# Effect of tidal curvature on dynamics of accelerated probes

Hari K<EMAIL_ADDRESS>Dawood Kothawala<EMAIL_ADDRESS>Centre
for Strings, Gravitation and Cosmology, Department of Physics, Indian
Institute of Technology Madras, Chennai 600 036, India

###### Abstract

We obtain a remarkable semi-analytic expression concerning the role of purely
tidal curvature on accelerated probes, revealing some novel insights into the
role of absolute vs. tidal acceleration in the response of such probes. The
key quantity we evaluate is the relation between geodesic ($\tau_{\rm geod}$)
and proper time ($\tau_{\rm acc}$) intervals between events on the probe
trajectory. This is obtained as a covariant power series in curvature using a
combination of analytical and numerical tools. A serendipitous observation
then reveals that one can exactly sum all terms involving the purely tidal
component ${\mathscr{E}}_{n}=R_{abcd}\varepsilon^{ab}\varepsilon^{cd}$ of
curvature, with $\varepsilon^{ab}$ the bi-normal to the plane of motion:

$\tau_{\rm
geod}=\frac{2}{\sqrt{{-\mathscr{E}}_{n}}}\sinh^{-1}\Biggl{[}\sqrt{\frac{-{\mathscr{E}}_{n}}{a^{2}-{\mathscr{E}}_{n}}}\sinh\left(\frac{1}{2}\sqrt{a^{2}-{\mathscr{E}}_{n}}\;\tau_{\rm
acc}\right)\Biggl{]}$

For classical clocks, the above result represents an interesting closed form
contribution of tidal curvature to the differential ageing of twins in the
classic Twin paradox. For quantum probes, it gives a thermal contribution to
the detector response with a modified Unruh temperature

$[k_{\rm
B}T]_{{\mathscr{E}}_{n}}=\frac{\hbar\sqrt{a^{2}-{\mathscr{E}}_{n}}}{2\pi}$

As an operational tool, the computational framework we present and the
corresponding results should find applications to a wide range of physical
problems that involve measurements and observations by use of accelerated
probes in curved spacetimes.

###### pacs:

04.60.-m

##  I Introduction

Physical effects in accelerated frames of reference have historically been a
bedrock for some of the most important insights into classical as well as
quantum physics. One often uses the results arrived at in accelerated frames
in Minkowski spacetime to gain understanding of physical phenomena in a curved
spacetime, gravitational time dilation and Hawking radiation being a couple of
textbook examples which one can intuitively understand by studying accelerated
clocks and quantum probes, respectively, in Minkowski spacetime (the latter,
of course, if the well known Unruh effect). However, by its very nature, the
principle of equivalence which one often uses to gain such an understanding
does not give any insight whatsoever into the role of curvature in these
phenomena. Curvature manifests itself through tidal acceleration, and this
acceleration is to be ignored if one is to use the principle of equivalence.
Moreover, for probes modelled as point systems, tidal forces can not possibly
have any influence, and hence one might think that one need only worry about
them when dealing with extended systems.

In absence of any general, exact results concerning the effects of curvature,
the above issues can only be justified through perturbative expansions in
curvature, except in some very rare cases (such as (anti-)de Sitter
spacetimes) where the full effect of the background constant curvature on
accelerated probes can be evaluated.

In this paper, we show that, remarkably, one can capture in an analytic manner
at least the effect of components of the Riemann tensor in the plane of motion
of a uniformly accelerated trajectory. While there still remain terms which
can only be described perturbatively, the subset of terms which can be summed
to an analytic form yield some very important insights into the dynamics and
response of accelerated probes in curved spacetimes. As a special case, we
recover the well known results in (anti-)de Sitter spacetimes for which the
parts that can not be summed vanish, and for a physically interesting class of
motions in spherically symmetric spacetimes provided the derivatives of
curvature are small, though the curvature itself might not be.

Our main result follows from the mathematical structure of the (square of)
geodesic interval $\sigma(x,x^{\prime})^{2}$ between points on an accelerated
trajectory. We show that one can exactly sum all terms involving the purely
tidal component ${\mathscr{E}}_{n}=R_{abcd}\varepsilon^{ab}\varepsilon^{cd}$
of curvature, with $\varepsilon^{ab}$ the bi-normal to the plane of motion.
This summation gives the following analytic form for the contribution to
$-\sigma(x,x^{\prime})^{2}$

$\Biggl{\\{}\frac{2}{\sqrt{{-\mathscr{E}}_{n}}}\sinh^{-1}\Biggl{[}\sqrt{\frac{-{\mathscr{E}}_{n}}{a^{2}-{\mathscr{E}}_{n}}}\sinh\left(\frac{1}{2}\sqrt{a^{2}-{\mathscr{E}}_{n}}\;\tau_{\rm
acc}\right)\Biggl{]}\Biggl{\\}}^{2}$

where $\tau_{\rm acc}$ is the proper time along the accelerated curve. In
Minkowski spacetime, the above expression reduces to it’s well known Rindler
form

$\sigma(x,x^{\prime})^{2}=-\Biggl{\\{}\frac{2}{a}\sinh\left(\frac{1}{2}a\;\tau_{\rm
acc}\right)\Biggl{\\}}^{2}$

We then show that the above expression has important implications when one
considers detectors coupled of quantum fields in a curved spacetime. In
Minkowski spacetime, as Unruhunruh-acc has shown, such a detector will respond
to the quantum fluctutations of the field. In particular, if the field is in
Minkowski vacuum state, the quantum vacuum fluctuations as probed by an
accelerated detector manifest as thermal fluctuations at a temperature

$k_{\rm B}T=\frac{\hbar a}{2\pi}$

where $a$ is the acceleration. We will show that the contribution of
${\mathscr{E}}_{n}$ to the detector response is thermal part with temperature

$[k_{\rm
B}T]_{{\mathscr{E}}_{n}}=\frac{\hbar\sqrt{a^{2}-{\mathscr{E}}_{n}}}{2\pi}$

which again reduces to Unruh’s result for Minkowski spacetime, but is now
stated in a manner which is applicable to a spacetime for which
${\mathscr{E}}_{n}$ is not necessarily small.

The paper is structured as follows: In Sec II, we present a solution to the
following mathematical problem: Given a curve $\mathcal{C}$ on a curved
manifold, and a pair of points on $\mathcal{C}$, what is the relation between
the arc-length distance between these points and the geodesic (“chordal”)
distance between them, assuming a unique such geodesic exists? We present the
set-up to find this relation to arbitrary orders of background curvature and
acceleration of $\mathcal{C}$, and explicitly quote the result till $10^{\rm
th}$ order in arc-length. In Sec III, we apply this result to hyperbolic
motion in curved spacetime, and show that it admits a remarkable summation of
certain terms yielding a closed form expression for them. In Sec IV, a semi-
analytic proof for the closed form expression in maximally symmetric spacetime
is skecthed. In Sec V, we use the above results in classical twin paradox,
especially to (anti-)de Sitter and Schwarzschild spacetimes and to evaluate
the clicking rate of the Unruh-deWitt detector, commenting also on the the
role of the van Vleck determinant. Finally, we conclude by putting our results
and analysis in a broader context and discussing possible generalizations.

##  II Accelerated vs Geodesic motion in curved spacetimes

Figure 1: The geometric setup for the problem. Two events on the hyperbolic
trajectory $\mathcal{C}(\tau)\equiv z^{i}(\tau)$ are connected by a geodesic.
The relation between the corresponding proper times holds the key information
relevant for classical and quantum probes.

