}$ is
absolutely integrable and has a well-defined Fourier transform that is
continuous, real and even. Therefore it does not contribute to
$\hat{T}_{\text{odd}}$. Turning to $T_{\text{sing}}$, its leading singularity
is universal, essentially because all stationary worldlines resemble inertial
worldlines on sufficiently short timescales. The specific coefficient is fixed
by the Hadamard form and the definition of the energy density along the curve.
Meanwhile the coefficient $A$ carries information about the specific curve at
hand. The Fourier transform of $T_{\text{sing}}$, in our convention, is

$\hat{T}_{\text{sing}}(u)=\dfrac{1}{2\pi}(u^{3}+Au)\Theta(u),$ (3.16)

where $\Theta$ is the Heaviside distribution. Evidently $T_{\text{sing}}$ does
not contribute to the first term in (3), while the odd part of $\hat{T}$ is

$\hat{T}_{\text{odd}}(u)=\dfrac{1}{4\pi}(u^{3}+Au),$ (3.17)

recalling that $\hat{T}_{\text{reg}}$ is even. We now have $Q_{\text{even}}$
in the form

$\displaystyle Q_{\text{even}}(\alpha)$
$\displaystyle=\dfrac{1}{2\pi^{2}}\left[\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}{u}\,\hat{T}_{\text{reg}}(u)+\dfrac{1}{4\pi}\int_{0}^{\alpha}\mathop{}\\!\mathrm{d}{u}(u^{3}+Au)\right]$
$\displaystyle=\dfrac{1}{32\pi^{3}}(\alpha^{4}+2A\alpha^{2})+\dfrac{T_{\text{reg}}(0)}{2\pi},$
(3.18)

where we have again used the evenness of $\hat{T}_{\text{reg}}$ and the
Fourier inversion formula. Inserting (3) into (3.12) and using Parseval’s
theorem gives the QEI bound

$\int\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{1}{16\pi^{2}}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}\left(\lvert
g^{\prime\prime}(s)\rvert^{2}+2A\lvert g^{\prime}(s)\rvert^{2}+B\lvert
g(s)\rvert^{2}\right),$ (3.19)

where $B=16\pi^{2}T_{\text{reg}}(0)$.

The upshot of this analysis is a direct route to the QEI once the point-split
expression $T$ is obtained; all that is needed is to isolate the appropriate
values of $A$ and $T_{\text{reg}}(0)$, avoiding the need to compute $\hat{T}$
explicitly. This apparent royal road is made possible because of the special
structure of the Minkowski vacuum two-point function for the massless scalar
field in four dimensions – closely related to Huygens’ principle. A similar
analysis for a QI on the Wick square can be found in Appendix C.

##  4 Computation of the point-split energy density

In this section we establish that the point-split energy density along
stationary worldlines obeys equations (3.14) and (3.15), and also that
$T_{\text{sing}}$ and $T_{\text{reg}}$ have the properties mentioned above,
with one exception that will be treated by examining the six prototypical
cases in Section 5.

Let $\gamma$ be any stationary worldline with
$\dot{\gamma}(s)=\exp(sM)\dot{\gamma}(0)$ and $\dot{\gamma}(0)$ a future-
pointing unit timelike vector. Suppose that

$e_{a}(s)=\exp(sM)e_{a}(0)\qquad(0\leq a\leq 3)$ (4.1)

is an adapted frame on $\gamma$ satisfying (2.3). In general there may be many
possible adapted tetrads of this type. However, if $\tilde{e}_{a}(s)$ is any
other then it is related to $e_{a}(s)$ by a rigid rotation, i.e.,
$\tilde{e}_{0}(s)=e_{0}(0)$ and
$\tilde{e}_{i}(0)=R_{i}^{\phantom{i}j}e_{j}(0)$ (summing $j$ over $1,2,3$),
where $\delta^{im}R_{i}^{\phantom{i}j}R_{m}^{\phantom{m}n}=\delta^{mn}$, $\det
R=1$. This must be true for some $R$ at $s=0$, and extends to all $s$ as both
tetrads evolve under $\exp(sM)$.

Next, recall that the vacuum $2$-point function may be given as a
distributional limit

$G_{0}(x,x^{\prime})=\lim_{\epsilon\to 0+}F(\sigma_{\epsilon}(x,x^{\prime}))$
(4.2)

where $F(z)=1/(4\pi^{2}z)$ and

$\sigma_{\epsilon}(x,x^{\prime})=-\eta_{\mu\nu}(x-x^{\prime}-i\epsilon\partial_{t})^{\mu}(x-x^{\prime}-i\epsilon\partial_{t})^{\nu}$
(4.3)

is the regulated signed squared geodesic separation of $x$ and $x^{\prime}$.
As usual, we have identified Minkowski spacetime with its tangent spaces at
all points.

Distributional derivatives may be taken under the limit in (4.2), giving

$\tfrac{1}{2}(\nabla_{\mu}\otimes 1)G_{0}(x,x^{\prime})=-\lim_{\epsilon\to
0+}F^{\prime}(\sigma_{\epsilon}(x,x^{\prime}))(x-x^{\prime}-i\epsilon\partial_{t})_{\mu}$
(4.4)

and

$\tfrac{1}{2}(\nabla_{\mu}\otimes\nabla_{\nu})G_{0}(x,x^{\prime})=\lim_{\epsilon\to
0+}\left(F^{\prime}(\sigma_{\epsilon}(x,x^{\prime}))\eta_{\mu\nu}-2F^{\prime\prime}(\sigma_{\epsilon}(x,x^{\prime}))(x-x^{\prime}-i\epsilon\partial_{t})_{\mu}(x-x^{\prime}-i\epsilon\partial_{t})_{\nu}\right).$
(4.5)

Contracting with $e_{a}(x)^{\mu}e_{a}(x^{\prime})^{\nu}$ (without summing on
$a$) and pulling back to the worldline, we find

$\displaystyle\tfrac{1}{2}((\nabla_{e_{a}}\otimes\nabla_{e_{a}})G_{0})(\gamma(s),\gamma(s^{\prime}))$
$\displaystyle=\lim_{\epsilon\to
0+}F^{\prime}(\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime})))C_{a}(s,s^{\prime})$
$\displaystyle\qquad\qquad+\lim_{\epsilon\to
0+}2F^{\prime\prime}(\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime})))D_{a}(s,s^{\prime})D_{a}(s^{\prime},s).$
(4.6)

(note the order of variables in the last two factors in the second term) where

$C_{a}(s,s^{\prime})=\eta_{\mu\nu}e_{a}^{\mu}(s)e_{a}^{\nu}(s^{\prime}),\qquad
D_{a}(s,s^{\prime})=(\gamma(s)-\gamma(s^{\prime}))_{\mu}e_{a}^{\mu}(s).$ (4.7)

Under a change of frame from $e_{a}$ to $\tilde{e}_{a}$ as described above,
one has $\tilde{C}_{0}=C_{0}$, $\tilde{D}_{0}=D_{0}$, while
$\tilde{D}_{i}=R_{i}^{\phantom{i}j}D_{j}$ and
$\tilde{C}_{i}(s,s^{\prime})=R_{i}^{\phantom{i}j}R_{i}^{\phantom{i}k}\eta_{\mu\nu}e_{j}^{\mu}(s)e_{k}^{\nu}(s^{\prime})$.
By orthogonality, this implies that
$\sum_{i=1}^{3}\tilde{C}_{i}(s,s^{\prime})=\sum_{i=1}^{3}C_{i}(s,s^{\prime})$
and
$\sum_{i=1}^{3}\tilde{D}_{a}(s,s^{\prime})\tilde{D}_{a}(s^{\prime},s)=\sum_{i=1}^{3}D_{a}(s,s^{\prime})D_{a}(s^{\prime},s)$.

In Appendix A, we give some further details to justify the above
distributional manipulations and prove the following result, where $\kappa$,
$\tau$ and $\upsilon$ are the curvature invariants of $\gamma$ as described in
Section 2.

######  Lemma.

(a) With the choice of tetrad just described, $C_{a}(s,s^{\prime})$ and
$D_{a}(s,s^{\prime})$ are translationally invariant, depending only on
$s-s^{\prime}$. There are entire analytic functions $G_{a}$ and $H_{a}$ such
that

$C_{a}(s,s^{\prime})=G_{a}(\kappa^{2}(s-s^{\prime})^{2}),\qquad
D_{a}(s,s^{\prime})D_{a}(s^{\prime},s)=-(s-s^{\prime})^{2}H_{a}(\kappa^{2}(s-s^{\prime})^{2}),$
(4.8)

where, in the limit $z\to 0$,

$\sum_{a=0}^{3}G_{a}(z)=-2+\frac{\tau^{2}+\upsilon^{2}}{\kappa^{2}}z+\frac{(\kappa\tau)^{2}-(\tau^{2}+\upsilon^{2})^{2}}{\kappa^{4}}z^{2}+\mathcal{O}(z^{3}),$
(4.9)

and

$\sum_{a=0}^{3}H_{a}(z)=1+\frac{z}{12}+\frac{\kappa^{2}+19\tau^{2}}{360\kappa^{2}}z^{2}+\mathcal{O}(z^{3}).$
(4.10)

(b) The signed square geodesic separation of points along $\gamma$ obeys

$\sigma_{0}(\gamma(s),\gamma(s^{\prime}))=-(s-s^{\prime})^{2}\Upsilon(\kappa^{2}(s-s^{\prime})^{2}),$
(4.11)

where $\Upsilon$ is entire analytic with

$\Upsilon(z)=1+\frac{1}{12}z+\frac{\kappa^{2}-\tau^{2}}{360\kappa^{2}}z^{2}+\mathcal{O}(z^{3})$
(4.12)

as $z\to 0$. Furthermore, for $z\in[0,\infty)$, $\Upsilon(z)$ is real with
$\Upsilon(z)\geq 1$.

The Lemma now allows us to compute the point-split energy density by
evaluating the right-hand side of (4) and summing over $a$. We use the fact
(explained in Appendix A) that

$\lim_{\epsilon\to
0+}\frac{(s-s^{\prime})^{2j}}{\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime}))^{k}}=\frac{(-1)^{k}}{\Upsilon(\kappa^{2}(s-s^{\prime})^{2})^{k}}\lim_{\epsilon\to
0+}\frac{1}{(s-s^{\prime}-i\epsilon)^{2(k-j)}},$ (4.13)

where the limits are taken in the sense of distributions, as is the
multiplication by a smooth prefactor on the right-hand side. If $j=k$, the
distributional limit on the right-hand side may be replaced by unity. In
particular, when calculating $T(s,s^{\prime})$ from (4), the factor
$(s-s^{\prime})^{2}$ in $D_{a}(s,s^{\prime})D_{a}(s^{\prime},s)$ cancels a
factor of $(s-s^{\prime}-i\epsilon)^{2}$ in the denominator, as $\epsilon\to
0+$. The upshot is that

$T(s,s^{\prime})=-\frac{1}{4\pi^{2}}\lim_{\epsilon\to
0+}\frac{K(\kappa^{2}(s-s^{\prime})^{2})}{(s-s^{\prime}-i\epsilon)^{4}},\qquad\text{where}\qquad
K(z)=\sum_{a=0}^{3}\left(\frac{G_{a}(z)}{\Upsilon(z)^{2}}-4\frac{H_{a}(z)}{\Upsilon(z)^{3}}\right)$
(4.14)

is a meromorphic function that is analytic in a neighbourhood of the positive
real axis (on which $\Upsilon$ is bounded away from zero).

The singular part is easily isolated by splitting off the first two terms of
the Taylor series for $K$ from the remainder, which carries a leading factor
of $(s-s^{\prime})^{4}$ that cancels the denominator in the limit $\epsilon\to
0+$. Similarly, the $\mathcal{O}(z)$ part of the Taylor series partly cancels
the denominator. Thus,
$T(s,s^{\prime})=T_{\text{sing}}(s-s^{\prime})+T_{\text{reg}}(s-s^{\prime})$
with

$T_{\text{sing}}(s)=-\frac{1}{4\pi^{2}}\lim_{\epsilon\to
0+}\frac{K(0)}{(s-i\epsilon)^{4}}-\frac{1}{4\pi^{2}}\lim_{\epsilon\to
0+}\frac{\kappa^{2}K^{\prime}(0)}{(s-i\epsilon)^{2}},$ (4.15)

and

$T_{\text{reg}}(s)=-\frac{\kappa^{4}}{4\pi^{2}}J((\kappa
s)^{2}),\qquad\text{where}\qquad J(z)=\frac{K(z)-K(0)-K^{\prime}(0)z}{z^{2}}$
(4.16)

is analytic on a neighbourhood of the positive real axis, so $J((\kappa
s)^{2})$ is smooth for $s\in\mathbb{R}$.

