# A remark on the quasilocal calculation of tidal heating: energy transfer
through the quasilocal surface

Albert Huber<EMAIL_ADDRESS>
(UAS Technikum Wien - Department of Applied Mathematics and Physics,
Höchstädtplatz 6, 1200 Vienna, Austria)

###### Abstract

In this note, using the quasilocal formalism of Brown and York, the flow of
energy through a closed surface containing a gravitating physical system is
calculated in a way that augments earlier results on the subject by Booth and
Creighton. To this end, by performing a variation of the total gravitational
Hamiltonian (bulk plus boundary part), it is shown that associated tidal
heating and deformation effects generally are larger than expected. This is
because the aforementioned variation leads to previously unrecognized
correction terms, including a bulk-to-boundary inflow term that does not
appear in the original calculation of the time derivative of the Brown-York
energy and leads to corrective extensions of Einstein’s quadrupole formula in
the large sphere limit.

Key words: quasilocal energy, gravitational Hamiltonian, tidal heating, bulk-
to-boundary inflow

## Introduction

The influence of tidal deformation and heating effects on a nearly isolated
gravitating systems due to external fields was successfully treated and
described by Booth and Creighton in [6]. Instead of using pseudotensorial
techniques, which were more popular at the time [11, 24, 28], the authors used
the Brown-York quasilocal formalism to address the problem. Specifically, in
order to describe the interaction of a spatially and temporally bounded
gravitating system with an external tidal field, they performed a variation of
the boundary part of the total gravitational Hamiltonian, which allowed them
to calculate energy and momentum fluxes on a quasilocal surface and thus
determine the tidal work done by an external gravitational field in the so-
called ’buffer zone’. Taking advantage of the fact that the associated
quasilocal energy expression coincides with the Arnowitt-Deser-Misner energy
at spatial infinity [2, 8, 23] and the Trautman-Bondi-Sachs energy at null
infinity [4, 7, 25, 29], the authors provided two interesting applications of
their formulae.

First, the authors calculated the energy flux due to gravitational waves
through a quasilocal surface in the wave zone (near infinity) to obtain the
correct mass loss for a system radiating gravitational waves.

Second, the authors calculated the tidal heating of a self-gravitating body
interacting with an external tidal field in the local asymptotic reference
frame of the body. While this effect typically occurs for arbitrary pairs of
bodies moving around each other in non-circular orbits, the example discussed
in [6] is one that is familiar from the solar system, namely the tidal
deformation and associated tidal heating of the Gallilean moon Io by Jupiter,
which is generally held responsible for the strong volcanic activity on
Jupiter’s satellite.

As will be shown in the second section of this work (after a brief overview of
the essentials of the considered quasilocal geometric framework in the first
section), particularly with respect to the latter application, the analysis of
Booth and Creighton can be extended in one particular respect: The mass-energy
transfer through the quasilocal surface can alternatively be calculated by
varying the total gravitational Hamiltonian (bulk plus boundary parts), which,
in contrast to the results originally obtained by the authors, leads to the
emergence of correction terms, that is, a confined stress-energy term and a
bulk-to-boundary inflow term that combines the dynamical degrees of freedom of
the bulk with those of the boundary. Although this does not change the
validity of the original results of Booth and Creighton (the authors
considered a case in which the correction terms become zero), it is
nevertheless argued in the present work that especially the latter term must
be taken into account in the quasilocal calculation of more violent tidal
deformations and heating effects than those treated in [6], especially in
cases of tidal deformations of celestial bodies revolving in circular motion
around each other, where there is considerable mass and/or radiation transfer
through the quasilocal surface (possibly due to the collapse of one of the
bodies). To illustrate this, a recap of Booth’s and Creighton’s results on the
tidal heating of isolated bodies is given in the third section of this work,
followed by a discussion of the extent to which these results are insufficient
to describe fluctuations of the interior field of tidally deformed bodies that
lead to nonnegligible backreactions to the exterior fields of these objects.
In this context, specific ideas from the theory of extended irreversible
thermodynamics are used as the basis for the discussion. Apart from that,
another comparatively simple geometrical example is discussed, namely the
mass-energy inflow through the quasilocal surface of a spatially and
temporally bounded gravitating system, caused by gravitoelectromagnetic (GEM)
fields. Using this example, it is argued that corrections to the time
derivative of the Brown-York mass occur and lead to corrections to the
Einstein quadrupole formula in the large sphere limit; corrections that should
find application in Einstein-Hilbert gravity not only in the special cases
discussed in this paper, but in many cases of interest, including the
description of, for example, tidal deformation and tidal heating effects
caused by accretion phenomena or merger processes in relativistic N-body
systems.

##  1 Geometric Setting and quasilocal Action

As a prerequisite for later considerations, this section provides a brief
introduction to the quasilocal framework discussed in [5, 6] and review some
of the important aspects of the underlying geometric model. To that end, a
spacetime $(\mathcal{M},g)$ with manifold $\mathcal{M}=M\cup\partial M$ shall
be considered, which is foliated by a family of $t=const.$-hypersurfaces
$\\{\Sigma_{t}\\}$. Evolving continuously between two fixed instants of time
$\Sigma_{1}$ and $\Sigma_{2}$, this spacetime shall be assumed to be
temporally and spatially bounded. That is to say, it shall be assumed to be
bounded by a timelike hypersurface $\mathcal{B}$ in such way that there holds
$\partial M=\Sigma_{1}\cup\Sigma_{2}\cup\mathcal{B}$, where none of the
embedded boundary parts shall strictly be assumed to be smooth; however, the
timelike portion $\mathcal{B}$ shall be assumed to be connected for the sake
of simplicity. Moreover, the timelike boundary part shall be assumed to be
foliated by a collection of two-surfaces $\\{\Omega_{t}\\}$ such that
$\mathcal{B}=\\{\underset{t}{\cup}\Omega_{t}:\,t_{1}\leq t\leq t_{2}\\}$. As a
result of the fixings made, it is ensured that both spatial boundary parts
$\Sigma_{1}$ and $\Sigma_{2}$ are given in such a way that they contain a pair
of two-surfaces $\Omega_{1}$ and $\Omega_{2}$, respectively, which bifurcate
$\Sigma_{1}$ and $\Sigma_{2}$ locally.

The worldtube thus obtained divides spacetime into two distinct parts, an
interior part $\mathcal{M}$ and an exterior part $\bar{\mathcal{M}}$; and that
in such a way that the interior part $\mathcal{M}$ is contained in the
Lorentzian manifold $\bar{\mathcal{M}}$ of an associated ambient spacetime
$(\text{$\mathcal{\bar{M}}$},\bar{g})$ such that
$\mathcal{\mathcal{M}}\subset\bar{\mathcal{M}}$ and hence
$\Sigma_{1},\Sigma_{2},\mathcal{B}\subset\bar{\mathcal{M}}$.

Given this specific geometric setting, one can now select a time evolution
vector field $t^{a}$ which can be decomposed in components perpendicular and
parallel to the leaves of the spacelike foliation of the local spacetime
$(\mathcal{M},g)$. This means that $t^{a}$ can be written \- everywhere except
at the timelike boundary $\mathcal{B}$ \- in the form $t^{a}=Nn^{a}+N^{a}$,
where $N$ is the lapse function, $N^{a}$ is the shift vector field and $n^{a}$
represents a forward-pointing timelike unit vector field orthogonal to the
folia of $(\mathcal{M},g)$. At the timelike boundary $\mathcal{B}$, however,
the same vector field must be decomposed alternatively in the form
$t^{a}=\mathcal{N}v^{a}+\mathcal{N}^{a}$, where $\mathcal{N}$ is the boundary
lapse function, $\mathcal{N}^{a}$ is the boundary shift vector field and
$v^{a}$ is a forward-pointing timelike unit vector field being orthogonal to
$\Omega_{t}$ and tangent to $\mathcal{B}$. By the selection of a time flow
vector field, a time parameter $t$ can now be fixed by solving the relation
$L_{t}t=1$ (according to which $L_{t}$ represents the Lie derivative along
$t^{a}$).