Consider a curve $\mathcal{C}$ in an arbitrary spacetime, described by
functions $z^{i}(\tau)$ in some coordinate system. Here, $\tau$ is the proper
time parameter, $u^{i}(\tau)=\mathrm{d}z^{i}/\mathrm{d}\tau$ the four
velocity, and $a^{i}=\nabla_{\bm{u}}u^{i}$ the acceleration. Consider two
points $p_{0},p$ separated by an interval $\Delta\tau$ along the curve, and
assume $p$ to be in the geodesically convex normal neighbourhood of $p_{0}$.
There will then be a unique geodesic $\mathcal{G}_{\Delta\tau}$ connecting
$p_{0}$ and $p$ (with proper time parameter $\lambda$); see Fig. 1. We wish to
find the relation between the proper length $\tau_{\rm geod}=\Delta\lambda$ of
this geodesic and $\tau_{\rm acc}=\Delta\tau$. To do so, we set up Riemann
normal coordinates (RNC) at $p_{0}$, with ${\widehat{x}}^{i}(p_{0})=0$. Then,
by definition of RNCpoisson ,
${\widehat{x}}^{a}(p)=(\Delta\lambda){\widehat{t}}^{a}(0;\Delta\tau)$, where
${\widehat{t}}^{a}(0;\Delta\tau)$ is the tangent vector at $p_{0}$ with
$\eta_{ab}{\widehat{t}}^{a}(0;\Delta\tau){\widehat{t}}^{b}(0;\Delta\tau)=-1$.

To find the relation between $\Delta\lambda$ and $\Delta\tau$, we solve the
equation for the trajectory ${\widehat{z}}^{i}(\tau)$ of $\mathcal{C}$ in RNC,

$\displaystyle\frac{\mathrm{d}^{2}{\widehat{z}}^{i}}{\mathrm{d}\tau^{2}}+{\widehat{\Gamma}}^{i}_{\phantom{i}bc}\frac{\mathrm{d}{\widehat{z}}^{b}}{\mathrm{d}\tau}\frac{\mathrm{d}{\widehat{z}}^{c}}{\mathrm{d}\tau}={\widehat{a}}^{i}$
(1)

where ${\widehat{a}}^{i}$ the four acceleration, and
${\widehat{\Gamma}}^{i}_{\phantom{i}bc}(p_{0})=0$ (property of RNC). Then,
setting $\tau=0$ at $p_{0}$, we can equate

$\displaystyle{\widehat{z}}^{a}(-\Delta\tau)={\widehat{x}}^{a}(p)=(\Delta\lambda)\;{\widehat{t}}^{a}(0;\Delta\tau)$
(2)

to obtain

$\displaystyle(\Delta\lambda)^{2}=\eta_{ab}{\widehat{z}}^{a}(-\Delta\tau){\widehat{z}}^{b}(-\Delta\tau)$
(3)

To evaluate the LHS, we write a series solution of Eq. (1) as

$\displaystyle{\widehat{z}}^{k}(\tau)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{\infty}\frac{\tau^{n}}{n!}\left[\frac{\mathrm{d}^{n}{\widehat{z}}^{k}}{\mathrm{d}\tau^{n}}\right]_{\tau=0}$
(4) $\displaystyle=$
$\displaystyle{\widehat{u}}^{k}(0)\tau+{\widehat{a}}^{k}(0)\frac{\tau^{2}}{2}$
(6)
$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}=}+\left(\dot{\widehat{a}}^{k}(0)-\dot{\widehat{\Gamma}}^{a}_{\phantom{a}bc}(0){\widehat{u}}^{b}(0){\widehat{u}}^{c}(0)\right)\frac{\tau^{3}}{6}\ldots$

where the derivatives on the RHS are computed by successive differentiations
of Eq. (1). The argument “$0$” above means everything is evaluated at $p_{0}$,
and the overdot indicates differentiation along the trajectory. To clean up
the above expression, we express the proper time derivatives of
$\dot{\widehat{a}}^{k}$ in terms of covariant derivatives along
${\widehat{u}}^{k}$; for example,
$\dot{\widehat{a}}^{k}={\widehat{\nabla}}_{\bm{\widehat{u}}}{\widehat{a}}^{k}-{\widehat{\Gamma}}^{k}_{\phantom{k}ij}{\widehat{u}}^{i}{\widehat{u}}^{j}$.
with
${\widehat{\nabla}}_{\bm{\widehat{u}}}\equiv{\widehat{u}}^{i}{\widehat{\nabla}}_{i}$.
This introduces more Christoffel symbol terms, making the computation a
laborious task that is best done using a symbolic package. To simplify
analysis, we will make the following assumptions:

(i) The motion is hyperbolic, and

(ii) Derivatives of curvature, $\nabla_{i}R_{abcd}$, are “small”. Note that we
are not assuming the curvature itself to be small, only that it changes slowly
in the domain of interest.

The above assumptions are reasonable given that we are interested in obtaining
leading curvature corrections to uniformly accelerated systems; we will
discuss the relaxations of these in the final section. Even with these
assumptions, the calculation is computationally expensive, particularly since,
as we will show, the leading curvature corrections themselves appear only at
$O(\tau^{6})$! All our numerical computations have been done using
CADABRAcadabra-kp-lb .

We now give the mathematical simplifications that arise under the above
assumptions:

(i) Hyperbolic motion: The mathematical description of uniformly accelerated
hyperbolic motion in curved spacetime was first given by Rindlerrindler ,
essentially by imposing the Serret-Frenet formulaebook-synge to spacetime
curves. The hyperbolic motion is described by the equations

$\displaystyle{\nabla}_{\bm{u}}u^{k}$ $\displaystyle=$ $\displaystyle
an^{k}\hskip 7.11317pt;\hskip 7.11317pt{\nabla}_{\bm{u}}a=0$ (7)
$\displaystyle{\nabla}_{\bm{u}}n^{k}$ $\displaystyle=$ $\displaystyle au^{k}$
(8)

where $a^{2}=a_{i}a^{i}$ and $n^{i}=a^{i}/a$. These conditions are stated more
elegantly in terms of the Fermi derivative

$\frac{D_{F}}{\mathrm{d}\tau}\bm{u}=\frac{D_{F}}{\mathrm{d}\tau}\bm{n}=0$

where, by definition,
$\frac{D_{F}}{\mathrm{d}\tau}\bm{v}=\nabla_{\bm{u}}\bm{v}+(\bm{v}\cdot\bm{u})\bm{a}-(\bm{v}\cdot\bm{a})\bm{u}$.
This has important implication that $\bm{n}$ can be used as one of the basis
vectors while describing $\mathcal{C}$ in Fermi normal coordinates based on
it. This, in turn, means that one can use $a$ as one of the spatial
coordinates, a fact familiar from its many appearances in the study of thermal
properties of static horizons.

The Rindler conditions immediately help us take care of all covariant
derivatives of $a^{i}$ along $u^{i}$ in Eq. (6), since they imply

$\displaystyle({\nabla}_{\bm{u}})^{p}a^{k}$ $\displaystyle=$ $\displaystyle
a^{2p}u^{k}\hskip 23.47343pt(p=1,3,5,\ldots)$ (9)
$\displaystyle({\nabla}_{\bm{u}})^{p}a^{k}$ $\displaystyle=$ $\displaystyle
a^{2p-1}n^{k}\hskip 14.22636pt(p=2,4,6,\ldots)$ (10)

(ii) Ignoring derivatives of Riemann: If one ignores all the derivatives of
Riemann, the metric in RNC can be expressed as Katanaev :

$\displaystyle{\widehat{g}}_{ab}$ $\displaystyle=$
$\displaystyle\eta_{ab}+\frac{1}{2}\sum\limits_{k=1}^{\infty}(-1)^{k}\frac{2^{2k+2}}{(2k+2)!}\mathbb{M}^{m_{1}}_{a}\mathbb{M}^{m_{2}}_{m_{1}}....\mathbb{M}^{m_{(k-1)}}_{b}$
(11)

where the matrix $\mathbb{M}$ is defined by

$\displaystyle{\mathbb{M}}^{i}_{j}=R^{i}_{\phantom{i}ajb}{\hat{x}}^{a}{\hat{x}}^{b}$
(12)

It is easy to identify the series above as related to the series expansion of
$\sin x/x$, yielding

$\displaystyle{\widehat{g}}_{ab}$ $\displaystyle=$
$\displaystyle[{\sin{\mathbb{M}}}/{\mathbb{M}}]^{i}_{(a}\eta_{b)i}$ (13)

This also gives a closed form expression for the van Vleck determinant
$\Delta(0,{\hat{x}})$ in RNC, which we will need later:

$\displaystyle\Delta(0,{\hat{x}})=\sqrt{{\rm
det}[{\mathbb{M}}/{\sin{\mathbb{M}}}]}$ (14)

This follows from the fact that, in RNC,
$\Delta(0,{\hat{x}})=1/\sqrt{-{\hat{g}}}$.