Using the Lemma, we may read off that $K(0)=-6$, thus establishing (3.15),
with $A=\kappa^{2}K^{\prime}(0)$. Meanwhile, $T_{\text{reg}}(s)$ is smooth,
even, and real-valued for $s\in\mathbb{R}$. Provided that
$K(z)=\mathcal{O}(z)$ as $z\to\infty$ on the real axis, we find that
$T_{\text{reg}}(s)=\mathcal{O}(s^{-2})$ as $s\to\infty$, which completes the
properties needed in Section 3. Furthermore,

$T_{\text{reg}}(0)=-\frac{J(0)\kappa^{4}}{4\pi^{2}}=-\frac{K^{\prime\prime}(0)\kappa^{4}}{8\pi^{2}}.$
(4.17)

Note that if we had used the tetrad $\tilde{e}$ instead, the function $K$
would be unchanged, owing to the remarks before the Lemma. Thus the QEIs
obtained from $\tilde{e}_{a}$ and $e_{a}$ are identical.

These results now provide a calculational method to determine the QEI along
stationary worldlines. Starting from the generator $M\in\mathfrak{so}(1,3)$
and the initial $4$-velocity $u(0)$, choose a tetrad as described at the start
of this section, and compute the proper acceleration
$\kappa=\sqrt{-\eta(Mu(0),Mu(0))}$. The translational invariance of $C_{a}$
and $D_{a}$ means that they can be calculated conveniently as

$C_{a}(s,s^{\prime})=\eta_{\mu\nu}e_{a}^{\mu}(s-s^{\prime})e_{a}^{\nu}(0),\qquad
D_{a}(s,s^{\prime})=-(\gamma(s^{\prime}-s)-\gamma(0))_{\mu}e_{a}^{\mu}(0),$
(4.18)

from which $G_{a}$ and $H_{a}$ are easily obtained. The function $\Upsilon$ is
computed directly from the Lorentz interval between $\gamma(0)$ and
$\gamma(s)$. Then construct $K(z)$ according to (4.14) and check that
$K(z)=\mathcal{O}(z)$ as $z\to\infty$. Then the QEI along $\gamma$ is given by
(3.19), with constants

$A=\kappa^{2}K^{\prime}(0),\qquad B=-2\kappa^{4}K^{\prime\prime}(0).$ (4.19)

The constants $A$ and $B$ can be computed from the first few terms of the
Taylor expansions of $\sum_{a}G_{a}$, $\sum_{a}H_{a}$ and $\Upsilon$, given in
(4.9), (4.10) and (4.12) respectively. After a calculation, one finds

$K(z)=-6+z\dfrac{\kappa^{2}+\tau^{2}+\upsilon^{2}}{\kappa^{2}}-z^{2}\dfrac{1}{360\kappa^{4}}\left(3\kappa^{4}+62\kappa^{2}\tau^{2}+30(\kappa^{2}+\tau^{2}+\upsilon^{2})^{2}\right)+\mathcal{O}(z^{3}),$
(4.20)

from which the formulae (1.9) and (1.10) follow immediately. Nonetheless, this
is perhaps not the most illuminating calculation and also does not provide a
check that $K(z)=O(z)$ for large real $z$, which was assumed above. For these
reasons, and their own intrinsic interest, we will also provide explicit
calculations in Section 5 that together cover all possible stationary
worldlines.

##  5 QEIs for the prototypical stationary worldlines

We have now established the general QEI for stationary worldlines in Minkowski
spacetime, assuming a technical condition on the growth of $K$. In this
section, we reduce the problem of computing the QEI for a general stationary
worldline to six prototypical cases, which will be treated in turn. These
calculations follow the method of Section 4 and result in explicit formulae
for $K$. In this way it is seen that the growth condition holds in all cases
and we also obtain a check on the Taylor series calculations in Appendix B.

We have already discussed the fact that any stationary worldline may be
brought into one of the six standard forms by a Poincaré transformation,
without changing the curvature invariants. Owing to Poincaré invariance of the
vacuum state, and because Poincaré invariance maps an adapted tetrad of the
form $e_{a}(s)=\exp(sM)e_{a}(0)$ along the original curve to a tetrad with the
same properties on the new one, the point-split energy density obtained by the
method of Section 4 is exactly the same for the two worldlines, which
accordingly share the same QEI bound.

The QEIs for the prototypical stationary worldlines are now given in turn.
Most of the computations that follow were conducted using the computer algebra
system Maple.

###  5.1 Trivial subgroup: inertial motion

For the inertial worldline $\gamma(s)=(s,0,0,0)$, we employ the adapted tetrad
$\partial_{t},\partial_{x},\partial_{y},\partial_{z}$, which is constant along
$\gamma$, leading immediately to the relations $C_{0}(s,s^{\prime})=1$,
$C_{i}(s,s^{\prime})=-1$ for $i=1,2,3$, while
$D_{0}(s,s^{\prime})=s-s^{\prime}$, $D_{i}(s,s^{\prime})=0$ for all
$s,s^{\prime}$. It follows that $G_{0}=H_{0}\equiv 1$, $G_{i}\equiv-1$,
$H_{i}\equiv 0$. Furthermore, $\Upsilon\equiv 1$ because
$\sigma_{0}(\gamma(s),\gamma(s^{\prime}))=-(s-s^{\prime})^{2}$. Hence
$K\equiv-6$ and one finds $T(s,s^{\prime})=T_{\text{sing}}(s-s^{\prime})$
where

$T_{\text{sing}}(s)=\lim_{\epsilon\to 0+}\frac{3}{2\pi^{2}(s-i\epsilon)^{4}}.$
(5.1)

Consequently $T_{\text{reg}}$ vanishes identically, and we may read off
immediately that $A=B=0$, reproducing QEI (1.4) by substituting into (3.19),
and in agreement with (1.9) and (1.10). Of course these results are easily
obtained by direct differentiation of the two-point function; our purpose here
is to show how they follow from formulae in Section 4.

Alternatively, we may proceed by taking the Fourier transform

$\hat{T}_{\text{sing}}(u)=u^{3}\Theta(u)/(2\pi),$ (5.2)

from which we obtain $Q(\alpha)=\alpha^{4}\Theta(\alpha)/(16\pi^{3})$ by
(1.3), leading to (1.4) as discussed in the introduction.

###  5.2 Hyperbolic subgroups: linear acceleration

We consider a uniformly linearly accelerated worldline

$\gamma(s)=(a^{-1}\sinh as,\ a^{-1}\cosh as,\ 0,\ 0),$ (5.3)

whose velocity evolves under the $1$-parameter group of $tx$-boosts
$\dot{\gamma}^{\mu}(s)=H^{\mu}_{\phantom{\mu}\nu}(s)\dot{\gamma}^{\nu}(0)$,
where

$H^{\bullet}_{\phantom{\bullet}\bullet}(s)=\begin{pmatrix}\cosh as&\sinh
as&0&0\\\ \sinh as&\cosh as&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}=\exp\begin{pmatrix}0&as&0&0\\\ as&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}$ (5.4)

and $0\neq a\in\mathbb{R}$ is fixed. Noting that the initial velocity and its
first two derivatives are $\dot{\gamma}(0)=\partial_{t}$,
$\ddot{\gamma}(0)=a\partial_{x}$, $\ddot{\gamma}(0)=a^{2}\partial_{t}$, we
obtain an adapted tetrad by choosing the tetrad
$\partial_{t},\partial_{x},\partial_{y},\partial_{z}$ at $s=0$, and applying
the prescription $e_{a}^{\mu}(s)=H^{\mu}_{\phantom{\mu}\nu}(s)e_{a}^{\nu}(0)$
to find

$e_{0}(s)=\cosh as\partial_{t}+\sinh as\partial_{x},\quad e_{1}(s)=\sinh
as\partial_{t}+\cosh as\partial_{x},\quad e_{2}(s)=\partial_{y},\quad
e_{3}(s)=\partial_{z}.$ (5.5)

Straightforward calculation, following the method of Section 4, gives

$K(a^{2}s^{2})=-\frac{3(as)^{4}}{8\sinh^{4}(as/2)}$ (5.6)

and hence

$T(s,s^{\prime})=\lim_{\epsilon\to
0+}\frac{3a^{4}(s-s^{\prime})^{4}\operatorname{cosech}^{4}(a(s-s^{\prime})/2)}{32\pi^{2}(s-s^{\prime}-i\epsilon)^{4}},$
(5.7)

which may be simplified to

$T(s,s^{\prime})=\lim_{\epsilon\to
0+}\dfrac{3a^{4}}{32\pi^{2}}\operatorname{cosech}^{4}\left(a(s-s^{\prime}-i\epsilon)/2\right).$
(5.8)

Here, we have used the general fact that $\lim_{\epsilon\to
0+}g(x)f(x-i\epsilon)=\lim_{\epsilon\to 0+}g(x-i\epsilon)f(x-i\epsilon)$ in
the sense of distributions, when $f$ is analytic in a strip
$Z=\\{x-iy:x\in\mathbb{R},~{}0<y<y_{0}\\}\subset\mathbb{C}$ with $\sup_{z\in
Z}|f(z)(\Im z)^{N}|<\infty$ for some $N>0$ and $g$ is analytic on $Z$ and
continuous on $Z\cup\mathbb{R}$.

As the function $K(z)$ evidently decays rapidly as $z\to\infty$ on the real
axis, the method of Section 4 allows us to read off the QEI from the
derivatives of $K(z)$ at $z=0$ according to (4.19). Using

$K(z)=\frac{3z^{2}}{8\sinh^{4}(\sqrt{z}/2)}=-6+z-\frac{11}{120}z^{2}+O(z^{3}),$
(5.9)

we find $A=a^{2}$ and $B=11a^{4}/30$, in agreement with (1.9) and (1.10) using
the invariants from Table 1 and reproducing the result (1.5) from [17]. In
that reference, the point-split energy density (5.8) was found by a direct
calculation. Writing $T(s,s^{\prime})=T(s-s^{\prime})$, the Fourier transform
yields

$\hat{T}(u)=\frac{u^{3}-a^{2}u}{2\pi(1-e^{-2\pi u/a})}$ (5.10)

and by using the last expression in (3), a calculation gives

$\displaystyle Q_{\text{even}}(\alpha)$
$\displaystyle=\dfrac{1}{32\pi^{3}}\left(\alpha^{4}+2a^{2}\alpha^{2}+\dfrac{11}{30}a^{4}\right),$
(5.11)

from which (1.5) follows on inserting the above expression into (3.12) and
using Parseval’s theorem.

###  5.3 Hyperbolic subgroups: the catenary

Now consider the catenary

$\gamma(s)=(a^{-1}\cosh\chi\,\sinh as,a^{-1}\cosh\chi\,\cosh
as,-s\sinh\chi,0),$ (5.12)

for constant $a\neq 0$, with initial velocity

$\dot{\gamma}^{\bullet}(0)=(\cosh\chi,0,-\sinh\chi,0),$ (5.13)

and second and third derivatives

$\ddot{\gamma}^{\bullet}(0)=(0,a\cosh\chi,0,0),\qquad\dddot{\gamma}^{\bullet}(0)=(a^{2}\cosh\chi,0,0,0).$
(5.14)

The velocity evolves under the hyperbolic subgroup (5.4). Writing
$C=\cosh{\chi}$ and $V=\sinh{\chi}$, the tetrad

$\displaystyle e_{0}^{\bullet}(s)$
$\displaystyle=(C\cosh{as},C\sinh{as},-V,0),$ $\displaystyle\
e_{1}^{\bullet}(s)$ $\displaystyle=(\sinh{as},\cosh{as},0,0),$ $\displaystyle
e_{2}^{\bullet}(s)$ $\displaystyle=(-V\cosh{as},-V\sinh{as},C,0),$
$\displaystyle\ e_{3}^{\bullet}(s)$ $\displaystyle=(0,0,0,1)$ (5.15)

is adapted to $\gamma$ with
$e_{a}^{\mu}(s)=H^{\mu}_{\phantom{\mu}\nu}(s)e_{a}^{\nu}(0)$. A calculation
results in the formula

$K(z)=-\frac{4V^{2}(\operatorname{sinhc}^{2}(r)+v^{2})\sinh^{2}(r)+2(4C^{2}-1)\operatorname{sinhc}^{2}(r)-16V^{2}\operatorname{sinhc}(2r)+2v^{2}(4C^{2}-3)}{C^{4}(\operatorname{sinhc}^{2}(r)-v^{2})^{3}}$
(5.16)

where $v=\tanh\chi$, $r=\sqrt{z}/(2\cosh\chi)$ and
$\operatorname{sinhc}(x)=\sinh(x)/x$ is the hyperbolic version of the
$\operatorname{sinc}$ function. Note that we need not specify a branch for the
square root as it always appears in the argument of an even entire function,
and also that $K(z)\to 0$ as $z\to\infty$ in $\mathbb{R}$. The series
expansion is

$K(z)=-6+\frac{2C^{2}-1}{C^{2}}z-\frac{185C^{4}-182C^{2}+30}{360C^{4}}z^{2}+O(z^{3})$
(5.17)

and as $\kappa=aC$ we may read off $A=a^{2}(2C^{2}-1)=a^{2}\cosh{2\chi}$ and
$B=(185C^{4}-182C^{2}+30)a^{4}/90$. It is straightforward that these values
agree with (1.9) and (1.10) using the curvature invariants for this case. In
particular, the resulting QEI is compatible with a constant negative energy
density of

$-T_{\text{reg}}(0)=-\frac{(185\cosh^{4}\chi-182\cosh^{2}\chi+30)a^{4}}{1440\pi^{2}}$
(5.18)

along the worldline (5.12). As would be expected, the QEI for linear
acceleration is obtained in the limit $\chi\to 0$, but for $\chi\neq 0$, we
have $-T_{\text{reg}}(0)<-11a^{4}/480\pi^{2}$, and the QEI bound is consistent
with a strictly more negative constant energy density than is the case for the
linearly accelerated worldline with the same value of $a$.

