Surface gravity from tidal acceleration

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

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

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 $l^{\mu}$

This reparametrization changes the surface gravity as

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.

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]

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]

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]

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

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]).

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

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

So, for this spacetime

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.

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]

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

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]

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

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].

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]

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

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

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

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

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

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

Integrating this equation results in the tidal surface gravity

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

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.

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:

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

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

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

Exploiting the cyclic property of the curvature tensor

and the definition of the conformal Killing vector

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

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

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

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

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].