Using the assumptions (i) and (ii) above, we obtain

$\displaystyle\dot{\widehat{z}^{i}}(0)$
$\displaystyle=u^{i}\rvert_{p_{0}}=u^{i}(0)$ (15)
$\displaystyle\ddot{\widehat{z}^{i}}(0)$
$\displaystyle=\left[an^{i}-\Gamma^{i}_{mk}u^{m}u^{k}\right]_{p_{0}}=\left[an^{i}\right]_{p_{0}}$
$\displaystyle\dddot{\widehat{z}^{i}}(0)$
$\displaystyle=\left[a\,u^{k}{\partial}_{k}n^{i}-\Gamma^{i}_{mj,k}u^{k}u^{m}u^{j}-2\Gamma^{i}_{mj}(u^{k}{\partial}_{k}u^{j})\right]_{p_{0}}$
$\displaystyle=\left[a\,\nabla_{\bm{u}}n^{i}-3\Gamma^{i}_{km}u^{k}a^{m}-\Gamma^{i}_{mj,k}u^{k}u^{m}u^{j}+\Gamma^{i}_{mj}\Gamma^{j}_{kl}u^{m}u^{k}u^{l}\right]_{p_{0}}$
$\displaystyle=\left[a\,\nabla_{\bm{u}}n^{i}-\Gamma^{i}_{mj,k}u^{k}u^{m}u^{j}\right]_{p_{0}}$

Higher derivatives can be similarly obtained. We now have all the tools to
explicitly evaluate Eq. (3). Before giving the results for arbitrary curved
spacetime, we give the exact results for maximally symmetric spacetimes.

We must mention that there exists an alternate route to the above derivation
using the properties of Synge’s World function (one-half of the squared
geodesic distance) as given in Ref. book-jls . However, this method becomes
increasingly difficult at higher order. Our method based on RNC, in
conjunction with CADABRA, is easier to implement.

###  II.1 Maximally symmetric spacetimes

For maximally symmetric spacetimes, the Riemann tensor is
$R_{abcd}=\Lambda(g_{ac}g_{bd}-g_{ad}g_{bc})$, with $\Lambda$ constant. It is
more convenient to work with the so called embedding coordinates $X^{i}$, with
$Z^{i}(\tau)$ representing $\mathcal{C}$ in these coordinates (for
convenience, we denote all quantities in embedding coordinates by capitalised
versions of their corresponding symbols in RNC). The metric is given by book-
weinberg

$\displaystyle
g_{ab}=\eta_{ab}+\frac{\Lambda}{1-\Lambda\eta_{ij}X^{i}X^{j}}\eta_{ac}\eta_{bd}X^{c}X^{d}$
(16)

and the Christoffel connections are $\Gamma^{a}_{\phantom{a}bc}=\Lambda
X^{a}g_{bc}$. Therefore, $\Gamma^{a}_{\phantom{a}bc}U^{b}U^{c}=-\Lambda X^{a}$
and $\Gamma^{a}_{\phantom{a}bc}U^{b}A^{c}=0$, which enormously simplifies the
conditions Eq. (8). They become

$\displaystyle{\dot{U}}^{k}$ $\displaystyle=$ $\displaystyle A^{k}+\Lambda
X^{k}$ (17) $\displaystyle{\dot{A}}^{k}$ $\displaystyle=$ $\displaystyle
A^{2}U^{k}$ (18) $\displaystyle{\ddot{U}}^{k}$ $\displaystyle=$
$\displaystyle\left(A^{2}+\Lambda\right)U^{k}$ (19)

from which one can easily show that

$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}\tau}\right)^{2p+1}A^{k}$
$\displaystyle=$ $\displaystyle a^{2}q^{2p}\;U^{k}\hskip
23.47343pt(p=0,1,2,\ldots)$
$\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}\tau}\right)^{2p\phantom{+1}}A^{k}$
$\displaystyle=$ $\displaystyle a^{2}q^{2(p-1)}\;(A^{k}+\Lambda X^{k})\hskip
14.22636pt(p=1,2,3,\ldots)$

where $q=\sqrt{a^{2}+\Lambda}$. Plugging this into Eq. (6), the trajectory is
easily obtained as

$\displaystyle Z^{k}(\tau)=\frac{\sinh q\tau}{q}U^{k}(0)+\frac{\cosh
q\tau-1}{q^{2}}A^{k}(0)$ (20)

The corresponding trajectory in RNC is then obtained by the mapping between
RNC and embedding coordinates (see Ref hk-dk for a proof):

$X^{i}=\Delta^{-1/{(D-1)}}{\hat{x}}^{i}$ (21)

where $\Delta=\Delta(0,{\hat{x}})$ is the van Vleck determinant, and $D$ the
dimension of spacetime. This, when plugged into Eq. (3), finally gives (after
some simplifications) the relation between $\tau_{\rm geod}=\Delta\lambda$ and
$\tau_{\rm acc}=\Delta\tau$

$\displaystyle\tau_{\rm geod}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda}\Biggl{[}\sin^{-1}\left\\{\sqrt{-\Lambda\tau_{\rm
geod,\eta}^{2}(1+\frac{1}{4}\Lambda\tau_{\rm
geod,\eta}^{2})}\right\\}\Biggl{]}^{2}$ (23) $\displaystyle=$
$\displaystyle\frac{4}{\Lambda}\Biggl{[}\sinh^{-1}\left\\{\sqrt{\frac{\Lambda}{q^{2}}}\sinh\left(\frac{q\,\tau_{\rm
acc}}{2}\right)\right\\}\Biggl{]}^{2}$ $\displaystyle\tau_{\rm geod,\eta}^{2}$
$\displaystyle=$ $\displaystyle\frac{4}{q^{2}}\sinh^{2}\left({\frac{q\tau_{\rm
acc}}{2}}\right)$ (24)

(The second equality above follows from standard identities associated with
hyperbolic functions.) Note that

$\lim\limits_{\Lambda\to 0}\tau_{\rm
geod}^{2}=\frac{4}{a^{2}}\sinh^{2}\left({\frac{a\tau_{\rm acc}}{2}}\right)$

which is the familiar expression for Rindler trajectories in Minkowski
spacetime. In fact, $\tau_{\rm geod,\eta}^{2}$ has precisely the form
corresponding to Rindler motion in Minkowski spacetime, but with acceleration
$q=\sqrt{a^{2}+\Lambda}$.

With future anticipation, we highlight the periodicity

$\tau_{\rm geod}^{2}\left[\tau_{\rm acc}\right]=\tau_{\rm
geod}^{2}\left[\tau_{\rm acc}+(2\pi i){q^{-1}}\right]$ (25)

This periodicity is key to thermal properties that a probe, put on an
accelerated trajectory, will assign to the quantum field which is in inertial
vacuum state.

