# Surface gravity from tidal acceleration

Pravin Kumar Dahal<EMAIL_ADDRESS>School of Mathematical &
Physical Sciences, Macquarie University

###### Abstract

Surface gravity plays a pivotal role in the characterization of black holes
and also in formulating the laws of black hole thermodynamics. Despite its
significance, defining surface gravity in general spacetimes is a nontrivial
task and thus has multiple definitions that lack equivalence in dynamical
scenarios. This paper reviews different notions of dynamical surface gravity
and then proposes a new definition based on tidal acceleration, which is an
alternative way of characterizing spacetime curvature. By integrating tidal
acceleration from the horizon to infinity, we could retrieve surface gravity
in familiar situations of Schwarzschild and Kerr spacetimes. We outline a
generic procedure for calculating surface gravity and substantiate our
proposal by investigating the surface gravity of stationary spacetimes and
reproducing the results from the literature. Furthermore, we examine surface
gravity in nonstationary spacetimes, with a focus on Vaidya spacetime as a
model of evaporating black holes.

##  I Introduction

Surface gravity is a crucial concept in black hole physics, and its importance
is evident in various aspects, including the laws of black hole thermodynamics
[1, 2, 3]. It roughly gives a force required to hold the particle of unit mass
at the horizon by an observer at infinity (by means of an infinitely long and
massless string) [2, 4]. However, defining surface gravity, in general, is not
straightforward and several definitions of surface gravity exist in the
literature, which are not equivalent in dynamical spacetimes [5]. One of the
commonly used definitions of surface gravity is the Killing horizon surface
gravity, which measures the inaffinity on the Killing horizon, that is, the
failure of the geodesic Killing vector to be affinely parametrized. However,
this definition is not applicable to spacetimes where the notion of the
Killing horizon fails to exist. The Kodama vector [6] provides an alternative
approach to defining surface gravity, which is always available in spherical
symmetry. The Kodama-Hayward definition of surface gravity, proposed by
Hayward [7], agrees with surface gravity on the horizon of a Reissner-
Nordstrom black hole but not with other dynamical constructs. This definition
is commonly used in the Hamilton-Jacobi approach to studying Hawking radiation
for time-dependent horizons [8, 9]. For a comprehensive list of the
definitions of surface gravity, including their domains of applicability,
advantages, and limitations, please see the references above. We will discuss
some of these definitions below and provide additional references. We want to
mention that owing to limitations in its existing definitions, the appropriate
definition of surface gravity in non-trivial backgrounds is still an open
question.

Geodesic deviation refers to the way in which the paths of two nearby
particles can diverge or converge due to the curvature of space-time. This
deviation in geodesic trajectories of test particles can be attributed to the
tidal force, resulting in tidal acceleration between them [11, 4]. Thus, in
general relativity, tidal acceleration experienced by two neighbouring test
particles when they are separated by a small distance is a manifestation of
the curvature of spacetime or gravity. Therefore it is interesting to explore
how surface gravity can be obtained from the expression for tidal
acceleration.

Let us consider timelike geodesics whose tangent at the points on the Cauchy
surface is a timelike vector field $l^{\alpha}$. Then the inward radial
component of the rate of change of the vector joining any two nearby geodesics
that are radially displaced from one another on the Cauchy surface is given by
$R_{\alpha\beta\mu\nu}l^{\alpha}n^{\beta}l^{\mu}n^{\nu}$, where
$R_{\alpha\beta\mu\nu}$ is the Riemann tensor, and $n^{\alpha}$ is the unit
normal to the Cauchy surface. As one intuitively expects, this geodesic
deviation is governed by the effective total mass of the isolated system, and
one might also expect $R_{\alpha\beta\mu\nu}n^{\beta}n^{\nu}$ to contain
information about this mass [12]. This makes the objective of extracting
surface gravity from tidal acceleration reasonable, and we perform this task
here.

The Hawking effect [24] is a well-known phenomenon in the study of black
holes, which implies that a distant inertial observer would see the collapsing
source radiate. The e-folding relation between the Eddington-Finkelstein
coordinate $u$ and the Kruskal coordinate $U$ is necessary to describe the
Hawking effect, which suggests that the spacetime curvature and the likely
position of the formation of the horizon are essential features for Hawking
radiation [13]. The essential features, namely spacetime curvature is
incorporated in the quantity tidal acceleration, which can be integrated from
the horizon to infinity, showing that the Hawking effect has its origin at the
horizon and is perceived by the observer at infinity. Thus, expressing
Hawking’s temperature in terms of tidal acceleration is desirable as it
reveals particle production as a feature of curved spacetime. Hawking’s
temperature, in some familiar cases, is proportional to the surface gravity
[24, 9]. In that sense, it is natural to expect the relation for surface
gravity in terms of the tidal force.

The effects of particle production in the gravitational field have been
closely examined and compared with particle production in an electromagnetic
field [4, 20]. Although there is still no universal consensus on the radiation
from the uniformly accelerated charge, the notion of the relative acceleration
between the observer and the charge causing the emission of the radiation [21,
22, 23] is widely accepted. The relative acceleration causes curvature on the
electric field lines generating a stress and the energy required to overcome
this stress is emitted as the radiation. Accordingly, one can demand an
analogous mechanism for the emission of radiation in gravitational field,
which, although is observer dependent, fundamentally depends on the spacetime
curvature itself.

This article is organized as follows. We will review dynamical surface gravity
in Section II. Then in Section III, we will discuss about the calculation of
the surface gravity from tidal acceleration. Specifically, section III.1
presents a couple of specific examples to demonstrate that surface gravity can
be recovered from tidal acceleration. Subsequently, in Section III.2, we
provide a formal definition of surface gravity and establish its connection to
the usual acceleration due to gravity in the Newtonian limit. Using our
proposal, we outline the generic procedure for calculating surface gravity in
Section III.3. Additionally, we present an illustration using the general
spherically symmetric metric in Schwarzschild coordinates. We delve into the
realm of stationary spacetimes in Section III.4, for which surface gravity is
unambiguously defined. Reproducing well-established results, in this case,
serves as a pivotal benchmark for assessing the validity and effectiveness of
our proposal. In Section III.5, we extend the application of our definition to
the domain of nonstationary spacetimes. We specifically focus on Vaidya
spacetime, which serves as an excellent model of evaporating black holes [10].
Finally, in Section IV, we discuss our results and conclude this article.

We consider the metric $g_{\mu\nu}$ of signature $\left(-,+,+,+\right)$ and
take $G=c=1$. Similarly, depending on the convenience, we denote the covariant
derivative by $\nabla_{\mu}$ or by $D/Dx^{\mu}$. $\gamma$ denotes the timelike
geodesic curve parameterized by $\tau$, $(l^{\mu},n^{\mu},u^{\mu},v^{\mu})$
denotes the parallel propagated tetrad along this curve (except in Section II)
and $e^{\mu}_{~{}i}=(n^{\mu},u^{\mu},v^{\mu})$. Greek letters denote tensor
indices, and Roman letters denote tetrad indices; in particular, $\hat{\tau}$
and $\hat{\rho}$ denote the tetrad indices associated with the components
$l^{\mu}$ and $n^{\mu}$, respectively. The sign convention used here is
adopted by [11].