###  5.4 Parabolic subgroups: the semicubical parabola

We now consider the semicubical parabola

$\gamma(s)=\left(s+\dfrac{1}{6}a^{2}s^{3},\dfrac{1}{6}a^{2}s^{3},0,\dfrac{1}{2}as^{2}\right),$
(5.19)

for constant $a\neq 0$, whose velocity evolves as
$\dot{\gamma}^{\mu}(s)=P^{\mu}_{\phantom{\mu}\nu}(s)\dot{\gamma}(0)$ with
$\dot{\gamma}(0)=\partial_{t}$, where $P^{\mu}_{\phantom{\mu}\nu}$ was defined
in (2.14). From the initial derivatives $\dot{\gamma}(0)=\partial_{t}$,
$\ddot{\gamma}(0)=a\partial_{z}$,
$\dddot{\gamma}(0)=a^{2}(\partial_{t}+\partial_{x})$ one sees that the initial
tetrad $e_{0}(0)=\partial_{t}$, $e_{1}(0)=\partial_{z}$,
$e_{2}(0)=\partial_{x}$, $e_{3}(0)=\partial_{y}$ determines an adapted tetrad

$\displaystyle e_{0}^{\bullet}(s)$
$\displaystyle=\left(1+\tfrac{1}{2}(as)^{2},\tfrac{1}{2}(as)^{2},0,as\right),$
$\displaystyle e_{1}^{\bullet}(s)$ $\displaystyle=\left(as,as,0,1\right),$
$\displaystyle e_{2}^{\bullet}(s)$
$\displaystyle=\left(-\tfrac{1}{2}(as)^{2},1-\tfrac{1}{2}(as)^{2},0,-as\right),$
$\displaystyle e_{3}^{\bullet}(s)$ $\displaystyle=(0,0,1,0),$ (5.20)

at general proper time $s$ obeying
$e_{a}^{\mu}(s)=P^{\mu}_{\phantom{\mu}\nu}(s)e_{a}^{\nu}(0)$.

Straightforward calculation now gives

$K(z)=-\frac{6-z/2+5z^{2}/36}{(1+z/12)^{3}},$ (5.21)

with

$K(z)=-6+2z-\frac{37}{72}z^{2}+O(z^{3})$ (5.22)

as $z\to 0$ and $K(z)=O(z^{-1})$ for $z\to\infty$. Thus, the point-split
energy density is

$T(s,s^{\prime})=\lim_{\epsilon\to
0+}\frac{3-a^{2}(s-s^{\prime})^{2}/4+5a^{4}(s-s^{\prime})^{4}/72}{\pi^{2}(s-s^{\prime}-i\epsilon)^{4}(1+a^{2}(s-s^{\prime})^{2}/12)^{3}}$
(5.23)

and (4.19) gives $A=2a^{2}$ and $B=37a^{4}/18$, in agreement with (1.9) and
(1.10). Thus the QEI along a semicubical parabola is

$\int\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{1}{16\pi^{2}}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}\left(\lvert
g^{\prime\prime}(s)\rvert^{2}+4a^{2}\lvert
g^{\prime}(s)\rvert^{2}+\dfrac{37}{18}a^{4}\lvert g(s)\rvert^{2}\right),$
(5.24)

for any Hadamard state $\omega$. The long-time scaling limit of the above QEI
is then

$\liminf_{\lambda\xrightarrow{}\infty}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g_{\lambda}(s)|^{2}\langle\mathbf{:}T_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{37}{288\pi^{2}}a^{4},$
(5.25)

where as usual we choose $g$ with unit $L^{2}$-norm. The QEI is therefore
compatible with a constant negative energy density $-37a^{4}/(288\pi^{2})$
along the semicubical parabola. As one would expect, the QEI reduces to the
inertial case as $a\to 0$.

In fact the QEI (5.24) can also be obtained by a different method. Writing
$T(s,s^{\prime})=T(s-s^{\prime})$, the Fourier transform may be computed by
contour methods as

$\hat{T}(u)=\dfrac{1}{2\pi}\left[\left(\dfrac{2u^{2}}{\sqrt{12}}+\dfrac{7\lvert
u\rvert}{8}a+\dfrac{15}{8\sqrt{12}}a^{2}\right)ae^{-\lvert
u\rvert\sqrt{12}/a}+\left(u^{3}+2ua^{2}\right)\Theta(u)\right].$ (5.26)

The calculation is considerably simplified if one first replaces powers of
$s-s^{\prime}$ in (5.23) by powers of $s-s^{\prime}-i\epsilon$. To find
$Q_{\text{even}}(\alpha)$, we note that
$\hat{T}_{\text{odd}}(u)=(u^{3}+2ua^{2})/(4\pi)$, and also that the integral
of $\hat{T}$ over $(-\infty,0]$ may be evaluated in terms of
$\Gamma$-functions. After manipulation, the formula (3) gives

$\displaystyle Q_{\text{even}}(\alpha)$
$\displaystyle=\dfrac{1}{4\pi^{3}}\int_{-\infty}^{0}\left(\dfrac{2u^{2}}{\sqrt{12}}+\dfrac{7\lvert
u\rvert}{8}a+\dfrac{15}{8\sqrt{12}}a^{2}\right)ae^{-\lvert
u\rvert\sqrt{12}/a}\mathop{}\\!\mathrm{d}{u}+\dfrac{1}{8\pi^{3}}\int_{0}^{\alpha}\left(u^{3}+2ua^{2}\right)\mathop{}\\!\mathrm{d}{u}$
$\displaystyle=\dfrac{1}{32\pi^{3}}\alpha^{4}+\dfrac{a^{2}}{8\pi^{3}}\alpha^{2}+\dfrac{37a^{4}}{576\pi^{3}}.$
(5.27)

Inserting this expression in (3.12) and using Parseval’s theorem we reproduce
(5.24).

###  5.5 Elliptic subgroups: uniform rotation

Next, consider the uniformly rotating worldline

$\gamma(s)=\left(s\cosh\chi,0,r\cos{\omega s},r\sin\omega s\right),$ (5.28)

where the radius $r>0$ and proper angular velocity $\omega\neq 0$ together fix
the rapidity $\chi=\sinh^{-1}(r\omega)$. In this case, the velocity evolves
under rotations in the $yz$-plane as
$\dot{\gamma}^{\mu}(s)=R^{\mu}_{\phantom{\mu}\nu}(s)\dot{\gamma}^{\nu}(0)$,
where

$R^{\bullet}_{\phantom{\bullet}\bullet}(s)=\begin{pmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&\cos\omega s&-\sin\omega s\\\ 0&0&\sin\omega s&\cos\omega
s\end{pmatrix}=\exp\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&-\omega s\\\
0&0&\omega s&0\end{pmatrix}.$ (5.29)

Meanwhile, the initial velocity and its first two derivatives are

$\displaystyle\dot{\gamma}^{\bullet}(0)$ $\displaystyle=\left(C,0,0,V\right)$
$\displaystyle\ddot{\gamma}^{\bullet}(0)$
$\displaystyle=\left(0,0,-V\omega,0\right)$
$\displaystyle\dddot{\gamma}^{\bullet}(0)$
$\displaystyle=\left(0,0,0,-V\omega^{2}\right),$

where we have written $C=\cosh\chi$ and $V=r\omega=\sinh\chi$. Then
$e_{0}^{\bullet}(0)=(C,0,0,V)$, $e_{1}^{\bullet}(0)=(0,0,-1,0)$,
$e_{2}^{\bullet}(0)=(-V,0,0,-C)$, $e_{3}^{\bullet}(0)=(0,1,0,0)$, defines an
adapted tetrad at $s=0$, which can be extended along $\gamma$ by
$e_{a}^{\mu}(s)=R^{\mu}_{\phantom{\mu}\nu}(\omega s)e_{a}^{\nu}(0)$ to give

$\displaystyle e_{0}^{\bullet}(s)$ $\displaystyle=(C,0,-V\sin{\omega
s},V\cos{\omega s}),$ $\displaystyle e_{1}^{\bullet}(s)$
$\displaystyle=(0,0,-\cos{\omega s},-\sin{\omega s}),$ $\displaystyle
e_{2}^{\bullet}(s)$ $\displaystyle=(-V,0,C\sin{\omega s},-C\cos{\omega s}),$
$\displaystyle e_{3}^{\bullet}(s)$ $\displaystyle=(0,1,0,0).$ (5.30)

A calculation gives

$K(z)=\frac{4C^{2}\sin^{2}(\theta)(1+v^{2}\operatorname{sinc}^{2}(\theta))-2(4C^{2}-3)v^{2}\operatorname{sinc}^{2}(\theta)+16V^{2}\operatorname{sinc}(2\theta)+2(4C^{2}-1)}{C^{4}(1-v^{2}\operatorname{sinc}^{2}(\theta))^{3}},$
(5.31)

where $\theta=\sqrt{z}/(2\sinh(\chi))$, with series expansion

$K(z)=-6+\frac{2\cosh^{2}\chi-1}{\sinh^{2}\chi}z-\frac{185\cosh^{4}\chi-188\cosh^{2}\chi+33}{360\sinh^{4}\chi}z^{2}+O(z^{3}).$
(5.32)

As $\kappa=r\omega^{2}=\omega\sinh\chi$ we read off
$A=\omega^{2}\cosh(2\chi)=(2(r\omega)^{2}+1)\omega^{2}$ and

$B=\frac{\omega^{4}(185\cosh^{4}\chi-188\cosh^{2}\chi+33)}{90}=\frac{\omega^{4}(30+182(r\omega)^{2}+185(r\omega)^{4})}{90},$
(5.33)

which may be substituted into (1.8) to obtain the QEI in this case. In
particular, the QEI is compatible with a constant negative energy density of

$-T_{\text{reg}}(0)=-\frac{\omega^{4}(30+182(r\omega)^{2}+185(r\omega)^{4})}{1440\pi^{2}}$
(5.34)

along the worldline. While the point-split energy density may be written down
in terms of $K$, we do not know of any closed-form expression for its
transform. Thus the method of Sections 3 and 4 is the only available way to
compute this QEI.

Note that the QEI reduces to the inertial case if $\omega\to 0$ with $r$ fixed
– indeed, even if $r=o(\omega^{-2})$. One might initially be surprised that it
does not reduce in the same way when $r\to 0+$ with $\omega$ fixed. The
explanation is that the torsion of the curve does not vanish in this limit,
even though the curvature $\kappa$ does. This neatly illustrates the influence
of higher curvature invariants on the QEI bound.

