# Generalized Misner-Sharp Energy in $f(R)$ Gravity

Rong-Gen Cai<EMAIL_ADDRESS>Key Laboratory of Frontiers in Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O.
Box 2735, Beijing 100190, China  Department of Physics, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan  Li-Ming Cao<EMAIL_ADDRESS>Department
of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan  Ya-Peng Hu
<EMAIL_ADDRESS>Key Laboratory of Frontiers in Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735,
Beijing 100190, China  Graduate School of the Chinese Academy of Sciences,
Beijing 100039, China  Nobuyoshi Ohta<EMAIL_ADDRESS>Department of
Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan

###### Abstract

We study generalized Misner-Sharp energy in $f(R)$ gravity in a spherically
symmetric spacetime. We find that unlike the cases of Einstein gravity and
Gauss-Bonnet gravity, the existence of the generalized Misner-Sharp energy
depends on a constraint condition in the $f(R)$ gravity. When the constraint
condition is satisfied, one can define a generalized Misner-Sharp energy, but
it cannot always be written in an explicit quasi-local form. However, such a
form can be obtained in a FRW universe and for static spherically symmetric
solutions with constant scalar curvature. In the FRW universe, the generalized
Misner-Sharp energy is nothing but the total matter energy inside a sphere
with radius $r$, which acts as the boundary of a finite region under
consideration. The case of scalar-tensor gravity is also briefly discussed.

PACS numbers: 04.20.Cv, 04.50.+h, 04.70.Dy

††preprint: KU-TP 036

##  I Introduction

A gravitational field has certainly an associated energy. However, it is a
rather difficult task to define energy for a gravitational field in general
relativity. A local energy density of gravitational field does not make any
sense because the energy-momentum pseudo-tensor of gravitational field, which
explicitly depends on metric and its first derivative, will vanish due to the
strong equivalence principle at any point of spacetime in a locally flat
coordinate NonLocal1 ; NonLocal2 ; Szabados . In general relativity, however,
there exist two well-known definitions of total energy; one is the Arnowitt-
Deser-Misner (ADM) energy $E_{ADM}$ at spatial infinity ADM , and the other is
the Bondi-Sachs (BS) energy $E_{BS}$ at null infinity Bondi  describing an
isolated system in an asymptotically flat spacetime.

Due to the absence of the local energy density of gravitational field, it is
tempting to define some meaningful quasi-local energy, which is defined on a
boundary of a given region in spacetime. Indeed, it is possible to properly
define such quasi-local energies. Some useful definitions for quasi-local
energy exist in the literature, for instance, Brown-York energy York , Misner-
Sharp energy Misner , Hawking-Hayward energy hawking ; hayward  and Chen-
Nester energy Chen , etc. A nice review on this issue can be found in Szabados
. In this article, we focus on the Misner-Sharp energy.

The Misner-Sharp energy $E$ is defined in a spherically symmetric spacetime.
Various properties of the Misner-Sharp energy are discussed in some detail by
Hayward in Hayward ; Hayward1 . For example, the following properties are
established. In the Newtonian limit of a perfect fluid, the Misner-Sharp
energy $E$ yields the Newtonian mass to leading order and the Newtonian
kinetic and potential energy in the next order. For test particles, the
corresponding Hajicek energy is conserved and has the behavior appropriate to
energy in the Newtonian and special-relativistic limits. In the small-sphere
limit, the leading term in $E$ is the product of volume and the energy density
of the matter. In vacuo, the Misner-Sharp energy $E$ reduces to the
Schwarzschild energy. At null and spatial infinity, $E$ reduces to the BS and
ADM energies, respectively. In particular, it is shown that the conserved
Kodama current produces the conserved charge $E$.

In a four-dimensional, spherically-symmetric spacetime with metric

$ds^{2}=h_{ab}dx^{a}dx^{b}+r^{2}(x)d\Omega_{2}^{2},$ (1)

where $a=0$, $1$, $x^{a}$ is the coordinate on a two-dimensional spacetime
$(M^{2},h_{ab})$ and $d\Omega_{2}^{2}$ denotes the line element for a two-
dimensional sphere with unit radius, the Misner-Sharp energy $E$ can be
defined as

$E(r)=\frac{r}{2G}\left(1-h^{ab}\partial_{a}r\partial_{b}r\right).$ (2)

With this energy, the Einstein equations can be rewritten as

$dE=A\Psi_{a}dx^{a}+WdV,$ (3)

where $A=4\pi r^{2}$ is the area of the sphere with radius $r$ and $V=4\pi
r^{3}/3$ is its volume, $W$ is called work density defined as
$W=-h^{ab}T_{ab}/2$ and $\Psi$ energy supply vector, $\Psi_{a}=T_{a}^{\
b}\partial_{b}r+W\partial r_{a}$, with $T_{ab}$ being the projection of the
four-dimensional energy-momentum tenor $T_{\mu\nu}$ of matter in the normal
direction of the 2-dimensional sphere. The form (3) is called “unified first
law” Hayward2 ; Hayward3 . Projecting this form along a trapping horizon, one
is able to arrive at the first law of thermodynamics for dynamical black hole

$\langle dE,\xi\rangle=\frac{\kappa}{8\pi G}\langle dA,\xi\rangle+W\langle
dV,\xi\rangle,$ (4)

where $\xi$ is a projecting vector and
$\kappa=\frac{1}{2\sqrt{-h}}\partial_{a}(\sqrt{-h}h^{ab}\partial_{b}r)$ is
surface gravity on the trapping horizon. Defining $\delta Q=\langle
A\Psi,\xi\rangle=TdS$, we can derive entropy formula associated with apparent
horizon in various gravity theories Cai-Cao1 ; Cai-Cao2 ; CCHK . Indeed, the
Minser-Sharp energy plays an important role in connection between the Einstein
equations and first law of thermodynamics in FRW cosmological setup Cai-Cao1 ;
Cai-Cao2 ; Cai-Kim ; CCHK ; AC07  and black hole setup Pad .