##  II Dynamical surface gravity

An observer at infinity from the background of a stationary black hole
observes the thermal radiation (also known as Hawking radiation) of
temperature $\kappa/2\pi$, $\kappa$ being the black hole’s surface gravity
[24, 1, 4]. However, unlike in stationary spacetimes, there is no unambiguous
definition of surface gravity in dynamical spacetimes [26, 25]. This is
because there is no Killing vector field that is timelike at infinity in such
spacetimes. This vector field and the associated Killing horizon is the basis
for the definition of the (Killing horizon) surface gravity in stationary
spacetimes. Various proposals have been made for generalizing surface gravity
to dynamical cases, which reduce to the existing definition in stationary
spacetimes. Even for sufficiently slowly evolving horizons with properties
sufficiently close to their classical counterparts, these generalizations are
practically indistinguishable [28].

###  II.1 Inaffinity surface gravity

As horizon is the basis for the definition of surface gravity, some notion of
quasi-local horizon is necessary for its generalization in dynamical
spacetimes. One could use the familiar notion of marginally trapped surface
which is a compact spacelike two-surface with vanishing expansion of future
directed outgoing (or ingoing) null geodesics [27, 25]. For outgoing/ingoing
null geodesics $l^{\mu}$ (or $n^{\mu}$), marginally outer trapped surface is
given by the condition $\theta_{l}=0$, where

$\theta_{l}=h^{\mu\nu}\nabla_{\mu}l_{\nu},\quad
h_{\mu\nu}=g_{\mu\nu}+l_{\mu}n_{\nu}+l_{\nu}n_{\mu}.$ (1)

Then, analogous to the Killing horizon, we can define surface gravity for the
non-Killing horizon from the fact that $l^{\mu}$ is typically non-affinely
parametrized, even though it is not necessarily the horizon generator. As the
expansion of outgoing null curves on the marginally outer trapped surface
vanishes, these curves are locally constant radius orbits as they cross it

$l^{\nu}\nabla_{\nu}l^{\mu}=\kappa l^{\mu}.$ (2)

This definition of surface gravity depends only on the location of the quasi-
local horizon and is thus quasi-local. However, its value changes on changing
the parametrization of the null vector $l^{\mu}$

$l^{\mu}\to
l^{\prime\mu}=\frac{dx^{\mu}}{d\lambda^{\prime}}=l^{\mu}\frac{d\lambda}{d\lambda^{\prime}}.$
(3)

This reparametrization changes the surface gravity as

$\kappa\to\kappa^{\prime}=\frac{d\lambda}{d\lambda^{\prime}}\kappa+l^{\nu}\nabla_{\nu}\left(\frac{d\lambda}{d\lambda^{\prime}}\right).$
(4)

There always exists a choice of parameter such that $l^{\mu}$ is affinely
parametrized and $\kappa=0$. There are several ways to fix the parameter to
obtain the unique surface gravity [28, 26, 29], and we will discuss a couple
of them here. Note that the parametrization of the Killing vector is fixed by
the proper time of a stationary observer at infinity.

####  II.1.1 Fodor et al. proposal

Fodor et al. [30] proposes the normalization condition $n^{\mu}t_{\mu}=-1$ for
ingoing null geodesics, where $t^{\mu}$ is the asymptotic time translation
Killing vector. So, this definition is restricted to the spherically symmetric
asymptotically flat spacetimes. Outgoing null geodesics $l^{\mu}$ is uniquely
determined in a non-local manner by this normalization procedure and the
relation

$l^{\nu}\nabla_{\nu}l_{\mu}=\kappa l_{\mu},$ (5)

can be used to define the surface gravity. As $t^{\mu}$ can be expanded in
terms of $l^{\mu}$ and $n^{\mu}$ in spherical symmetry, the normalization
condition is equivalent to $l^{\mu}n_{\mu}=A(t,r)$, where $A(t,r)$ is an
arbitrary function. Readers are also referred to the Ref. [31] for the
explicit calculations of inaffinity surface gravity. Note that, differing
normalization by a constant factor does not affect the value of surface
gravity.

####  II.1.2 Aberu-Visser proposal

Let us illustrate the normalization condition proposed by Aberu and Visser
[32] using general spherically symmetric metric

$ds^{2}=-e^{2h(t,r)}f(t,r)dt^{2}+\frac{1}{f(t,r)}dr^{2}+r^{2}d\Omega^{2}.$ (6)

Corresponding null vectors satisfying the normalization condition
$l^{\mu}n_{\mu}=f/2$, which is proportional to the squared norm of the Kodama
vector $K^{\mu}=e^{-h}(1,0,0,0)$, are

$\displaystyle l^{\mu}=$ $\displaystyle\frac{1}{2}(e^{-h},f,0,0),$ (7)
$\displaystyle n^{\mu}=$
$\displaystyle-\frac{1}{2}\left(e^{-h},-f,0,0\right).$ (8)

Using Eq. (5), one could calculate the inaffinity surface gravity

$\kappa_{1}=\frac{1}{2}\left(\frac{\partial f}{\partial r}+f\frac{\partial
h}{\partial r}\right).$ (9)

Ref. [33] argues that this surface gravity is not the correct even in the
static limit, as it misses a factor of $e^{h}$. This argument is based on the
calculation of Hawking temperature in static situations using the standard
Wick rotation, demanding the absence of any conical singularity and
interpreting in terms of a periodicity in imaginary time. Again, null vectors
satisfying the normalization condition $l^{\mu}n_{\mu}=e^{2h}f/2$, which is
proportional to the squared norm of the asymptotic time translation Killing
vector $t^{\mu}=(1,0,0,0)$, are

$\displaystyle l^{\mu}=$ $\displaystyle\frac{1}{2}(1,e^{h}f,0,0),$ (10)
$\displaystyle n^{\mu}=$
$\displaystyle-\frac{1}{2}\left(1,-e^{h}f,0,0\right).$ (11)

As above, one could calculate the inaffinity surface gravity from Eq. (5)

$\kappa_{2l}=\frac{1}{2}\left(e^{h}\frac{\partial f}{\partial
r}+2e^{h}f\frac{\partial h}{\partial r}+\frac{\partial h}{\partial t}\right).$
(12)

We could also calculate the inaffinity surface gravity corresponding to the
null vector $n^{\mu}$ by using the same formula (5)

$\kappa_{2n}=\frac{1}{2}\left(e^{h}\frac{\partial f}{\partial
r}+2e^{h}f\frac{\partial h}{\partial r}-\frac{\partial h}{\partial t}\right).$
(13)

The Aberu-Visser proposal is to average the surface gravity due to both future
and past pointing null vectors

$\kappa_{2}=\frac{\kappa_{2l}+\kappa_{2n}}{2}=\frac{1}{2}\left(e^{h}\frac{\partial
f}{\partial r}+2e^{h}f\frac{\partial h}{\partial r}\right).$ (14)

This quantity has the correct factor of $e^{h}$ and thus gives the desired
static limit (at the horizon).