###  5.6 Loxodromic subgroups

Finally, we study the loxodromic worldline

$\gamma^{\bullet}(s)=(Ca^{-1}\sinh(as),Ca^{-1}\cosh(as),V\omega^{-1}\cos(\omega
s),V\omega^{-1}\sin(\omega s)),$ (5.35)

where $C=\cosh\chi$, $V=\sinh\chi$ for fixed $\chi\neq 0$, $a\neq 0$ and
$\omega\neq 0$. This worldline undergoes both rotation in the $yz$-plane at
constant proper angular velocity $\omega$ and constant distance $|V/\omega|$
from the $x$-axis, while undergoing uniform acceleration in the $x$-direction.
The velocity evolves as
$\dot{\gamma}^{\mu}(s)=L_{a,\omega}{}^{\mu}_{\phantom{\mu}\nu}(s)\dot{\gamma}^{\nu}(0)$,
where

$L_{a,\omega}{}^{\bullet}_{\phantom{\bullet}\bullet}(s)=\begin{pmatrix}\cosh
as&\sinh as&0&0\\\ \sinh as&\cosh as&0&0\\\ 0&0&\cos\omega s&-\sin\omega s\\\
0&0&\sin\omega s&\cos\omega s\end{pmatrix}=\exp\begin{pmatrix}0&as&0&0\\\
as&0&0&0\\\ 0&0&0&-\omega s\\\ 0&0&\omega s&0\end{pmatrix}.$ (5.36)

It can be checked that

$\displaystyle e_{0}^{\bullet}(s)$
$\displaystyle=(C\cosh{as},C\sinh{as},-V\sin{\omega s},V\cos{\omega s}),$
$\displaystyle e_{1}^{\bullet}(s)$
$\displaystyle=(Ca\kappa^{-1}\sinh{as},Ca\kappa^{-1}\cosh{as},-V\omega\kappa^{-1}\cos{\omega
s},-V\omega\kappa^{-1}\sin{\omega s}),$ $\displaystyle e_{2}^{\bullet}(s)$
$\displaystyle=(-V\cosh{as},-V\sinh{as},C\sin{\omega s},-C\cos{\omega s}),$
$\displaystyle e_{3}^{\bullet}(s)$
$\displaystyle=(V\omega\kappa^{-1}\sinh{as},V\omega\kappa^{-1}\cosh{as},Ca\kappa^{-1}\cos{\omega
s},Ca\kappa^{-1}\sin{\omega s})$ (5.37)

defines an adapted tetrad for $\gamma$, obeying
$e_{a}^{\mu}(s)=L_{a,\omega}{}^{\mu}_{\phantom{\mu}\nu}(s)e_{a}^{\nu}(0)$,
while the calculation of $K$ by computer algebra produces

$\displaystyle K(z)$
$\displaystyle=\frac{1}{(C^{2}\operatorname{sinhc}^{2}(ar)-V^{2}\operatorname{sinc}^{2}(\omega
r))^{3}}\left(16C^{2}V^{2}\operatorname{sinc}(2\omega
r)\operatorname{sinhc}(2ar)\right.$
$\displaystyle\qquad\left.+4(C^{2}\sin^{2}(\omega
r)-V^{2}\sinh^{2}(ar))(V^{2}\operatorname{sinc}^{2}(\omega
r)+C^{2}\operatorname{sinhc}^{2}(ar))\right.$
$\displaystyle\qquad\qquad\left.-2V^{2}(C^{2}+3V^{2})\operatorname{sinc}^{2}(\omega
r)-2C^{2}(3C^{2}+V^{2})\operatorname{sinhc}^{2}(ar)\right),$ (5.38)

where $r=\sqrt{z}/(2\sqrt{C^{2}a^{2}+V^{2}\omega^{2}})$. For large real $z$,
it is easily seen that

$K(z)\sim-4V^{2}(ar)^{2}/(C^{4}\operatorname{sinhc}^{2}(ar))\to 0$ (5.39)

as $z\to\infty$ in $\mathbb{R}$. Meanwhile, the Taylor expansion about $z=0$
reads

$\displaystyle K(z)$
$\displaystyle=-6+\frac{(a^{2}+\omega^{2})(C^{2}+V^{2})}{C^{2}a^{2}+V^{2}\omega^{2}}z-\frac{z^{2}}{360(C^{2}a^{2}+V^{2}\omega^{2})^{2}}\left((33a^{4}+60a^{2}\omega^{2}+30\omega^{4})C^{4}\right.$
$\displaystyle\qquad+\left.(122a^{4}+250a^{2}\omega^{2}+122\omega^{4})(CV)^{2}+(33\omega^{4}+60a^{2}\omega^{2}+30a^{4})V^{4}\right)+O(z^{3})$

so $A=(C^{2}+V^{2})(a^{2}+\omega^{2})$, while $B$ is given by

$\displaystyle 90B$
$\displaystyle=(3a^{4}+30(a^{2}+\omega^{2})^{2})C^{4}+(3\omega^{4}+(30(a^{2}+\omega^{2})^{2})V^{4}+(122(a^{2}+\omega^{2})^{2}+6a^{2}\omega^{2})C^{2}V^{2}$
$\displaystyle=3(C^{2}a^{2}+V^{2}\omega^{2})^{2}+62(a^{2}+\omega^{2})^{2}(CV)^{2}+30(a^{2}+\omega^{2})^{2}(C^{2}+V^{2})^{2},$
(5.40)

in which the last term is $30A^{2}$. These values are easily expressed in
terms of curvature invariants. Using (2.19) and $C^{2}-V^{2}=1$ one has

$\kappa^{2}(\tau^{2}+\upsilon^{2})=(a^{2}+\omega^{2})^{2}(CV)^{2}+(a\omega)^{2}=(V^{2}a^{2}+C^{2}\omega^{2})(C^{2}a^{2}+V^{2}\omega^{2})=\kappa^{2}(V^{2}a^{2}+C^{2}\omega^{2}),$
(5.41)

from which the identity

$\kappa^{2}+\tau^{2}+\upsilon^{2}=(a^{2}+\omega^{2})(C^{2}+V^{2})=A$ (5.42)

follows directly, in agreement with (1.9). Using this in (5.40) together with
(2.19) we see that $B$ takes the form (1.10). We see that the QEI is
compatible with a constant negative energy density of $-T_{\text{reg}}(0)$
along the worldline (2.18), where

$T_{\text{reg}}(0)=\dfrac{185(a^{2}+\omega^{2})^{2}C^{4}-(182a^{4}+370a^{2}\omega^{2}+188\omega^{4})C^{2}+33\omega^{4}+60a^{2}\omega^{2}+30a^{4}}{1440\pi^{2}}$
(5.43)

and we have used $V^{2}=C^{2}-1$. Note that the QEI does not reduce to the
hyperbolic QEI in the limit $\chi\to 0$ with $a$ and $\omega$ fixed. This is
because the hypertorsion has a nonzero limit
$\upsilon\to\operatorname{sgn}(a)\omega$, even though the torsion vanishes and
the curvature tends to $a$. Nonetheless, it is easily seen from (5.40) that
$90B\geq 33a^{4}$ and hence that $-T_{\text{reg}}(0)<-11a^{4}/(480\pi^{2})$,
so that the QEI for loxodromic worldlines can be consistent with a more
negative constant value of the energy density than the linearly accelerated
worldline with the same value of $a$.

##  6 Summary and discussion

In this paper we have succeeded in giving an exact closed form expression
(1.8)–(1.10) for the QEI for the massless scalar field on any stationary
worldline in four-dimensional Minkowski spacetime. This was achieved by a
novel method that circumvented the need to take Fourier transforms of the
point-split energy density along the worldline, and which reduced the problem
to the computation of certain Taylor coefficients of functions determined by a
tetrad adapted to the worldline. In addition, we have given explicit
calculations for the six prototypical classes of stationary trajectory,
obtaining agreement with our general result (and also verifying a technical
condition needed for the general analysis). The resulting QEI bound depends
only on the curvature, torsion and hypertorsion of the worldline. We have also
conducted – in Appendix C – a parallel exercise for a quantum inequality on
the Wick square. A scaling analysis (see (1.11)) shows how these bounds take a
universal form on timescales short in relation to the curvature scales, from
which they then deviate at longer timescales. In the infinite time limit, they
would all allow the field to exhibit a constant negative energy density (or
zero in the inertial case).

Our results complement those of Kontou and Olum [37, 38], who computed an
absolute QEI [19] in an approximation of spacetimes where the curvature was
weak. There, the worldline was taken to be a geodesic. Our present results
indicate the corrections that should enter at leading order when that
assumption is dropped. (We reemphasise that our results are exact for massless
fields in Minkowski spacetime on stationary trajectories.)

To conclude, we first mention various potential extensions of our work and
then return to the question of whether the long-time limits of the QEI are
saturated by physical states of the field. Starting with extensions, we expect
that our general method would extend fairly directly to stationary worldlines
in any even-dimensional Minkowski spacetimes, leading to closed form results
in terms of the appropriate curvature invariants. In odd dimensions, the
vacuum two-point function involves noninteger powers of the geodesic
separation, which adds an extra complication. It would be interesting to
investigate this case in more detail. (For higher-dimensional treatment of the
Unruh detector response in higher dimensions, which would be related to the
Wick QI in these cases, see [33], and for specific calculations relating to
the detailed balance definition of Unruh temperature along stationary
worldlines in $4$-dimensions, see [29].) Next, massive fields typically have
QEI bounds that are exponentially suppressed relative to the massless ones.
Here, we do not expect that our method would easily produce closed-form
results, but again, it would be worth investigating, as would the situation
for higher spin fields.

Finally, we consider the extent to which the long term average bounds can be
attained. In the case of inertial worldlines this is obvious: the long-term
average value of zero is attained in the Minkowski vacuum state. For uniformly
accelerated curves it was noted in [17] that the bound (1.6) is attained by
the Rindler vacuum for the right wedge $x>|t|$ in Minkowski spacetime. It is
useful to put this in a broader context. Adopting coordinates
$t=\xi\sinh\chi$, $x=\xi\cosh\chi$, the Rindler wedge $x>|t|$ of Minkowski
spacetime has metric
$\xi^{2}\mathop{}\\!\mathrm{d}{\chi}^{2}-\mathop{}\\!\mathrm{d}{\xi}^{2}-\mathop{}\\!\mathrm{d}{y}^{2}-\mathop{}\\!\mathrm{d}{z}^{2}$,
and any curve $\chi\mapsto(a\chi,1/a,y_{0},z_{0})$ with $a>0$ is a curve of
proper acceleration $a$ in proper time parameterisation. Moreover, the energy
density measured by an observer moving on a curve of constant $\xi$, in the
thermal state of temperature $\beta^{-1}$ with respect to the coordinate
$\chi$, is

$\langle{:}T_{\mu\nu}u^{\mu}u^{\nu}{:}\rangle_{\beta}=\frac{(4\pi^{2}-\beta^{2})(33\beta^{2}+12\pi^{2})}{1440\pi^{2}\beta^{4}\xi^{4}},$
(6.1)

reducing to

$\langle{:}T_{\mu\nu}u^{\mu}u^{\nu}{:}\rangle_{\infty}=-\frac{11}{480\pi^{2}\xi^{4}}$
(6.2)

for the Rindler ground state. At $\beta=2\pi$, the thermal state on Rindler
spacetime is precisely the restriction of the Minkowski vacuum to the right
wedge, which is why the energy density vanishes. Because most references
(e.g., [7, 9, 4]) only discuss the conformally coupled stress-energy tensor
(the ‘new improved’ stress tensor) and [17] only considered the ground state
without giving details, the relevant calculations are briefly reviewed in
Appendix D. On restriction to the curve $\xi=1/a$ we see that all these states
have constant energy density consistent with (1.6) (see Fig. 1) and that this
bound is attained by the Rindler ground state.

$\beta$$\rho$$-11$$0$$2\pi$$4\pi$$6\pi$$8\pi$$10\pi$$12\pi$$14\pi$ Figure 1:
Plot of
$\rho=(480\pi^{2}\xi^{4})\langle{:}T_{\mu\nu}u^{\mu}u^{\nu}{:}\rangle_{\beta}$
on a curve of constant $\xi$, against $\beta$. The dotted line corresponds to
the QEI bound (1.6), which is attained as $\beta\to\infty$, corresponding to
the Rindler ground state.

One should note that the Rindler ground state (and indeed all the $\beta$-KMS
states other than the special case $\beta=2\pi$) is not defined on all of
Minkowski, but just on the wedge $x>|t|$. The obvious divergence of the
stress-energy tensor as $\xi\to 0+$ shows that the state cannot be extended as
a Hadamard state beyond the wedge. The reason they satisfy the Minkowski QEI
is because this QEI is local and covariant – see [17] for a discussion and
many similar calculations, and [13] for a more abstract viewpoint inspired by
[5]. Nonetheless, it remains open as to whether equality in (1.6) can be
attained by a Hadamard state defined on all of Minkowski; our conjecture is
that one can find global Hadamard states that approximate the Rindler ground
state sufficiently well that the bound (1.6) is satisfied in a limiting sense.
These issues will be addressed elsewhere.

Turning to the remaining stationary worldlines, the QEI is again consistent
with a constant strictly negative energy density and we can again ask whether
the bound is attained in any sense. Letaw and Pfautsch [42] considered the
problem of quantising the field in coordinates associated with the various
stationary worldlines and seeking an appropriate ground state. For the
inertial, uniformly rotating, and semicubical parabolic worldlines, they
concluded that the resulting state was precisely the Minkowski vacuum state.
This means that we have no obvious candidate state associated with the
uniformly rotating and semicubical parabolic worldlines with negative energy
density. On the other hand, the catenary (5.12) and loxodromic worldlines
(2.18) both result in a Rindler vacuum state on the $x>|t|$ wedge, which is
the causal hull of the worldline in question. One may compute the energy
density along these curves in the Rindler vacuum, using the renormalised
stress energy tensor given in Appendix D, yielding constant energy densities
$-(14\cosh^{2}\chi+19)a^{4}/(1440\pi^{2}\cosh^{4}\chi)$ in each case. This
value is strictly greater than $-11a^{4}/(480\pi^{2})$ for $\chi\neq 0$, which
is greater than the most negative constant energy density consistent with the
QEIs in these cases (see the remarks at the end of sections 5.3 and 5.6). Thus
they are are consistent with the QEIs but do not saturate them.

It therefore remains an open and intriguing question, whether there are
(sequences of) Hadamard states that attain these QEI bounds (in a limiting
sense). Resolving this question, and its analogues in 2+1 dimensions, may have
relevance to proposed experiments to detect the Unruh effect using a laser
beam whose intersection with a Bose-Einstein condensate follows a uniformly
rotating worldline [30].