Next, it may be assumed that there exists an outward-pointing spacelike unit
normal to $\mathcal{B}$, henceforth denoted by $u^{a}$, which is usually non-
orthogonal to the timelike generator $n^{a}$, but orthogonal to another
timelike unit vector field $v^{a}$ that is tangent to $\mathcal{B}$. At the
same time, there is a spacelike vector field $s^{a}$ which is orthogonal to
the timelike generator $n^{a}$. Due to the non-orthogonality of the two vector
fields $n^{a}$ and $u^{a}$, there must exist a scalar field $\eta\equiv
u_{a}n^{a}$, which vanishes only for the special case of a timelike boundary
which is exactly orthogonal to all leaves of its spacelike foliation. This
scalar field also occurs in the relation $s_{a}v^{a}\equiv-\eta$ and in the
decomposition relations $v^{a}=\lambda(n^{a}-\eta u^{a})$,
$s^{a}=\lambda(u^{a}+\eta n^{a})$ and $n^{a}=\lambda(v^{a}+\eta s^{a})$,
$u^{a}=\lambda(s^{a}-\eta v^{a})$, whose validity follows directly from that
of the normalization conditions
$u^{a}u_{a}=s^{a}s_{a}=-n^{a}n_{a}=-v^{a}v_{a}=1$, where
$\lambda=\frac{1}{\sqrt{1+\eta^{2}}}$ is a boost parameter. This boost
parameter also occurs in the relation $\mathcal{N}=\lambda N$ between the
standard lapse function $N$ and the boundary lapse function
$\mathcal{\mathcal{N}}$, which occur in the volume elements of spacetime and
that of the embedded spacelike boundary hypersurface, i.e.
$\sqrt{-g}=N\sqrt{h}$ and $\sqrt{-\gamma}=\mathcal{N}\sqrt{q}$. These
relations, in turn, apply with respect to the determinant of the four-metric
$g_{ab}$ of $(\mathcal{M},g)$, its induced three-metrics
$h_{ab}=g_{ab}+n_{a}n_{b}$ on $\Sigma_{t}$ and $\gamma_{ab}=g_{ab}-u_{a}u_{b}$
on $\mathcal{B}$ and the two-metric
$q_{ab}=g_{ab}+n_{a}n_{b}-s_{a}s_{b}=g_{ab}-u_{a}u_{b}+v_{a}v_{b}$ that is
induced by $g_{ab}$ on $\Omega_{t}$.

Ultimately, as a byproduct of considering all these definitions, Hayward’s
quasilocal action [14]

$S_{G}[g]=\frac{1}{16\pi}(\underset{M}{\int}R\omega_{g}+\underset{\Sigma_{1}}{2\overset{\Sigma_{2}}{\int}}K\omega_{h}+2\underset{\mathcal{B}}{\int}\Theta\omega_{\gamma}+\underset{\Omega_{1}}{2\overset{\Omega_{2}}{\int}}\sinh^{-1}\eta\omega_{q})$
(1)

can finally be set up, where, in this context, $R=g^{ab}R_{ab}$ denotes the
Ricci scalar of the spacetime $(\mathcal{M},g)$ and $K=h^{ab}K_{ab}$ and
$\Theta=\gamma^{ab}\Theta_{ab}$ term the scalar extrinsic curvatures of the
spacelike folia $\Sigma_{t}$ and the timelike boundary $\mathcal{B}$ of
$(\mathcal{M},g)$. Moreover, as can be seen, the different volume elements
$\omega_{g}:=\sqrt{-g}d^{4}x$, $\omega_{h}:=\sqrt{h}d^{3}x$,
$\omega_{\gamma}:=\sqrt{-\gamma}dtd^{2}x$ and $\omega_{q}:=\sqrt{q}d^{2}x$
enter the definition of this quasilocal action.

Following closely the steps taken in [5, 8], one finds that Hayward’s action
can be re-written in a slightly different form. More specfically, using the
decomposition relation
$R=^{(3)}R+K_{ab}K^{ab}-K^{2}+2\nabla_{b}(Kn^{b}-a^{b})$ for the Ricci scalar,
the the Hamiltonian and momentum constraints
$\mathcal{H}=\frac{8\pi}{\sqrt{h}}(P_{ab}P^{ab}-\frac{1}{2}P^{2})-\frac{\sqrt{h}}{16\pi}{}^{(3)}R$
and $\mathcal{H}{}_{a}=-2D_{b}P_{\;a}^{b}$, which are defined with respect to
the fields $P^{ab}=\frac{\sqrt{h}}{16\pi}(K^{ab}-h^{ab}K)$,
$K_{ab}=\frac{1}{2N}(\dot{h}_{ab}-2D_{(a}N_{b)})$ and
$\frac{\sqrt{-g}}{16\pi}(K_{ab}K^{ab}-K^{2})=P_{ab}\dot{h}^{ab}-\frac{8\pi
N}{\sqrt{h}}(P_{ab}P^{ab}-\frac{1}{2}P^{2})-2P_{ab}D^{a}N^{b}$, as well as the
identities $k=\lambda(\Theta+\eta K-a_{b}u^{b}+\lambda(v\nabla)\eta)$,
$Nk-(K_{ab}-Kh_{ab})N^{a}s^{b}=\mathcal{N}\mathfrak{K}-\lambda(\mathcal{N}\nabla)\eta-(\mathcal{N}\nabla)v_{a}u^{a}$,
in relation to which
$\mathfrak{K}=q^{ab}\mathfrak{K}_{ab}=q^{ab}\nabla_{a}u_{b}$ represents the
extrinsic curvature of the spacelike intersection surfaces $\Omega_{t}$ formed
in relation to $u^{a}$, one finds that Hayward’s action can be rewritten in
the form

$\displaystyle S_{G}[g]$ $\displaystyle=\int
dt\underset{\Sigma_{t}}{\int}d^{3}x[P_{ab}h^{ab}-N\mathcal{H}-\mathcal{H}_{a}N^{a}]+$
(2) $\displaystyle+\frac{1}{8\pi}\int
dt\underset{\Omega_{t}}{\int}\dot{\omega}_{q}\sinh^{-1}\eta+\int
dt\underset{\Omega_{t}}{\int}d^{2}x[\mathcal{N}\mathfrak{h}+\mathfrak{h}_{a}\mathcal{N}^{a}],$

where $\mathfrak{h}=\frac{\sqrt{q}}{8\pi}\mathfrak{K}$,
$\mathfrak{h}_{a}=-\frac{\sqrt{q}}{8\pi}u^{c}q_{\;a}^{b}\nabla_{b}v_{c}$ and
the overdot denotes the Lie derivative with respect to $t^{a}$.

As a result of these conversions of Hayward’s quasilocal action, the
generalized gravitational Hamiltonian

$H=\underset{\Sigma_{t}}{\int}(N\mathcal{H}+\mathcal{H}_{a}N^{a})d^{3}x-\underset{\Omega_{t}}{\int}(\mathcal{N}\mathfrak{h}+\mathfrak{h}_{a}\mathcal{N}^{a})d^{2}x$
(3)

is obtained, whose precise form was first determined by Brown and York in the
course of their field-theoretical generalization of Hamilton-Jacobi theory
[8]. Given with respect to the canonical variables $(h^{ab},P_{ab})$ and
$(\sqrt{q},\sinh^{-1}\eta)$, this generalized gravitational Hamiltonian
provides a feasible quasilocal description of the total amount of energy-
momentum of a gravitational field with spatially and temporally finite
dimensions, which represents an alternative to more traditional
pseudotensorial descriptions of the subject111For a detailed review, see for
instance [27]..