Note that the original form (2) for the Misner-Sharp energy is applicable for
Einstein gravity without cosmological constant in four dimensions, thus it is
tempting to give corresponding forms for the case with a cosmological constant
and/or in other gravity theories. Indeed, a generalized form is given for
Gauss-Bonnet gravity and more general Lovelock gravity in Maeda ; Maeda1 . In
particular, we would like to mention here that Gong and Wang in GW  introduce
a modified Misner-Sharp energy and discuss its relation to horizon
thermodynamics. With the generalized Misner-Sharp energy, it is shown that the
Clausius relation $\delta Q=TdS$ indeed gives correct entropy formula for
Lovelock gravity Cai-Cao1 ; CCHK .

Recently, a kind of modified gravity theories, $f(R)$, whose Lagrangian is a
function of curvature scalar $R$, has attracted a lot of attention. A main
motivation is to explain the observed accelerated expansion of the universe
without introducing the exotic dark energy with a large negative pressure. For
a review on $f(R)$ gravity, see Sotiriou . Of course, $f(R)$ gravity is a
simple generalization of Einstein gravity; when $f(R)=R$, it goes back to
Einstein theory. However, $f(R)$ is quite different from another
generalization of Einstein gravity, Lovelock gravity. The equations of motion
of the latter do not contain more than second-order derivatives, while the
equations of motion for the former do. In addition, let us notice that in some
sense, the $f(R)$ gravity is quite similar to scalar-tensor gravity, a
generalization of Einstein gravity again.

In this paper we are mainly concerned with the question whether there exists a
similar Misner-Sharp energy for $f(R)$ gravity in a spherically symmetric
spacetime. For this goal, we will take two methods, which are basically
equivalent, in fact. One is called integration method, and the other is
conserved charge method associated with the Kodama current. The integration
method is introduced in a previous paper of ours CCHK  for the case of
radiation matter in Lovelock gravity. We find that existence of a generalized
Misner-Sharp energy is not trivial for $f(R)$ gravity. Its existence depends
on a constraint. Once the constraint is satisfied, we could have a generalized
Misner-Sharp energy. Otherwise, the answer is negative. The same situation
happens for the scalar-tensor gravity theory.

The organization of the paper is as follows. In Sec. II, as a warm-up
exercise, we derive the generalized Misner-Sharp energy in Gauss-Bonnet
gravity by using the integration method and by generalizing the discussion in
CCHK  to more general matter content. In Sec. III, we discuss the generalized
Misner-Sharp energy in $f(R)$ gravity by the integration method and conserved
charge method, respectively. Sec. IV is devoted to investigating some special
cases, homogeneous and isotropic FRW cosmology and static spherically
symmetric case. In these cases the generalized Misner-Sharp energy has a
simple form. The conclusion and some discussions are given in Sec. V. In the
appendix, we briefly discuss the generalized Misner-Sharp energy for scalar-
tensor gravity in a FRW universe.

##  II Generalized Misner-Sharp energy in Gauss-Bonnet gravity: integration
method

The equations of motion of Gauss-Bonnet gravity can be written down as

$G_{\mu\nu}+\alpha H_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu},$ (5)

where

$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu},$ $\displaystyle H_{\mu\nu}$
$\displaystyle=$ $\displaystyle 2(RR_{\mu\nu}-2R_{\mu\alpha}R_{\nu}^{\
\alpha}-2R^{\alpha\beta}R_{\mu\alpha\nu\beta}+R_{\mu}^{\
\alpha\beta\gamma}R_{\nu\alpha\beta\gamma})-\frac{1}{2}g_{\mu\nu}L_{GB},$ (6)

and $\alpha$ is a coupling constant with dimension of length squared. The
Gauss-Bonnet term is
$L_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$.

Consider an $n$-dimensional spherically symmetric spacetime of metric in the
double-null form

$ds^{2}=-2e^{-\varphi(u,v)}dudv+r^{2}(u,v)\gamma_{ij}dz^{i}dz^{j},$ (7)

where $\gamma_{ij}$ is the metric on an $(n-2)$-dimensional constant curvature
space $K^{n-2}$ with its sectional curvature $k=\pm 1,0$, and the two-
dimensional spacetime spanned by two null coordinates $(u,v)$ and its metric
are denoted as $(M^{2},h_{ab})$. Thus, the equations of gravitational field
(5) can be written explicitly as Maeda1

$\displaystyle-\frac{8\pi G}{n-2}rT_{uu}$ $\displaystyle=$
$\displaystyle(r_{,uu}+\varphi_{,u}r_{,u})\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]},$
$\displaystyle-\frac{8\pi G}{n-2}rT_{vv}$ $\displaystyle=$
$\displaystyle(r,_{vv}+\varphi,_{v}r,_{v})\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]},$
$\displaystyle\frac{8\pi G}{n-2}r^{2}T_{uv}$ $\displaystyle=$ $\displaystyle
rr,_{uv}+(n-3)r,_{u}r,_{v}+\frac{n-3}{2}ke^{-\varphi}+\frac{{\tilde{\alpha}}}{2r^{2}}[(n-5)k^{2}e^{-\varphi}+4rr,_{uv}(k+2e^{\varphi}r,_{u}r,_{v})$
(8)
$\displaystyle+4(n-5)r,_{u}r,_{v}(k+e^{\varphi}r,_{u}r,_{v})]-\frac{n-1}{2}{\tilde{\Lambda}}r^{2}e^{-\varphi},$

where ${\tilde{\alpha}}=(n-3)(n-4)\alpha,$
${\tilde{\Lambda}}=2\Lambda/[(n-1)(n-2)].$

