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IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class. Quantum Grav. 25 (2008) 075017 (11pp) doi:10.1088/0264-9381/25/7/075017

Energymomentum complexes in f(R) theories ofgravity

Tuomas Multamaki1, Antti Putaja1, Iiro Vilja1 and Elias C Vagenas2

1 Department of Physics, University of Turku, FIN-20014 Turku, Finland2 Research Center for Astronomy and Applied Mathematics, Academy of Athens,Soranou Efessiou 4, GR-11527, Athens, Greece

E-mail: [email protected], [email protected], [email protected] and [email protected]

Received 4 January 2008, in final form 22 February 2008

Published 18 March 2008Online at stacks.iop.org/CQG/25/075017

Abstract

Despite the fact that modified theories of gravity, in particular the f(R) gravity

models have attracted much attention in previous years, the problem of the

energy localization in the framework of these models has not been addressed.

In the present work the concept of energymomentum complexes is presented

in this context. We generalize the LandauLifshitz prescription of calculating

the energymomentum complex to the framework of f(R) gravity. As an

important special case, we explicitly calculate the energymomentum complex

for the Schwarzschildde Sitter metric for a general f(R) theory as well as for

a number of specific, popular choices of f(R).

PACS numbers: 04.20.Cv, 04.70.s

1. Introduction

It has been almost a century since the birth of general relativity and there are still problems that

remain unsolved. The energymomentum localization is one of them which till today is treated

as a vexed problem. Much attention has been devoted to this problematic issue. Einstein

was the first who tried to solve it by introducing the methodology of energymomentum

pseudotensors. He presented the first such prescription [1] and after that a plethora of different

energymomentum prescriptions were proposed [27]. All these prescriptions were restrictedto compute the energy as well as the momenta distributions in quasi-Cartesian coordinates.

Mller was the first to present an energymomentum prescription which could be utilized in

any coordinate system [8].

The idea of energymomentum pseudotensors was gravely criticized for several reasons

[912] (actually one of the drawbacks was the aforesaid use of quasi-Cartesian coordinates

which was solved by Mllers prescription). Firstly, although a symmetric and locally

conserved object, its nature is nontensorial and thus its physical interpretation seemed

0264-9381/08/075017+11$30.00 2008 IOP Publishing Ltd Printed in the UK 1
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Class. Quantum Grav. 25 (2008) 075017 T Multamaki et al

obscure [13]. Second, different energymomentum complexes could yield different energy

distributions for the same gravitational background [14, 15]. Third, energymomentum

complexes were local objects while it was commonly believed that the proper energy

momentum of the gravitational field was only total, i.e. it cannot be localized [ 16]. For a

long period of time the idea of energymomentum pseudotensors was relinquished.The approach of energymomentum pseudotensors for the thorny problem of

energymomentum localization was rejuvenated in 1990 by Virbhadra and collaborators

[1726]. Since then, numerous works have been performed on computing the energy and

momenta distributions of different gravitational backgrounds using several energymomentum

prescriptions (for a recent list of references see [27]). In 1996 Aguirregabiria, Chamorro and

Virbhadra [28] showed that five different energymomentum complexes yield the same energy

distribution for any KerrSchild class metric. Additionally, their results were identical with the

results of Penrose [29] and Tod[30] using the notion of quasi-local mass. Many attempts since

then have been performed to give new definitions of quasi-local energy in general relativity

[3236]. Considerable efforts have also been performed in constructing superenergy tensors

[37]. Motivated by the works of Bel [3840] and independently of Robinson [41], many

investigations have been carried out in this field [4246].In 1999 Chang, Nester and Chen [47] proved that every energymomentum complex is

associated with a Hamiltonian boundary term. Therefore, the energymomentum complexes

can be considered as quasi-local, boundary condition dependent conserved quantities. Finally,

it should be pointed out that though a long way has been trodden, the solution to the problem

of energymomentum localization in the framework of general relativity is way ahead.

Another challenge to the development of physical theory of gravitation is that the plethora

of observational data collected recently indicates that our universe is undergoing an accelerated

expansion. Motivated by this observational evidence, we have been on a long hunt for the

explanation for this speed-up. Until today, three possible reasons have been presented. Two

of them, namely the cosmological constant and the quintessence field, are developed in the

framework of general relativity. The third one is developed in the framework of alternative

theories of gravity. In particular, the simplest among the aforesaid models are that in which

EinsteinHilbert action is modified by an additional term.Modified theories of gravity, especially the f(R) gravity models that replace the Einstein

Hilbert action of general relativity (henceforth abbreviated to GR) with an arbitrary function

of the curvature scalar, have been extensively studied in recent years (see e.g. [4856] and

references therein). The challenges in constructing viable models in the light of cosmological

constraints (see e.g. [5759] and references therein), instabilities [6062], solar system

constraints (see e.g. [6366] and references therein) and evolution large-scale perturbations

[6769] are now known. The solar system constraints are a major obstacle to most theories

[7073] but they can be completely removed by certain types of models [58, 66, 7476].