##  III General curved spacetimes

For general spacetimes, even with assumptions (i) and (ii), the computation
becomes unwieldy pretty fast, and we therefore employ CADABRA for the same. To
$O((\Delta\tau)^{10})$, the relation between $\Delta\lambda=\tau_{\rm geod}$
and $\Delta\tau=\tau_{\rm acc}$ is finally obtained (after a laborious
computation) as:

$\displaystyle\tau_{\rm geod}^{2}$ $\displaystyle={\tau^{2}_{\rm
acc}}+\frac{1}{12}a^{2}{\tau^{4}_{\rm
acc}}+\frac{1}{360}\left(a^{4}+3a^{2}{\mathscr{E}}_{n}\right){\tau^{6}_{\rm
acc}}+\frac{1}{20160}\left(a^{6}+17a^{2}R_{\bullet
0n0}R^{\bullet}\,_{0n0}+18a^{4}{\mathscr{E}}_{n}\right){\tau^{8}_{\rm acc}}$
$\displaystyle+\frac{1}{1814400}\left(a^{8}+81a^{6}{\mathscr{E}}_{n}+675a^{4}R_{\bullet
0n0}R^{\bullet}\,_{0n0}+336a^{4}R_{\bullet
nn0}R^{\bullet}\,_{nn0}+155a^{2}R^{\bullet}\,_{0n0}R_{\bullet 0\bigstar
0}R^{\bigstar}\,_{0n0}\right){\tau^{10}_{\rm acc}}+O(\tau^{12}_{\rm acc})$
(26) $\displaystyle=\tau^{2}_{\rm acc}+\frac{1}{12}a^{2}{\tau}^{4}_{\rm
acc}+\frac{1}{360}\left(a^{4}+3a^{2}{\mathscr{E}}_{n}\right){\tau}^{6}_{\rm
acc}+\frac{1}{20160}\left(a^{6}+17a^{2}{\mathscr{E}}_{n}^{2}+18a^{4}{\mathscr{E}}_{n}\right){\tau}^{8}_{\rm
acc}$
$\displaystyle+\frac{1}{1814400}\left(a^{8}+81a^{6}{\mathscr{E}}_{n}+339a^{4}{\mathscr{E}}_{n}^{2}+155a^{2}{\mathscr{E}}_{n}^{3}\right){\tau}^{10}_{\rm
acc}+O(\tau^{12}_{\rm acc})+\mathscr{R}_{A}$ (27)

where, for better readability, we have used following convenient notations:
the symbols “$0$”, “$n$” on Riemann indicates contraction of the corresponding
indices with $u^{i},n^{i}$, with
${\mathscr{E}}_{n}=R_{0n0n}=R_{abcd}u^{a}n^{b}u^{c}n^{d}$ and so on. Indices
which are summed over are indicated by $\bullet,\bigstar,$etc. The first
equality is our first main result, whose extensions and generalizations we
will discuss in the last section. The second equality represents a remarkable
structure following from the first one. To arrive at it, we extract all the
terms in the first equality which depend purely on ${\mathscr{E}}_{n}$ by
expanding the summations; for instance, $R_{\bullet
nn0}R^{\bullet}\,_{nn0}=-{\mathscr{E}}_{n}^{2}+R_{Ann0}R^{A}\,_{nn0}$, with
$A$ representing an index in the space orthogonal to $u^{i}$ and $n^{i}$.
Hence, $\mathscr{R}_{A}$ collectively represents all terms that have at least
one Riemann tensor with at least one index which is neither $0$ nor $n$. Our
main observation then follows from the power series representation of the
following analytic function:

$\displaystyle\Biggl{[}\frac{2}{\sqrt{r_{0}}}\sinh^{-1}\left\\{\sqrt{\frac{r_{0}}{1+r_{0}}}\sinh\left(\frac{1}{2}\sqrt{1+r_{0}}w\right)\right\\}\Biggl{]}^{2}=w^{2}$
$\displaystyle+$
$\displaystyle\frac{w^{4}}{12}+\frac{1}{360}(1-3r_{0})w^{6}+\frac{\left(17r_{0}^{2}-18r_{0}+1\right)w^{8}}{20160}$
(28) $\displaystyle+$
$\displaystyle\frac{\left(-155r_{0}^{3}+339r_{0}^{2}-81r_{0}+1\right)w^{10}}{1814400}+O\left(w^{12}\right)$

which, on comparison with Eq. (27), yields

$\displaystyle\tau_{\rm geod}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{{-\mathscr{E}}_{n}}}\sinh^{-1}\Biggl{[}\sqrt{\frac{-{\mathscr{E}}_{n}}{a^{2}-{\mathscr{E}}_{n}}}\sinh\left(\frac{\sqrt{a^{2}-{\mathscr{E}}_{n}}\,\tau_{\rm
acc}}{2}\right)\Biggl{]}$ (29) $\displaystyle\hskip 7.11317pt+\mathscr{R}_{A}$

We must highlight that the above expression is exact, and it is very much
possible that the terms represented by $\mathscr{R}_{A}$ also admit some sort
of summation to a closed form; we have, however, not been able to do so as
yet, and will comment on it in the final section.

Spherically symmetric spacetimes:

Remarkably, our results also cover the important class of spherical symmetric
spacetimes, with metric:

$\displaystyle\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+h(r)\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}$
(30)

where $f(r)$ and $h(r)$ are arbitrary functions, and $\mathrm{d}\Omega^{2}$ is
the line element of the $2-$sphere. Due to spherical symmetry, and the
maximally symmetric $2-$sphere part, the Riemann tensor of this class of
spacetimes has a unique decomposition, as can be easily shown by a direct
generalization of the form given for Schwarzschild metric in Ref. candelas .
For arbitrary choice of $f(r)$ and $h(r)$, the Riemann tensor is decomposed
as,

$\displaystyle R_{\textsf{abcd}}$ $\displaystyle=$
$\displaystyle\frac{ff^{\prime}h^{\prime}+h\left[f^{\prime
2}-2ff^{\prime\prime}\right]}{4f^{2}h^{2}}\left(g_{\textsf{ac}}g_{\textsf{bd}}-g_{\textsf{ad}}g_{\textsf{bc}}\right)$
$\displaystyle R_{ABCD}$ $\displaystyle=$
$\displaystyle\frac{h-1}{r^{2}h}\left(g_{AC}g_{BD}-g_{AD}g_{BC}\right)$
$\displaystyle R_{\textsf{a}A\textsf{b}B}$ $\displaystyle=$
$\displaystyle\frac{1}{2rh}\;g^{\prime}_{\textsf{ab}}g_{AB}$

where, indices $\textsf{a},\textsf{b}\in\\{t,r\\}$, indices
$A,\,B\in\\{\theta,\phi\\}$ and prime is the derivative with respect to the
radial coordinate $r$.

In these spacetimes, consider hyperbolic motion in the $t-r$ plane; then the
unit vectors $u^{i}$ and $a^{i}$ are in $t-r$ plane. In order to have a non-
zero value for $\mathscr{R}_{A}$, there must be at least one Riemann tensor
component in $t,r,A$ direction, which do not exists. A more formal proof of
this is given in the next section, for a maximally symmetric spacetimes. The
same argument, however, works in spherical symmetry as well. This will
eventually imply ${\mathscr{R}}_{A}=0$ and hence give

$\displaystyle\tau_{\rm geod}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{{-\mathscr{E}}_{n}}}\sinh^{-1}\Biggl{[}\sqrt{\frac{-{\mathscr{E}}_{n}}{a^{2}-{\mathscr{E}}_{n}}}\sinh\left(\frac{\sqrt{a^{2}-{\mathscr{E}}_{n}}\,\tau_{\rm
acc}}{2}\right)\Biggl{]}$ (31) $\displaystyle+$ terms involving $\nabla
R_{abcd}$

For the special case of static observers at $r=R,\theta,\phi=$ constant (and
$g(r)=1/f(r)$), ${\mathscr{E}}_{n}=f^{\prime\prime}(R)/2|$. For (A-)dS
spacetimes, with $f(r)=1-\Lambda r^{2}$, this reproduces the exact result
derived earlier, while for Schwarzschild/Reissner-Nordstrom, the above gives a
distinct contribution which, to the best of our knowledge, has not appeared in
the literature. If there exists a radius at which $a$ diverges, this will lead
approximately to the Rindler result in Minkowski spacetime, but at an
arbitrary radius, the additional curvature term might become significant.