The essential point of the integration method is that, similar to the case of
Einstein gravity (3), one assumes the equations (8) of gravitational field can
be cast into the form

$dE_{eff}=A\Psi_{a}dx^{a}+WdV,$ (9)

where $A=V_{n-2}^{k}r^{n-2}$ and $V=V_{n-2}^{k}r^{n-1}/(n-1)$ are area and
volume of the $(n-2)$-dimensional space with radius $r$, and energy supply
vector $\Psi$ and energy density $W$ are defined on $(M^{2},h_{ab})$ as in the
case of Einstein gravity. The right hand side in (9) can be explicitly
expressed as

$A\Psi_{a}dx^{a}+WdV=A(u,v)du+B(u,v)dv,$ (10)

where

$\displaystyle A(u,v)$ $\displaystyle=$ $\displaystyle
V_{n-2}^{k}r^{n-2}e^{\varphi}(r,_{u}T_{uv}-r,_{v}T_{uu}),$ (11) $\displaystyle
B(u,v)$ $\displaystyle=$ $\displaystyle
V_{n-2}^{k}r^{n-2}e^{\varphi}(r,_{v}T_{uv}-r,_{u}T_{vv}).$ (12)

With the equations in (8), we can express $A$ and $B$ in terms of geometric
quantities as

$\displaystyle A(u,v)$ $\displaystyle=$ $\displaystyle\frac{V_{n-2}^{k}}{8\pi
G}e^{\varphi}(n-2)r^{n-4}\Big{\\{}\frac{e^{-\varphi}}{2r^{2}}r,_{u}[-(n-1){\tilde{\Lambda}}r^{4}+(n-3)r^{2}(k+2e^{\varphi}r,_{u}r,_{v})$
$\displaystyle+(n-5){\tilde{\alpha}}(k+2e^{\varphi}r,_{u}r,_{v})^{2}+2e^{\varphi}r^{3}r,_{uv}+4e^{\varphi}{\tilde{\alpha}}r(k+2e^{\varphi}r,_{u}r,_{v})r,_{uv}]$
$\displaystyle+rr,_{v}\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]}(\varphi,_{u}r,_{u}+r,_{uu})\Big{\\}},$
$\displaystyle B(u,v)$ $\displaystyle=$ $\displaystyle\frac{V_{n-2}^{k}}{8\pi
G}e^{\varphi}(n-2)r^{n-4}\Big{\\{}\frac{e^{-\varphi}}{2r^{2}}r,_{v}[-(n-1){\tilde{\Lambda}}r^{4}+(n-3)r^{2}(k+2e^{\varphi}r,_{u}r,_{v})$
(13)
$\displaystyle+(n-5){\tilde{\alpha}}(k+2e^{\varphi}r,_{u}r,_{v})^{2}+2e^{\varphi}r^{3}r,_{uv}+4{\tilde{\alpha}}re^{\varphi}(k+2e^{\varphi}r,_{u}r,_{v})r,_{uv}]$
$\displaystyle+rr,_{u}\Big{[}1+\frac{2{\tilde{\alpha}}}{r^{2}}(k+2e^{\varphi}r,_{u}r,_{v})\Big{]}(\varphi,_{v}r,_{v}+r,_{vv})\Big{\\}}.$

Now we try to derive the generalized Misner-Sharp energy by integrating the
equation (9). Clearly, if it is integrable, the following integrable condition
has to be satisfied

$\frac{\partial A(u,v)}{\partial v}=\frac{\partial B(u,v)}{\partial u}.$ (14)

It is easy to check that $A$ and $B$ given in (13) indeed satisfy the
integrable condition (14). Thus directly integrating (9) gives the
generalizing Misner-Sharp energy

$\displaystyle E_{eff}$ $\displaystyle=$ $\displaystyle\int
A(u,v)du+\int\Big{[}B(u,v)-\frac{\partial}{\partial v}\int A(u,v)du\Big{]}dv$
(15) $\displaystyle=$ $\displaystyle\frac{(n-2)V_{n-2}^{k}r^{n-3}}{16\pi
G}[-{\tilde{\Lambda}}r^{2}+(k+2e^{\varphi}r,_{u}r,_{v})+{\tilde{\alpha}}r^{-2}(k+2e^{\varphi}r,_{u}r,_{v})^{2}].$

Note that here the second term in the first line of (15) in fact vanishes and
we have fixed an integration constant so that $E_{eff}$ reduces to the Misner-
Sharp energy in Einstein gravity when ${\tilde{\alpha}}=0$. In addition, the
generalized Misner-Sharp energy can be rewritten in a covariant form

$\displaystyle E_{eff}$ $\displaystyle=$
$\displaystyle\frac{(n-2)V_{n-2}^{k}r^{n-3}}{16\pi
G}[-{\tilde{\Lambda}}r^{2}+(k-h^{ab}D_{a}rD_{b}r)+{\tilde{\alpha}}r^{-2}(k-h^{ab}D_{a}rD_{b}r)^{2}].$
(16)

This is the generalized Misner-Sharp energy given by Maeda and Nozawa in
Maeda1  through Kodama conserved charge method.