An important characteristic of the given generalized Hamiltonian is the fact
that it splits up into a bulk part
$H_{Bulk}=\underset{\Sigma_{t}}{\int}(N\mathcal{H}+\mathcal{H}_{a}N^{a})d^{3}x$
and a boundary part
$H_{Boundary}=-\underset{\Omega_{t}}{\int}(\mathcal{N}\mathfrak{h}+\mathfrak{h}_{a}\mathcal{N}^{a})d^{2}x$,
where, as already indicated, the boundary part $H_{Boundary}$ coincides in an
asymptotically flat spacetime in the so-called large sphere limit with well-
known candidates for the total quasilocal stress-energy-momentum of the
gravitational field itself - with that of Arnowitt, Deser and Misner at
spacelike infinity and that of Bondi and Sachs at null infinity.

Given the exact form of the quasilocal gravitational Hamiltonian $(3),$ the
quasilocal charge

$E=\underset{\Sigma_{t}}{\int}(N\varepsilon-
p_{a}N^{a})\omega_{h}-\underset{\Omega_{t}}{\int}(\mathcal{N}e-j_{a}\mathcal{N}^{a})\omega_{q}$
(4)

can be derived by using Einstein’s equations

$G_{ab}=8\pi T_{ab}.$ (5)

As may be noted, the quasilocal charge obtained again consists of two parts; a
bulk part $E_{Bulk}=\underset{\Sigma_{t}}{\int}(N\varepsilon-
p_{a}N^{a})\omega_{h}$ and a boundary part
$E_{Boundary}=-\underset{\Omega_{t}}{\int}(\mathcal{N}e-j_{a}\mathcal{N}^{a})\omega_{q}$.

This all is of significance in that the results obtained so far form the
starting point for Booth and Creighton’s quasilocal calculation of tidal
heating, which will be briefly discussed in the third and final section of
this paper. However, for the analysis of the results of the now upcoming
second section of this work, it is important to note that the autors
considered only the boundary part of expression $(4)$ as the basis for their
quasilocal calculation of tidal heating. More specifically, an important step
in their calculation was to determine the temporal variation of said boundary
part, for which they obtained the result

$\frac{dW}{dt}=\underset{\Omega_{t}}{\int}(-\dot{\mathcal{N}}e+j_{a}\dot{\mathcal{N}}^{a}+\frac{\mathcal{N}}{2}s^{ab}\dot{q}_{ab})\omega_{q}.$
(6)

As will be shown in the next section, the calculation yields a different
result when the full quasilocal expression $(4)$ (and only its boundary part)
is considered. Specifically, it is shown that the temporal variation of the
quasilocal charge $(4)$ leads to correction terms that do not appear in the
original calculation of Booth and Creighton, but must typically be included
when considering non-vanishing stress-energy tensors. As is argued later in
the third and final section of this work, this may prove to be of relevance
not only for the determination of tidal heating effects, but also for other
applications in the quasilocal approach to general relativity.

##  2 Gravitational Hamiltonian and quasilocal Energy Flux

With the results of the previous section at hand, the next step will be to
derive the exact form of the correction terms mentioned in the introduction.
For this purpose, a temporal variation of the total functional $E$ (bulk plus
boundary) shall be calculated in the following by Lie dragging it along the
flow of the time evolution vector field $t^{a}$. As a basis for doing this,
the decomposition relations $T_{ab}=\varepsilon
n_{a}n_{b}+n_{a}p_{b}+p_{a}n_{b}+S_{ab}$ and
$\tau_{ab}=ev_{a}v_{b}+v_{a}j_{b}+j_{a}v_{b}+s_{ab}$ will be used, whose
validity implies that $T_{ab}t^{a}n^{b}=N\varepsilon-p_{a}N^{a}$ and
$\tau_{ab}t^{a}v^{b}=\mathcal{N}e-j_{a}\mathcal{N}^{a}$ holds true with
respect to any given pair of (totally general) stress-energy and surface
stress-energy tensors $T_{ab}$ and $\tau_{ab}$. As may be noted in this
context, the fields $p_{a}=h_{a}^{\>c}T_{cb}n^{b}$ and
$j_{a}=q_{a}^{\>c}\tau_{cb}v^{b}$ represent the mass-energy and surface mass-
energy fluxes and $S_{ab}=h_{a}^{\>c}h_{b}^{\>d}T_{cd}$ and
$s_{ab}=q_{a}^{\>c}q_{b}^{\>d}\tau_{cd}$ represent the stress and surface
stress tensors of matter fields located within the bounded spacetime region
$\mathcal{M}\subset\mathcal{\bar{M}}$.

Assuming that $\mathsf{\mathfrak{L}}_{t}$ denotes the induced Lie-derivative
at $\Sigma_{t}$ pointing along $t^{a}$, it proves to be instrumental for the
calculation of the time derivative of $E$ to note that

$\displaystyle\mathsf{\mathfrak{L}}_{t}[\omega_{h}(N\varepsilon-p_{a}N^{a})]$
$\displaystyle=\omega_{h}\mathsf{\mathfrak{L}}_{t}(N\varepsilon-p_{a}N^{a})+$
$\displaystyle+\omega_{h}(NK+D_{b}N^{b})(N\varepsilon-p_{a}N^{a})$ (7)

holds true with respect to any spacetime $(\mathcal{M},g)$ whose geometric
structure permits consideration of the quasilocal approach described in the
very first section. Moreover, it proves beneficial to heed another
observation: The local conservation laws $\nabla_{a}T_{\;c}^{a}\cdot
Nn^{c}=\nabla_{a}T_{\;c}^{a}\cdot N^{c}=0$, in combination with the
decompositions of $T_{ab}$ and $\tau_{ab}$ mentioned above, lead to the
generalized continuity and contracted generalized Euler equations

$\mathsf{\mathfrak{L}}_{t}\varepsilon=(ND)\varepsilon-\varepsilon NK-
ND_{a}p^{a}-2(pD)N-NS_{ab}K^{ab}$ (8)

and

$\mathsf{\mathfrak{L}}_{t}p_{a}\cdot
N^{a}=(ND)(p_{a}N^{a})-ND_{b}S_{\;a}^{b}N^{a}-S_{\;a}^{b}N^{a}D_{b}N-NKp_{a}N^{a}-\varepsilon(ND)N;$
(9)

equations that were first published (though in terms of some slightly
different conventions) many years ago in [32]. For notational clarification,
it may be noted here that the definitions $(ND):=N^{a}D_{a}$ and
$(pD):=p^{a}D_{a}$ have been used in the present context.