_Acknowledgements_ CJF thanks Alexander Strohmaier and Valter Moretti for
useful conversations concerning the Hörmander pseudo-topologies, and Aron Wall
for posing an interesting direction for further study. The work of JT was in
part funded by an EPSRC studentship at the University of Sheffield and a
summer studentship from the University of York. We thank Elizabeth Winstanley
for a reading of the manuscript and some helpful suggestions.

##  Appendix A Details on the method

We give further details on the method described in Section 4 and prove the
Lemma stated there. Some aspects are treated using techniques of microlocal
analysis – we will be rather brief on those details, referring the reader to
appropriate literature, while indicating the structure of the argument.

To start, we observe that, for $\epsilon>0$, $F(\sigma_{\epsilon}(x,0))$ can
be written

$F(\sigma_{\epsilon}(x,0))=\int\frac{\mathrm{d}^{3}\bm{k}}{(2\pi)^{3}}\frac{e^{-\|\bm{k}\|\epsilon-
ik\cdot x}}{2\|\bm{k}\|},$ (A.1)

where $k_{\bullet}=(\|\bm{k}\|,\bm{k})$, $x^{\bullet}=(t,\bm{x})$. Thus for
any $\varphi\in C_{0}^{\infty}(\mathbb{R}^{4})$, the distribution
$u_{\epsilon}(x)=\varphi(x)F(\sigma_{\epsilon}(x,0))$ has Fourier transform

$\hat{u}_{\epsilon}(k^{\prime})=\int\frac{\mathrm{d}^{3}\bm{k}}{(2\pi)^{3}}\frac{e^{-\epsilon\|\bm{k}\|}}{2\|\bm{k}\|}\hat{\varphi}(k^{\prime}-k).$
(A.2)

As $\hat{\varphi}$ decays faster than inverse polynomials and
$k\in\mathcal{N}^{+}$, where $\mathcal{N}^{+/-}$ is the bundle of future/past-
pointing null covectors, it may be shown that $F(\sigma_{\epsilon}(x,0))$
converges in $\mathscr{D}^{\prime}_{\mathcal{N}^{+}}(\mathbb{R}^{4})$ with
respect to the Hörmander pseudo-topology [35]. It follows from this that the
vacuum $2$-point function $G_{0}(x,x^{\prime})$ is the limit of
$F(\sigma_{\epsilon}(x,x^{\prime}))=F(\sigma_{\epsilon}(x-x^{\prime},0))$ in
$\mathscr{D}^{\prime}_{\mathcal{N}^{+}\times\mathcal{N}^{-}}(\mathbb{R}^{4}\times\mathbb{R}^{4})$
and has wavefront set
$\operatorname{WF}(G_{0})\subset\mathcal{N}^{+}\times\mathcal{N}^{-}$, as is
also known on general grounds because the state is Hadamard [49].

These facts have various consequences. First, the pull-back of (any derivative
operator acting on) $G_{0}$ by
$\varphi:(s,s^{\prime})\mapsto(\gamma(s),\gamma(s^{\prime}))$ is well-defined
because the set of normals to $\varphi$ does not intersect
$\operatorname{WF}(G_{0})$, essentially because timelike and null vectors
cannot be orthogonal – see [11] for details. Consequently the pull-back is
well-defined by standard results explained in Chapter 8 of [35] and has
wavefront set contained in
$\varphi^{*}\operatorname{WF}(G_{0})\subset\varphi^{*}(\mathcal{N}^{+}\times\mathcal{N}^{-})=\Gamma\times(-\Gamma)$,
where $\Gamma=\mathbb{R}\times(0,\infty)\subset T^{*}\mathbb{R}$. Moreover,
$\varphi^{*}G_{0}$ is the limit in
$\mathscr{D}^{\prime}_{\Gamma\times(-\Gamma)}(\mathbb{R}\times\mathbb{R})$ of
$\varphi^{*}F\circ\sigma_{\epsilon}$ as $\epsilon\to 0+$, which justifies
taking the pull-back under the $\epsilon\to 0+$ limits in (4). Similar
arguments apply to the convergence of
$F^{\prime}(\sigma_{\epsilon}(x,x^{\prime}))$ and
$F^{\prime\prime}(\sigma_{\epsilon}(x,x^{\prime}))$ as $\epsilon\to 0+$.

Next, recall that the stationary worldline $\gamma$ has velocity
$u=\dot{\gamma}$ evolving according to $u(s)=\exp(sM)u(0)$, for
$M\in\mathfrak{so}(1,3)$ with dimensions of inverse time, and that the right-
handed tetrad $e_{a}(s)$ obeys $e_{a}(s)=\exp(sM)e_{a}(0)$, with
$u(s)=e_{0}(s)$, $\dot{u}(s)\in\operatorname{span}\\{e_{1}(s)\\}$, and
$\ddot{u}(s)\in\operatorname{span}\\{e_{0}(s),e_{1}(s),\linebreak[0]e_{2}(s)\\}$.
The Cartesian coordinates of $\gamma(s)$, and components of $e_{a}(s)$ are
evidently real analytic in $s$. We extend $e_{a}$ to a smooth tetrad in a
neighbourhood of $\gamma$ in an arbitrary fashion. Recall that the functions
$C_{a}$ and $D_{a}$ are defined, in index-free notation, by

$C_{a}(s,s^{\prime})=\eta(e_{a}(s),e_{a}(s^{\prime})),\qquad
D_{a}(s,s^{\prime})=\eta(\gamma(s)-\gamma(s^{\prime}),e_{a}(s)).$ (A.3)

We now prove the lemma needed in Section 4, which we restate for convenience.

######  Lemma.

(a) With the choice of tetrad just described, $C_{a}(s,s^{\prime})$ and
$D_{a}(s,s^{\prime})$ are translationally invariant, depending only on
$s-s^{\prime}$. There are entire analytic functions $G_{a}$ and $H_{a}$ such
that

$C_{a}(s,s^{\prime})=G_{a}(\kappa^{2}(s-s^{\prime})^{2}),\qquad
D_{a}(s,s^{\prime})D_{a}(s^{\prime},s)=-(s-s^{\prime})^{2}H_{a}(\kappa^{2}(s-s^{\prime})^{2}),$
(A.4)

where in the limit $z\to 0$,

$\sum_{a=0}^{3}G_{a}(z)=-2+\frac{\tau^{2}+\upsilon^{2}}{\kappa^{2}}z+\frac{(\kappa\tau)^{2}-(\tau^{2}+\upsilon^{2})^{2}}{\kappa^{4}}z^{2}+\mathcal{O}(z^{3}),$
(A.5)

and

$\sum_{a=0}^{3}H_{a}(z)=1+\frac{z}{12}+\frac{\kappa^{2}+19\tau^{2}}{360\kappa^{2}}z^{2}+\mathcal{O}(z^{3}).$
(A.6)

(b) The signed square geodesic separation of points along $\gamma$ obeys

$\sigma_{0}(\gamma(s),\gamma(s^{\prime}))=-(s-s^{\prime})^{2}\Upsilon(\kappa^{2}(s-s^{\prime})^{2}),$
(A.7)

where $\Upsilon$ is entire analytic with
$\Upsilon(z)=1+\tfrac{1}{12}z+\tfrac{1}{360}(1-\tau^{2}/\kappa^{2})z^{2}+\mathcal{O}(z^{3})$
as $z\to 0$. Furthermore, for $z\in[0,\infty)$, $\Upsilon(z)$ is real with
$\Upsilon(z)\geq 1$.

###### Proof.

(a) For inertial worldlines, $e_{a}(s)$ is constant and the result holds
trivially with $G_{0}(z)\equiv 1$, $G_{i}(z)\equiv 1$, $H_{0}(z)\equiv-1$,
$H_{i}(z)\equiv 0$. From now on we may assume that $\kappa$ is nonzero.

It follows from (4.1) that

$C_{a}(s,s^{\prime})=\eta(\exp(s^{\prime}M)e_{a}(0),\exp(sM)e_{a}(0))=\eta(e_{a}(0),\exp((s-s^{\prime})M)e_{a}(0)),$
(A.8)

so $C_{a}$ depends only on $s-s^{\prime}$. As every component of the matrix
$\exp(sM)$ is analytic, and because $C_{a}(s,s^{\prime})=C_{a}(s^{\prime},s)$,
we deduce that $C_{a}(s,s^{\prime})=G_{a}(\kappa^{2}(s-s^{\prime})^{2})$ for
dimensionless entire analytic functions $G_{a}$.

Next, observe that

$\frac{\partial}{\partial
s^{\prime}}D_{a}(s,s^{\prime})=-\eta(e_{0}(s^{\prime}),e_{a}(s))=-\eta(e_{0}(0),\exp((s-s^{\prime})M)e_{a}(0)).$
(A.9)

Integrating with respect to $s^{\prime}$ and using $D_{a}(s,s)=0$ we may
deduce that $\kappa D_{a}(s,s^{\prime})$ is a dimensionless entire analytic
function of $(s-s^{\prime})\kappa$. Again using $D_{a}(s,s)=0$ and because
(A.9) gives $\partial D_{0}/\partial s^{\prime}|_{s^{\prime}=s}=-1$ and
$\partial D_{i}/\partial s^{\prime}|_{s^{\prime}=s}=0$ for $i=1,2,3$, we have

$D_{0}(s,s^{\prime})=(s-s^{\prime})\left(1+\mathcal{O}((\kappa(s-s^{\prime}))^{2})\right),\qquad
D_{i}(s,s^{\prime})=\kappa^{-1}\mathcal{O}((\kappa(s-s^{\prime}))^{2}),$
(A.10)

where we have also used the fact that
$D_{0}(s,s^{\prime})=-D_{0}(s^{\prime},s)$ as a consequence of (A.9). Because
$D_{a}(s,s^{\prime})D_{a}(s^{\prime},s)$ is invariant under interchange of $s$
and $s^{\prime}$, we now have

$D_{a}(s,s^{\prime})D_{a}(s^{\prime},s)=-(s-s^{\prime})^{2}H_{a}(\kappa^{2}(s-s^{\prime})^{2})$
(A.11)

for dimensionless entire analytic functions $H_{a}$. The Taylor series of
$G_{a}$, $H_{a}$ and their sums, are computed up to second order in Appendix
B.

(b) Next, we study the geodesic separation between $\gamma(s)$ and
$\gamma(s^{\prime})$. We note that

$\frac{\partial}{\partial
s}\sigma_{0}(\gamma(s),\gamma(s^{\prime}))=-2D_{0}(s,s^{\prime})$ (A.12)

depends only on $s-s^{\prime}$, so
$\sigma_{0}(\gamma(s),\gamma(s^{\prime}))=\Sigma(s-s^{\prime})+f(s^{\prime})$
and on considering $s=s^{\prime}$ we find that $f$ is constant and may be
absorbed into $\Sigma$, which is also seen to be even. The first terms in its
Taylor expansion are easily found: $\Sigma(0)=0$, while

$\Sigma^{\prime\prime}(s-s^{\prime})=-2\eta(u(s),u(s^{\prime})),\qquad\Sigma^{(4)}(s-s^{\prime})=2\eta(\dot{u}(s),\dot{u}(s^{\prime})),\qquad\Sigma^{(6)}(s-s^{\prime})=-2\eta(\ddot{u}(s),\ddot{u}(s^{\prime}))$
(A.13)

giving

$\Sigma^{\prime\prime}(0)=-2,\qquad\Sigma^{(4)}(0)=-2\kappa^{2},\qquad\Sigma^{(6)}(0)=-2\kappa^{2}(\kappa^{2}-\tau^{2})$
(A.14)

using (2.9). Accordingly, we have established (A.7), the analyticity of
$\Upsilon$, and also the expansion

$\Upsilon(z)=1+\frac{z}{12}+\frac{\kappa^{2}-\tau^{2}}{360\kappa^{2}}z^{2}+\mathcal{O}(z^{3})$
(A.15)

as $z\to 0$. Finally, as $\gamma(0)$ and $\gamma(s)$ are connected by a smooth
timelike curve, the timelike geodesic that connects them maximises proper
time. Thus $-\sigma_{0}(\gamma(s),\gamma(0))\geq s^{2}$ for all
$s\in\mathbb{R}$ and consequently, $\Upsilon(z)\geq 1$ for $z\in[0,\infty)$,
which concludes the proof. ∎

Finally, we explain how the identity (4.13) may be proved. First note that

$\displaystyle\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime}))$
$\displaystyle=\sigma_{0}(\gamma(s),\gamma(s^{\prime}))+2i\epsilon(\gamma^{0}(s)-\gamma^{0}(s^{\prime}))+\epsilon^{2}$
$\displaystyle=-(s-s^{\prime})^{2}\Upsilon(\kappa^{2}(s-s^{\prime})^{2})+2i\epsilon(\gamma^{0}(s)-\gamma^{0}(s^{\prime}))+\epsilon^{2}$
$\displaystyle=-(s-s^{\prime}-i\epsilon)^{2}\Upsilon(\kappa^{2}(s-s^{\prime})^{2})+\epsilon\Psi(s,s^{\prime})+\epsilon^{2}\Xi(s,s^{\prime})$

for smooth (indeed analytic) functions $\Psi$ and $\Xi$. Let $S$ be the
difference between the distribution on the left-hand side of (4.13) and the
distribution on the right-hand side. Then, using the fact that $\Upsilon$ is
nonvanishing on the real axis, $S$ takes the form

$S(s,s^{\prime})=\lim_{\epsilon\to
0+}\sum_{r=1}^{2k}\frac{\epsilon^{r}S_{r}(s,s^{\prime})}{\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime}))^{k}(s-s^{\prime}-i\epsilon)^{2k}}$
(A.16)

for smooth functions $S_{r}\in C^{\infty}(\mathbb{R}^{2})$ ($1\leq r\leq 2k$).
All that is needed now is to show that the distributional limit

$\lim_{\epsilon\to
0+}\frac{1}{\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime}))^{k}(s-s^{\prime}-i\epsilon)^{2k}}$
(A.17)

exists, whereupon $S$ must vanish due to the strictly positive powers of
$\epsilon$ in (A.16). The required result now follows from the sequential
continuity of the distributional product with respect to the Hörmander pseudo-
topology (Theorem 2.5.10 in [34]), and the fact that both
$1/\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime}))$ and

$\frac{1}{s-s^{\prime}-i\epsilon}=i\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{k}\,e^{-ik(s-s^{\prime}-i\epsilon)}$
(A.18)

have limits as $\epsilon\to 0+$ in
$\mathscr{D}_{\Gamma\times(-\Gamma)}^{\prime}(\mathbb{R}^{2})$, where, as
before, $\Gamma=\mathbb{R}\times(0,\infty)\subset\dot{T}^{*}\mathbb{R}$.