##  III Generalized Misner-Sharp energy in $f(R)$ gravity: the general case

In this section, we first try to derive the generalized Misner-Sharp energy in
$f(R)$ gravity by using the integration method. Then we consider the conserved
charge method. Here we consider the four-dimensional case with spherical
symmetry, and the line element is

$ds^{2}=-2e^{-\varphi(u,v)}dudv+r^{2}(u,v)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (17)

The action of the $f(R)$ gravity in the metric formalism is

$S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}f(R)+S_{matter},$ (18)

Varying the action with respect to metric yields equations of gravitational
field

$f_{R}R_{\mu\nu}-\frac{1}{2}fg_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}f_{R}+g_{\mu\nu}\square
f_{R}=8\pi GT_{\mu\nu},$ (19)

where $f_{R}=df(R)/dR,$ and $T_{\mu\nu}$ is the energy-momentum tensor for
matter field from $S_{matter}$. Note that the field equations also can be
rewritten in the form

$G_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{1}{f_{R}}\Big{[}\frac{1}{2}g_{\mu\nu}(f-Rf_{R})+\nabla_{\mu}\nabla_{\nu}f_{R}-g_{\mu\nu}\square
f_{R}\Big{]}+\frac{8\pi G}{f_{R}}T_{\mu\nu}.$ (20)

In this case, the right hand side can be regarded as an effective energy-
momentum tensor.

###  III.1 Integration method

In the metric (17), the field equations (19) can be explicitly expressed as

$\displaystyle 8\pi GT_{uu}$ $\displaystyle=$
$\displaystyle-2f_{R}\frac{\varphi,_{u}r,_{u}+r,_{uu}}{r}-f_{R},_{uu}-\varphi,_{u}f_{R},_{u},$
$\displaystyle 8\pi GT_{vv}$ $\displaystyle=$
$\displaystyle-2f_{R}\frac{\varphi,_{v}r,_{v}+r,_{vv}}{r}-f_{R},_{vv}-\varphi,_{v}f_{R},_{v},$
$\displaystyle 8\pi GT_{uv}$ $\displaystyle=$ $\displaystyle
f_{R}\varphi_{,uv}-2f_{R}\frac{r,_{uv}}{r}+\frac{1}{2}fe^{-\varphi}+f_{R},_{uv}+\frac{2r,_{u}f_{R},_{v}+2r,_{v}f_{R},_{u}}{r}.$
(21)

In this case, following the method discussed in the previous section, we
obtain

$\displaystyle A(u,v)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}e^{\varphi}(r,_{u}T_{uv}-r,_{v}T_{uu})$ $\displaystyle=$
$\displaystyle\frac{r^{2}e^{\varphi}}{2G}\Big{[}r,_{u}\Big{(}f_{R}\varphi_{,uv}-2f_{R}\frac{r,_{uv}}{r}+\frac{1}{2}fe^{-\varphi}+f_{R},_{uv}+\frac{2r,_{u}f_{R},_{v}+2r,_{v}f_{R},_{u}}{r}\Big{)}$
$\displaystyle+r,_{v}\Big{(}2f_{R}\frac{\varphi,_{u}r,_{u}+r,_{uu}}{r}+f_{R},_{uu}+\varphi,_{u}f_{R},_{u}\Big{)}\Big{]},$
$\displaystyle B(u,v)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}e^{\varphi}(r,_{v}T_{uv}-r,_{u}T_{vv})$ (22) $\displaystyle=$
$\displaystyle\frac{r^{2}e^{\varphi}}{2G}\Big{[}r,_{v}\Big{(}f_{R}\varphi_{,uv}-f_{R}\frac{2r,_{uv}}{r}+\frac{1}{2}fe^{-\varphi}+f_{R},_{uv}+\frac{2r,_{u}f_{R},_{v}+2r,_{v}f_{R},_{u}}{r}\Big{)}$
$\displaystyle+r,_{u}\Big{(}2f_{R}\frac{\varphi,_{v}r,_{v}+r,_{vv}}{r}+f_{R},_{vv}+\varphi,_{v}f_{R},_{v}\Big{)}\Big{]}.$

Checking the integrable condition, however, unlike the case of Gauss-Bonnet
gravity, we find that it is not always satisfied for the $f(R)$ gravity:

$\displaystyle\frac{\partial A(u,v)}{\partial v}-\frac{\partial
B(u,v)}{\partial u}$ $\displaystyle=$
$\displaystyle-r^{2}e^{\varphi}[(\varphi,_{u}r,_{u}+r,_{uu})(f_{R},_{vv}+\varphi,_{v}f_{R},_{v})-(\varphi,_{v}r,_{v}+r,_{vv})$
(23) $\displaystyle(f_{R},_{uu}+\varphi,_{u}f_{R},_{u})]/2G.$

If the right hand side of the above equation vanishes, in principle, one is
able to obtain a generalized Misner-Sharp energy by integrating (9). On the
other hand, if the integrable condition is not satisfied, one is not able to
rewrite the form $Adu+Bdv$ as a total differential form, which implies that
generalized Misner-Sharp energy does not exist in this case. Now we assume
that the integrable condition is satisfied, that is to say, the right hand
side of the equation (23) vanishes. Thus, we can obtain the generalized
Misner-Sharp energy for the $f(R)$ gravity as

$\displaystyle E_{eff}\ $ $\displaystyle=$ $\displaystyle\int
A(u,v)du+\int\Big{[}B(u,v)-\frac{\partial}{\partial v}\int A(u,v)du\Big{]}dv$
(24) $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1+2e^{\varphi}r,_{u}r,_{v})f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)+re^{\varphi}(f_{R},_{u}r,_{v}+f_{R},_{v}r,_{u})\Big{]}$
$\displaystyle-\frac{1}{2G}\int\Big{[}f_{R},_{u}e^{\varphi}(r^{2}r,_{v}),_{u}+f_{R},_{u}(r-\frac{1}{6}r^{3}R)+f_{R},_{v}r^{2}(r,_{u}e^{\varphi}),_{u}\Big{]}du$
$\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
$\displaystyle-\frac{1}{2G}\int\Big{[}f_{R},_{u}e^{\varphi}(r^{2}r,_{v}),_{u}+f_{R},_{u}\Big{(}r-\frac{1}{6}r^{3}R\Big{)}+f_{R},_{v}r^{2}(r,_{u}e^{\varphi}),_{u}\Big{]}du,$

where we have used
$R=2[\frac{1}{r}+e^{\varphi}(2\frac{r_{,u}r_{,v}}{r^{2}}-\varphi_{,uv}+4\frac{r_{,uv}}{r})]$
and $f_{,u}=f_{R}R_{,u}$. We see that $E_{eff}$ reduces to the Misner-Sharp
energy in the Einstein gravity when $f_{R}=1$. Unfortunately, we see from (24)
that due to the existence of the integration in (24), we cannot arrive at an
explicit quasi-local energy for the general case. In the next section,
however, we will show that the integration can be carried out in some special
cases. Before that, we will first obtain the same result using the conserved
charge method in the next subsection.

