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arXiv:0705.3882v1 [gr-qc] 26 May 2007

Viscous Spacetime Fluid and Higher Curvature Gravity

S. C. Tiwari

Institute of Natural Philosophy

c/o 1 Kusum Kutir Mahamanapuri,Varanasi 221005, India

The Einstein field equation as an equation of state of a thermodynamical system of spacetime is

reconsidered in the present Letter. We argue that a consistent interpretation leads us to identify

scalar curvature and cosmological constant terms representing the bulk viscosity of the spacetime

fluid. Since Einstein equation itself corresponds to a near-equilibrium state in this interpretation

invoking f(R) gravity for nonequilibrium thermodynamics is not required. A logically consistent

generalization to include the effect of so called ’tidal forces’ due to the Riemann curvature is pre-

sented. A new equation of state for higher curvature gravity is derived and its physical interpretation

is discussed.

PACS numbers: 04.70.Dy, 04.20.Cv

Higher curvature Lagrangians have been discussed as

logically possible constructions for gravitation since long

[1] , however current interest in such theories is inspired

by various shades of quantum gravity/effective field the-

ory [2]. Amongst them f(R) gravity, where f is a non-

linear function of the scalar curvature R, is considered as

an attractive model to explain the observed cosmic accel-

eration phase [3]. The conflict of the metric formulation

of f(R) gravity with the solar system observations has

been claimed to get resolved in the Palatini formalism

[4], however it has been argued that this theory results

in the appreciable deviations from the microphysics (e.g.

electron-electron scattering experiments), and violation

of the equivalence principle [5]. It may be asked: Is there

an alternative approach to higher curvature gravity?

Departing from the variational principle approach, re-

cently a thermodynamical derivation of the f(R) gravity

has been proposed [6] generalizing the previous deriva-

tion of the Einstein field equation [7]. Thermodynamics

of spacetime in [7] is motivated by the black hole ther-

modynamics: it is assumed that similar to the black hole

entropy formula a universal entropy density α per unit

horizon area for all local Rindler horizons could be de-

fined, and the Clausius relation TdS = δQ of equilibrium

thermodynamics holds for all local horizons. Here the

heat flow across the horizon δQ is defined to be the boost

energy of the matter and T is the Unruh temperature.

Jacobson then argues that since entropy is interpreted

as horizon area the Clausius relation would be satisfied

provided the spacetime curvature in the presence of mat-

ter is such that the Einstein field equation holds. In [6]

this approach is applied to the assumed entropy density

proportional to f(R), and it is found that local energy

conservation entails entropy production term, and hence

nonequilibrium thermodynamics of spacetime. Further

the higher curvature equation of state obtained is shown

to be identical with the field equation derived from the

f(R) Lagrangian.

Obviously rederivation of the known field equations by

itself is not of much significance unless new insights are

gained. Unfortunately the paradigm of field theory has

overshadowed this promising thermodynamic approach

in [6, 7].

divergence law for the matter stress tensor is satisfied

in a conformally related spacetime, ˜ gµν= f(R)gµν with

no need to postulate bulk viscosity production term as

is done by Eling et al in [6], and therefore the issue of

nonequilibrium thermodynamics becomes trivial. Note

that this result is independent of the unimodular per-

spective suggested in [8].

In this Letter spacetime as a thermodynamical system

in the spirit of Jacobson’s approach is considered with

the two-fold aim: 1) to gain new insights into the na-

ture of spacetime assumed to be some kind of fluid at

large scale, and 2) to motivate nonequilibrium thermo-

dynamics in the case considered in [7] and generalize it

to include the effect of the so called ’tidal force’ caused

by the Riemann curvature of spacetime [9] and derive a

new higher curvature equation of state.

First we argue that a consistent equation of state inter-

pretation of the Einstein field equation demands a careful

reanalysis of the local equilibrium condition: the Φ term

in Eq.(6) below while easily interpreted in the field the-

oretic setting, seems to require a different meaning since

local energy conservation is assumed to be a form of first

law of thermodynamics [6]. This term in the Einstein

field equation in the absence of matter stress tensor is

suggested to represent a dissipative contribution to the

stress tensor of vacuum. Analogous to the bulk viscosity

for the dissipative fluid [10], Φ term, therefore signifies

viscous spacetime, and the Einstein equation already cor-

responds to a quasi-equilibrium system.

Next we recall Bondi’s argument [11] noted in [8] that

the essence of observable gravitation is that relative ac-

celeration varies at each spacetime point. It would mean

that even the assumption of local thermodynamic equilib-

rium for local Rindler horizons would, in principle, fail.