It is widely known that when a new theory is introduced, it is expected that this new theory

will successfully answer all already-solved (in the framework of the old theory) problems.

Moreover, it is anticipated that this new theory will be able to address, alleviate and finally

solve problems that the existing old theory cannot. Following this line of thought, we addresshere, to our knowledge, for the very first time the problematic issue of energymomentum

localization in the context off(R) gravity models which as mentioned above intend to replace

GR. We take the first steps in this direction and consider energymomentum complexes within

f(R) gravity models.

The remainder of the paper is as follows. In section 2, we present the basic equations

and formalism off(R) gravity. In sections 3 and 4, the LandauLifshitz energymomentum

complex and the Schwarzschildde Sitter (henceforth abbreviated as SdS) metric, or the SdS

2


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black hole background, are reviewed. In section 5, we extend the concept of the Landau

Lifshitz energymomentum complex into the framework of f(R) theories. As a special case,

we compute the energymomentum complex of the SdS metric for a number of commonly

considered f(R) theories. In the final section we summarize the results and present our

conclusions.

2. f(R) gravity formalism

The action for f(R) gravity is

S =

d4xg

1

16 Gf(R) + Lm

, (1)

where the standard EinsteinHilbert action is replaced by a general function of scalar curvature

f(R). The corresponding field equations (in the metric approach) are found by varying with

respect to the metric g and read as

F(R)R

12

f (R)g

F(R) + gF(R)

=8 GTm (2)

where Tm is the standard minimally coupled stressenergy tensor and F(R) df/dR.In contrast to the standard Einsteins equations from the EinsteinHilbert action, the field

equations are now of higher order in derivatives.

Contracting the field equations gives

F(R)R 2f(R) + 3F(R) = 8G( 3p), (3)where we have assumed that we can describe the stressenergy tensor with a perfect fluid.

From the contracted equation it is clear that in vacuum, any constant scalar curvature metric

with R = R0 is a solution of the contracted equation as long as F (R0)R0 = 2f (R0). Ingeneral, the whole set of field equations is solved exactly by the SdS metric [77] (for a more

recent work on spherically symmetric solutions of modified field equations in f(R) gravity

see also [78])

ds2 = B dt2 B1 dr 2 r 2 d2 r2 sin2 d2 (4)with

B(r) = 1 2Mr

r2

3. (5)

The scalar curvature for this metric is R0 = 4. Hence any f(R) theory, including thestandard general relativity, satisfying the constant curvature condition F (R0)R0 = 2f (R0)has the SdS (black hole) metric as an exact solution. We will return to this important special

case in a later section.

3. The LandauLifshitz energymomentum complex

In general, the energymomentum complex (henceforth abbreviated as EMC) carriescoordinate-dependent information on the energy content of the gravitational and matter

fields. It sums up the energies of the matter fields through the energymomentum tensor

T (henceforth abbreviated as EM), and that of the gravitational field through the energy

momentum pseudotensor t (henceforth abbreviated as EMPT), which depends on the

coordinate system used to describe the system. The EMPT cannot be defined uniquely

and a number of suggestions, with different mathematical properties, exist. All of them lead

to conserved quantities of the gravitational theory.

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The most straightforward conserved quantity is the integrated EMC over the three-

dimensional space integral

EEMC

= B(0,r)d3x 00 (6)

which represents both the energy of the gravitational field and that of matter inside the

coordinate volume B(0, r). In the case of a black hole, it consists of two parts: the black hole

mass Mand the energy stored in the gravitational field t00, therefore

EEMC = M +

B(0,r)

d4xgt00. (7)

In the construction of Landau and Lifshitz [2], one looks for an object, , that is

antisymmetric in its indices since then = 0 due to the covariant continuity equation,

which in a locally Minkowskian coordinate system simplified to

T = 0. (8)

HenceT = . (9)

Einsteins equations give

T = 12

R 1

2Rg

, (10)

where 2 = 8 G (we have set c = 1), and in the locally Minkowskian coordinates Riccitensor and Ricci scalar read as

R = 12

gg g ( g + g g g) (11)and

R = g gg= (gg gg)g. (12)

Using these expressions in equation (10), one can rewrite the EM tensor as [2]

T = , (13)where

= 122

1

(g)

x[(g)(gg gg )]. (14)

In a locally Minkowskian coordinate system g = 0 and hence one can define(g)T h H (15)

where two new tensors, so-called superpotentials,

h = (g) = 122

x[(g)(gg gg )] (16)

H = 122

[(g)(gg gg)] (17)

have been defined.