By suitably combining these relations and applying the inverse Leibniz rule
with the idea of producing a total divergence term, one obtains the result

$\displaystyle
N\mathsf{\mathfrak{L}}_{t}\varepsilon-\mathsf{\mathfrak{L}}_{t}p_{a}\cdot
N^{a}$ $\displaystyle=D_{a}[(N\varepsilon-
p_{b}N^{b})N^{a}-N^{2}p^{a}+NS_{\;b}^{a}N^{b}]-$ (10)
$\displaystyle-(NK+D_{a}N^{a})(N\varepsilon-
p_{b}N^{b})-NS_{ab}(NK^{ab}+D^{(a}N^{b)}).$

Therefore, in further consequence, one finds

$\displaystyle\mathsf{\mathfrak{L}}_{t}[\omega_{h}(N\varepsilon-p_{a}N^{a})]$
$\displaystyle=\omega_{h}(\mathsf{\mathfrak{L}}_{t}N\varepsilon-
p_{a}\mathsf{\mathfrak{L}}_{t}N^{a}+\frac{N}{2}S_{ab}\mathsf{\mathfrak{L}}_{t}h^{ab})+$
(11) $\displaystyle+\omega_{h}D_{a}[(N\varepsilon-
p_{b}N^{b})N^{a}-N^{2}p^{a}+NS_{\;b}^{a}N^{b}].$

By applying Gauss’ law in the next step and using the constraint relation
$\mathcal{D}_{a}\tau_{\>b}^{a}=\pi^{a}=\gamma_{\>b}^{a}T_{ac}u^{c}$, in
respect to which
$\mathcal{D}_{a}\tau_{\>b}^{a}=\gamma_{\>b}^{d}\gamma_{\>c}^{e}\nabla_{e}\tau_{\,d}^{c}$
applies, the temporal variation of the functional $E$ can now
straightforwardly be calculated, leading to the result

$\frac{dE}{dt}=\underset{\Sigma_{t}}{\int}(\dot{N}\varepsilon-
p_{a}\dot{N}^{a}+\frac{N}{2}S_{ab}\dot{h}^{ab})\omega_{h}-\underset{\Omega_{t}}{\int}(\dot{\mathcal{N}}e-j_{a}\dot{\mathcal{N}}^{a}+\frac{\mathcal{N}}{2}s_{ab}\dot{q}^{ab}-\mathfrak{X})\omega_{q}.$
(12)

Here, it should be noted that to obtain this very integral expression, the
definitions $s_{ab}=q_{a}^{\>c}q_{b}^{\>d}\tau_{cd}$,
$b^{a}\equiv(v\nabla)v^{a}$, $\dot{N}=\mathsf{\mathfrak{L}}_{t}N$,
$\dot{N}^{a}=\mathsf{\mathfrak{L}}_{t}N^{a}$,
$\dot{h}^{ab}=\mathsf{\mathfrak{L}}_{t}h^{ab}$ as well as
$\dot{\mathcal{N}}=\mathcal{L}_{t}\mathcal{N}$,
$\dot{\mathcal{N}}^{a}=\mathcal{L}_{t}\mathcal{N}^{a}$,
$\dot{q}^{ab}=\mathcal{L}_{t}q^{ab}$ have been used, where $\mathcal{L}_{t}$
denotes the induced Lie-derivative at $\Omega_{t}$ pointing along $t^{a}$.

The integrand $\mathfrak{X}$ occurring in the boundary part of expression
$(12)$ consists of two parts, i.e.

$\mathfrak{X}=\mathfrak{X}_{1}+\mathfrak{X}_{2},$ (13)

where

$\mathfrak{X}_{1}=\mathcal{N}^{2}\pi_{a}v^{a}+\mathcal{N}\pi_{a}\mathcal{N}^{a}$
(14)

represents the confined stress-energy term and

$\mathfrak{X}_{2}=-\frac{\eta}{\lambda}\mathcal{N}^{2}\varepsilon+\eta\mathcal{N}^{2}p_{a}v^{a}+\eta\mathcal{N}p_{a}\mathcal{N}^{a}-\frac{\mathcal{N}^{2}}{\lambda}p_{a}u^{a}+\lambda\mathcal{N}S_{ab}v^{a}u^{b}+\lambda
S_{ab}\mathcal{N}^{a}u^{b}$ (15)

represents a bulk-to-boundary inflow term, as may be concluded from the fact
that the bulk expressions $\varepsilon$, $p_{a}$ and $S_{ab}$ enter into its
definition222Note that the bulk fields $\varepsilon$, $p_{a}$ and $S_{ab}$
can, of course, be re-expressed on the boundary in terms of associated
boundary fields. However, in order to highlight the fact that these fields can
be used to interpret $\underset{\Omega_{t}}{\int}\mathfrak{X}_{2}\omega_{q}$
as a bulk-to-boundary inflow, the form of $\varepsilon$, $p_{a}$ and $S_{ab}$
has been intentionally left unchanged at this point.. For the sake of
clarification, it may be noted here that the relations
$N_{a}s^{a}=-\eta\mathcal{N}$, $N\varepsilon-
p_{a}N^{a}=\frac{\mathcal{N}}{\lambda}\varepsilon-\mathcal{N}p_{a}v^{a}-p_{a}\mathcal{N}^{a}$,
$N^{2}p_{a}s^{a}=\frac{\mathcal{N}^{2}}{\lambda}p_{a}u^{a}$ and
$S_{ab}s^{a}N^{b}=\lambda\mathcal{N}S_{ab}v^{a}u^{b}+\lambda
S_{ab}\mathcal{N}^{a}u^{b}$ have been used in order to obtain this particular
form of equation $(15)$. The corresponding integral term that occurs in
relation $(12)$, which results from converting the total divergence appearing
in equation $(11)$, is equal to the net flux of quasilocal bulk energy passing
through $\Omega_{t}$ being transferred from the gravitating physical system
under consideration (through the spatial boundary) to its environment.

Upon closer inspection of the full boundary part of integral expression
$(12)$, it can further be observed that said part of the expression (on
account of its physical units) can be interpreted as a power functional of the
form $\mathcal{P}=\underset{\Omega_{t}}{\int}\mathcal{I}\omega_{q}$, where the
corresponding integrand reads

$\mathcal{I}=\dot{\mathcal{N}}e-j_{a}\dot{\mathcal{N}}^{a}+\frac{\mathcal{N}}{2}s_{ab}\dot{q}^{ab}-\mathfrak{X}.$
(16)

Sure enough, however, such an interpretation only truly makes sense if
electromagnetic and/or gravitational radiation passes through the quasilocal
surface $\Omega_{t}$; a case in which the quantity $\mathcal{I}$ typically can
be identified as the intensity of the total radiant energy escaping from the
system into the environment. But as shall be discussed in greater detail in
the next section, where applications will be considered, the integral
$\underset{\Omega_{t}}{\int}\mathfrak{X}_{2}\omega_{q}$ cannot be neglected.

Before that, however, equation $(12)$ shall first be considered from a
different angle in order to also illuminate the geometric aspect of the chosen
approach. For this purpose, one may rewrite $(12)$ using the field equations
of the theory, which yields the result

$\displaystyle\frac{dH}{dt}$
$\displaystyle=\underset{\Sigma_{t}}{\int}(\dot{N}\mathcal{H}-\mathcal{H}_{a}\dot{N}^{a}+\frac{N}{2}\mathcal{Q}_{ab}\dot{h}^{ab})d^{3}x-$
(17)
$\displaystyle-\underset{\Omega_{t}}{\int}(\dot{\mathcal{N}}\mathfrak{h}-\mathfrak{h}_{a}\dot{\mathcal{N}}^{a}+\frac{\mathcal{N}}{2}\mathfrak{Q}_{ab}\dot{q}^{ab}-\sqrt{q}\mathfrak{X})d^{2}x$

for the temporal variation of the gravitational Hamiltonian. As may be noted,
the definitions

$\displaystyle\mathcal{Q}_{ab}$
$\displaystyle=\frac{\sqrt{h}}{8\pi}\\{{}^{(3)}R_{ab}-2K_{ac}K_{\;b}^{c}+KK_{ab}-\frac{1}{N}\left[\dot{K}_{ab}+(ND)K_{ab}+D_{a}D_{b}N\right]-$
$\displaystyle-\frac{1}{2}h_{ab}\left({}^{(3)}R+K^{2}-K_{cd}K^{cd}-\frac{2}{N}\left[\dot{K}+(ND)K+D_{a}D^{a}N\right]\right)\\}$
(18)

and

$\mathfrak{Q}_{ab}=\frac{\sqrt{q}}{8\pi}(\mathfrak{K}_{ab}-(\mathfrak{K}-b_{a}u^{a})q_{ab})$
(19)

have been used in the present context. Moreover, as may also be noted,
relation $(18)$ has been used in combination with the constraint equations of
the theory to rewrite relations $(13)$ and $(14)$ in the form