However the gravitation is so weak that the correction

due to Riemann curvature is proposed to be incorporated

retaining key elements of the Jacobson’s approach in the

following. The notion of local Rindler horizon is used to

define heat flow and entropy change in [7]. The equiv-

alence principle enables locally flat spacetime at each

spacetime point, and for an infinitesimal 2-surface ele-

Moreover we have shown [8] that covariant

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ment through the point one can assume vanishing shear

and expansion to the past (or inside) of the plane. The

past horizon, the local Rindler horizon is assumed instan-

taneously stationary. This idea helps in introducing an

approximate local boost Killing vector field which can be

related with the horizon tangent vector and affine param-

eter along the null geodesic. Now instead of employing

the Jacobi deviation equation for relative acceleration, we

assume that the imprint of the tidal force is carried by

the congruence of the null geodesics via a correction term

dependent on the Riemann curvature. Both nonuniform

temperature and viscosity lead to dissipation; since we

do not know how to define these quantities here instead

of the entropy change we calculate change in the thermo-

dynamic potential F = ST. Heat flow and change in the

horizon area are calculated for local Rindler horizons to

lowest order in the affine parameter as is done in [7]. A

correction term proportional to the Riemann curvature

is introduced in the change δF. Equating δQ and δF,

and imposing covariant divergence law for matter stress

tensor finally give the desired equation of state.

Letter concludes with a brief discussion on the physical

interpretation and prospects of this equation.

Briefly the main ideas of [7] are as follows. The black

hole formula, namely the proportionality between en-

tropy and the horizon area, is assumed to hold for all local

Rindler horizons at each spacetime point of the manifold

M. Causal horizon at a point p is specified by a space-like

2-surface B, and the boundary of the past of B comprises

of the congruences of null geodesics. Assuming vanishing

shear and expansion at p the past horizon of B is called

local Rindler horizon. The energy flux across the hori-

zon is used for heat energy, and calculated in terms of

the boost energy of matter: define an approximate boost

Killing vector field χµfuture pointing on the causal hori-

zon, and related with the horizon tangent vector kµand

affine parameter λ by χµ= −aλkµ. The heat flux to the

past of B is given by

The

δQ =

?

TµνχµdΣν

(1)

The integral is taken over a small region of pencil of gen-

erators of the inside past horizon terminating at p. If area

element is dA then dΣν= kνdλdA, and Eq.(1) becomes

δQ = −a

?

TµνkµkνλdλdA(2)

Change in the horizon area is given in terms of the ex-

pansion of the congruence of null geodesics generating

the horizon δA =

?θdλdA. The expansion of the null

geodesics generating the horizon is given by the Ray-

chaudhuri equation

dθ

dλ= −θ2

2− σµνσµν− Rµνkµkν

(3)

Assuming vanishing shear and neglecting θ2term we get

the solution

θ = −λRµνkµkν

(4)

Assuming universal entropy density α per unit horizon

area it is straightforward to calculate the entropy change

δS = −α

?

RµνkµkνλdλdA (5)

The condition that the Clausius relation is satisfied for

all null vectors kµ, and making use of the Unruh tem-

perature ¯ ha/2π gives

Rµν+ Φgµν= (2π/¯ hα)Tµν

(6)

The unknown function Φ is determined using the covari-

ant divergence law for the stress tensor and contracted

Bianchi identity; Einstein equation with a cosmological

constant (CC) Λ is obtained. Here Newton’s gravita-

tional constant is identified as G = 1/4¯ hα, and the func-

tion Φ is obtained to be

Φ = −R

2+ Λ

(7)

In the light of thermodynamic derivation a logically

consistent physical interpretation of the equation of state

is proposed here. In the Clausius relation integrands of

(2) and (5) have been equated for all null vectors to

obtain Eq.(6).Since Tµν is the matter stress tensor,

from the integral form itself it is logical to infer that

Rµν is proportional to the stress of assumed spacetime

fluid. What does Φ term represent? In the derivation Φ

term is needed to satisfy the first law of thermodynam-

ics (in the form of local energy conservation), and hence

it should correspond to an additional stress tensor for a

dissipative process in Eq.(6). Noting its formal similarity

with the bulk viscosity stress tensor for an isotropic fluid

proportional to the tensor δij, see Eq.(8.4.42) in [10] it

seems reasonable to identify Φ term in Eq.(6) as bulk vis-

cosity tensor of the spacetime. Deeper insight is gained

analysing the CC term. Setting matter stress tensor zero

and assuming a vanishing CC the Ricci flat spacetime

becomes indistinguishable from Gµν = Rµν−R

as R = 0. Thus vacuum Einstein equation with van-

ishing CC could be interpreted as an equation of state

of a perfect fluid. The presence of matter is suggested

to cause a dissipative process leading to a nonvanishing

scalar curvature, i.e. a bulk viscosity to the spacetime;

geometrically scalar curvature arises via the contracted

Bianchi identity so that the Einstein tensor satisfies the

covariant divergence law. Here R has been given a ther-

modynamic significance.