##  Appendix B Taylor series calculation

We compute the Taylor series of both $G_{a}$ and $H_{a}$ up to second order,
using equations (4.7), (4.8) and (4.18). Recalling that
$C_{a}(s,s^{\prime})=G_{a}(\kappa^{2}(s-s^{\prime})^{2})$, one can expand the
right hand side into a Taylor series in $s-s^{\prime}$ about the point
$s-s^{\prime}=0$ and then differentiate to yield

$\displaystyle-\frac{1}{2\kappa^{2}}\frac{\partial^{2}C_{a}}{\partial
s\partial s^{\prime}}$
$\displaystyle=G^{\prime}_{a}(0)+3\kappa^{2}(s-s^{\prime})^{2}G_{a}^{\prime\prime}(0)+\mathcal{O}((s-s^{\prime})^{4})$
(B.1)
$\displaystyle\frac{1}{12\kappa^{4}}\frac{\partial^{4}C_{a}}{\partial^{2}s\partial^{2}s^{\prime}}$
$\displaystyle=G_{a}^{\prime\prime}(0)+\mathcal{O}((s-s^{\prime})^{2})$ (B.2)

as $s-s^{\prime}\to 0$. Differentiating equation (4.7) and setting
$s=s^{\prime}=0$, one easily finds

$G^{\prime}_{a}(0)=-\frac{\eta(\dot{e}_{a}(0),\dot{e}_{a}(0))}{2\kappa^{2}},\qquad
G^{\prime\prime}_{a}(0)=\frac{\eta(\ddot{e}_{a}(0),\ddot{e}_{a}(0))}{12\kappa^{4}}$
(B.3)

by equating powers of $s-s^{\prime}$. The derivatives of the $e_{a}$ can be
read off from the generalized Frenet-Serret equations (2.5) and its
derivatives (2.8), allowing us to express $G_{a}^{\prime}(0)$ and
$G_{a}^{\prime\prime}(0)$ in terms of curvature invariants.

An easy computation shows that

$G_{a}^{\prime}(0)=\dfrac{1}{2}\eta_{0a}+\dfrac{\kappa^{2}-\tau^{2}}{2\kappa^{2}}\eta_{1a}-\dfrac{\tau^{2}+\upsilon^{2}}{2\kappa^{2}}\eta_{2a}-\dfrac{\upsilon^{2}}{2\kappa^{2}}\eta_{3a}$
(B.4)

and

$G_{a}^{\prime\prime}(0)=\dfrac{\kappa^{2}-\tau^{2}}{12\kappa^{2}}\eta_{0a}+\dfrac{\tau^{2}\upsilon^{2}+(\kappa^{2}-\tau^{2})^{2}}{12\kappa^{4}}\eta_{1a}-\dfrac{\kappa^{2}\tau^{2}-(\tau^{2}+\upsilon^{2})^{2}}{12\kappa^{4}}\eta_{2a}+\upsilon^{2}\dfrac{\tau^{2}+\upsilon^{2}}{12\kappa^{4}}\eta_{3a},$
(B.5)

where $\eta(e_{a}(0),e_{b}(0))=\eta_{ab}$ by orthogonality of the tetrad
field. Reconstructing $G_{a}$ using a Taylor series therefore yields

$\displaystyle G_{a}(z)$
$\displaystyle=\eta_{aa}+\dfrac{1}{2\kappa^{2}}z\left(\eta_{0a}\kappa^{2}-\eta_{1a}(\tau^{2}-\kappa^{2})-\eta_{2a}(\upsilon^{2}+\tau^{2})-\eta_{3a}\upsilon^{2}\right)$
$\displaystyle\quad+\dfrac{z^{2}}{24\kappa^{4}}\left(\eta_{0a}\kappa^{2}(\kappa^{2}-\tau^{2})+\eta_{1a}(\tau^{2}\upsilon^{2}+(\kappa^{2}-\tau^{2})^{2})-\eta_{2a}(\kappa^{2}\tau^{2}-(\tau^{2}+\upsilon^{2})^{2})+\eta_{3a}\upsilon^{2}(\tau^{2}+\upsilon^{2})\right)$
$\displaystyle\quad+\mathcal{O}(z^{3}).$ (B.6)

Summing, we obtain

$\sum_{a=0}^{3}G_{a}(z)=-2+\frac{\tau^{2}+\upsilon^{2}}{\kappa^{2}}z+\frac{(\kappa\tau)^{2}-(\tau^{2}+\upsilon^{2})^{2}}{\kappa^{4}}z^{2}+\mathcal{O}(z^{3})$
(B.7)

as $z\to 0$.

Applying exactly the same methodology to $H_{a}$, one writes
$E_{a}(s,s^{\prime})=D_{a}(s,s^{\prime})D_{a}(s^{\prime},s)$ so that

$\displaystyle E_{a}(s,s^{\prime})$
$\displaystyle=-(s-s^{\prime})^{2}H_{a}(\kappa^{2}(s-s^{\prime})^{2})$
$\displaystyle=-(s-s^{\prime})^{2}H_{a}(0)-\kappa^{2}(s-s^{\prime})^{4}H_{a}^{\prime}(0)-\tfrac{1}{2}\kappa^{4}(s-s^{\prime})^{6}H_{a}^{\prime\prime}(0)+O((s-s^{\prime})^{8}).$
(B.8)

Differentiation yields

$\displaystyle\dfrac{\partial^{2}E_{a}}{\partial s\partial s^{\prime}}$
$\displaystyle=2H_{a}(0)+12\kappa^{2}(s-s^{\prime})^{2}H_{a}^{\prime}(0)+15\kappa^{4}(s-s^{\prime})^{4}H_{a}^{\prime\prime}(0)+\mathcal{O}((s-s^{\prime})^{6})$
(B.9)
$\displaystyle\dfrac{\partial^{4}E_{a}}{\partial^{2}s\partial^{2}s^{\prime}}$
$\displaystyle=-24\kappa^{2}H_{a}^{\prime}(0)-180\kappa^{4}(s-s^{\prime})^{2}H_{a}^{\prime\prime}(0)+\mathcal{O}((s-s^{\prime})^{4})$
(B.10)
$\displaystyle\dfrac{\partial^{6}E_{a}}{\partial^{3}s\partial^{3}s^{\prime}}$
$\displaystyle=360\kappa^{4}H_{a}^{\prime\prime}(0)+\mathcal{O}((s-s^{\prime})^{4}),$
(B.11)

from which $H_{a}(0)$, $H_{a}^{\prime}(0)$ and $H_{a}^{\prime\prime}(0)$ can
be obtained differentiating equation (4.18) using Leibniz’ rule and
subsequently setting $s=s^{\prime}=0$. It is easily verifiable that this
yields

$\displaystyle H_{a}(0)$
$\displaystyle=\left[\eta(\dot{\gamma}(0),e_{a}(0))\right]^{2}=\left[\eta(e_{0}(0),e_{a}(0))\right]^{2},$
(B.12) $\displaystyle H_{a}^{\prime}(0)$
$\displaystyle=-\dfrac{1}{4\kappa^{2}}[\eta(\ddot{\gamma}(0),e_{a}(0))]^{2}+\dfrac{1}{3\kappa^{2}}\eta(\dot{\gamma}(0),e_{a}(0))\eta(\dddot{\gamma}(0),e_{a}(0)),$
(B.13) $\displaystyle H_{a}^{\prime\prime}(0)$
$\displaystyle=\dfrac{1}{18\kappa^{4}}[\eta(\dddot{\gamma}(0),e_{a}(0))]^{2}-\dfrac{1}{12\kappa^{4}}\eta(\ddot{\gamma}(0),e_{a}(0))\eta(\gamma^{(4)}(0),e_{a}(0))$
$\displaystyle\qquad+\dfrac{1}{30\kappa^{4}}\eta(\dot{\gamma}(0),e_{a}(0))\eta(\gamma^{(5)}(0),e_{a}(0)),$
(B.14)

and after some straightforward computation,

$\displaystyle H_{a}(0)$ $\displaystyle=\eta_{0a}$ (B.15) $\displaystyle
H_{a}^{\prime}(0)$ $\displaystyle=\dfrac{1}{3}\eta_{0a}+\dfrac{1}{4}\eta_{1a}$
(B.16) $\displaystyle H_{a}^{\prime\prime}(0)$
$\displaystyle=(\eta_{0a})^{2}\left(\dfrac{1}{18}+\dfrac{\kappa^{2}-\tau^{2}}{30\kappa^{2}}\right)-\dfrac{\kappa^{2}-\tau^{2}}{12\kappa^{2}}(\eta_{1a})^{2}+\dfrac{\tau^{2}}{18\kappa^{2}}(\eta_{2a})^{2}$
$\displaystyle=\eta_{0a}\left(\dfrac{1}{18}+\dfrac{\kappa^{2}-\tau^{2}}{30\kappa^{2}}\right)+\dfrac{\kappa^{2}-\tau^{2}}{12\kappa^{2}}\eta_{1a}-\dfrac{\tau^{2}}{18\kappa^{2}}\eta_{2a}$
(B.17)

using the fact that $(\eta_{0a})^{2}=\eta_{0a}$ and
$(\eta_{ia})^{2}=-\eta_{ia}$ for $i=1,2,3$, as can be explicitly seen in the
calculation of $H_{a}^{\prime\prime}(0)$. Reconstructing $H_{a}$ using a
Taylor series, one obtains

$H_{a}(z)=\eta_{0a}+\dfrac{1}{12}z\left(4\eta_{0a}+3\eta_{1a}\right)\\\
+\dfrac{1}{360\kappa^{2}}z^{2}\left(\eta_{0a}(10\kappa^{2}+6(\kappa^{2}-\tau^{2}))+15\eta_{1a}(\kappa^{2}-\tau^{2})-10\eta_{2a}\tau^{2}\right)+\mathcal{O}(z^{3}),$
(B.18)

and summing,

$\sum_{a=0}^{3}H_{a}(z)=1+\frac{z}{12}+\frac{\kappa^{2}+19\tau^{2}}{360\kappa^{2}}z^{2}+\mathcal{O}(z^{3}).$
(B.19)

##  Appendix C Wick square

In this Appendix we show how a quantum inequality for the Wick square can be
obtained along stationary trajectories. This is a simpler calculation than the
one used for the energy density and we shall be relatively brief.

Recall that the general QEI involves a (sum of) pull-backs of a suitable
differential operator acting on the two-point function,

$T(s,s^{\prime})=\langle\mathcal{Q}\phi(\gamma(s))\mathcal{Q}\phi(\gamma(s^{\prime}))\rangle_{\omega_{0}}=((\mathcal{Q}\otimes\mathcal{Q})G_{0})(\gamma(s),\gamma(s^{\prime})).$
(C.1)

For a quantum inequality on the Wick square, the operator $\mathcal{Q}$ can be
simply identified as the identity, so $T(s,s^{\prime})$ can be written in this
case as

$T(s,s^{\prime})=G_{0}(\gamma(s),\gamma(s^{\prime})).$ (C.2)

Using the results of Section 4 and in particular, equation (4.13), the two-
point function can be neatly expressed as

$T(s,s^{\prime})=\lim_{\epsilon\to
0+}\dfrac{1}{4\pi^{2}\sigma_{\epsilon}(\gamma(s),\gamma(s^{\prime}))}=-\lim_{\epsilon\to
0+}\dfrac{1}{4\pi^{2}(s-s^{\prime}-i\epsilon)^{2}}\left[\Upsilon\left(\kappa^{2}(s-s^{\prime})^{2}\right)\right]^{-1}.$
(C.3)

As $\Upsilon(\kappa^{2}s^{2})\geq 1$ for $s\in\mathbb{R}$ by the Lemma, the
entire function $\Upsilon(z)$ is nonvanishing on the real axis, and
$\Upsilon(z)^{-1}$ is therefore analytic in a neighbourhood of the real axis.
Using (4.12) we may write $\Upsilon(z)^{-1}=1+zJ(z)$, where $J$ is also
analytic in a neighbourhood of the real axis, with $J(0)=-1/12$. Because
$0<1+zJ(z)\leq 1$ for $z\geq 0$, we may deduce that $0\leq-J(z)<1/z$ for
$z>0$.