###  II.2 Hayward-Kodama surface gravity

In dynamical spherically symmetric spacetimes, Kodama vector provides a
generalization to the Killing vector and is parallel to the Killing vector,
whenever it exists. Let $r$ be the areal radius of a 4-dimensional general
spherically symmetric spacetime (for example, in Eq. (6)) and $\epsilon_{ab}$,
{$a,b\\}=\\{0,1$} be the volume element of the surface orthogonal to the
2-sphere of symmetry. Then the Kodama vector could be defined as [6]

$K^{a}=\epsilon^{ab}\nabla_{b}r,\quad K_{\theta}=0=K_{\phi}.$ (15)

As the Kodama vector is divergence free $\nabla_{a}K^{a}=0$, there exists a
conserved current $J_{a}=G_{ab}K^{b}$. The charge associated with this current
is the Misner-Sharp mass [34].

Using the Kodama vector, Hayward defined the surface gravity on the marginally
trapped surface (introduced in Eq. (1)) as [6, 35]

$K^{a}\left(\nabla_{b}K_{a}-\nabla_{a}K_{b}\right)=\kappa K_{b}.$ (16)

Because of the uniqueness of the Kodama vector, Hayward-Kodama surface gravity
is unique. Using Eq. (15), we can write this expression for the surface
gravity differently as [7]

$\kappa=\frac{1}{2}~{}\mathrm{div}~{}(\mathrm{grad}~{}r),$ (17)

where the divergence and gradient is on the 2-dimensional space orthogonal to
the 2-sphere of symmetry. For the general spherically symmetric metric of Eq.
(6), we have the Kodama vector $K^{\mu}=e^{-h}(1,0,0,0)$. So, the Hayward-
Kodama surface garvity is

$\kappa=\frac{1}{2}\frac{\partial f}{\partial r}.$ (18)

As mentioned below Eq. (9), this quantity is missing the factor of $e^{h}$ to
be correct even in the static limit.

There is also the notion of trapping gravity proposed by Hayward [3]. Trapping
gravity does not give the correct result for Reissner-Nordström black holes
[28]. As a natural generalization of the trapping gravity in spherical
symmetry, Mukhoyama and Hayward also proposed another definition of surface
gravity [36]. We will not discuss these in detail here (see Refs. [28, 26, 3,
36, 37]).

###  II.3 Peeling surface gravity

Peeling off properties is the approximate exponential relation (near the
horizon) between the affine parameters on the null generators of past and
future null infinity. Refs. [5, 13, 38] argue that Hawking temperature is
related to this property of null geodesics that actually reach future null
infinity. Associated with the peeling off properties is peeling surface
gravity.

In general spherically symmetric spacetimes, radial null geodesics satisfy

$\frac{dr}{dt}\approx\pm 2\kappa\left(r-r_{H}(t)\right)+{\cal
O}\left(r-r_{H}(t)\right)^{2},$ (19)

where $\kappa$ is the peeling surface gravity. More explicitly, for the metric
of Eq. (6), radial null geodesics take the form

$\displaystyle\frac{dr}{dt}=$ $\displaystyle\pm e^{h}f$ $\displaystyle=$
$\displaystyle\pm\left(e^{h}\frac{\partial f}{\partial r}+e^{h}f\frac{\partial
h}{\partial r}\right)\left(r-r_{H}(t)\right)+{\cal
O}\left(r-r_{H}(t)\right)^{2}.$ (20)

So, for this spacetime

$\kappa=\frac{1}{2}\left(e^{h}\frac{\partial f}{\partial
r}+e^{h}f\frac{\partial h}{\partial r}\right).$ (21)

Hence, the peeling surface gravity contains the correct factor of $e^{h}$ to
give the desired static limit.

Surface gravity of dynamical spacetimes that are conformal transforms of
stationary spacetimes can be calculated by the procedure outlined in Ref. [39]
(see also, Ref. [40, 41]). It states that surface gravity is invariant under
conformal transformations of stationary spacetimes (with nonvanishing
conformal factors). There are works on the calculation of various notions of
surface gravity in momentum dependent Schwarzschild black hole [42] and
Einstein-aether black holes [43], which are classes of Lorentz invariance
violating theories [44]. Similarly, Ref. [46] has used the peeling surface
gravity to calculate the Hawking temperature of noncommutative Schwarzschild
black hole using the tunnelling procedure. Also, calculations of the surface
gravity of black holes in Horndeski theory have been performed in Ref. [45]
and on Lovelock gravity in Ref. [47]. The comparison of different surface
gravity definitions in different regimes may provide a clue to the most useful
notion of surface gravity in general spacetimes.

Various authors have also proposed new procedures for calculating surface
gravity, which have so far been tested only for stationary spacetimes. For
example, Ref. [48] presents a definition of surface gravity in terms of the
invariant four-volume of a black hole (which is the volume of a spacetime
interior to the horizon) that has formed from a gravitational collapse in a
finite proper time. Similarly, Ref. [49] defines surface gravity in terms of
the two-dimensional expansion of rate of the geodesic congruence as seen by
freely falling observers crossing the horizon. Additionally, Ref. [50]
proposes a method for calculating the surface gravity from the entropic force
on holographic screens [51] for spherically symmetric black holes.

##  III Tidal surface gravity

###  III.1 Schwarzschild and Kerr spacetimes

In order to demonstrate the reader that the notion of surface gravity could be
obtained from the expression for the tidal acceleration, we provide an
illustration for the Schwarzschild and Kerr spacetimes, which are two of the
most important solutions of the Einstein field equations.

Let us first consider the Schwarzschild spacetime, which describes the
gravitational field of a non-rotating spherically symmetric mass. For a
comoving observer separated by a distance $\zeta^{\hat{\rho}}$ in the radial
direction, the tidal acceleration is given by [11]

$\frac{D^{2}\zeta^{\hat{\rho}}}{D\tau^{2}}=-R_{\hat{\tau}\hat{\rho}\hat{\tau}\hat{\rho}}\zeta^{\hat{\rho}}=\frac{2M}{r^{3}}\zeta^{\hat{\rho}},$
(22)

where $M$ is the black hole mass, $r$ is the radial coordinate, and $\tau$ is
the proper time along the trajectory of the particles. We can consider the
case where the initial separation $\zeta^{\hat{\rho}}$ is infinitesimal in the
radial direction $dr$, which gives the infinitesimal tidal acceleration in the
radial direction $d\kappa$. We can then calculate the tidal acceleration by
integrating this infinitesimal quantity from the horizon to infinity

$\kappa=\int d\kappa=\int_{2M}^{\infty}\frac{2M}{r^{3}}\delta r=\frac{1}{4M}.$
(23)

Thus, we have recovered the usual surface gravity of Schwarzschild black hole.