What is the physical significance of Λ ? In the light of

its appearance in combination with R in (7) this would

correspond to the bulk viscosity treating the whole Φ

term as the bulk viscosity stress tensor of the spacetime.

Thus empty spacetime with nonvanishing CC would be

like a viscous spacetime fluid. In Jacobson’s approach the

thermal behavior of quantum vacuum in flat spacetime is

an important ingredient. The time translation symme-

try in Rindler wedge corresponds to the boost symme-

try of the Minkowskian spacetime; quantum vacuum in

2gµν= 0

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Minkowskian spacetime as observed in the uniformly ac-

celerating frame acquires thermal properties of a thermal

bath of real particles with the Unruh temperature [12].

Assuming that quantum vacuum fluctuations would be

significant only at Planck length scales a universal en-

tropy per unit horizon area and Unruh temperature have

been assumed in [7]. This is a reasonable assumption,

however in the light of thermodynamic derivation it may

be asked: Does there exist averaged out observable ef-

fect of vacuum fluctuations? There is an interesting phe-

nomenon in quantum optics [13] : atomic motion in light

radiation in the so called optical molasses bears resem-

blance with that of a particle in a viscous fluid. Analo-

gous to this, envisaging gravitational molasses having ori-

gin in quantum fluctuations in the spacetime bulk viscos-

ity associated with CC could be attributed to this effect.

Another possibility is, following suggestions in the liter-

ature, to treat CC as the energy density of quantum vac-

uum fluctuations, Λ = 8πGV0/c4where V0is the vacuum

expectation value. In this case, nonzero CC would cause

dissipative process similar to matter energy and constant

scalar curvature would determine the bulk viscosity of

spacetime. We emphasize that both CC and quantum

vacuum possess rather speculative character in general

relativity, and the microscopic nature of the spacetime

is unknown to us, hence the preceding discussion though

quite plausible, also remains speculative.

The question of bulk viscosity for the Einstein equa-

tion is also discussed in [6]; the expected bulk viscosity of

3¯ hα/4π is obtained from the f(R) equation in the Ein-

stein frame. Authors argue it to be incorrect, and note

the ambiguity of sign. However negative bulk viscosity

could be related with the acausal teleological boundary

conditions for the black hole [14]. The stretched-horizon

formalism developed in [14] gives useful insights on black

hole physics; in particular, energy and momentum con-

servation laws for the membrane resemble with those of

viscous fluid. The present interpretation that the Ein-

stein equation represents a viscous spacetime fluid could

be viewed as a generalization, albeit a radical one, of the

membrane paradigm of [14].

Thermodynamic derivation of the Einstein equation

based on arbitrary spacelike 2-surfaces has been recently

established [15].It is, therefore, reasonable to assert

that the preceding physical interpretation is of general

validity. Clearly viscosity term implies that the system

is not in thermodynamic equilibrium; moreover higher

curvature, e.g. f(R) gravity, is not needed to motivate

nonequilibrium thermodynamics. The present interpre-

tation however does offer an effective procedure to in-

clude the effect of Riemann curvature as a correction

to viscous stress tensor. Returning to the fluid anal-

ogy, in general the dissipative coupling coefficient ηijkl

is a fourth rank viscosity tensor [10], and has symmetry

of indices similar to curvature tensor Rµναβ.

ing that the correction due to the Riemann curvature is

proportional to Rµσνρthe product with the Ricci tensor

keeping in mind the symmetry of the indices would be

Assum-

RσρRµσνρ. Thus we construct the simplest form of δF

δF = −α¯ ha

2π

?