##  IV A Semi-analytic proof

The result above was obtained from numerical computations in CADABRA, combined
with the observation that the coefficients match exactly with the power series
expansion of certain combination of hyperbolic functions. This combination is
by no means trivial, and it is only through a fortuitous circumstance that we
were able to arrive at it. However, one notes that the final expression, Eq.
(29), is similar to the expressions Eq. (23) obtained in the case of maximal
symmetry, apart from the $\mathscr{R}_{A}$ term. This allows us to give the
following semi-analytic proof for our expression.

From Eq. (13) and the conditions Eq. (8), it is easy to see that terms in
$\tau_{\rm geod}$ which are higher order in Riemann appear in the form
$R_{abc\bullet}R^{\bullet}\,_{ghk}\ldots R^{\bigstar}\,_{def}$ with the free
indices dotted with $n^{a}$’s and $u^{a}$’s. Consider such a term with $k+1$
factors on Riemann tensors. We write this expression in a general form as

$\displaystyle{\bar{Q}}_{abcdef}$ $\displaystyle=$ $\displaystyle
R_{abcm_{1}}Q^{m_{1}}_{m_{k}}[k-1]R^{m_{k}}\,_{def}$ $\displaystyle{\rm
where\;\;\;\;}Q^{m_{1}}_{m_{k}}[k-1]$ $\displaystyle=$
$\displaystyle\prod^{k-1}_{i=1}R^{m_{i}}\,_{p_{(2i-1)}p_{2i}m_{(i+1)}}$

(All the $p_{i}$’s are dotted with $u^{a}$ or $n^{a}$, and hence we do not
display them on $Q^{a}_{b}[k-1]$.) This tensor has the property
$Q^{m_{1}}_{m_{k}}[k-1]=R^{m_{1}}\,_{p_{1}p_{2}m_{2}}Q^{m_{2}}_{m_{k}}[k-2]$
which is obvious from above. Now, in maximal symmetry, due to the structure of
Riemann, one has the following fact: since
$(a,b,c,d,e,f,p_{i})\in\left\\{0,n\right\\}$, we must also have
$m_{1},m_{k}\in\left\\{0,n\right\\}$, otherwise ${\bar{Q}}_{abcdef}=0$.
Therefore, for $R^{m_{1}}\,_{p_{1}p_{2}m_{2}}$, the indices,
$m_{1},p_{1},p_{2}\in\\{0,n\\}$, which in turn implies that
$m_{2}\in\\{0,n\\}$ in maximally symmetry. Since the tensor
$Q^{m_{1}}_{m_{k}}[k-1]$ obeys the above recursion relation, all other product
inside this tensor will be the same form.

Therefore, we have shown that in maximal symmetry, the degeneracy arising
because of the fact that $R^{A}\,_{nnA}=R^{A}\,_{00A}=R^{n}\,_{00n}=\Lambda$
is broken in presence of acceleration, and terms other than $R_{0n0n}$ do not
contribute to the series. We have therefore proved that in a generic curved
spacetime, all the terms in $\tau_{\rm geod}$ which have all the Riemann
tensor indices belonging to $0$ or $n$, must sum to the expression in maximal
symmetry with $\Lambda\to(-R_{0n0n})$.

##  V Applications

###  V.1 The classical twin paradox

The mapping of our problem to the classical version of the twin paradox is
immediate from Fig. 1, read as follows: The twins separate at $p_{1}$, with
one of them moving on the hyperbolic trajectory $\mathcal{C}$ by using some
external force, and the other following a geodesic. Provided the twins meet,
the expression we have derived directly gives the relation between their ages.
This, of course, is a somewhat unrealistic setup since the relative Lorentz
boost $\gamma_{\rm rel}=-\bm{u}\cdot\bm{t}\neq 1$, implying a non-zero
relative velocity when they meet. One can presumably remedy this by smoothing
the accelerated trajectory near the meeting points, which would involve
introducing a mild time dependence in $a$, with only a tiny modification to
the above expression.

The twin paradox problem for (anti)de Sitter spacetime is addressed in Ref.
twin-1 ; Ads , and the case of Schwarzschild spacetime is taken up in Ref.
twin-2 . However, these references work with specific accelerated
trajectories, while our result is more general and applicable to any
hyperbolic trajectory, with $a$ arbitrary.

de Sitter spacetime: Consider the twins, one accelerating with constant
acceleration and other one stationary in de Sitter spacetime. The hyperbolic
trajectory of the accelerated twin has been discussed by Rindler rindler , and
has been used in Ref. twin-1 to discuss the twin paradox in de Sitter. The
trajectory, in standard FLRW coordinates, is given by

$\displaystyle t(\tau_{\rm
acc})=\frac{1}{H}\ln\left(\frac{\Psi}{2q^{2}e^{q\tau_{\rm acc}}}\right);\quad
r(\tau_{\rm acc})=\frac{a\left(e^{q\tau_{\rm acc}}-1\right)^{2}}{\Psi}$

where, $H$ is the Hubble constant, $\Psi$ is defined as
$\Psi=H(q+H)e^{2q\tau_{\rm acc}}+2a^{2}e^{q\tau_{\rm acc}}-H(q-H)$ and $q$ is
same as defined before, with $H^{2}=\Lambda$. The geodesic distance in de
Sitter can be found using $\tau_{\rm
geo}^{2}=H^{-2}\cos^{-1}\left[\left({\eta^{2}+\eta\prime^{2}-\ell^{2}}\right)/{2\eta\eta\prime}\right]^{2}$
where $\eta$ is the conformal time and $\ell^{2}$ is the squared spatial
distance. It is not difficult to verify, by writing the trajectory in terms of
conformal time and substituting in the equation for geodesic distance, we
recover Eq. (23) as expected. The simplicity of the expressions presented we
have derived, vis a vis the more cluttered ones when restricting to special
coordinate systems, is worth highlighting. The geometric insights it provides
is an added bonus.

Figure 2: Comparing the relation between the $\tau_{\rm acc}$ and $\tau_{\rm
geod}$ for for static observer and radial geodesic observers in Schwarzschild
($r_{s}=2GM/c^{2}$). The parameters involved are the radius of the static
observer, $r_{0}$, and radius of the turning point for the geodesic observer,
$r_{\rm tp}$. Left plot is for $r_{0}=1.2r_{s},r_{\rm tp}=0.5r_{0}$, and the
right one for $r_{0}=5r_{s},r_{\rm tp}=2r_{0}$

anti - de Sitter spacetime: For anti-de Sitter spacetime, $\Lambda<0$ and some
interesting features emerge. The negative curvature constant will impose a
bound to the value of $\tau_{\rm acc}$. This can be demonstrated using Eq.
(23). Assume $a>\sqrt{|\Lambda|}$, then, for $\Lambda<0$, the RHS of Eq. (23)
will have $\sinh^{-1}\to\sin^{-1}$ and $\tau_{\rm acc}$ will be bounded due to
the bound on the range of function. Thus, for $a>\sqrt{|\Lambda|}$, the proper
time of acceleration will have a bound as,

$\displaystyle\tau_{\rm acc}$ $\displaystyle\leq$
$\displaystyle\frac{2}{\sqrt{a^{2}-|\Lambda|}}\sinh^{-1}\Biggl{[}\sqrt{\frac{a^{2}-|\Lambda|}{|\Lambda|}}\Biggl{]}\leq\frac{2}{\sqrt{|\Lambda|}}$

It is easy to verify that the upper bound here is consistent with the fact
that the maximum geodesic length in anti-de Sitter spacetime is
${\pi}/{\sqrt{|\Lambda|}}$ Ads .