It is important to note that there are numerous trivial ways of obtaining
$1/4M$ in Schwarzschild spacetime. Therefore, one may not pursue this idea of
extracting surface gravity seriously unless verifying the procedure for some
nontrivial example. For this reason, we turn to the Kerr spacetime, which
describes the gravitational field of a rotating black hole. Consider a
comoving observer moving along the axis of rotation and separated by a
distance $\zeta^{\hat{\rho}}$ in the radial direction. For this observer, the
tidal acceleration is given by [14]

$\frac{D^{2}\zeta^{\hat{\rho}}}{d\tau^{2}}=-R_{\hat{\tau}\hat{\rho}\hat{\tau}\hat{\rho}}\zeta^{\hat{\rho}}=\frac{2Mr}{(r^{2}+a^{2})^{3}}(r^{2}-3a^{2})\zeta^{\hat{\rho}},$
(24)

where $a$ is the spin parameter of the black hole. Again, we can consider the
case where the initial separation $\zeta^{\hat{\rho}}$ is infinitesimal in the
radial direction $dr$, which gives the infinitesimal tidal acceleration in the
radial direction $d\kappa$. The tidal acceleration results from the
integration of this infinitesimal quantity from the horizon to infinity

$\displaystyle\kappa=\int_{r_{+}}^{\infty}\frac{2Mr}{(r^{2}+a^{2})^{3}}(r^{2}-3a^{2})\delta
r=\frac{\sqrt{M^{2}-a^{2}}}{a^{2}+r_{+}^{2}},$ (25)

where $r_{+}=M+\sqrt{M^{2}-a^{2}}$ is the radius of the event horizon. This
gives the familiar surface gravity for Kerr black hole. In taking the example
of Kerr metric, we have not gone outside our domain of spherical symmetry.
This is due to the fact that the event horizon of the Kerr metric is itself
spherical, and the calculations are restricted to the equatorial plane where
the equations of motion are very similar to those for spherical symmetry.
Principal trajectories of Kerr spacetime, in general, are also independent of
$\theta$ coordinate [15].

###  III.2 Formal developments

In this section, we develop a generic formalism to calculate the tidal surface
gravity from the tidal acceleration between two freely falling particles in
curved spacetimes. Consider a pair of such particles separated by a distance
$\zeta^{\rho}$, the relative acceleration they experience can be expressed as
[16]

$\frac{D^{2}\zeta^{\mu}}{D\tau^{2}}=-R^{\mu}_{\nu\lambda\rho}\zeta^{\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau},$
(26)

where $R^{\mu}_{\nu\lambda\rho}$ is the Riemann tensor.

To facilitate the calculation, we introduce a parallel propagated orthonormal
tetrad $(l^{\mu},n^{\mu},u^{\mu},v^{\mu})$, such that $l^{\mu}=dx^{\mu}/d\tau$
is the tangent vector to the trajectory of one of the particles. In this
frame, the tidal acceleration takes the form

$k^{\mu}_{~{}~{};\nu}l^{\nu}=-R^{\mu}_{~{}~{}\nu\lambda\rho}\zeta^{\lambda}l^{\nu}l^{\rho},$
(27)

where $k^{\mu}:=D\zeta^{\mu}/D\tau$.

We can define an infinitesimal scalar
$d\kappa_{i}=e^{\mu}_{~{}~{}i}k_{\mu;\nu}l^{\nu}$ corresponding to the
infinitesimal four-separation $\zeta^{\rho}$, which allows us to write this
equation as

$d\kappa_{i}=R_{\mu\nu\lambda\rho}\zeta^{\lambda}l^{\nu}l^{\rho}e^{\mu}_{~{}~{}i}.$
(28)

Writing this scalar in the four-dimensional form is trivial as we can see from
this equation that $d\kappa_{0}=0$ identically.

Let us write an arbitrary infinitesimal separation $\zeta^{\mu}$ between two
particles momentarily at rest in the local inertial frame of an infalling
observer as

$\zeta^{i}=\left(-A_{2}\delta r,A_{3}\delta\theta,A_{4}\delta\phi\right),$
(29)

for some arbitrary functions $A_{2}$, $A_{3}$ and $A_{4}$. We note that
$\zeta^{i}$ is tangent to the spacelike geodesic starting from an arbitrary
point of $\gamma$ and orthogonal to it

$e^{\mu}_{~{}~{}i}\zeta^{i}l_{\mu}\Big{|}_{\gamma}=0,$ (30)

where $\gamma$ is the worldline of one of the particles. To obtain this
equality, we have used the relation

$\zeta^{\mu}=A_{1}l^{\mu}\delta t+e^{\mu}_{~{}~{}i}\zeta^{i}\\\
=A_{1}l^{\mu}\delta t-A_{2}n^{\mu}\delta
r+A_{3}u^{\mu}\delta\theta+A_{4}v^{\mu}\delta\phi,$ (31)

for some arbitrary function $A_{1}$. So, by definition, $\zeta^{i}$ constitute
the Fermi normal coordinates [17, 18], while $\zeta^{\mu}$ still gives an
arbitrary infinitesimal separation.

Let us utilize the observer dependence property of surface gravity, which
enables us to select a foliation generated by the vectors $(u^{\mu},v^{\mu})$.
The normal to this two-surface is given by the vector $(l^{\mu},n^{\mu})$.
Thus, for $\zeta^{\mu}$ defined normal to this foliation, we could write
$\zeta^{\mu}=B_{1}n^{\mu}+B_{2}l^{\mu}$, for arbitrary functions $B_{1}$ and
$B_{2}$. By comparing this expression with Eq. (31) gives
$\zeta^{\mu}=A_{1}l^{\mu}\delta t-A_{2}n^{\mu}\delta r$. Substituting this
form of $\zeta^{\mu}$ back into Eq. (28) yields the infinitesimal quantity

$d\kappa_{i}=-A_{2}R_{i\hat{\tau}\hat{\rho}\hat{\tau}}\delta
r=-A_{2}R_{\hat{\tau}i\hat{\tau}\hat{\rho}}\delta r.$ (32)

Integrating this equation results in the tidal surface gravity

$\kappa_{i}|_{\infty}-\kappa_{i}|_{r}=-\int_{r}^{\infty}A_{2}R_{\hat{\tau}i\hat{\tau}\hat{\rho}}\delta
r.$ (33)

We thus obtain an invariant notion of surface gravity that depends on the
curvature of the spacetime. Also note that

${\cal
K}=\frac{R_{\mu\nu\lambda\rho}n^{\lambda}l^{\nu}l^{\rho}n^{\mu}}{(g_{\lambda\mu}g_{\rho\nu}-g_{\lambda\nu}g_{\rho\mu})n^{\lambda}l^{\nu}l^{\rho}n^{\mu}}=-R_{\hat{\tau}\hat{\rho}\hat{\tau}\hat{\rho}},$
(34)

is the Gaussian curvature of the two-surface generated by the vectors
$(l^{\nu},n^{\nu})$. The use of this quantity to define an invariant
gravitational density originated from the work of Synge [19]. Here, the
Gaussian curvature arises from the fact that in defining the surface gravity,
we take a radial/principal trajectory from infinity to the horizon. This
trajectory defines a curvature on two-dimensions $(l^{\nu},n^{\nu})$.

Among the components $\kappa_{i}$, it is important to note that for the case
of spherical symmetry, which we are restricting ourselves to here, only
$\kappa_{2}$ is relevant. This is due to the fact that, among the components
of the curvature scalar $R_{\hat{\tau}i\hat{\tau}\hat{\rho}}$, only
$R_{\hat{\tau}\hat{\rho}\hat{\tau}\hat{\rho}}$ is nonvanishing for the case of
spherical symmetry.