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In a general coordinate system, equation (15) is no longer valid and one defines a new

object t such that

(

g)(T + t)

h

x

. (18)

The new object, namely the EMPT t, is straightforwardly computed in a general coordinate

system employing equation (18) since T canbe expressed in terms of thegeometric quantities

by using Einsteins equation, i.e. equation (10), and h is given in equation (16).

Carrying out this somewhat lengthy but routine exercise, one obtains [ 2]

tLL =

1

2 2

2

(gg gg)

+ gg

+

+ g g

+

+ gg

. (19)

The LandauLifshitz EMC, i.e.

LL , can now be evaluated either as a sum of the EM and theEMPT, namely

LL = (g)

tLL + T

(20)

or directly using equation (18) which is now written as

LL h

x . (21)

4. The Schwarzschildde Sitter metric

In GR, the empty space solution outside a static spherically symmetric mass distribution in

a universe with a cosmological constant is the SdS metric, or the SdS black hole metric. In

spherically symmetric coordinates, it reads outside the mass distribution as (in units where

G = 1) given in equations (4) and (5). Mis the total mass and is the cosmological constant.Due to the fact that some EMPTs are calculated in Cartesian coordinates, we need to re-express

the SdS black hole metric in Cartesian terms. The metric (4) then reads as

ds2 = B dt2 B1x2 + y2 + z2

x2 + y2 + z2dx2 x

2 + B1y2 + z2

x2 + y2 + z2dy2 x

2 + y2 + B1z2

x2 + y2 + z2dz2

(B1 1)2yz

x2 + y2 + z2dy dz (B

1 1)2xyx2 + y2 + z2

dx dy (B1 1)2xz

x2 + y2 + z2dx dz. (22)

For this metric, but not for the metric in spherically symmetric coordinates, the determinant

reads as g = 1. This feature is used regularly in the following as outer factors like gequal unity.

The EMPTs corresponding to the SdS solution are straightforwardly calculable (in aCartesian coordinate system). Working in a space where r > 0 and thus EMPTs and EMCs

coincide, we obtain for the 00-component of the LandauLifshitz EMPT

(g)t00 = 22

36M2 + 12Mr 3 + r4(r2 9)r 2(6M + r(r2 3))2

= 22

9r4 + (6M + r3)2r2(r3 3r + 6M)2 . (23)

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For comparison, one can easily repeat this exercise for the other well-known EMPTs: we

find that the Weinberg EMPT is equal to that of Landau and Lifshitz, the Einstein and Tolman

EMPT is

t00 = 2

, (24)

and the Mller EMPT is

t00 = 2

2. (25)

It is evident that the EMPT of the SdS metric strongly depends on the chosen

construction, even though EMPTs derivedin differentprescriptions can sometimesbe identical.

Furthermore, their behavior far from the mass source is wildly different. Therefore, by

looking at only the functional form of a single EMC, it is obscure what is the precise physical

interpretation of it, even though it represents a conserved quantity. However, we emphasize

that the different EMCs differ by a divergence term related to the boundary conditions of the

physical situation [47].

5. EMPT in f(R) theories of gravity

Among the different EMPTs studied in the literature, LandauLifshitzs and Weinbergs

prescriptions appear to be the most straightforwardly suitable for extending into f(R) gravity

theories. Here we consider extending the LandauLifshitzs prescription and leave others for

future work.

In f(R) theories, one can write the field equations as

T =1

2

f(R)R

1

2gf(R) DD f(R) + gf(R)

. (26)

Like in GR, the covariant continuity equation holds, i.e. DT = 0 [82], suggesting that

one should write the rhs of equation (26) as a divergence of an object antisymmetric in its

indices, i.e. in a form h .

Following Landaus and Lifshitzs prescription and considering a locally Minkowskian

coordinate system at a given point we obtain

2T = f(R)R 12

g f(R) + (gg gg ) f(R)= f(R)G + 1

2g(f(R)R f (R)) + [(gg gg ) f(R)]

= [f(R)2 + (gg gg)f(R) R] f(R)2+ 1

2g(f(R)R f (R))

=[f

(R)2 + (gg

gg)f(R) R] +

12

g (f(R)R

f (R)), (27)

where is that defined in (14). It is noteworthy that the term f(R)2 vanishes in

the locally Minkowskian coordinate system, because is linear in the first derivatives

of the metrics. We are partially able to write the rhs of the field equations, namely

equation (27), as a divergence. The remaining term, absent in GR, remains problematic in a

general case without a clear method which would enable us to write it as a four divergence.