###  III.2 Conserved charge method

In a spherically symmetric spacetime, one can define a Kodama vector. The
energy-momentum tensor together with the Kodama vector can lead to a conserved
current, whose corresponding conserved charge is just the Misner-Sharp energy
in Einstein gravity. In Gauss-Bonnet gravity, Maeda and Nozawa Maeda1  obtain
the generalized Misner-Sharp energy with help of this method. In this section
we would like to see whether the conserved current method leads to a
generalized Misner-Sharp energy for the $f(R)$ gravity.

For a spherically symmetric spacetime, one can define the Kodama vector as
Kodama ; Sasaki

$K^{\mu}=-\epsilon^{\mu\nu}\nabla_{\nu}r,$ (25)

where $\epsilon_{\mu\nu}=\epsilon_{ab}(dx^{a})_{\mu}(dx^{b})_{\nu}$, and
$\epsilon_{ab}$ is the volume element of $(M^{2},h_{ab})$. For the spherically
symmetric spacetime (17), we have

$K^{\mu}=e^{\varphi}r,_{v}\Big{(}\frac{\partial}{\partial
u}\Big{)}^{\mu}-e^{\varphi}r,_{u}\Big{(}\frac{\partial}{\partial
v}\Big{)}^{\mu}.$ (26)

Conservation of the energy-momentum tensor for matter fields $T_{\mu\nu}$ in
(19) guarantees that the left hand side of the equation (19) is also
divergence-free, which can be easily checked by using the identity

$(\square\nabla_{\nu}-\nabla_{\nu}\square)F=R_{\mu\nu}\nabla^{\mu}F,$ (27)

where $F$ is an arbitrary scalar function. With the Kodama vector, define an
energy current as

$J^{\mu}=-T_{\nu}^{\mu}K^{\nu}.$ (28)

However, we find that unlike in the cases of Einstein gravity and Gauss-Bonnet
gravity Maeda1 , the energy current defined in (28) is not always divergence-
free for the $f(R)$ gravity except the case with condition

$\nabla_{\mu}\nabla_{\nu}f_{R}\nabla^{\mu}K^{\nu}=0.$ (29)

Namely, if the constraint equation (29) is satisfied, the energy current is
divergence-free

$\nabla_{\mu}J^{\mu}=0.$ (30)

In this case, we can define an associated conserved charge

$Q_{J}=\int_{\Sigma}J^{\mu}d\Sigma_{\mu},$ (31)

where $\Sigma$ is some hypersurface and
$d\Sigma_{\mu}=\sqrt{-g}dx^{v}dx^{\lambda}dx^{\rho}\delta_{\mu v\lambda\rho}$
is a directed surface line element on $\Sigma$. By using the line element in
(17) and equations in (21), we obtain

$\displaystyle Q_{J}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}J^{\mu}d\Sigma_{\mu}$ (32) $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
$\displaystyle-\frac{1}{2G}\int\Big{[}f_{R},_{u}e^{\varphi}(r^{2}r,_{v}),_{u}+f_{R},_{u}\Big{(}r-\frac{1}{6}r^{3}R\Big{)}+f_{R},_{v}r^{2}(r,_{u}e^{\varphi}),_{u}\Big{]}du,$

where we have chosen the hypersurface $\Sigma$ with a given $v$. One can
immediately see that the charge $Q_{J}$ is precisely the generalized Misner-
Sharp energy $E_{eff}$ given by the integration method in (24). Again, this is
not a satisfying situation since we cannot express the generalized Misner-
Sharp energy in a true quasi-local form.

##  IV Generalized Misner-Sharp energy in f(R) gravity: special cases

The existence of the integration in (24) is painful. An interesting question
is whether it will be absent in some special cases. The answer is positive. We
will here discuss two special cases. One is the homogeneous and isotropic FRW
universe and the other is the static spherically symmetric spacetime with
constant scalar curvature.

###  IV.1 FRW Universe

Consider the metric

$ds^{2}=-dt^{2}+e^{2\psi(t,\rho)}d\rho^{2}+r^{2}(t,\rho)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (33)

In this metric, the Kodama vector is

$K^{a}=e^{-\psi}r,_{\rho}\Big{(}\frac{\partial}{\partial
t}\Big{)}^{a}-e^{-\psi}r,_{t}\Big{(}\frac{\partial}{\partial\rho}\Big{)}^{a}.$
(34)

Following the same procedure, we can rewrite the equation in (10) as

$A\Psi_{a}dx^{a}+WdV=A(t,\rho)dt+B(t,\rho)d\rho.$ (35)

where

$\displaystyle A(t,\rho)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}e^{-2\psi}(T_{t\rho}r,_{\rho}-T_{\rho\rho}r,_{t}),$ $\displaystyle
B(t,\rho)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}(T_{tt}r,_{\rho}-T_{t\rho}r,_{t}).$ (36)

With the equations of gravitational field of the $f(R)$ gravity, $A$ and $B$
can be expressed in terms of geometric quantities. One can then arrive at the
generalized Misner-Sharp energy