The definition of surface gravity presented above relies on the use of two
orthogonal vectors, namely $(l^{\mu},n^{\mu})$. In a four-dimensional
spacetime ${\cal V}$, we can include two additional orthogonal vectors,
completing the tetrad as $(l^{\mu},n^{\mu},u^{\mu},v^{\mu})$. Petrov D
spacetimes allow for a general procedure to solve the parallel transport
equations [53, 52]. Such spacetimes allow a unique $1+1+2$ decomposition, such
that the first two components align with the principal vectors and the
remaining two-surface is the surface orthogonal to the principal vectors [54].
Therefore, the definition of surface gravity given above yields a unique
result in these spacetimes.

####  III.2.1 Newtonian Limit

In order to analyze the Newtonian limit of our definition of surface gravity,
we first introduce the concept of Fermi normal coordinates. The geodesic curve
$\gamma$ with tangent $l^{\mu}$ representing the motion of a test particle
together with the tetrad $e^{\mu}_{~{}~{}i}=(n^{\mu},u^{\mu},v^{\mu})$ form a
comoving local reference frame. The position of a point of a star with respect
to the chosen reference frame can be described by a three-vector
$\vec{\zeta}=\sum_{i=1}^{3}\zeta^{i}e_{i}$, and in the Fermi normal
coordinates $T,\zeta^{i}$, with $T=\hat{\tau}$, the line element up to
quadratic order can be written in the form [55]

$ds^{2}=-\left(1+R_{\hat{\tau}i\hat{\tau}j}\zeta^{i}\zeta^{j}\right)dT^{2}-\frac{4}{3}R_{\hat{\tau}ijk}\zeta^{i}\zeta^{k}dTd\zeta^{j}\\\
+\left(\delta_{ij}-\frac{1}{3}R_{ikjl}\zeta^{k}\zeta^{l}\right)d\zeta^{i}d\zeta^{j}.$
(35)

In the Newtonian limit, where the gravitational field could be approximated
from the weak static source, the metric component $g_{00}$ can be written in
terms of the Newton potential as $g_{00}=-(1+2\Phi)$, and we obtain the
component of the curvature tensor $R_{\hat{\tau}i\hat{\tau}j}=\Phi_{,ij}$. In
this limit, the geodesic deviation equation takes the form [4]

$\int\frac{d^{2}\zeta_{i}}{d\tau^{2}}=-\int
R_{\hat{\tau}i\hat{\tau}j}\zeta^{j}=-\Phi_{,i}.$ (36)

We thus have the tidal surface gravity $\kappa_{i}=-\int
R_{\hat{\tau}i\hat{\tau}j}\zeta^{j}=-\Phi_{,i}$, which coincides with the
acceleration due to gravity in the Newtonian limit.

###  III.3 General Structure of the Deviation Vector

The deviation vector given in Eq. (31), in a particular choice of spacetime
foliation, contains two arbitrary and undetermined functions. In this section,
we will adopt the approach presented in Ref. [56] to discuss the general
structure of the deviation vector, in order to constrain these undetermined
functions.

We consider a family of geodesics, given by $x^{\alpha}=x^{\alpha}(\tau,s)$,
where the parameter $s$ labels the individual geodesic, parameterized by
proper time $\tau$. The tangent vectors to these curves are defined as
$l^{\alpha}=\partial x^{\alpha}/\partial\tau$ and the deviation vector as
$\zeta^{\alpha}=\delta x^{\alpha}=\partial x^{\alpha}/\partial s$. The
covariant derivatives along the curves are denoted by $D_{\tau}$ and $D_{s}$,
respectively.

The evolution and deformation of the deviation vector along the geodesic is
given by the following equation:

$D_{\tau}\zeta^{\alpha}=\zeta^{\beta}\nabla_{\beta}l^{\alpha}.$ (37)

This equation follows from the fact that second partial derivatives with
respect to $\tau$ and $s$ commute.

As a consequence of this equation and of the fact that
$l^{\alpha}\nabla_{\beta}l_{\alpha}=0$, we have

$l_{\alpha}D_{\tau}\zeta^{\alpha}=D_{\tau}l_{\alpha}\zeta^{\alpha}=0\implies
l_{\alpha}\zeta^{\alpha}=\mathrm{Const.}$ (38)

In the following calculations, we will use these two properties of the
deviation vector to determine the surface gravity explicitly.

####  III.3.1 Example in (t,r) coordinates

In this section, we use the approach outlined in the previous section to
calculate the deviation vector in (t,r) coordinates.

Assuming that the deviation vector has the form given in Eq. (31), simplified
for the particular choice of spacetime foliation, i.e.,

$\zeta^{\mu}=\left(-A_{2}\dot{r}n^{\mu}+A_{1}\dot{t}l^{\mu}\right)d\tau,$ (39)

we first impose the condition $\zeta^{\alpha}l_{\alpha}=\mathrm{Const.}$,
which gives the relation

$A_{1}\dot{t}=\mathrm{Const.}={\cal D}.$ (40)

Next, we calculate $l^{\mu}\nabla_{\mu}\zeta^{\alpha}$ and
$\zeta^{\mu}\nabla_{\mu}l^{\alpha}$ and use the equation for the evolution and
deformation of the deviation vector (Eq. (37)) to obtain

$n^{\alpha}l^{\mu}\nabla_{\mu}\left(A_{2}\dot{r}\right)=A_{2}\dot{r}n^{\mu}\nabla_{\mu}l^{\alpha}.$
(41)

Simplifying this expression yields the value for $A_{2}$, which is required
for the calculation of the surface gravity:

$A_{2}=\frac{1}{\dot{r}}\exp\left(\int
n^{\alpha}n^{\mu}\nabla_{\mu}l_{\alpha}d\tau\right).$ (42)

Thus, we have demonstrated a generic procedure for determining the deviation
vector in non-stationary spacetimes and used this procedure to obtain a
specific expression for the deviation vector in (t,r) coordinates. This form
of deviation vector is necessary for calculating the surface gravity in non-
stationary spacetimes.

###  III.4 Application to stationary spacetimes

####  III.4.1 General setup

Before delving into the specifics of stationary spacetimes, it is useful to
establish a calculation that holds for general spherical symmetry. To begin
our calculation, we define principal vectors $l^{\mu}$ and $n^{\mu}$ as

$\displaystyle l^{\mu}=$ $\displaystyle(\dot{t},\dot{r},0,0),$ (43)
$\displaystyle n^{\mu}=$
$\displaystyle\left(\frac{1}{e^{h}f}\dot{r},e^{h}f\dot{t},0,0\right),$ (44)

where

$-e^{2h(t,r)}f(t,r)\dot{t}^{2}+\frac{1}{f(t,r)}\dot{r}^{2}=-1.$ (45)

Then, we solve for the curvature component
$R_{\hat{\rho}\hat{\tau}\hat{\rho}\hat{\tau}}$ in its frame. This yields the
expression

$R_{\hat{\rho}\hat{\tau}\hat{\rho}\hat{\tau}}=R_{\mu\nu\alpha\beta}l^{\mu}n^{\nu}l^{\alpha}n^{\beta}\\\
=\frac{e^{-h}}{2}\frac{\partial}{\partial
r}\left(e^{-h}\frac{\partial(e^{2h}f)}{\partial
r}\right)+\frac{e^{-h}}{2}\frac{\partial}{\partial
t}\left(e^{-h}f^{-2}\frac{\partial f}{\partial t}\right),$ (46)

which we can further simplify using the assumption that $f$ and $e^{h}$ depend
on time only through the Misner-Sharp mass $M(t,r)$ [57, 58]. We work in the
semiclassical limit, where we assume small evaporation rate and thus neglect
terms of the form $(\partial M/\partial t)^{2}$ and $\partial^{2}M/\partial
t^{2}$. We then integrate to obtain

$\kappa=-\int_{r_{H}}^{\infty}A_{2}\frac{e^{-h}}{2}\frac{\partial}{\partial
r}\left(e^{-h}\frac{\partial(e^{2h}f)}{\partial r}\right)\delta r,$ (47)

where $r_{H}$ is the event/apparent horizon radius and the general form of
$A_{2}$ is given in Eq. (42). This expression gives the tidal surface gravity
for general spherically symmetric spacetimes in Schwarzschild coordinates,
where a particular choice of foliation is adopted. It is worth noting,
however, that a different choice of foliation would be preferred when
calculating surface gravity in advanced/retarded Eddington-Finkelstein
coordinates (appendix A provides a detailed calculation of the surface gravity
in advanced coordinates).