We can, however, proceed in an important special case where the scalar curvature is a constant,

R = R0. The SdS black hole metric belongs to such a class of metrics.6


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5.1. The LandauLifshitz energymomentum complex for a metric with constant scalar

curvature

For a metric with a constant scalar curvature, R = R0, equation (27) simplifies to

T = [f(R0) ] + 122

g(f(R0)R0 f (R0))

= [f(R0) ] +1

62(g

x gx )(f(R0)R0 f (R0))

= [f(R0) +1

62(gx gx )(f(R0)R0 f (R0))] (28)

so that the generalized LandauLifshitz superpotential takes the form

h = f(R0) +1

62(gx gx )[f(R0)R0 f (R0)]. (29)

The EMPT t in a general coordinate system defined in the LandauLifshitz prescription can

now be read out from the expression for the EMC (remember that g=

1)

T + t h . (30)Hence the generalized LandauLifshitz EMC reads as

= f(R0)LL +1

6 2[f(R0)R0 f (R0)](g x gx ), (31)

where LL is the LandauLifshitz EMC evaluated in the framework of GR (see equation (21)).

The 00-component reads as

00 = f(R0)00LL +1

62

f(R0)R0 f (R0)

(g

00x g0x0)

=f(R0)

00

LL+

1

62(f(R0)R0

f (R0))(i g

00xi + 3g00). (32)

Equation (31) is a general formula valid for any f(R) theory when the studied metric has

constant scalar curvature. The standard GR result is recovered when f(R) = R.

5.2. Energymomentum complex of the SdS metric of some f(R) models

Using equation (31) we can compute the EMC of the SdS metric in a general f(R) theory:

00 = f(R0)00LL +1

62(f(R0)R0 f (R0))

rB (r) + 3B(r)

= f(R0)00LL +1

62(f(R0)R0 f (R0))

3 4M

r 5

3r 2

= 22

9r4 + (6M + r3)2r 2(r3 3r + 6M)2 f

(R0) +1

6 2(f(R0)R0 f (R0))

3 4M

r 5

3r2

.

(33)

This result is valid for any f(R) theory that has the SdS metric as a vacuum solution

i.e. any theory which satisfies the vacuum equation f(R0)R0 2f (R0) = 0. Again notethat when f(R) = R we recover the standard, i.e. GR, form of LandauLifshitz EMPT,equation (23), as expected.

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An important special case encompassing popular choices of f(R) is a generic action

function

f(R) = R (1)n1 aRn

+ (1)m1bRm, (34)

where n and m are positive integers and a, b are any real numbers. This form of function f(R)is widely used in cosmological context. In this case the generalized LandauLifshitz EMC

takes the form

00 = 2(1+2n)n

9r 22

r[(12M 9r + 5r3)(a(1 + n) + b(m 1)(4)m+n)]

9 [9r4 + (6M + r3)2][an + (4)n(4 + bm(4)m)]

(6M 3r + r3)2

. (35)

For the form off(R) considered above, i.e. equation (34), and recalling that for the SdS

metric we have R0 = 4, constant curvature condition can be written as(4)n+1 = a(n + 2) + b(m 2)(4)m+n. (36)

In the special case where m=

2 or b=

0 we obtain

a = (4)n+1

n + 2. (37)

Note that not all type (34) models are cosmologically viable. It is known that those vacuum

solutions with R0 such that f(R0) > 0 (note our sign convention) are inherently unstable

[60] and therefore not suitable for the cosmological model.

Particularly the often used model off(R) theory of gravity is

f(R) = R 4

R R 2, (38)

which has a stable vacuum whenever > 1/(3

32). The 00-component of the

corresponding generalized LandauLifshitz EMC for this model is written as

00 = 118r 22R0

r

R 30 24 12M 9r + 5r 3

369r4 + (6M + r3)2

4 + R20 2R 30

R0(6M 3r + r3)2

, (39)

which for the special case of the SdS black hole metric with the cosmological vacuum

R0 =

32, reduces to

00 = 2 + 3

32

18

3r 22

5

4

34r4 + 32r(4M 3r)

72 [192

3M2 + 482Mr3 362r 4 +

34r6]

(24M 12r +

32r3)2

. (40)

Another cosmologically interesting f(R) gravity model includes also logarithmic

dependence on curvature. Thus it reads

f(R) = R + (1)m1cRm d ln |R|

k

, (41)

where its parameters are related to the cosmological constant by the constant curvature

condition written now as

d + (4)mcm = 2

(4)mc + 2 + dln

4

k

. (42)

8


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systems are another example of a relevant open question. We hope to address these issues in

future work.

Acknowledgments

TM is supported by the Academy of Finland. AP acknowledges support from the Academy

of Finland under project no. 8111953.

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