Schwarzschild: In the case of Schwarzschild geometry, the simplest scenario
would be to consider motion in $t-r$ plane, and take for the accelerated twin
the one at rest at $r=r_{0},\theta,\phi=$ constant. The second twin can then
be set on the radial geodesic that goes out from $r=r_{0}$, turns around, and
reaches back to $r=r_{0}$ to meet the static twin. Fixing the values for
$r_{0}$ and the turning point one can relate the $\tau_{\rm geod}$ and
$\tau_{\rm acc}$ twin-2 . The Fig. 3 shows the comparison of this, as given,
for example, in twin-2 , with the result we have obtained. The deviations of
our result from the exact one is due to derivatives of the Riemann tensor,
which our formula does not capture. This is evident when the accelerated
observer is very near to the event horizon, $r_{s}={2GM}/{c^{2}}$ of the black
hole and the turning point of the geodesic is very large. The derivatives of
the Riemann tensor will matter for such instances. For a stationary observer
at $r\gg r_{0}$, our result is closer to the exact relation.

###  V.2 Quantum probes - the Unruh-de Witt detector

In the case of Minkowski spacetime, the Unruh-Dewitt detector is coupled to a
real scalar field in the Minkowski vacuum. The detector is accelerated and the
transition probability of the detector to detect particles in the accelerated
frame of the Minkowski vacuum is obtained. This will give a planckian
spectrum. A broad discussion of this can be found in Ref. book-birrell . For
generalising this a curved spacetime, we consider the detector coupled to a
real scalar field $\phi$ in an arbitrary Hadamard state. The Hadamard state in
arbitrary curved spacetime is equivalent to the Minkowski vacuum in the
Minkowski spacetime. Instead of considering the interaction from past infinity
to future infinity, the switching functionud-det-1 is used to limit the
interaction to take place for some finite time. The interaction Hamiltonian
for the detector is given by $H_{int}(\tau)=c\,m(\tau)\chi(\tau)\phi(x(\tau))$
where $\tau$ is the proper time along the accelerated trajectory, $c$ is the
coupling constant, $m$ is the detector’s monopole moment and $\chi(\tau)$ is
the switching function. Due to the presence of the switching function, the
interaction only takes place when $\chi$ is positive definite. The joint
system of detector being in the ground state and particle in arbitrary
Hadamard state is given by, $\left|\sf
H\right\rangle\otimes\left|0\right\rangle$. Then using the perturbation
theory, the probability of the detector in $\left|1\right\rangle$ is given by,
$\mathcal{P}=c^{2}\left|\left\langle
0\right|m(0)\left|1\right\rangle\right|^{2}\mathcal{F}(\omega)$ where,
$\omega$ is the difference in energy between the states and
$\mathcal{F}(\omega)$ is the response function given by

$\mathcal{F}(\omega)=\int^{\infty}_{-\infty}\,\mathrm{d}\tau\int^{\infty}_{-\infty}\,\mathrm{d}\tau^{\prime\prime}\,\chi(\tau)\chi(\tau^{\prime\prime})e^{-i\omega\left(\tau-\tau^{\prime\prime}\right)}G^{+}(\tau^{\prime\prime},\tau)$
(33)

where $G^{+}$ is the Wightmann two-point function of the field. The
straightforward way to account for the switching is to use a window function
and a simplest way of doing this is choosing $\chi(\tau)$ as,

$\chi(\tau)=\Theta(\tau-\tau_{0})\Theta(\tau^{\prime}-\tau)$ (34)

where, $\tau_{0}$ is the proper time when the detector is switched on and
$\tau^{\prime}$ is the proper time when detector is read (at the base point).
Using this window function, the response function becomes,

$\mathcal{F}(\omega)=\int^{\tau^{\prime}}_{\tau_{0}}\,\mathrm{d}\tau\int^{\tau^{\prime}}_{\tau_{0}}\,\mathrm{d}\tau^{\prime\prime}\,e^{-i\omega\left(\tau-\tau^{\prime\prime}\right)}G^{+}(\tau^{\prime\prime},\tau)$
(35)

Using the coordinate transformation $u=\tau,\,s=\tau-\tau^{\prime\prime}$ for
$\tau^{\prime\prime}<\tau$, $u=\tau-\tau^{\prime\prime},\,s=\tau$ for
$\tau<\tau^{\prime\prime}$ and the property of Wightmann function
$G^{+}(\tau,\tau^{\prime\prime})=G^{+^{*}}(\tau^{\prime\prime},\tau)$, one can
obtain the response function as in Ref. ud-det-2 ,

$\mathcal{F}(\omega)=2\int^{\tau^{\prime}}_{\tau_{0}}\,\mathrm{d}u\int^{u-\tau_{0}}_{0}\mathrm{d}s\,\text{Re}\left[e^{-i\omega
s}G^{+}(u,u-s)\right]$ (36)

The more physically significant quantity is the transition rate which defined
as the time derivative of $\mathcal{F}(\omega)$ given by,

$\dot{\mathcal{F}}(\omega)=2\int^{\tau^{\prime}-\tau_{0}}_{0}\mathrm{d}s\,\text{Re}\left[e^{-i\omega
s}G^{+}(\tau^{\prime},\tau^{\prime}-s)\right]$ (37)

The Wightmann two-point function for the Hadamard state will be given by,
$G^{+}(x,x^{\prime}):=\left\langle\sf
H\middle|\hat{\phi}(x)\hat{\phi}(x^{\prime})\middle|\sf H\right\rangle$ where
$x$ is the coordinate position. The projection of this Wightmann function
along the accelerated trajectory
$G^{+}(\tau,\tau^{\prime})=G^{+}(x,x^{\prime})$, is taken to find the
transition rate. The Wightmann two-point function in a Hadamard state is given
bywightmann-fn ,

$G^{+}(x,x^{\prime})=\frac{1}{4\pi^{2}}\left(\frac{\Delta^{\frac{1}{2}}(x,x^{\prime})}{\sigma_{\epsilon}^{2}(x,x^{\prime})}+v(x,x^{\prime})\ln\left[\sigma_{\epsilon}^{2}(x,x^{\prime})\right]\right)$
(38)

where $\Delta(x,x^{\prime})$ is the van Vleck determinant, $\epsilon$ is a
small positive parameter, $\sigma_{\epsilon}^{2}(x,x^{\prime})$ is the Synge
world functionbook-jls with $i\epsilon$ prescription given as,
$\sigma_{\epsilon}^{2}(x,x^{\prime}):=\sigma^{2}(x,x^{\prime})+2i\epsilon\left[T(x)-T(x^{\prime})\right]$,
where $T(x)$ is increasing global time function and $v(x,x^{\prime})$ is
polynomial function of $\sigma^{2}(x,x^{\prime})$.