####  III.4.2 Stationary spacetimes

Now, we apply the previous results to stationary spacetimes. Specifically, we
use Eq. (47), which follows identically from Eq. (46) without any
approximation. This is because, in stationary spacetimes, the time derivative
terms present in the final equality of Eq. (46) identically vanishes. To
determine $A_{2}$, we consider the deviation vector $\zeta^{\mu}$ defined in
Eq. (39)

$\zeta^{\mu}=\left(A_{1}\dot{t}^{2}-A_{2}\frac{\dot{r}^{2}}{e^{h}f},A_{1}\dot{t}\dot{r}-A_{2}e^{h}f\dot{t}\dot{r},0,0\right)d\tau.$
(48)

where Eqs. (43) and (44) are used for simplification. Demanding that
$\zeta^{\mu}$ be a timelike Killing vector gives $A_{1}=A_{2}e^{h}f$. As we
are using a Killing vector as a deviation vector, it is necessary to show that
the Killing vector also satisfies the tidal force equation, which we do in
appendix B. Substituting this relation back into the equation for
$\zeta^{\mu}$ and setting $A_{2}=e^{h}$, we get

$\zeta^{\mu}=\left(1,0,0,0\right)d\tau,$ (49)

which satisfies the timelike Killing vector condition. We use this expression
for $A_{2}$ into Eq. (47) to obtain

$\kappa=-\frac{e^{-h}}{2}\frac{\partial(e^{2h}f)}{\partial
r}\bigg{|}_{r=r_{H}},$ (50)

where, we have assumed asymptotic flatness at infinity. Further simplification
gives

$\kappa=-\frac{e^{h}}{2}\left(2f\frac{\partial h}{\partial r}+\frac{\partial
f}{\partial r}\right)\bigg{|}_{r=r_{H}}=-\frac{e^{h}}{2}\frac{\partial
f}{\partial r}\bigg{|}_{r=r_{H}},$ (51)

which exactly matches the result for the (Peeling) surface gravity given in
Ref. [32]. In obtaining the last equality, we have applied the same
approximation used in the given reference. Specifically, we have taken
$\partial h/\partial r$ to be finite, such that the expression containing it
can be neglected with respect to the remaining terms in the equation (this
assumption is not made anywhere else).

####  III.4.3 More on Thermodynamics

The study of black hole thermodynamics is of great importance for
understanding the nature of gravity and its relationship with thermodynamics.
In this section, we will use the generalized expression for surface gravity
derived in the previous section to study the laws of thermodynamics for black
holes.

The surface gravity of a black hole plays a crucial role in black hole
thermodynamics, as it is directly related to the Hawking temperature, which
governs the thermal properties of the black hole. In the previous section, we
derived a generalized expression for the surface gravity, which can be written
alternatively in the form of an antisymmetric second rank tensor $F_{\rho\mu}$
as (see appendix B for the derivation)

$\kappa=F_{\rho\mu}l^{\rho}n^{\mu},$ (52)

where

$F_{\rho\mu}=\frac{1}{2}\left(\zeta_{\mu;\rho}-\zeta_{\rho;\mu}\right).$ (53)

Using this expression, we can write the first law of black hole thermodynamics
as

$M=\frac{1}{4\pi}\lim_{S_{t}\to\infty}\oint_{S_{t}}F_{\rho\mu}l^{\rho}n^{\mu}\sqrt{\sigma}d^{2}\theta,$
(54)

where $\sigma_{AB}$, $A,B=1,2$ is the metric on the two-surface $S_{t}$. The
first law gives a relation for the change in energy of a black hole in terms
of its change in area and surface gravity. We thus have the Komar formula
relating the mass of the black hole to the flux of the Komar current, which is
conserved along the horizon [2].

If we consider that the boundary $S_{t}$ of the hypersurface $\Sigma$ consists
of $\partial\Sigma$, we can rewrite this relation for mass as [59, 60]:

$M=\frac{1}{4\pi}\int_{\Sigma}F^{\rho\mu}_{~{}~{}~{}~{};\mu}l_{\rho}\sqrt{\gamma^{(\Sigma)}}d^{3}x+\frac{1}{4\pi}\oint_{\partial\Sigma}F_{\rho\mu}l^{\rho}n^{\mu}\sqrt{\sigma}d^{2}\theta,$
(55)

where $d\Sigma_{\mu}=l_{\rho}\sqrt{\gamma^{(\Sigma)}}d^{3}x$ is the surface
element of $\Sigma$. The first term on the right-hand side of this equation
can be simplified using the equation for the Killing vector:

$F^{\rho\mu}_{~{}~{}~{}~{};\mu}l_{\rho}=-R_{\beta\alpha}l^{\beta}\xi^{\alpha}.$
(56)

Using this equation, we get

$M=-\frac{1}{4\pi}\int_{\Sigma}R_{\beta\alpha}l^{\beta}\xi^{\alpha}\sqrt{\gamma^{(\Sigma)}}d^{3}x+\frac{1}{4\pi}\oint_{\partial\Sigma}F_{\rho\mu}l^{\rho}n^{\mu}\sqrt{\sigma}d^{2}\theta,$
(57)

which gives the first law for the stationary spacetimes.