We can now split the pulled back two-point function into its singular and
regular parts as
$T(s,s^{\prime})=T_{\text{sing}}(s-s^{\prime})+T_{\text{reg}}(s-s^{\prime})$,
where

$T_{\text{sing}}(s)=-\dfrac{1}{4\pi^{2}}\lim_{\epsilon\to
0+}\dfrac{1}{(s-i\epsilon)^{2}},$ (C.4)

and

$T_{\text{reg}}(s)=-\dfrac{J(\kappa^{2}s^{2})}{4\pi^{2}}\lim_{\epsilon\to
0+}\dfrac{\kappa^{2}s^{2}}{(s-i\epsilon)^{2}}=-\dfrac{\kappa^{2}J(\kappa^{2}s^{2})}{4\pi^{2}},$
(C.5)

with $T_{\text{reg}}(0)=\kappa^{2}/(48\pi^{2})$. Here we used the identity
$\lim_{\epsilon\to 0+}x^{2}/(x-i\epsilon)^{2}=\lim_{\epsilon\to
0+}(x-i\epsilon)^{2}/(x-i\epsilon)^{2}=1$ of distributional limits, because
$g(z)=z^{2}$ is entire, while $f(z)=z^{-2}$ is analytic in the open lower
half-plane $Z\subset\mathbb{C}$ and obeys $\sup_{z\in Z}\lvert f(z)(\Im
z)^{2}\rvert=1$ (see the argument below equation (5.8)).

Observing that the two-point function given above is translationally
invariant, we can use the bound given by (1.2) and (1.3) and thus write

$\int\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}(\mathcal{Q}\phi)^{2}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{\alpha}\lvert\hat{g}(\alpha)\rvert^{2}Q_{\text{even}}(\alpha)$
(C.6)

where

$Q_{\text{even}}(\alpha)=\dfrac{1}{2\pi^{2}}\left[\int_{-\infty}^{0}\hat{T}(u)\mathop{}\\!\mathrm{d}{u}+\int_{0}^{\alpha}\hat{T}_{\text{odd}}(u)\mathop{}\\!\mathrm{d}{u}\right].$
(C.7)

The Fourier transform of $T_{\text{sing}}$ is easily shown to be
$\hat{T}_{\text{sing}}(u)=\tfrac{u}{2\pi}\Theta(u)$. Again, $T_{\text{reg}}$
is smooth, real and even on $\mathbb{R}$, decaying like $\mathcal{O}(s^{-2})$
as $\lvert s\rvert\to\infty$ because of the decay of $J$. Evidently
$T_{\text{reg}}$ does not contribute to $\hat{T}_{\text{odd}}$ as
$T_{\text{reg}}$ is absolutely integrable and has a well defined, continuous,
real and even Fourier transform. In this case, $T_{\text{sing}}$ is actually
universal; the information relating to the specific worldline is encoded in
$T_{\text{reg}}$, as can also be seen below in Eq. (C.10). Clearly,
$\hat{T}_{\text{sing}}$ does not contribute to the first term in (C.7) and,
recalling that $T_{\text{reg}}$ is even, the odd part of $\hat{T}$ is

$\hat{T}_{\text{odd}}(u)=\dfrac{u}{4\pi},$ (C.8)

and so $Q_{\text{even}}$ is given in the form

$\displaystyle Q_{\text{even}}(\alpha)$
$\displaystyle=\dfrac{1}{2\pi^{2}}\left[\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}{u}\,\hat{T}_{\text{reg}}(u)+\dfrac{1}{4\pi}\int_{0}^{\alpha}\mathop{}\\!\mathrm{d}{u}\
u\right]=\dfrac{1}{16\pi^{3}}\alpha^{2}+\dfrac{T_{\text{reg}}(0)}{2\pi}.$
(C.9)

In direct analogy to the analysis of the energy density, the evenness of
$T_{\text{reg}}$ and the Fourier inversion formula have been used. Inserting
this into (C.6) gives the QI bound

$\int\mathop{}\\!\mathrm{d}{s}|g(s)|^{2}\langle\mathbf{:}\phi^{2}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{1}{8\pi^{2}}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}\left(\lvert
g^{\prime}(s)\rvert^{2}+C\lvert g(s)\rvert^{2}\right).$ (C.10)

where $C=8\pi^{2}T_{\text{reg}}(0)=\kappa^{2}/6$.

Considering the scaling behaviour, using the same test function
$g_{\lambda}(s)=\lambda^{-1/2}g(\lambda/s)$ as in the case for the QEI (1.11),
one can easily verify that

$\int\mathop{}\\!\mathrm{d}{s}|g_{\lambda}(s)|^{2}\langle\mathbf{:}\phi^{2}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{\|g^{\prime}\|^{2}}{8\pi^{2}\lambda^{2}}-\dfrac{\kappa^{2}\|g\|^{2}}{48\pi^{2}},$
(C.11)

where again $\|g\|^{2}$ denotes the $L^{2}$-norm of the function $g$. Taking
the limit $\lambda\to\infty$ yields the following formula,

$\liminf_{\lambda\xrightarrow{}\infty}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}{s}|g_{\lambda}(s)|^{2}\langle\mathbf{:}\phi^{2}\mathbf{:}\rangle_{\omega}(\gamma(s))\geq-\dfrac{\kappa^{2}}{48\pi^{2}}$
(C.12)

when considering the functions $g$ such that $\|g\|^{2}=1$. Physically, since
one can interpret $12\langle\mathbf{:}\phi^{2}\mathbf{:}\rangle$ as the square
of a local temperature [6], states with negative expected Wick square are
regarded as being locally out of equilibrium. The above bound therefore
quantifies the extent to which the thermal interpretation may fail uniformly
along these worldlines, in terms of their proper acceleration. This raises an
intriguing question as to whether there are states that would saturate this
bound – something quite relevant to the Unruh experiments discussed in Section
6.

In relation to the Unruh effect, a study of the detailed balance temperature
obtained from the excitation of an Unruh-DeWitt detector carried along
stationary worldlines can be found in [29]. Here the quantum field is assumed
to be in the vacuum state, and the temperature depends not only on the
curvature invariants but also on the energy gap of the detector. Although this
is a different focus from our results, which concern averages of the Wick
square in arbitrary Hadamard states, there are technical similarities, because
the pulled back vacuum Wightman function plays a key role in both. It would be
interesting to understand whether some of the methods described here can be
used to corroborate the numerical results of [29].

##  Appendix D Computation of the renormalised stress-tensor for thermal and
ground states on Rindler spacetime

The Feynman propagator for a thermal state at inverse temperature $\beta$ of
the massless scalar field in Minkowski spacetime was given by Dowker [9] and
the Wightman functions (including for higher spin) can be found in [45].
Adopting coordinates $t=\xi\sinh\chi$, $x=\xi\cosh\chi$, the Rindler wedge
$x>|t|$ of Minkowski spacetime has metric
$\xi^{2}\mathop{}\\!\mathrm{d}{\chi}^{2}-\mathop{}\\!\mathrm{d}{\xi}^{2}-\mathop{}\\!\mathrm{d}{y}^{2}-\mathop{}\\!\mathrm{d}{z}^{2}$,
and any curve $\chi\mapsto(a\chi,1/a,y_{0},z_{0})$ with $a>0$ is a curve of
proper acceleration $a$ in proper time parameterisation. Given two points
$x=(\chi,\xi,y,z)$ and
$x^{\prime}=(\chi^{\prime},\xi^{\prime},y^{\prime},z^{\prime})$, write

$\alpha(x,x^{\prime})=\cosh^{-1}\left(\frac{\xi^{2}+(\xi^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}{2\xi\xi^{\prime}}\right),$
(D.1)

whereupon the Wightman function
$G_{\beta}(x,x^{\prime})=\langle\phi(x)\phi(x^{\prime})\rangle_{\beta}$ for
the temperature $\beta^{-1}$ KMS state with respect to the coordinate $\chi$
is

$G_{\beta}(x,x^{\prime})=\frac{1}{4\pi\beta\xi\xi^{\prime}\sinh\alpha(x,x^{\prime})}\left(\frac{\sinh(2\pi\alpha(x,x^{\prime})/\beta)}{\cosh(2\pi\alpha(x,x^{\prime})/\beta)-\cosh(2\pi(\chi-\chi^{\prime}-i\epsilon)/\beta)}\right).$
(D.2)

The $\beta=2\pi$ case coincides with the restriction of the Minkowski vacuum
state to the wedge, while the zero temperature limit has Wightman function

$G_{\infty}(x,x^{\prime})=-\frac{\alpha(x,x^{\prime})}{4\pi^{2}\xi\xi^{\prime}\sinh\alpha(x,x^{\prime})(\alpha(x,x^{\prime})^{2}-(\chi-\chi^{\prime}-i\epsilon)^{2})}.$
(D.3)

To obtain the renormalised (minimally coupled) stress-energy tensor, we first
apply suitable derivatives to $G_{\beta}-G_{2\pi}$ and take the limit
$x^{\prime}\to x$, obtaining

$\langle{:}(\nabla_{\mu}\phi)(x)(\nabla_{\nu}\phi)(x){:}\rangle_{\beta}=\frac{4\pi^{2}-\beta^{2}}{1440\pi^{2}\beta^{4}\xi^{4}}\left((16\pi^{2}+14\beta^{2})\hat{u}_{\mu}\hat{u}_{\nu}+30\beta^{2}\hat{a}_{\mu}\hat{a}_{\nu}-(4\pi^{2}+11\beta^{2})\eta_{\mu\nu}\right),$
(D.4)

where, at spacetime position $x$,
$\hat{u}^{\mu}=\xi^{-1}(\partial_{\chi})^{\mu}$ is the $4$-velocity of the
curve through $x$ with constant $\xi$, $y$ and $z$, and
$\hat{a}^{\mu}=(\partial_{\xi})^{\mu}$ is the unit spacelike vector parallel
to the $4$-acceleration of this curve. Consequently,

$\langle{:}T_{\mu\nu}{:}\rangle_{\beta}=\frac{4\pi^{2}-\beta^{2}}{1440\pi^{2}\beta^{4}\xi^{4}}\left((16\pi^{2}+14\beta^{2})\hat{u}_{\mu}\hat{u}_{\nu}+30\beta^{2}\hat{a}_{\mu}\hat{a}_{\nu}-(4\pi^{2}-19\beta^{2})\eta_{\mu\nu}\right)$
(D.5)

and the result for Rindler ground state is obtained by taking
$\beta\to\infty$, giving

$\langle{:}T_{\mu\nu}{:}\rangle_{\infty}=-\frac{1}{1440\pi^{2}\xi^{4}}\left(14\hat{u}_{\mu}\hat{u}_{\nu}+30\hat{a}_{\mu}\hat{a}_{\nu}+19\eta_{\mu\nu}\right).$
(D.6)

Computing the energy density on curves of constant $\xi$ yields (6.1).