Thus the pull-back of the Wightmann function onto the accelerated trajectory
which is the detector worldline, requires the relation between the proper time
$\tau$ on the worldline and the squared geodesic distance
$\sigma^{2}(x,x^{\prime})$ between the considered time interval. The effect
due to the form of window function and the regularisation are discussed more
in Refs. ud-det-1 ; ud-det-2 . We will be focusing on the effects due to
curvature in $\sigma^{2}(x,x\prime)$. Using Eq. (14) and our result,

$\displaystyle\frac{\Delta^{\frac{1}{2}}(x,x^{\prime})}{\sigma^{2}(x,x^{\prime})}$
$\displaystyle=$ $\displaystyle\frac{-{\mathscr{E}}_{n}\left(\sqrt{{\rm
det}[{\mathbb{M}}/{\sin{\mathbb{M}}}]}\right)^{1/2}}{4\Biggl{(}\sinh^{-1}\Biggl{[}\sqrt{\frac{-{\mathscr{E}}_{n}}{a^{2}-{\mathscr{E}}_{n}}}\sinh\left(\frac{\sqrt{a^{2}-{\mathscr{E}}_{n}}\,\tau_{\rm
acc}}{2}\right)\Biggl{]}\Biggl{)}^{2}}$ (39) $\displaystyle+$
$\displaystyle{\mathscr{R}}_{A}+\textrm{terms depending on $\nabla R_{abcd}$}$

The pole structure of the response function will not be affected by
$\Delta(0,{\hat{x}})$, since $\Delta(0,{\hat{x}})=1+O(\tau_{\rm acc}^{2})$. A
computation similar to the one for Rindler motion in Minkowski spacetime then
yields

$\displaystyle\dot{\mathcal{F}}(\omega;\tau)$ $\displaystyle=$
$\displaystyle\frac{\omega}{2\pi}\Biggl{[}{\exp\left(\frac{\hbar\omega}{[k_{\rm
B}T]_{{\mathscr{E}}_{n}}}\right)-1}\Biggl{]}^{-1}$ (40) $\displaystyle+$
$\displaystyle(\textrm{terms depending on ${\mathscr{R}}_{A},\nabla
R_{abcd}$})$

with

$\displaystyle[k_{\rm
B}T]_{{\mathscr{E}}_{n}}=\frac{\hbar}{2\pi}\sqrt{a^{2}-{\mathscr{E}}_{n}}$
(41)

The above result is exact for (anti) de-Sitter spacetimes, with
${\mathscr{E}}_{n}=-\Lambda$ since $\sigma^{2}(x,x^{\prime})$ is exact. In the
last section, we will elucidate the geometrical connection between our result
and the well known derivations based on global embeddings (GEMS). Note that
the above thermal contribution can also be understood directly in terms of the
identity $\tau_{\rm geod}^{2}\left[\tau_{\rm acc}\right]=\tau_{\rm
geod}^{2}\left[\tau_{\rm acc}+(2\pi i){q^{-1}}\right]$ satisfied by our result
to the leading order, ignoring $\mathscr{R}_{A}$. This immediately implies the
periodicity in Euclidean time of the two-point function $G^{+}(x,x^{\prime})$
(to this order), implying a temperature $k_{B}T={\hbar q}/{2\pi}$ ud-temp ;
ud-ds1 ; ud-ds2 .

Apart from (A-)dS, ${\mathscr{R}}_{A}$ vanishes for several other spacetimes
of physical significance for specific hyperbolic motion. Two relevant examples
are:

1\. Spherically symmetric spacetimes: For static observes in spherically
symmetric spacetimes with $g^{rr}=-g_{00}=f(r)$ in standard coordinates
$(t,r,\theta,\phi)$, we have ${\mathscr{R}}_{A}=0$ and
${\mathscr{E}}_{n}=f^{\prime\prime}(r)/2|_{p_{0}}$, and the above result
becomes

$\displaystyle[k_{\rm
B}T]_{{\mathscr{E}}_{n}}=\frac{\hbar}{2\pi}\sqrt{a^{2}-\frac{f^{\prime\prime}(r)}{2}\Biggl{|}_{p_{0}}}+O(\nabla
R_{abcd})$ (42)

2\. FLRW spacetimes: For hyperbolic motion in FLRW universes ($k=0$) with
scale factor $a_{1}(t)$, once again we have ${\mathscr{R}}_{A}=0$ and
${\mathscr{E}}_{n}=-{\ddot{a}}_{1}(t)/a_{1}(t)|_{p_{0}}$.

$\displaystyle[k_{\rm
B}T]_{{\mathscr{E}}_{n}}=\frac{\hbar}{2\pi}\sqrt{a^{2}+\frac{{\ddot{a}}_{1}(t)}{a_{1}(t)}\Biggl{|}_{p_{0}}}+O(\nabla
R_{abcd})$ (43)

There might many more examples of hyperbolic trajectories in well-known
spacetimes for which ${\mathscr{R}}_{A}=0$ due to symmetry, in which case our
result captures the full Riemann tensor modification modulo any effects due to
derivatives of Riemann.

Let us also comment on the $a\to 0$ limit of the final result. For
$-{\mathscr{E}}_{n}>0$, this leaves a finite temperature proportional to
$a_{0}=\sqrt{-{\mathscr{E}}_{n}}$, whose interpretation remains unclear, since
there is no natural choice for $\bm{n}$ when $a=0$. Note, however, that the
difference $\sqrt{a^{2}+a_{0}^{2}}-a_{0}$ has the right limits; it goes as
$a^{2}/2a_{0}$ for $a\ll a_{0}$, and as $a$ for $a\gg|a_{0}|$. It is not clear
to us if this difference has any physical relevance.

We end this section with a comment on a couple of papers (Refs. afshordi )
which give a “prescription” to attribute temperature to quantum field theories
in curved spacetime based on the structure of the two-point function.
Essentially, the curvature correction in this prescription arises by Taylor
expanding the van Vleck determinant in the numerator of the two-point function
to $O(\tau_{\rm acc}^{2})$, and combining it with the pure acceleration term
of the same order in the denominator. The prescription then “reads-off” an
effective temperature $\sqrt{a^{2}-R_{ab}u^{a}u^{b}+w}$, where $w$ is the
contribution from the state dependent term in the two point function. This is
clearly different from the temperature that can be identified from the thermal
part of the detector response, although both yield the same result for
maximally symmetric spacetimes. In computing the detector response, it is
clear that the van Vleck determinant does not contribute at all to the
coincidence limit poles of the two-point function, neither does it contribute
to the residues (see also ud-det-1 ). The only divergence(s) in $\Delta(x,y)$
arise(s) at the caustics, and these are infrared divergences.

##  VI Implications and Discussion

Measurements and observations in the accelerated frames are tools to
understand some of the most subtle aspects of classical and quantum physics
such as the notion of inertia, Mach principle, the equivalence principle,
Unruh effect. The general rule of thumb, when using the results derived in
accelerated frames in Minkowski spacetime to deduce aspects of gravitational
field, is the following: As long as $a\gg\sqrt{|R_{abcd}|}$, where
$|R_{abcd}|$ is typical magnitude of the curvature tensor, one expects an
accelerated frame of reference to mimic effects of a gravitational field to
the lowest order. Even though such a condition seems to make sense, it is too
stringent since one is ignoring the fact that acceleration $a^{i}$ defines a
spacelike direction, and hence components of Riemann tensor in the
$\bm{u}\bm{\wedge}\bm{n}$ plane might lead to interesting effects, and even
non-trivial, effects. Very often, when one talks about effects such as, say,
the Unruh effect, in curved spacetime, one often imagines the additional
curvature contribution coming because of modification to dynamical equations,
such as the d’ Alembartian and hence $G^{+}(x,x^{\prime})$. To the best of our
knowledge, the dependence on curvature arising due to kinematical aspects of
the trajectory is hardly ever considered. In more precise terms, the curvature
dependence inherent in the relation

$\sigma^{2}(z^{i}(\tau),z^{i}(\tau^{\prime}))\to\sigma^{2}(\tau^{\prime}-\tau,R_{abcd})$

is hardly ever discussed. This dependence is independent of the dynamical
equations in the problem one is studying and, as we have shown here, is
crucial as it contributes in an important, non-trivial manner to standard
results. Another way of proceeding with the computation would have been to use
identities associated with Synge’s World function bi-scalar book-jls ;
however, this method becomes cumbersome beyond fourth order, and, at this
order, one does not encounter any curvature terms at all.