###  III.5 Spacetimes conformal to stationary spacetimes

To compute the tidal forces in these spacetimes, we will first show that in
the limit of our approximation where we neglect terms of the form $(\partial
M/\partial t)^{2}$ and $\partial^{2}M/\partial t^{2}$, conformal Killing
vectors also satisfy the tidal force equation. For this, we use the
commutation relation for the covariant derivative

$\zeta_{\sigma;\rho;\mu}-\zeta_{\sigma;\mu;\rho}=-R^{\lambda}_{\sigma\rho\mu}\zeta_{\lambda}.$
(58)

Exploiting the cyclic property of the curvature tensor

$\zeta_{\sigma;\rho;\mu}-\zeta_{\sigma;\mu;\rho}+\zeta_{\mu;\sigma;\rho}-\zeta_{\mu;\rho;\sigma}+\zeta_{\rho;\mu;\sigma}-\zeta_{\rho;\sigma;\mu}=0,$
(59)

and the definition of the conformal Killing vector

$\zeta_{\nu;\mu}+\zeta_{\mu;\nu}={\cal B}g_{\mu\nu},$ (60)

for some arbitrary function ${\cal B}$, we obtain the following equation

$2\zeta_{\mu;\rho;\sigma}=-{\cal B}_{;\mu}g_{\sigma\rho}+{\cal
B}_{;\rho}g_{\mu\sigma}+{\cal
B}_{;\sigma}g_{\rho\mu}-2R^{\lambda}_{\sigma\rho\mu}\zeta_{\lambda}.$ (61)

From this equation, one can infer that conformal Killing vectors do not
satisfy tidal force equation, in general

$2\zeta_{\mu;\rho;\sigma}e^{\mu}_{~{}~{}i}l^{\rho}l^{\sigma}=-2R^{\lambda}_{\sigma\rho\mu}\zeta_{\lambda}e^{\mu}_{~{}~{}i}l^{\rho}l^{\sigma}+{\cal
B}_{;\mu}e^{\mu}_{~{}~{}i}.$ (62)

However, $\zeta^{\mu}-{\cal D}^{\mu}$, where $2{\cal
D}_{\mu;\rho;\sigma}={\cal B}_{;\mu}g_{\rho\sigma}$ satisfies the tidal force
equation

$\left(\zeta_{\mu}-{\cal
D}_{\mu}\right)_{;\rho;\sigma}e^{\mu}_{~{}~{}i}l^{\rho}l^{\sigma}=-R^{\lambda}_{\sigma\rho\mu}\zeta_{\lambda}e^{\mu}_{~{}~{}i}l^{\rho}l^{\sigma}.$
(63)

Hence, the surface gravity for spacetimes conformal to stationary spacetimes,
in general, could be determined from the fact that in such spacetimes,
$\zeta_{\mu}-{\cal D}_{\mu}$ is the deviation vector. However, as ${\cal B}$
vanishes for stationary spacetimes, for small evaporation/accretion rates, the
function ${\cal B}=\zeta^{\mu}_{~{}~{};\mu}/2$, introduced above, could be
expanded in terms of $\partial M/\partial t$. We thus have, for appropriate
normalization of the conformal Killing vector

$\frac{\partial{\cal B}}{\partial x^{\mu}}={\cal O}\left(\frac{\partial
M}{\partial t}\right)^{2}+{\cal O}\left(\frac{\partial^{2}M}{\partial
t^{2}}\right).$ (64)

So, neglecting terms containing $(\partial M/\partial t)^{2}$ and
$\partial^{2}M/\partial t^{2}$, ${\cal B}$ would be constant, and the
conformal Killing vector would satisfy the tidal force equation. These generic
lines of reasoning will be illustrated through a specific example of advanced
Vaidya spacetime, whose near horizon provides an excellent limit to the
general spherically symmetric evaporating black holes [10].

####  III.5.1 Specific example of Vaidya

The Vaidya spacetime has been extensively studied in the context of
gravitational collapse and black hole physics. In this paper, we consider the
Vaidya spacetime in advanced Eddington-Finkelstein coordinates

$ds^{2}=-f(v,r)dv^{2}+2dvdr+r^{2}d\Omega^{2}.$ (65)

We focus on the linear Vaidya case, where the metric is conformal to a static
metric with a Killing vector that gives the temperature in the original
spacetime. Accounting for the conformal factor gives the temperature of Vaidya
[39].

We will now calculate the surface gravity of linear Vaidya using conformal
Killing vectors. The linear Vaidya metric could be written alternatively as
conformal to the static spacetime

$ds^{2}=e^{-2\alpha T/r_{0}}\bigg{(}-\left(1-\frac{r_{0}}{R}+\frac{2\alpha
R}{r_{0}}\right)dT^{2}+2dTdR\\\ +R^{2}d\Omega^{2}\bigg{)},$ (66)

where $\alpha$ and $r_{0}$ are constants. The metric is conformal to the
static spacetime. The transformation relations between the advanced $(v,r)$
and the static $(T,R)$ coordinates are given by

$\displaystyle\alpha v=$ $\displaystyle r_{0}-r_{0}e^{-\alpha T/r_{0}},\qquad
dv=e^{-\alpha T/r_{0}}dT$ $\displaystyle r=$ $\displaystyle Re^{-\alpha
T/r_{0}},\qquad dr=e^{-\alpha T/r_{0}}\left(dR-\frac{\alpha
R}{r_{0}}dT\right).$ (67)

The conformal Killing vector of the Vaidya metric is given by

$\zeta^{\mu}=-\left(\frac{M}{M^{\prime}},r,0,0\right).$ (68)

In static $(T,R)$ coordinates, it becomes

$\zeta^{\mu}=\left(\frac{r_{0}}{\alpha},0,0,0\right).$ (69)

From this, we see that the conformal factor ${\cal B}=\mathrm{Const.}$, from
which we could write ${\cal D}=0$. To match the deviation vector given in Eq.
(78) with the conformal Killing vector, we should have the following values of
$A_{1}$ and $A_{2}$

$A_{1}=\left(A_{2}-\frac{r}{f\dot{v}\dot{r}-\dot{r}^{2}}\right)f,\quad
A_{2}=\frac{r\dot{v}}{\dot{r}}-\frac{M}{M^{\prime}}.$ (70)

Also in $(T,R)$ coordinates, we could write

$A_{2}\dot{r}=\frac{r_{0}}{\alpha}e^{-2\alpha T/r_{0}}\dot{R}.$ (71)

We can calculate the tidal acceleration between two freely moving particles
momentarily at rest in the local inertial frame of an infalling observer using
the formula

$\frac{D^{2}\zeta^{\hat{\rho}}}{d\tau^{2}}=-R_{\hat{\tau}\hat{\rho}\hat{\tau}\hat{\rho}}\zeta^{\hat{\rho}}=\frac{2M}{r^{3}}\zeta^{\hat{\rho}}.$
(72)

It should also be emphasized here that the component of the curvature tensor
in the local inertial frame freely falling with the observer
$R_{\hat{\tau}\hat{\rho}\hat{\tau}\hat{\rho}}$ is equal to the component of
the curvature tensor in the locally inertial static orthonormal frame
$R_{\hat{t}\hat{\rho}\hat{t}\hat{\rho}}$. Taking $\zeta^{\hat{\rho}}=-A_{2}dr$
from Eq. (31), we obtain the surface gravity

$\kappa=\int
d\kappa=\frac{\Omega\alpha}{r_{0}}\int_{2M}^{\infty}A_{2}\frac{2M}{r^{3}}\delta
r=\frac{1}{2M}\int\frac{r_{0}^{2}}{R^{3}}dR=\frac{1}{4M},$ (73)

where $\Omega^{2}=e^{2\alpha T/r_{0}}$ is the conformal factor, which
transforms the Vaidya metric into a static spacetime. We have rescaled the
surface gravity, as this is necessary to make the physical prediction
equivalent in two conformal frames [61]. Moreover, the prefactor
$\alpha/r_{0}$ is taking care of the normalization of the Killing vector in
static $(T,R)$ coordinates. The term containing $M^{\prime}$ is absent in the
surface gravity of Vaidya. Note that the term
$R_{\hat{\tau}\hat{\rho}\hat{\tau}\hat{\rho}}$ is coordinate independent and
the radial coordinate $r$ in Schwarzschild and Vaidya are the same. Hence,
although this result is convenient to obtain in $(v,r)$ coordinates, it would
be unchanged if we use Schwarzschild coordinates.