$\mathfrak{X}_{1}=\mathcal{N}^{2}\mathcal{D}_{a}\tau_{\;b}^{a}v^{b}+\mathcal{N}\mathcal{D}_{a}\tau_{\;b}^{a}\mathcal{N}^{a}$
(20)

and

$\displaystyle\mathfrak{X}_{2}$
$\displaystyle=-\frac{\eta}{\lambda}\mathcal{N}^{2}\mathcal{H}-\frac{1}{2}\eta\mathcal{N}^{2}\mathcal{H}_{a}v^{a}-\frac{1}{2}\eta\mathcal{N}\mathcal{H}_{a}\mathcal{N}^{a}-$
(21)
$\displaystyle+\frac{\mathcal{N}^{2}}{2\lambda}\mathcal{H}_{a}u^{a}+\lambda\mathcal{N}\mathcal{Q}_{ab}v^{a}u^{b}+\lambda\mathcal{Q}_{ab}\mathcal{N}^{a}u^{b}$

and therefore obtain with $(17)$ a purely geometrical expression that agrees
exactly with the time derivative of the quasilocal gravitational Hamiltonian.
Here, as may be noted, the use of Einstein’s equations (or rather the
constraint equations resulting from them) does not prove to be mandatory in
this context, but rather practical, since all calculations could also be
carried out exclusively with geometric quantities, which, of course, yields
exactly the same result in the end.

As may be noted, there is something both astonishing and appealing about
results $(12)$ and $(17)$, namely, that the occurring bulk and boundary
integral expressions look very similar; only the correction terms prevent an
exact similarity. But not only this similarity is interesting: From the
structure of $(17)$ it can be generally concluded that a change of the matter
content of a spatially and temporally bounded gravitating system always
depends on how the individual components of the metric evolve over time.
Moreover, it can be concluded that such a change also strongly depends on the
structure of the occurring correction terms, which can be interpreted
physically as a bounded mass energy distribution plus a non-vanishing bulk to
boundary inflow that combines the dynamical degrees of freedom of the bulk
with those of the boundary; even if it is assumed that the considered local
gravitational field is stationary. It can be expected that pseudotensorial
methods lead to similar results, although further studies of such an approach
may be in order.

Anyway, to illustrate the physical relevance of the derivations made, some
selected applications will now be discussed in the next section, with a
particular focus on possibly testable implications of the theory. In
particular, two effects will be examined in more detail, namely the loss of
quasilocal mass energy due to $i)$ tidal deformation and heating, and $ii)$
the emission of gravitational, electromagnetic, and/or gravitoelectromagnetic
radiation. With respect to the first type of effect, as it turns out, the
model proves consistent with the results of Booth and Creighton in the sense
that the derived correction terms play no role in calculating the loss of
quasilocal mass due to tidal heating. The reason for this, as will be shown,
is that the authors’ approach remains fully valid precisely when the source-
free Einstein equations are considered (as done in [6]), whereas it must very
well be extended if matter fields are present and thus the non-vacuum field
equations apply. However, this is exactly what is to be expected in the
presence of tidal effects which lead to perturbations not only of the external
but also of the internal field of a body. Something similar is also to be
expected, as shall be made clear below, in the second case to be discussed,
which is devoted to the combined emission of gravitational and electromagnetic
radiation by a gravitating physical system.

##  3 Tidal Heating, Gravitoelectromagnetism and further Applications

In view of the results derived in the previous section, the question naturally
arises whether or not the derived correction terms play a role for the
quasilocal calculation of tidal heating, and whether or not, therefore, the
original results of Booth and Creighton from [6] need to be modified or
generalized in some sense. Unsurprisingly, as will be explained below, the
authors’ approach turns out to be perfectly sound (in terms of the specific
assumptions made in their work). However, as shall also be made clear below,
said approach needs very well to be generalized in cases where quasilocal
methods are to be used to describe more violent tidal deformations and heating
effects than those discussed in [6], such as in cases of tidal deformations of
celestial bodies in $N$-body systems that are accompanied by substantial mass
and/or radiation transfer through the quasilocal surface, as occur for example
when one of the bodies collapses or pairs of bodies merge with each other.

Before discussing mass loss due to such effects, however, it is first
necessary to show that Booth’s and Creighton’s results occur as a special case
of those derived in the previous section. To show this, it is sufficient to
make a single assumption, namely that the considered spacetime
$(\mathcal{M},g)$ is Ricci-flat and therefore the on-shell vacuum field
equations

$R_{ab}=0$ (22)

are satisfied; a requirement that has the effect that the time derivative of
the bulk part appearing in $(12)$ becomes zero in the same way as the derived
correction terms, which further implies that this relation reduces to the form

$\frac{dE}{dt}=-\underset{\Omega_{t}}{\int}(\dot{\mathcal{N}}e-j_{a}\dot{\mathcal{N}}^{a}+\frac{\mathcal{N}}{2}s_{ab}\dot{q}^{ab})\omega_{q}.$
(23)

Consequently, however, by utilizing the fact that
$\dot{q}^{ab}=-q^{ac}q^{bd}\dot{q}_{cd}$ holds in the given context, one finds
that relations $(6)$, $(12)$ and $(23)$ all coincide in the case that the
source-free Einstein equations are satisfied, thus making it clear that the
results of the work of Booth and Creighton occur as a special case of the
results derived in section two of this work. Based on this, it can be further
concluded that the temporal variation of the full Hamiltonian (bulk plus
boundary) needs to be considered only when matter fields are present and
spacetime is not Ricci-flat, while in the opposite case it is sufficient to
consider the boundary part of the gravitational Hamiltonian, as was done in
[6].

Having clarified this, it shall now be briefly be reviewed how the obtained
formula can be used to describe tidal deformation and heating effects. To this
end, one may recall that Booth and Creighton used the Hartle-Thorne model [13,
28] as a basis for their quasilocal calculations, i.e., a perturbative vacuum
model that describes how the external field of a small self-gravitating
central body changes under the influence of an external tidal field. This very
model deals with a spacetime metric $g_{ab}$ of the form

$g_{ab}=\eta_{ab}+e_{ab},$ (24)

which is given with respect to the flat Minkowski metric $\eta_{ab}$, where
$|e_{ab}|\ll 1$ applies by definition. In rectangular coordinates, the
individual components of the perturbation tensor $e_{ab}$ take the form
$e_{00}\equiv\phi\equiv-\frac{2M}{r}-\frac{3Q_{ik}s^{i}s^{k}}{r}+r^{2}\mathcal{E}_{ik}s^{i}s^{k}$,
$e_{0j}\equiv
A_{j}\equiv-\frac{2}{r}\frac{dQ_{jk}}{dt}s^{k}+\frac{4r^{3}}{21}\frac{d\mathcal{E}_{jk}}{dt}s^{k}-\frac{10r^{3}}{21}\frac{d\mathcal{E}_{ik}}{dt}s^{i}s^{k}s_{j}$
and
$e_{ij}=(\frac{2M}{r}-\frac{3}{r^{3}}Q_{kl}s^{k}s^{l}+r^{2}\mathcal{E}_{kl}s^{k}s^{l})\delta_{ij}$,
where, in this regard, the field
$Q_{ik}=\int\varepsilon(x_{i}x_{k}-\frac{1}{3}r^{2}\delta_{ik})d^{3}x$
represents the tracefree and symmetric quadrupole moment of the self-
gravitating body and $\mathcal{E}_{ik}=\mathcal{R}_{i0k0}$ the electric part
of the Weyl tensor, which coincides with the Riemann curvature tensor in the
given case. The radial unit vector occurring in this context is given by
$s^{i}=\frac{x^{i}}{r}$, where $r$ is the radial distance from the center of
the isolated body as measured in its local asymptotic rest frame.