$\displaystyle E_{eff}$ $\displaystyle=$ $\displaystyle\int
B(t,\rho)d\rho+\int\Big{[}A(t,\rho)-\frac{\partial}{\partial t}\int
B(t,\rho)d\rho\Big{]}dt$ (37) $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{r^{2}}{6}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
$\displaystyle+\frac{1}{2G}\int\Big{\\{}f_{R,_{\rho}}\Big{[}(-e^{-2\psi}r^{2}r,_{\rho}\psi,_{\rho}+e^{-2\psi}r^{2}r,_{\rho\rho}-r^{2}r,_{t}\psi,_{t})-r(1+r,_{t}^{2}-e^{-2\psi}r,_{\rho}^{2})+\frac{1}{6}r^{3}R\Big{]}$
$\displaystyle\qquad+r^{2}f_{R,t}(\psi,_{t}r,_{\rho}-r,_{t\rho})\Big{\\}}d\rho,$

where the integrable condition is assumed to be satisfied

$\frac{\partial A(t,\rho)}{\partial\rho}-\frac{\partial B(t,\rho)}{\partial
t}=0\text{ }.$ (38)

Now we express the generalized Misner-Sharp energy in a FRW metric

$ds^{2}=-dt^{2}+\frac{a^{2}(t)d\rho^{2}}{1-k\rho^{2}}+r^{2}(t,\rho)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (39)

where $r(t,\rho)\equiv a(t)\rho$, and
$e^{\psi(t,\rho)}=\frac{a(t)}{\sqrt{1-k\rho^{2}}}$ corresponding to (33).
Because in the FRW universe, the Ricci scalar
$R=6(\frac{k}{a^{2}}+\frac{\overset{.}{a}^{2}}{a^{2}}+\frac{\overset{..}{a}}{a})$
just depends on time, we can check that the integrand in the final step in
(37) exactly vanishes. Thus, the generalized Misner-Sharp energy $E_{eff}$ in
this case can be explicitly expressed as

$\displaystyle E_{eff}$ $\displaystyle=$
$\displaystyle\frac{r}{2G}\Big{[}(1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{1}{6}r^{2}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\Big{]}$
(40) $\displaystyle=$
$\displaystyle\frac{r^{3}}{2G}\left(\frac{1}{r_{A}^{2}}f_{R}+\frac{1}{6}(f-f_{R}R)+H\partial_{t}f_{R}\right),$

where $r_{A}=1/\sqrt{H^{2}+\frac{k}{a^{2}}}$, which in fact, is the location
of apparent horizon of the FRW universe.

###  IV.2 Static spherically symmetric case

The general line element of a static spherically symmetric spacetime can be
written down as

$ds^{2}=-\lambda(r)dt^{2}+g(r)dr^{2}+r^{2}d\Omega_{2}^{2},$ (41)

where $\lambda$ and $g$ are two functions of the radial coordinate. In this
case, the Kodama vector is

$K^{\mu}=\frac{1}{\sqrt{g\lambda}}\left(\frac{\partial}{\partial
t}\right)^{\mu}.\\\ $ (42)

Using the static spherically symmetric metric, we can easily check that the
constraint (29) is naturally satisfied

$\nabla_{\mu}\nabla_{\nu}f_{R}\nabla^{\mu}K^{\nu}=0.\\\ $ (43)

Following the same procedure, we can rewrite the equation in (10) as

$dE_{eff}=A(t,r)dt+B(t,r)dr,$ (44)

where

$\displaystyle A(t,r)$ $\displaystyle=$ $\displaystyle\frac{4\pi
r^{2}}{g}(T_{tr}r,_{r}-T_{rr}r,_{t})=0,$ (45) $\displaystyle B(t,r)$
$\displaystyle=$ $\displaystyle\frac{4\pi
r^{2}}{\lambda}(T_{tt}r,_{r}-T_{tr}r,_{t})$ (46) $\displaystyle=$
$\displaystyle\frac{r^{2}}{2G}\left(\frac{1}{2}(f-f_{R}R)+\frac{1}{r^{2}}(1+\frac{rg^{{}^{\prime}}}{g^{2}}-\frac{1}{g})f_{R}+f_{R,r}(\frac{g^{{}^{\prime}}}{2g^{2}}-\frac{2}{rg})-\frac{1}{g}f_{R,rr}\right),$

where a prime denotes the derivative with respect to $r$. Integrating (44)
gives the generalized Misner-Sharp energy

$\displaystyle E_{eff}$ $\displaystyle=$ $\displaystyle\int
B(t,r)dr=\frac{r}{2G}\left((1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{r^{2}}{6}(f-f_{R}R)-rh^{ab}\partial_{a}f_{R}\partial_{b}r\right)$
(47) $\displaystyle-\frac{1}{2G}\int
dr(\frac{r^{2}g^{{}^{\prime}}}{2g^{2}}+r-\frac{r}{g}-\frac{1}{6}r^{3}R)f_{R,r}.$

Clearly the integral in (47) will be absent in two cases, one is
$\frac{r^{2}g^{{}^{\prime}}}{2g^{2}}+r-\frac{r}{g}-\frac{1}{6}r^{3}R=0$, the
other is $f_{R,r}=0$. We here consider the latter case. The trivial case with
$f(R)=R$ naturally satisfies the condition. In this case, $f_{R}=1$, and (47)
gives the Misner-Sharp energy. A little nontrivial case is that the solution
is a constant curvature one with scalar curvature $R=R_{0}=const.$ In that
case, $f_{R,r}=0$, and (47) reduces to

$E_{eff}=\frac{r}{2G}\left((1-h^{ab}\partial_{a}r\partial_{b}r)f_{R}+\frac{r^{2}}{6}(f-f_{R}R)\right).$
(48)

Note that here $R$, $f_{R}$ and $f$ are all constants. Compare to the Misner-
Sharp energy (2) and the (16), we can see clearly that this expression is
nothing but the generalized Misner-Sharp energy with a cosmological constant.
Here the effective Newtonian constant is $G/f_{R}$ and the effective
cosmological constant $\Lambda=-(f-Rf_{R})/(2f_{R})$.