## Bibliography

  * [1] Alcubierre, M.: The warp drive: hyper-fast travel within general relativity.  Classical and Quantum Gravity 11(5), L73–L77 (1994).  DOI 10.1088/0264-9381/11/5/001.  URL https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001
  * [2] Bell, J., Leinaas, J.: Electrons as accelerated thermometers.  Nuclear Physics B 212(1), 131–150 (1983).  DOI https://doi.org/10.1016/0550-3213(83)90601-6.  URL https://www.sciencedirect.com/science/article/pii/0550321383906016
  * [3] Born, M.: Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips.  Annalen der Physik 335(11), 1–56 (1909).  DOI 10.1002/andp.19093351102.  URL https://doi.org/10.1002/andp.19093351102
  * [4] Brown, M.R., Ottewill, A.C., Page, D.N.: Conformally invariant quantum field theory in static Einstein space-times.  Phys. Rev. D (3) 33(10), 2840–2850 (1986).  DOI 10.1103/PhysRevD.33.2840.  URL https://doi.org/10.1103/PhysRevD.33.2840
  * [5] Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: A new paradigm for local quantum physics.  Commun. Math. Phys. 237, 31–68 (2003).  DOI 10.1007/s00220-003-0815-7.  URL https://doi.org/10.1007/s00220-003-0815-7
  * [6] Buchholz, D., Schlemmer, J.: Local temperature in curved spacetime.  Classical Quantum Gravity 24(7), F25–F31 (2007).  DOI 10.1088/0264-9381/24/7/F01.  URL https://doi.org/10.1088/0264-9381/24/7/F01
  * [7] Candelas, P., Deutsch, D.: On the vacuum stress induced by uniform acceleration or supporting the ether.  Proc. Roy. Soc. London Ser. A 354(1676), 79–99 (1977).  DOI 10.1098/rspa.1977.0057.  URL https://doi.org/10.1098/rspa.1977.0057
  * [8] De Bièvre, S., Merkli, M.: The Unruh effect revisited.  Classical Quantum Gravity 23(22), 6525–6541 (2006).  DOI 10.1088/0264-9381/23/22/026.  URL https://doi.org/10.1088/0264-9381/23/22/026
  * [9] Dowker, J.S.: Thermal properties of Green’s functions in Rindler, de Sitter, and Schwarzschild spaces.  Phys. Rev. D 18, 1856–1860 (1978).  DOI 10.1103/PhysRevD.18.1856.  URL https://link.aps.org/doi/10.1103/PhysRevD.18.1856
  * [10] Epstein, H., Glaser, V., Jaffe, A.: Nonpositivity of the energy density in quantized field theories.  Nuovo Cimento (10) 36, 1016–1022 (1965).  DOI 10.1007/BF02749799.  URL https://doi.org/10.1007/BF02749799
  * [11] Fewster, C.J.: A general worldline quantum inequality.  Classical and Quantum Gravity 17(9), 1897–1911 (2000).  DOI 10.1088/0264-9381/17/9/302.  URL http://dx.doi.org/10.1088/0264-9381/17/9/302
  * [12] Fewster, C.J.: Energy inequalities in quantum field theory.  In: J.C. Zambrini (ed.) XIVth International Congress on Mathematical Physics. World Scientific, Singapore (2005).  DOI 10.48550/arXiv.math-ph/0501073.  URL https://doi.org/10.48550/arXiv.math-ph/0501073.  An expanded and updated version is available as math-ph/0501073
  * [13] Fewster, C.J.: Quantum energy inequalities and local covariance. II. Categorical formulation.  Gen. Relativity Gravitation 39, 1855–1890 (2007).  DOI 10.1007/s10714-007-0494-3.  URL https://doi.org/10.1007/s10714-007-0494-3
  * [14] Fewster, C.J.: Quantum energy inequalities.  In: F.S.N. Lobo (ed.) Wormholes, Warp Drives and Energy Conditions, _Fundamental Theories of Physics_ , vol. 189, pp. 215–254. Springer International Publishing (2017).  DOI 10.1007/978-3-319-55182-1˙10.  URL https://doi.org/10.1007/978-3-319-55182-1_10
  * [15] Fewster, C.J., Eveson, S.P.: Bounds on negative energy densities in flat spacetime.  Phys. Rev. D (3) 58(8), 084010, 6 (1998).  DOI 10.1103/PhysRevD.58.084010.  URL https://doi.org/10.1103/PhysRevD.58.084010
  * [16] Fewster, C.J., Kontou, E.A.: A semiclassical singularity theorem.  Classical Quantum Gravity 39(7), Paper No. 075028, 32 (2022).  DOI 10.1088/1361-6382/ac566b.  URL https://doi.org/10.1088/1361-6382/ac566b
  * [17] Fewster, C.J., Pfenning, M.: Quantum energy inequalities and local covariance. I. Globally hyperbolic spacetimes.  Journal of Mathematical Physics 47(8), 082303 (2006).  DOI 10.1063/1.2212669.  URL http://dx.doi.org/10.1063/1.2212669
  * [18] Fewster, C.J., Roman, T.A.: On wormholes with arbitrarily small quantities of exotic matter.  Phys. Rev. D (3) 72(4), 044023, 15 (2005).  DOI 10.1103/PhysRevD.72.044023.  URL https://doi.org/10.1103/PhysRevD.72.044023
  * [19] Fewster, C.J., Smith, C.J.: Absolute quantum energy inequalities in curved spacetime.  Ann. Henri Poincaré 9(3), 425–455 (2008).  DOI 10.1007/s00023-008-0361-0.  URL https://doi.org/10.1007/s00023-008-0361-0
  * [20] Fewster, C.J., Teo, E.: Bounds on negative energy densities in static space-times.  Phys. Rev. D 59, 104016 (1999).  DOI 10.1103/PhysRevD.59.104016.  URL http://dx.doi.org/10.1103/PhysRevD.59.104016
  * [21] Fewster, C.J., Teo, E.: Quantum inequalities and “quantum interest” as eigenvalue problems.  Phys. Rev. D (3) 61(8), 084012, 14 (2000).  DOI 10.1103/PhysRevD.61.084012.  URL https://link.aps.org/doi/10.1103/PhysRevD.61.084012
  * [22] Ford, L.H.: Quantum coherence effects and the second law of thermodynamics.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 364, 227 – 236 (1978).  DOI 10.1098/rspa.1978.0197.  URL https://doi.org/10.1098/rspa.1978.0197
  * [23] Ford, L.H.: Constraints on negative-energy fluxes.  Phys. Rev. D 43, 3972–3978 (1991).  DOI 10.1103/PhysRevD.43.3972.  URL https://link.aps.org/doi/10.1103/PhysRevD.43.3972
  * [24] Ford, L.H., Roman, T.A.: Averaged energy conditions and quantum inequalities.  Phys. Rev. D (3) 51(8), 4277–4286 (1995).  DOI 10.1103/PhysRevD.51.4277.  URL https://doi.org/10.1103/PhysRevD.51.4277
  * [25] Ford, L.H., Roman, T.A.: Quantum field theory constrains traversable wormhole geometries.  Phys. Rev. D (3) 53(10), 5496–5507 (1996).  DOI 10.1103/PhysRevD.53.5496.  URL https://doi.org/10.1103/PhysRevD.53.5496
  * [26] Ford, L.H., Roman, T.A.: The quantum interest conjecture.  Phys. Rev. D (3) 60(10), 104018, 8 (1999).  DOI 10.1103/PhysRevD.60.104018.  URL https://link.aps.org/doi/10.1103/PhysRevD.60.104018
  * [27] Fröb, M.B., Cadamuro, D.: A quantum energy inequality in the sine–gordon model (2022).  DOI 10.48550/ARXIV.2212.07377.  URL https://arxiv.org/abs/2212.07377
  * [28] Giulini, D.: The rich structure of Minkowski space.  In: V. Petkov (ed.) Minkowski Spacetime: A Hundred Years Later, pp. 83–132. Springer Netherlands, Dordrecht (2010).  DOI 10.1007/978-90-481-3475-5˙4.  URL https://doi.org/10.1007/978-90-481-3475-5_4.  ArXiv:0802.4345
  * [29] Good, M., Juárez-Aubry, B.A., Moustos, D., Temirkhan, M.: Unruh-like effects: effective temperatures along stationary worldlines.  J. High Energy Phys. (6), 059, 25 (2020).  DOI 10.1007/jhep06(2020)059.  URL https://doi.org/10.1007/jhep06(2020)059
  * [30] Gooding, C., Biermann, S., Erne, S., Louko, J., Unruh, W.G., Schmiedmayer, J., Weinfurtner, S.: Interferometric Unruh detectors for Bose-Einstein condensates.  Phys. Rev. Lett. 125(21), 213603, 6 (2020).  DOI 10.1103/physrevlett.125.213603.  URL https://link.aps.org/doi/10.1103/PhysRevLett.125.213603
  * [31] Hawking, S.W.: The occurrence of singularities in cosmology. I.  Proc. Roy. Soc. London Ser. A 294, 511–521 (1966).  DOI 10.1098/rspa.1966.0221.  URL https://doi.org/10.1098/rspa.1966.0221
  * [32] Herglotz, G.: Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper.  Annalen der Physik 336(2), 393–415 (1910).  DOI 10.1002/andp.19103360208.  URL https://doi.org/10.1002/andp.19103360208
  * [33] Hodgkinson, L., Louko, J.: How often does the Unruh-DeWitt detector click beyond four dimensions?  J. Math. Phys. 53(8), 082301, 22 (2012).  DOI 10.1063/1.4739453.  URL https://doi.org/10.1063/1.4739453
  * [34] Hörmander, L.: Fourier integral operators. I.  Acta Math. 127(1-2), 79–183 (1971).  DOI 10.1007/BF02392052.  URL https://doi.org/10.1007/BF02392052
  * [35] Hörmander, L.: The analysis of linear partial differential operators. I.  Classics in Mathematics. Springer-Verlag, Berlin (2003).  DOI 10.1007/978-3-642-61497-2.  URL https://doi.org/10.1007/978-3-642-61497-2.  Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]
  * [36] Kay, B., Wald, R.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon.  Physics Reports 207(2), 49–136 (1991).  DOI 10.1016/0370-1573(91)90015-E.  URL https://doi.org/10.1016/0370-1573(91)90015-E
  * [37] Kontou, E.A., Olum, K.D.: Quantum inequality for a scalar field with a background potential.  Phys. Rev. D 90, 024031 (2014).  DOI 10.1103/PhysRevD.90.024031.  URL https://link.aps.org/doi/10.1103/PhysRevD.90.024031
  * [38] Kontou, E.A., Olum, K.D.: Quantum inequality in spacetimes with small curvature.  Phys. Rev. D 91(10), 104005, 14 (2015).  DOI 10.1103/PhysRevD.91.104005.  URL https://link.aps.org/doi/10.1103/PhysRevD.91.104005
  * [39] Kontou, E.A., Sanders, K.: Energy conditions in general relativity and quantum field theory.  Classical Quantum Gravity 37(19), 193001, 63 (2020).  DOI 10.1088/1361-6382/ab8fcf.  URL https://dx.doi.org/10.1088/1361-6382/ab8fcf
  * [40] Kottler, F.: Über die Raumzeitlinien der Minkowski’schen Welt.  Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, Wien 121, 2a, 1659–1759 (1912).  URL https://viewer.acdh.oeaw.ac.at/viewer/image/MN_2Abt_a_121_2_1912/238/
  * [41] Letaw, J.R.: Stationary world lines and the vacuum excitation of noninertial detectors.  Phys. Rev. D 23, 1709–1714 (1981).  DOI 10.1103/PhysRevD.23.1709.  URL https://link.aps.org/doi/10.1103/PhysRevD.23.1709
  * [42] Letaw, J.R., Pfautsch, J.D.: Quantized scalar field in the stationary coordinate systems of flat spacetime.  Phys. Rev. D (3) 24(6), 1491–1498 (1981).  DOI 10.1103/PhysRevD.24.1491.  URL https://doi.org/10.1103/PhysRevD.24.1491
  * [43] Letaw, J.R., Pfautsch, J.D.: The stationary coordinate systems in flat spacetime.  J. Math. Phys. 23(3), 425–431 (1982).  DOI 10.1063/1.525364.  URL https://doi.org/10.1063/1.525364
  * [44] Moretti, V.: On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods.  Lett. Math. Phys. 111(5), Paper No. 130, 19 (2021).  DOI 10.1007/s11005-021-01464-4.  URL https://doi.org/10.1007/s11005-021-01464-4
  * [45] Moretti, V., Vanzo, L.: Thermal Wightman functions and renormalized stress tensors in the Rindler wedge.  Phys. Lett. B 375(1-4), 54–59 (1996).  DOI 10.1016/0370-2693(96)00223-7.  URL https://doi.org/10.1016/0370-2693(96)00223-7
  * [46] Morris, M.S., Thorne, K.S.: Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity.  Amer. J. Phys. 56(5), 395–412 (1988).  DOI 10.1119/1.15620.  URL https://doi.org/10.1119/1.15620
  * [47] Penrose, R.: Gravitational collapse and space-time singularities.  Phys. Rev. Lett. 14, 57–59 (1965).  DOI 10.1103/PhysRevLett.14.57.  URL https://doi.org/10.1103/PhysRevLett.14.57
  * [48] Pfenning, M.J., Ford, L.H.: The unphysical nature of “warp drive”.  Classical Quantum Gravity 14(7), 1743–1751 (1997).  DOI 10.1088/0264-9381/14/7/011.  URL https://doi.org/10.1088/0264-9381/14/7/011
  * [49] Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time.  Comm. Math. Phys. 179(3), 529–553 (1996).  DOI 10.1007/BF02100096.  URL https://doi.org/10.1007/BF02100096
  * [50] Shaw, R.: The subgroup structure of the homogeneous Lorentz group.  Quart. J. Math. Oxford Ser. (2) 21, 101–124 (1970).  DOI 10.1093/qmath/21.1.101.  URL https://doi.org/10.1093/qmath/21.1.101
  * [51] Synge, J.L.: Timelike helices in flat space-time.  Proc. Roy. Irish Acad. Sect. A 65, 27–42 (1967) (1967).  URL http://www.jstor.org/stable/20488646
  * [52] Unruh, W.G.: Notes on black-hole evaporation.  Phys. Rev. D 14, 870–892 (1976).  DOI 10.1103/PhysRevD.14.870.  URL https://link.aps.org/doi/10.1103/PhysRevD.14.870