Taking advantage of the gauge ambiguity in setting up a nearly Minkowskian
coordinate system and switching to spherical coordinates, it was shown in [6]
that the Hartle-Thorne metric $(24)$ can be converted into a slightly
different form. More specifically, by using the diffeomorphism freedom to make
the replacement $e_{ab}\rightarrow e_{ab}+2\partial_{(a}\xi_{b)}$ with respect
to the co-vector field $\xi_{j}=\alpha r^{-2}Q_{jk}s^{k}+\beta
r^{3}\mathcal{E}_{jk}s^{k}+\gamma r^{3}\mathcal{E}_{ik}s^{i}s^{k}s_{j}$, where
$\alpha$, $\beta$ and $\gamma$ are free constants of order one, it was shown
it the aforementioned work that the rate at which tidal work is performed on
an isolated body is given (modulo higher order terms) by the same formula as
in the post-Newtonian approximation of the theory, which had previously been
derived by using pseudotensorial techniques [11, 24]. Using here the shorthand
notation
$\mathfrak{Y}=\frac{1}{60}\frac{d}{dt}\left[2ar^{5}\mathcal{E}_{ik}\mathcal{E}^{ik}+2b\mathcal{E}_{ik}Q^{ik}-cr^{5}Q_{ik}Q^{ik}\right]$
in conjunction with the definitions
$a=8\beta\gamma+4\gamma^{2}-2\beta^{2}+4\gamma-2\beta-3,b=8\alpha\gamma-12\gamma+6\beta-2\alpha+3$
and $c=4\alpha^{2}-12\alpha-9$, the obtained formula reads

$\frac{dE}{dt}=\frac{1}{2}\mathcal{E}_{ik}\frac{dQ^{ik}}{dt}+\mathfrak{Y}.$
(25)

In this context, the $\frac{1}{2}\mathcal{E}_{ik}\frac{dQ^{ik}}{dt}$-term
specifies the irreversible (unrecoverable) part of the work done to deform and
heat the system (the dissipated energy is converted into heat), whereas the
second $\mathfrak{Y}$-term represents the reversible (recoverable) part of the
work being done to increase the potential energy of the system. Regarding
these two terms, the former irreversible one turns out to be the one of
primary interest, since $i)$ it is invariant under diffeomorphisms genererated
by $\xi_{j}$ (in contrast to the second reversible term) and $ii)$ it is the
same leading order term that was previously obtained through analogous
calculation using pseudotensorial instead of quasilocal methods (which,
surprisingly, all lead to the same expression regardless of the choice of
energy-momentum complex and gauge conditions [11, 24]; a situation previously
known only from the case of pseudotensors calculated from Kerr-Schild metrics
[22, 30, 31]). Although not really obvious at the time, this compatibility of
results is not very surprising from today’s viewpoint in that it has been
shown by Chang, Chen and Nester [10] that any energy-momentum pseudotensor
leads to an associated Hamiltonian boundary term, so that it becomes clear
that - within the quasilocal Hamiltonian framework - pseudotensors either give
rise to distinct Dirichlet type boundary conditions or to Neumann type
boundary conditions; all of which are algebraic in terms of the metric.
Consequently, however, since the tidal heating results obtained in the
literature appear to be independent of the choice of boundary conditions, it
is hardly surprising that the purely quasilocal calculation of Booth and
Creighton leads to very similar results as those of earlier works on the
subject that used the pseudotensorial approach. This is not least the case
because, in the approximate calculations mentioned, the precise form of the
boundary conditions is not essential, because the results obtained by using
different boundary conditions deviate from each other only in higher order.
Therefore, from a purely mathematical point of view, the good agreement of the
results proves to be completely natural.

However, what is striking in this context is that, in contrast to the results
of earlier pseudotensorial approaches, the quasilocal results are not only
completely general but also coordinate-independent, which is why the results
obtained could in principle also be used to determine tidal deformations and
heating effects in more sophisticated and physically challenging situations;
that is, in situations where not only fluctuations of the external
gravitational field of the body are considered, but also changes of the
interior field of the self-gravitating body are taken into account. After all,
it has to be kept in mind that the Thorne-Hartle model is not designed to
describe directly the tidal deformations of a self-gravitating body, but only
the backreaction on its external gravitational field. Backreactions to the
interior field of the body are not accounted for by the model, which is why it
can only be used to determine the effects of tidal heating if the overall
structural changes of the body are sufficiently small and thus negligible.

Situations in which disregarding such structural changes no longer seems
justified are those in which the shape, composition, and macroscopic thermal
state of a given body (or collection of bodies) change drastically during
gravitational interaction, e.g., when the body becomes heavily tidally
deformed and heated during interaction with another, more massive celestial
body, which may eventually lead to the collapse and disintegration of the body
and its accretion by its much more massive companion. In such a situation,
with both celestial bodies typically orbiting each other in circular motion,
it is reasonable to expect that matter is constantly escaping from the system
through the quasilocal surface of spacetime; at least as long as one assumes
(inspired by Newton’s theory of gravity) that the quasilocal surface
represents the body’s gravisphere, i.e., the spatially bounded sphere of
influence of the collapsing body under consideration. Certainly also the
collision of bodies - a process in which similarly strong tidal deformations
can occur - cannot be described on the basis of a vacuum model like that of
Hartle and Thorne.

It must be acknowledged, however, that describing the aforementioned
phenomenological effects (even at the numerical level) is an extremely
complicated venture; especially given the need to describe how the body’s
internal gravitational field changes such that the matter content of the
spatially and temporally bounded gravitational system changes relative to its
local physical environment.

Yet, despite the lack of a geometric model that could fully account for these
effects, it shall be argued below that the integral expression $(12)$ of the
previous section can indeed be used to predict the behavior of very general
types of matter distributions whose general form appears suitable to describe
(at least in principle) processes in which objects are very strongly tidally
deformed due to the phenomenological effects mentioned above. Moreover, it
will be argued why and to what extent the Hartle-Thorne approximation proves
insufficient to describe gravitational backreactions caused by isolated bodies
that fail to be in almost perfect (macroscopic) internal thermal equilibrium.

As a basis for the arguments used, the theory of extended irreversible
thermodynamics [15, 16, 18, 19, 20, 21] shall be used at this point, with the
main idea being that the corresponding physical framework is general enough to
consider the case where tidal heating of a body (or collection of bodies)
leads to the occurrence of heat flows and viscous stresses, and thus to non-
negligible backreactions on the internal geometric field of the body (or
bodies). In the mentioned theory, the matter distribution under consideration
is that of a fluid mixture. Such a matter distribution can be fully
characterized by three different quantities, i.e. the so-called thermal
energy-momentum tensor $T_{\;b}^{a}=\epsilon
w^{a}w_{b}+\varpi^{a}w_{b}+w^{a}\varpi_{b}+(p+\Pi)h{}_{\,b}^{a}+\Pi{}_{\,b}^{a}$,
a particle number current $N_{A}^{a}=n_{A}w^{a}+\nu_{A}^{a}$, defined with
respect to a number $A$ of species of particles, and a so-called covariant
entropy current $S^{a}=sw^{a}+\eta^{a}$, where, in this context, the scalar
fields $\epsilon$, $p$ and $s$ represent the energy density, pressure and
total entropy density of the fluid mixture, $n_{A}$ is the particle density of
a number of particle species $A$ and $\pi$ represents the viscous bulk
pressure. The vector fields $w^{a}$, $\varpi^{a},$ $\nu_{A}^{a}$ and
$\eta^{a}$ meanwhile represent the four-velocity vector of the system, an
energy flux current and so-called particle diffusion and heat conduction
fluxes, while the tensor field $\pi{}_{\,b}^{a}$ represents the trace-free
anistropic viscous stress-tensor333As may be noted, all of these fields,
including the thermal stress-energy tensor, could of course also be written
down in $3+1$-form, but as will be made clear below, specifying these
quantities in this form will prove unnecessary for the ensuing argument..