##  V Conclusion and discussion

The Misner-Sharp quasi-local energy plays a key role in understanding the
“unified first law”, the relation between the first law of thermodynamics and
dynamical equations of gravitational field and thermodynamics of apparent
horizon in FRW universe, etc. In this paper we studied the generalized Misner-
Sharp energy in $f(R)$ gravity by two approaches. One is the integration
method and the other is the conserved charge method. It turns out that in
general we cannot arrive at an explicit expression for the generalized Misner-
Sharp energy in a quasi-local form [see (24) and (37)], even assuming the
integrable condition (14) is satisfied. This situation is quite different from
the cases of Einstein gravity and Gauss-Bonnet gravity. This is certainly
related to the fact that for the $f(R)$ gravity, the energy current (28) is
not always divergence-free, while it does in Einstein and Gauss-Bonnet
gravities. The existence of the conserved current requires (29) is satisfied.

Some remarks on our results are in order.

(1) The relation between the two methods to derive the generalized Misner-
Sharp energy. We obtained the same generalized Misner-Sharp energy by
employing two methods: integration and conserved charge methods. At first
glance, these two methods looks different, but in fact, they are equivalent.
First let us notice that the constraint equation (23) has a relation to the
one (29):

$\frac{\partial A(u,v)}{\partial v}-\frac{\partial B(u,v)}{\partial
u}=-e^{-\varphi}r^{2}\nabla_{\mu}\nabla_{v}f_{R}\nabla^{\mu}K^{v}/2.$ (49)

Namely these two integrable conditions are equivalent. Second, substituting
the conserved current in (28) into (31), we can write the associated charge

$\displaystyle Q_{J}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}J^{\mu}d\Sigma_{\mu}$ (50) $\displaystyle=$
$\displaystyle\int 4\pi r^{2}e^{\varphi}(r,_{u}T_{uv}-r,_{v}T_{uu})du,$

where the integrand is precisely $A(u,v)$ in (22). Thus, we have finished our
proof of the equivalence of the two methods.

In addition, the useful components of
$K_{\mu\nu}\equiv\nabla_{\mu}K_{\upsilon}$ and
$F^{\mu\nu}\equiv\nabla^{\mu}\nabla^{\nu}F$ in coordinates
$(t,\rho,\theta,\phi)$ are

$\displaystyle K_{tt}$ $\displaystyle=$ $\displaystyle
e^{-\psi}(\psi,_{t}r,_{\rho}-r,_{t\rho}),~{}~{}K_{t\rho}=-e^{\psi}r,_{tt},~{}~{}K_{\rho
t}=-\partial_{\rho}(e^{-\psi}r,_{\rho})+\psi,_{t}e^{\psi}r,_{t},$
$\displaystyle K_{\rho\rho}$ $\displaystyle=$ $\displaystyle
e^{\psi}(\psi_{,t}r_{,\rho}-r_{,t\rho}),~{}~{}F^{tt}=F_{,tt},~{}~{}F^{t\rho}=F^{\rho
t}=-e^{-2\psi}F_{,\rho t}+e^{-2\psi}\psi_{,t}F_{,\rho},$ $\displaystyle
F^{\rho\rho}$ $\displaystyle=$ $\displaystyle
e^{-2\psi}[\partial_{\rho}(e^{-2\psi}F_{,\rho})+\psi_{,\rho}e^{-2\psi}F_{,\rho}-\psi_{,t}F_{,t}].$
(51)

With the help of those quantities, we can easily check that the constraint
equation (29) is satisfied for the FRW universe.

(2) The meaning of the generalized Misner-Sharp energy in FRW universe. To see
clearly this, let us write down the Friedmann equations of the $f(R)$ gravity

$\displaystyle H^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{6f_{R}}[(f_{R}R-f)-6H\partial_{t}f_{R}+16\pi
G\tilde{\rho}],$ $\displaystyle\overset{.}{H}-\frac{k}{a^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2f_{R}}[H\partial_{t}f_{R}-\partial_{t}\partial_{t}f_{R}-8\pi
G(\tilde{\rho}+\tilde{p})],$ (52)

where $\tilde{\rho}$ and $\tilde{p}$ are energy density and pressure of the
ideal fluid in the universe. With the first line in (52), we can easily see
that the generalized Misner-Sharp energy in (40) can be rewritten as

$E_{eff}=\tilde{\rho}V,$ (53)

where $V=4\pi r^{3}/3$ is the volume of a sphere with radius $r$. Therefore,
in fact, the generalized Misner-Sharp energy in the FRW universe is nothing
but the total matter energy within a sphere with radius $r$.

(3) Thermodynamics of apparent horizon in the $f(R)$ gravity. On the apparent
horizon of a FRW universe, the energy crossing the apparent horizon within
time interval $dt$ is Cai ; Cai-Kim

$\delta Q=dE_{eff}|_{r_{A}}=A(\tilde{\rho}+\tilde{p})Hr_{A}dt.$ (54)

Note that the horizon entropy of the $f(R)$ gravity is
$S=\frac{A}{4G}f_{R}=\pi r_{A}^{2}f_{R}/G$, while the temperature of the
apparent horizon is Cai-Kim ; Cai2 ; Li : $T=\frac{1}{2\pi r_{A}}$. Obviously,
the usual Clausius relation $\delta Q=TdS$ does not hold. On the other hand,
an internal entropy production is needed to balance the energy conservation,
$\delta Q=TdS+Td_{i}S$ with

$d_{i}S=\pi
r_{A}[Hr_{A}^{3}(H\partial_{t}f_{R}-\partial_{t}\partial_{t}f_{R})-\partial_{t}f_{R}r_{A}]/G.$
(55)

This is an effect of the non-equilibrium thermodynamics of spacetime Jacobson1
; Cai ; Elizalde ; Eling .