Moreover, despite being a coordinate-independent quantity, surface gravity is
actually observer-dependent. While working, for example, in advanced $(v,r)$
coordinates, the foliation generated by the $v=\mathrm{const.}$ surface is
usually preferred. However, it is worth mentioning that this choice of
foliation generally does not coincide with the foliation generated by the
$t=\mathrm{const.}$ surface in Schwarzschild $(t,r)$ coordinates.

##  IV Discussion and conclusions

We have reviewed various notions of dynamical surface gravity and presented a
method for calculating the surface gravity from tidal acceleration. The
calculation of tidal surface gravity depends on the spacetime curvature,
would-be-horizon and an observer. These are all the features surface gravity
should have if we want to relate it to the Hawking temperature. We have seen
that this procedure gives the correct surface gravity for the standard cases
of Schwarzschild and Kerr spacetimes. Moreover, we were able to retrieve the
(Peeling) surface gravity from the literature in the general case of
stationary spacetimes in spherical symmetry.

The term containing $M^{\prime}(v)$ does not appear in the surface gravity of
Vaidya shows that small dynamical effects do not contribute significantly to
the Hawking process [10]. To obtain this result we have integrated from the
conformal Killing horizon, which was shown to coincide with the Rindler
horizon. This integration from the conformal Killing horizon is reflected in
the normalization factor, because of which the time component of the conformal
Killing vector reduces to unity at the horizon (only the value at horizon
contributes to the surface gravity).

It is interesting to see how this notion of surface gravity generalizes to the
axially symmetric spacetimes. Only Petrov D spacetimes admit the additional
constant of motion, which allow complete integrability of geodesic equations
and analytic solution of the equation of parallel transport equation applied
to an orthonormal tetrad [52, 53]. These are the necessary quantities for
calculating tidal acceleration and hence surface gravity.

## Acknowledgements

PKD is supported by an International Macquarie University Research Excellence
Scholarship.

##  Appendix A Stationary spacetimes in (v,r) coordinates

Analogous to Eqs. (43) and (44), we define principal null vectors in $(v,r)$
coordinates as

$\displaystyle l^{\mu}=$ $\displaystyle(\dot{v},\dot{r},0,0),$ (74)
$\displaystyle n^{\mu}=$
$\displaystyle\left(\dot{v},-\dot{r}+e^{h_{+}}f\dot{v},0,0\right),$ (75)

where

$-e^{2h_{+}(v,r)}f(v,r)\dot{v}^{2}+2e^{h_{+}(v,r)}\dot{v}\dot{r}=-1.$ (76)

Then, with this choice of tetrad, we write the curvature component
$R_{\hat{\rho}\hat{\tau}\hat{\rho}\hat{\tau}}$ in its frame

$R_{\hat{\rho}\hat{\tau}\hat{\rho}\hat{\tau}}=R_{\mu\nu\alpha\beta}l^{\mu}n^{\nu}l^{\alpha}n^{\beta}\\\
=e^{-h_{+}}\left(\frac{1}{2}\frac{\partial}{\partial
r}\left(e^{h_{+}}\frac{\partial f}{\partial r}\right)+\frac{\partial}{\partial
r}\left(e^{h_{+}}\frac{\partial h}{\partial
r}f\right)+\frac{\partial^{2}h_{+}}{\partial v\partial r}\right).$ (77)

We thus can substitute this relation into Eq. (33) to obtain the tidal surface
gravity in $(v,r)$ coordinates. Now, in stationary spacetimes, the time
derivative terms present in the final equality of Eq. (77) identically
vanishes. To determine $A_{2}$, we consider the deviation vector $\zeta^{\mu}$
defined in Eq. (39) in $(v,r)$ coordinates

$\displaystyle\zeta^{\mu}=$
$\displaystyle\Bigg{(}A_{1}e^{h_{+}}\dot{v}^{2}-\frac{A_{1}}{f}\dot{v}\dot{r}-A_{2}\dot{v}\dot{r},$
$\displaystyle
A_{1}e^{h_{+}}\dot{v}\dot{r}-\frac{A_{1}}{f}\dot{r}^{2}+A_{2}\dot{r}^{2}-A_{2}e^{h_{+}}f\dot{v}\dot{r},0,0\Bigg{)}d\tau,$
(78)

where Eqs. (74) and (75) are used for simplification. The factor $e^{h(t,r)}$,
which appears from coordinate transformation, is absorbed into the coefficient
$A_{1}$, which is not important for later calculations. Demanding that
$\zeta^{\mu}$ be a timelike Killing vector gives $A_{1}=A_{2}f$. Substituting
this relation back into the equation for $\zeta^{\mu}$ and setting
$A_{2}=e^{h_{+}}$, we get

$\zeta^{\mu}=\left(1,0,0,0\right)d\tau,$ (79)

which satisfies the timelike Killing vector condition. We use this expression
for $A_{2}$ to obtain the following relation for the tidal surface gravity in
$(v,r)$ coordinates

$\kappa=-\frac{e^{-h_{+}}}{2}\frac{\partial(e^{2h_{+}}f)}{\partial
r}\bigg{|}_{r=r_{+}},$ (80)

where we have assumed asymptotic flatness at infinity.

##  Appendix B Tidal force and Killing vector

The Killing condition, $\zeta_{\mu;\nu}+\zeta_{\nu;\mu}=0$, can be combined
with the commutation rule for the covariant derivatives and the cyclic sum
rule for the curvature tensor to show that a Killing vector satisfies the
tidal force equation [16]

$\zeta_{\mu;\rho;\sigma}=-R^{\lambda}_{~{}~{}\sigma\rho\mu}\zeta_{\lambda}\implies\zeta_{\mu;\rho;\sigma}l^{\rho}l^{\sigma}=-R^{\lambda}_{~{}~{}\sigma\rho\mu}\zeta_{\lambda}l^{\rho}l^{\sigma}.$
(81)

So, the surface gravity, $d\kappa_{i}=e^{\mu}_{~{}~{}i}k_{\mu;\nu}l^{\nu}$,
takes the form

$d\kappa_{i}=\frac{1}{2}\zeta_{\mu;\rho;\nu}\left(l^{\rho}e^{\mu}_{~{}~{}i}-e^{\rho}_{~{}~{}i}l^{\mu}\right)l^{\nu}=R_{\mu\nu\alpha\beta}l^{\nu}l^{\beta}e^{\alpha}_{~{}~{}i}\zeta^{\mu}.$
(82)

In obtaining the second equality, we have used the Killing condition stated
above. As the vectors $l^{\rho}$ and $e^{\mu}_{~{}~{}i}$ are parallel
propagated, integrating this equation along the trajectory of $l^{\mu}$ yields
the expression for the surface gravity:

$\kappa_{i}=\frac{1}{2}\zeta_{\mu;\rho}\left(l^{\rho}e^{\mu}_{~{}~{}i}-e^{\rho}_{~{}~{}i}l^{\mu}\right),$
(83)

which is exactly the same as the expression given in Ref. [59].

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