Depending crucially on whether one wishes to specify these quantities in the
so-called energy or particle reference frames, different choices for some of
the vector fields can typically be made, namely either
$\nu_{A}^{a}=-\eta^{-1}q^{a}$, $\eta^{a}=\eta^{-1}\Theta q^{a}-Q^{a}$ and
$p^{a}=0$ or $\nu_{A}^{a}=0$, $\eta^{a}=\beta q^{a}-Q^{a}$ and $p^{a}=q^{a}$,
where $\beta=T^{-1}\equiv(\frac{\partial\varepsilon}{\partial s})^{-1}$ is the
inverse of the infamous Tolman-Ehrenfest temperature, $\eta$ is the
relativistic enthalpy or ’injection energy’ per particle, $\Theta$ is the so-
called thermal potential of the fluid mixture (relativistic chemical potential
per particle and per temperature) and $Q^{a}$ is a second order contribution
which may be interpreted as a generalized heat flux and for which different
approaches exist in the literature.

With respect to one of these choices, the physical behavior of fluid mixtures
can be specified exactly in almost equilibrium states, in which the
expressions given below must follow the differential laws

$\nabla_{a}T_{\;b}^{a}=0,\>\nabla_{a}N_{A}^{a}=0,\>\nabla_{a}S^{a}\geq 0.$
(26)

The respective laws are the laws of local conservation of energy and particle
number and the so-called Clausius-Duhem inequality, which is nothing more than
the differential form of the entropy law of thermodynamics.

All this is important for the subject of tidal heating for the following
reason: it makes a big difference for the form of the bulk part of the
integral relation $(12)$ whether the matter distribution under consideration
is in thermal equilibrium with its environment or not. More precisely, it
turns out that the bulk part of said relation is always zero (or, more
precisely, can be chosen to be zero) for fluid mixtures that are in perfect
thermal equilibrium, while this is not true for corresponding matter fields
outside thermal equilibrium. For such matter distributions, as it turns out,
the bulk part is always non-zero.

To see this, one may recall that the main equilibrium condition for
relativistic viscous fluids in extended irreversible thermodynamics (as well
as in classical theory) to be in thermal equilibrium is $\nabla_{a}S^{a}=0$; a
condition that requires the complete absence of dissipative mechanisms. For
this condition to be satisfied and for the corresponding system to actually
reach a local thermostatic equilibrium state, the heat exchange between the
fluids must come to a halt and the sum of all thermal potentials and chemical
reaction rates must approach zero. Besides that, the thermal energy-momentum
tensor and the particle number and entropy currents of the fluid mixture must
be the same as that of an ordinary ideal fluid, meaning that
$T_{\;b}^{a}=(\epsilon+p)w^{a}w_{b}+p\delta{}_{\,b}^{a}$,
$N_{A}^{a}=n_{A}\cdot w^{a}$ and $S^{a}=s\cdot w^{a}$. For this to hold, an
equilibrium equation of state of the form
$\epsilon+p=T(s+\underset{A}{\sum}\Theta{}_{A}n_{A})$ must be valid, where
each $\Theta_{A}$ represents a thermal potential associated with a species $A$
of particles (characterized in terms of the chemical potential
$\mu_{A}\equiv\frac{\partial\epsilon}{\partial n_{A}}$). Ultimately, moreover,
the motion of the fluids should be rigid in Born’ s sense, which means that
the orthogonal distances between the adjacent material world lines of the two
fluids must remain constant. In effect, this means that
$h_{a}^{\>c}h_{b}^{\>d}\nabla_{(c}w_{d)}=0$ must hold, which in turn means
that $\nabla_{(a}w_{b)}=0$ must be satisfied. However, from this it follows
that the matter field under consideration must, at perfect thermal
equilibrium, curve spacetime in such a way that a stationary gravitational
field is generated with a timelike Killing vector field $\xi^{a}$ that is
directly proportional to the four-velocity $w^{a}$, i.e. $\xi^{a}=\beta w^{a}$
with $\beta\equiv c\cdot N\equiv c\cdot\sqrt{-\xi_{a}\xi^{a}}$ with $c=const.$

And while it is also possible in principle for internal thermal equilibrium to
occur with respect to non-Killing observers, it is still reasonable to expect
that the variables $\dot{N}$, $\dot{N}^{a}$, and $\dot{h}^{ab}$ for fluid
mixtures in aequilibrio are very close to zero (with respect to the Killing
time parameter, which is admittedly the most natural choice). From this, in
turn, it becomes clear that in non-equilibrium situations the bulk part of
relation $(12)$ can only be neglected and the Thorne-Hartle model can only be
in agreement with the results of the quasilocal canonical framework of Brown
and York if the $3+1$-quantities $\varepsilon$, $p_{a}$, and $S_{ab}$
introduced in the previous section are all exactly or at least close to zero.

For quasilocal surfaces, which are located very far from the boundary of the
spatially finite internal gravitational field of the matter distribution under
consideration, one can safely assume that the mentioned quantities actually
become zero and thus for weak gravitational fields the Thorne-Hartle
approximation remains valid and leads to correct results. An important reason
for this is that dissipative heating of small, weakly gravitating bodies (such
as, for example, Jupiter’s satellite Io) it can be expected that only small
amounts of electromagentic radiation are emitted in the course of the tidal
deformation process. However, for gravitational interactions of more massive
objects, especially when a body is torn apart by tidal deformations or
otherwise becomes unstable during the interaction period (with the consequence
that not only the outer vacuum field of the body fluctuates, but also the
inner field, which can lead to the emission of non-negligible amounts of
electromagnetic radiation), this can no longer be expected to be true,
implying that the Hartle-Thorne approximation can no longer be used and it
becomes necessary to consider integral relation $(12)$ instead of relation
$(23)$. For certain types of matter accumulations, the structural changes in
the gravitational source may even cause mass-energy to flow out of the system
at sometimes very high velocities (think of relativistic jet flows in
accreting supermassive black holes). Such phenomena are typically expected
when bodies merge with each other; an effect that, like tidal heating, is
described in various places in the literature based on the linearized
approximation of general relativity [1, 3, 9, 10, 12, 17, 26].

Now, all this has an interesting consequence: the possibility of jointly
measuring gravitational and electromagnetic radiation emitted during violent
astrophysical events such as those described above suggests that the
corrections calculated in the second part of the present work can lead to
actual observable effects. More specifically, when considering relation (16),
it becomes clear that if the Brown-York expression is to be a physically
feasible candidate for gravitational mass-energy not only in theory, but also
in reallife situations, there should be a shift in the total intensity when
electromagnetic and gravitational radiation are emitted simultaneously from a
gravitational matter source. This effect should then be measurable. Moreover,
a comparable intensity shift should also occur when gravitoelectromagnetic
effects are measured.