(4) The case of scalar-tensor gravity. Indeed the $f(R)$ gravity is quite
similar to scalar-tensor gravity theory in some sense Sotiriou . Our
conclusion on the $f(R)$ gravity therefore also holds for scalar-tensor
gravity. In particular, the existence of a generalized Misner-Sharp energy has
to obey a constraint condition as well for the scalar-tensor theory. However,
in the FRW universe, a simple expression for the generalized Misner-Sharp
energy can be given, which can be seen in appendix A.

## Acknowledgments

YPH thanks D. Orlov for useful discussions. RGC and YPH are supported
partially by grants from NSFC, China (No. 10525060, No. 108215504 and No.
10975168) and a grant from MSTC, China (No. 2010CB833004). NO was supported in
part by the Grant-in-Aid for Scientific Research Fund of the JSPS No.
20540283, and also by the Japan-U.K. Research Cooperative Program. This work
is completed during RGC’s visit to Kinki University, Japan with the support of
JSPS invitation fellowship.

##  Appendix A Generalized Misner-Sharp energy of scalar-tensor theory in FRW
universe

The Lagrangian of a generic scalar-tensor gravity in 4-dimensional space-time
can be written as

$L=\frac{1}{16\pi}F(\phi)R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)+L_{m}.$
(56)

where we set Newtonian constant $G=1$, $F(\phi)$ is an arbitrary positive
continuous function of the scalar field $\phi$, $V(\phi)$ is its potential,
and $L_{m}$ denotes the Lagrangian of other matter fields. Varying the
associated action with respect to spacetime metric and the scalar field yields
equations of motion

$\displaystyle FG_{\mu\nu}+g_{\mu\nu}\square F-\nabla_{\mu}\nabla_{\nu}F$
$\displaystyle=$ $\displaystyle 8\pi(T_{\mu\nu}^{\phi}+T_{\mu\nu}^{m}),$ (57)
$\displaystyle\square\phi-V^{\prime}(\phi)+\frac{1}{16\pi}F^{\prime}(\phi)R$
$\displaystyle=$ $\displaystyle 0.$ (58)

where $T_{\mu\nu}^{m}$ is the energy-momentum tensor of matter fields, and
$T_{\mu\nu}^{\phi}$ is defined as

$T_{\mu\nu}^{\phi}=\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\Big{(}\frac{1}{2}g^{\rho\sigma}\partial_{\rho}\phi\partial_{\sigma}\phi+V(\phi)\Big{)}.$
(59)

Note that here $T_{\mu\nu}^{\phi}$ is not the energy-momentum tensor of the
scalar field. Similar to the case of f(R) gravity, we find that the current
$J^{\mu}=-T_{\nu}^{\mu(m)}K^{\nu}$ is not always divergence-free unless the
condition is satisfied

$(\nabla_{\mu}\nabla_{\nu}F+8\pi\partial_{\mu}\phi\partial_{\nu}\phi)\nabla^{\mu}K^{\nu}=0.$
(60)

However, we can easily check that the condition (60) can be always satisfied
for the FRW universe (39) by using (12). Some useful components of equations
of gravitational field (57) are given by

$\displaystyle 8\pi T_{tt}^{m}$ $\displaystyle=$ $\displaystyle
3F\Big{(}\frac{k}{a^{2}}+H^{2}\Big{)}+3H\overset{.}{F}-8\pi\Big{(}\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)},~{}~{}8\pi
T_{t\rho}^{m}=0,$ $\displaystyle 8\pi T_{\rho\rho}^{m}$ $\displaystyle=$
$\displaystyle\frac{a^{2}}{1-k\rho^{2}}\Big{[}-F\Big{(}\frac{k}{a^{2}}+H^{2}+\frac{2\overset{..}{a}}{a}\Big{)}-\overset{..}{F}-2H\overset{.}{F}+8\pi\Big{(}-\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)}\Big{]}.$
(61)

In this case, corresponding $A$ and $B$ in (35) for the scalar-tensor theory,
respectively, are

$\displaystyle A(t,\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}Hr^{3}\Big{[}F\Big{(}\frac{k}{a^{2}}+H^{2}+\frac{2\overset{..}{a}}{a}\Big{)}+\overset{..}{F}+2H\overset{.}{F}-8\pi\Big{(}-\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)}\Big{]},$
$\displaystyle B(t,\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\rho^{2}a^{3}\Big{[}3F\Big{(}\frac{k}{a^{2}}+H^{2}\Big{)}+3H\overset{.}{F}-8\pi\Big{(}\frac{1}{2}\overset{.}{\phi}^{2}+V\Big{)}\Big{]}.$
(62)

They obey the integrable condition (38). With these quantities we can obtain
the generalized Misner-Sharp energy of the scalar-tensor theory in the FRW
universe

$E_{eff}\ =\int
B(t,\rho)d\rho=\frac{r^{3}}{2}[F(\frac{k}{a^{2}}+H^{2})+H\overset{.}{F}-\frac{8\pi}{3}(\frac{1}{2}\overset{.}{\phi}^{2}+V)].$
(63)

Comparing this with the first equation in (61), one can immediately see that
the generalized Misner-Sharp energy is just the total matter energy in a
sphere with radius $r$ in the FRW universe.

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