A hint towards the importance of such a kinematical term comes from (anti) de
Sitter spacetime, wherein a temperature of $\sqrt{a^{2}+\Lambda}$ can be
associated with hyperbolic motion in a precise sense – an exact result with no
restriction on relation $a$ and $|\Lambda|$. In this paper, we have shown that
an analog of such a result holds in an arbitrary curved spacetime. As
illustrated in the paper, the geodesic interval between two points on a
hyperbolic curve can be expressed as a series in which all the terms involving
components of Riemann tensor in the plane of motion,
${\mathscr{E}}_{n}=R_{abcd}u^{a}n^{b}u^{c}n^{d}$, sum up exactly to an
analytic function. Remarkably, this function depends on acceleration only
through the parameter $q=\sqrt{a^{2}-{\mathscr{E}}_{n}}$. Being exact, such a
dependence eliminates any need for a restriction on relative magnitudes of $a$
and ${\mathscr{E}}_{n}$, thereby allowing one to read-off the interesting non-
pertubative effects arising from the combination. While one still does not
have any handle on off-the-plane components of Riemann, and must therefore
assume them to be small compared to $\\{a^{2},{\mathscr{E}}_{n}\\}$, the
analytic dependence on ${\mathscr{E}}_{n}$ already leads to insights which the
Taylor series itself would never have lead to.

Connection with GEMS: For (A)dS, our result is exact and reproduces the the
famous result by Deser and Levin ud-ds2 for (anti) de-Sitter spacetimes
considered as embedded sub-manifolds of a five dimensional flat spacetime (the
so called GEMS approach). We here briefly highlight the mathematical
connection between these results. We start with the Gauss-Codazzi equation,

$\displaystyle
R_{\textsf{ABCD}}h^{\textsf{A}}_{a}h^{\textsf{B}}_{b}h^{\textsf{C}}_{c}h^{\textsf{D}}_{d}=R_{abcd}+\varepsilon\left(K_{ad}K_{bc}-K_{ac}K_{bd}\right)$
(44)

where, $h^{\textsf{A}}_{a}$ is the projector, $K_{ab}$ is the extrinsic
curvature, $\varepsilon=\pm 1$ which depend on the magnitude of unit normal on
the hypersurface and indices A,B,C,D indicate the higher dimensional space.
Since the embedding is into a higher dimensional flat space(time), we have
$R_{\textsf{ABCD}}=0$, and hence

$\displaystyle R_{abcd}=\varepsilon\left(K_{ac}K_{bd}-K_{ad}K_{bc}\right)$
(45)

Moreover, for the standard embedding of (anti) de Sitter spacetimes in $D=5$,
we have $K_{ab}=-K_{uu}g_{ab}$ with $K_{uu}:=K_{ab}u^{a}u^{b}$. This
immediately implies $R_{abcd}u^{a}n^{b}u^{c}n^{d}=-\varepsilon K_{uu}^{2}$.
Now, we use the following result from differential geometry (see Ref. dk-
duality for a pedagogical proof)

$\displaystyle a_{5}^{2}=a^{2}+\varepsilon K_{uu}^{2}$ (46)

where, $a_{5}$ is the acceleration of $u^{a}$ in $5-d$ flat spacetime, $a$ is
it’s acceleration in the $4-d$ spacetime. The above identities immediately
yield

$\displaystyle a_{5}^{2}=a^{2}-R_{abcd}u^{a}n^{b}u^{c}n^{d}=q^{2}$ (47)

thereby providing a connection with our result. A similar analysis can be done
for spherically symmetric spacetimes, but then the embedding spacetime has
dimension greater than $5$. For a $6$ dimensional embedding of Schwarzschild,
for instance, one would have to compute both the extrinsic curvatures
(corresponding to the two normals) to establish a similar connection.

We conclude with a few comments that we hope will be relevant for future work:

  1. 1.

In deriving our results, we have ignored the derivatives of Riemann, but the
method we have described can be readily extended to incorporate these
derivatives. However, we do not expect any exact results to arise out of such
a study, nor will the result established here be affected by these additional
terms. Our expressions are exact for (A)dS, and provides a remarkably good fit
to the known cases of hyperbolic motion in Schwarzschild; Fig. 3 shows the
comparison for trajectories with same magnitude of acceleration in different
spacetimes.

Figure 3: Comparing the relation between the $\tau_{\rm acc}$ and $\tau_{\rm
geod}$ for different spacetimes. The values chosen are, $r_{0}=2r_{s}$,
$r_{\rm tp}=2r_{0}$ for Schwarzschild and $|\Lambda|={a^{2}}/{2}$ for de
Sitter and anti-de Sitter where $a$ is the magnitude of acceleration for
Schwarzschild observer.

  2. 2.

It would be nice to apply our result to spherically symmetric black hole
solutions in particular, and check how the resultant temperature there is
related to the Hawking temperature. In this context, a couple of remarks can
be made: First, if we consider static observers near a Killing horizon, then
$a$ diverges and hence dominates over $\mathscr{E}_{n}$. The temperature we
have obtained will then give the red-shifted Hawking temperature to leading
order. Second, we know that the Unruh effect relies crucially on Lorentz
invariance (LI). Ref. viqar-louko explicitly discusses this issue by pointing
out the non-trivial consequences that may arise from violations of LI, by
analyzing atoms modelled as UD detectors. In this context, it is worth
noticing that the analytic function appearing on the RHS of Eq. (29)
represents the full contribution of the boost invariant part of Riemann with
respect to boosts in the $\bm{u}\wedge\bm{n}$ plane. To see this, note that
the boost weight zero components of Riemann are $R_{0n0n},R_{0nAB},R_{A[0n]B}$
and $R_{ABCD}$. The last three do not appear in the expression for $\tau_{\rm
geod}$ for hyperbolic motion. Hence, the contribution we have extracted is
also the purely boost invariant part of Riemann tensor. It would nice to see
if any further insights can be gained from our result concerning detectors in
the background of a black hole emitting Hawking radiation smerlak .

  3. 3.

Our analysis brings into sharp focus the role of absolute vs. tidal
acceleration in the response of accelerated probes, and it therefore will
hopefully provide a first step towards quantifying the qualitative picture
often associated with vacuum phenomenon such as Schwinger, Unruh, or Hawking
effects – that of peeling of trajectories caused by a classical background
field (electric or gravitational).

  4. 4.

The analysis/results presented here can be generalised in several directions.
The most important generalisation would be to obtain a summed version for the
contribution of $\mathscr{R}_{A}$. Beyond this, one might generalize further
by including derivatives of curvature as well as deviations from hyperbolic
motion. However, we do not expect any analytic results in such cases for
arbitrary trajectories and/or curvature. The case of time dependent
acceleration is non-trivial even in Minkowski spacetime, with no closed form
expressions available. In our view, a much more insightful procedure would be
to use the method described in this paper to generalise to curved spacetime
the known cases of other stationary trajectories in Minkowski spacetime, in
particular the case of uniform rotation. The conditions for hyperbolic motion
will now need to be modified to incorporate rotation. Although this will
complicate the computations, such an extension might yield useful insights
into behavior of rotating detectors rotating-atom . Any closed form expression
for this case will surely also bring in information about additional
components of Riemann.

###### Acknowledgements.

HK would like to thank IIT, Madras and Ministry of Human Resources and
Development (MHRD), India for financial support.

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