To illustrate this, the mass-energy inflow through the quasilocal surface of a
spatially and temporally bounded gravitating system caused by
gravitoelectromagnetic fields shall be calculated next. Using for this purpose
a perturbative splitting of the metric of the form $(24)$, where the
perturbation field $e_{ab}$ is used to define the trace-reversed gravitaitonal
potential $\psi_{ab}=e_{ab}-\frac{1}{2}\eta_{ab}e$, which is subject to the
conditions $\partial^{a}\psi_{ab}=0$, $|\psi_{00}|\gg|\psi_{jk}|$ and
$|\psi_{0i}|\gg|\psi_{jk}|$, one finds that the linearized Einstein’s
equations reduce to the form

$\square\psi_{ab}=16\pi T_{ab}.$ (27)

Here, it can be assumed for simplicity’s sake that $T_{jk}=0$ for $j,k=1,2,3$
and therefore $S_{ab}$ applies. Moreover, it can be assumed that $s_{ab}=0$.

By defining next the fields $\psi_{00}(x)\equiv 2\phi(x)$ and
$\psi_{0k}(x)\equiv 2A_{k}(x)$, one finds that the linearized field equations
read

$\square\phi=16\pi\varepsilon,\;\square A_{k}=16\pi p_{k},$ (28)

where $k=1,2,3$ shall apply in the present context. The form of the solutions
of these Poisson equations is

$\phi(t,\vec{x})=\int\frac{\phi(t-|\vec{x}-\vec{y}|,\vec{y})}{|\vec{x}-\vec{y}|}d^{4}y$
(29)

and

$\vec{A}(t,\vec{x})=\int\frac{\vec{A}(t-|\vec{x}-\vec{y}|,\vec{y})}{|\vec{x}-\vec{y}|}d^{4}y.$
(30)

The resulting line element of the metric reads

$ds^{2}=-\left(1-2\phi\right)dt^{2}-4A_{k}dtdx^{k}+\left(1+2\phi\right)\delta_{jk}dx^{j}dx^{k}$
(31)

in Cartesian coordinates. The solutions obtained can be used to define
gravielectric and gravimagnetic field strengths
$\mathcal{\vec{E}}:=\frac{1}{2}\vec{\nabla}\phi$ and
$\mathcal{\vec{B}}:=\vec{\nabla}\times\vec{A}$, which in turn can be used to
set up the gravitoelectromagnetic pendants to the Maxwell equations and the
Lorentz force law, i.e. the following set of relations

$\displaystyle\vec{\nabla}\mathcal{\vec{E}}$ $\displaystyle=4\pi\varepsilon$
(32) $\displaystyle\vec{\nabla}\mathcal{\vec{B}}$ $\displaystyle=0$ (33)
$\displaystyle\vec{\nabla}\times\mathcal{\vec{E}}$
$\displaystyle=-\partial_{t}\mathcal{\vec{B}}$ (34)
$\displaystyle\vec{\nabla}\times\mathcal{\vec{B}}$
$\displaystyle=-4\pi\vec{p}+\partial_{t}\mathcal{\vec{E}}$ (35)

and

$\vec{F}=m\left(\mathcal{\vec{E}}+4\vec{v}\times\mathcal{\vec{B}}\right),$
(36)

which have to be met for the sake of consistency.

By considering the fact that $N=\sqrt{g^{tt}}=\sqrt{1+2\phi}\approx 1+\phi$,
one finds by neglecting higher order terms that relation $(12)$ takes the form

$\frac{dE}{dt}=\underset{\Sigma_{t}}{\int}(\dot{\phi}\varepsilon-
p_{k}\dot{A}^{k})\omega_{h}-\underset{\Omega_{t}}{\int}(\lambda\dot{\phi}e-j_{k}\dot{\mathcal{A}}^{k}-\mathfrak{X})\omega_{q},$
(37)

where
$\varepsilon=\frac{1}{8\pi}\left(\mathcal{\vec{E}}^{2}+\mathcal{\vec{B}}^{2}\right)$
and $p_{k}=\frac{1}{\pi}\epsilon_{kij}\mathcal{E}_{i}\mathcal{B}_{j}$ are the
GEM energy current density and the GEM poynting vector and
$\mathcal{A}^{b}=q_{\;c}^{b}A^{c}$ is the vector potential on the quasilocal
boundary surface. The crucial observation in this context is that the second
corrective part $\mathfrak{X}_{2}$ of the integrand $\mathfrak{X}$ is
different from zero. It reads

$\mathfrak{X}_{2}=-\lambda\eta\left(1+2\phi\right)\varepsilon+\lambda^{2}\eta\left(1+2\phi\right)p_{k}v^{k}+\lambda\eta\left(1+\phi\right)p_{k}\mathcal{A}^{k}-\left(1+2\phi\right)p_{k}u^{k}.$
(38)

Imagining now that the two-surface $\Omega_{t}$ is a sphere at infinity, it
becomes clear that the corresponding integral expression will also be
different form zero, i.e $\underset{\Omega_{t}\equiv
S_{\infty}}{\int}\mathfrak{X}_{2}\omega_{q}\neq 0$.

From this, however, it now follows that the quasilocal treatment of
gravitoelectromagnetic effects requires consideration of the corrections
resulting from the time derivative of the Brown-York mass given in the second
section of this paper. Similar corrections should also arise, as mentioned
earlier, when gravitational and electromagnetic radiation from a strongly
gravitating matter source are measured cooperatively, since also in such a
case a non-vanishing bulk-to-boundary energy inflow term should arise, and
hence corrections to Einstein’s quadrupole formula. The reason for this is
that in both cases Einstein’s field equations are satisfied for a non-
vanishing matter source, which is why the correction terms resulting from the
time derivative of the total quasilocal gravitational Hamiltonian cannot be
neglected in either case.

From a practical point of view, the calculated corrections should manifest
themselves in the form of a shift in the intensity of the measured radiation.
It is therefore to be expected that the presence of a corresponding detectable
intensity shift of incoming gravitational and electromagnetic radiation could
provide a practical test of the utility and feasibility of the Brown-York
expression as a reasonable candidate for the mass-energy of the gravitational
field.

## Conclusion

In the present work, previous results by Booth and Creighton on quasilocal
tidal heating were extended in that the mass-energy transfer through the
quasilocal surface was calculated in a different manner, namely by varying the
total gravitational Hamiltonian (bulk plus boundary parts) and not only the
boundary part. As it turned out, all the results obtained are fully compatible
with those of Booth’s and Creighton’s work if it is assumed that the vacuum
field equations of the theory are satisfied (as the authors actually did in
their work). If this is not the case, however, as was shown, correction terms
corrections resulting from the time derivative of the Brown-York mass must be
taken into account, one of which specifies how mass flows from the bulk into
the boundary. To demonstrate the feasibility of the results obtained, their
applicability to the geometric setting considered in Booth’s and Creighton’s
work on quasilocal tidal heating was demonstrated in the third section of the
work and a comparatively simple geometric example was treated, namely, the
mass-energy inflow through the quasilocal surface of a spatially and
temporally bounded gravitating system caused by gravitoelectromagnetic fields.
This simple example was used to show that the matter content of a spatially
and temporally bounded gravitating system can change with time such that mass-
energy flows through the quasilocal surface and thereby escapes from the
system. From this insight, it was concluded that the obtained results can be
used to describe tidal deformation and tidal heating effects in more
challenging physical situations, such as in the description of accretion
phenomena or merger processes in relativistic $N$-body systems.

Acknowledgements:

I want to greatly thank Ivan Booth for reading through and suggesting
improvements to the first draft of the manuscript. I would also like to thank
Friedrich Kupka for sharing his many insights into the phenomenology of
astrophysics.

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