[Rµν+β(RσρRµσνρ−R

2Rµν)]kµkνλdλdA

(8)

The last term in the square brackett in the integrand

corresponds to the viscosity as identified above in Eq.(7)

for the scalar curvature. This also makes the application

of covariant divergence law unambiguous. Using Clausius

relation and equating the integrands of (8) and (2) for all

null vectors we get

Rµν+ β(RσρRµσνρ−R

2Rµν) + Ψgµν=2π

¯ hαTµν

(9)

The unknown function Ψ is determined taking the co-

variant divergence of (9) and using the Bianchi identity

and vanishing divergence of matter stress tensor. Use is

made of following relations

(RµσνρRσρ):µ= (1

4RσρRσρ−1

2R:α

:α),ν+1

2(R;µν):µ(10)

(RRµν):µ= (R;µν):µ+ (R2

4

− R:α

:α),ν

(11)

The expression for Ψ is finally obtained to be

Ψ = −R

2+βR2

8

−βRσρRσρ

4

(12)

Substituting Ψ in (9) we get the desired higher curvature

equation of state

Gµν−βR

2(Rµν−R

4gµν)+βRσρ(Rµσνρ−Rσρ

4

gµν) =2π

¯ hαTµν

(13)

Perusal of the higher curvature field equations in the lit-

erature [2, 3, 4] shows that Eq.(13) is a new result. This

equation has some remarkable properties: 1) The arbi-

trary constant β could be adjusted to keep intact the es-

tablished physical tests of the Einstein equation since the

higher curvature effect appears as a correction to Gµν. 2)

The trace of (13) leads to the same result as that of pure

general relativity. Note also that the first two terms are

same as the ones in R2gravity in the Palatini formal-

ism which would make it to be in agreement with some

of the results of that theory. However the third term is

a new addition. And 3) In contrast to the higher cur-

vature field equations obtained from the action principle

which are fourth order derivative equations, Eq.(13) is a

second order derivative equation similar to that one ob-

tains in the Palatini version where the action is varied

taking metric and affine connection as independent vari-

ables. These characteristics strongly suggest viability of

Eq.(13) as higher curvature gravity theory.

It is well known that the question of gravitational en-

ergy in general relativity is quite complex [16], and even

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more so in higher curvature gravities [2, 17]. An interest-

ing consequence of (13) is that in the empty spacetime it

reduces to

Rµν+ βRσρ(Rµσνρ−Rσρ

4

gµν) = 0(14)

What is the physical significance of the second term in

Eq.(14)? In analogy with the traceless stress tensor of

electromagnetic field it is plausible to identify this term

to represent stress tensor of gravity radiation loss. Bondi

presented a lucid discussion [18] on the inductive and

wave transfer of gravitational energy in general relativ-

ity, and considered axially-symmetric vacuum solution of

Weyl and Levi-Civita for this purpose. Since the vacuum

defined by Eq.(14) has different nature, it would be inter-

esting to delineate inductive and wave energy transfers

in this case.

The problem of entropy production for f(R) gravity

has been extensively discussed in [19]. What would be

the entropy production in the present case? There is

another interesting question: Could one derive Eq.(13) in

the action formalism? We propose to investigate Palatini

formalism for the action used by Deser and Tekin [2]

elsewhere since the main theme of the present work is

thermodynamic approach to the spacetime.

In conclusion, we have argued that Einstein equation

in thermodynamic approach represents a viscous space-

time fluid, and derived a new equation of state for higher

curvature corrections.

I thank Prof. S. Deser for helpful correspondence, and

Prof. Rong-Gen Cai for drawing my attention to [19].

Library facility of Banaras Hindu University is acknowl-

edged.

[1] A. S. Eddington, The Mathematical Theory of Relativity

(C. U. P. 1924).

[2] J. Ellis et al, Phys. Rev. D 59,103503(1999); S. Deser and

B. Tekin, Phys. Rev. Lett. 89,101101(2002).

[3] S. M. Carroll et al, Phys. Rev. D 70,043528 (2004).

[4] D. N. Vollick, Phys. Rev. D 68, 063510 (2003); X. Meng

and P. Wang, Class. Quantum Grav. 22, 23 (2005).

[5] E. E. Flanagan, Phys. Rev. Lett. 92, 071101 (2004); G.

J. Olmo, Phys. Rev. Lett. 98, 061101 (2007).

[6] C. Eling, R. Guedens, and T. Jacobson, Phys. Rev. Lett.

96,121301(2006)

[7] T. Jacobson, Phys. Rev. Lett. 75,1260 (1995)

[8] S. C. Tiwari, arXiv: gr-qc/0612099

[9] S. W. Hawking and G. F. R. Ellis, The Large Scale Struc-

ture of Spacetime (C.U.P. 1973).

[10] P. M. Chaikin and T. C. Lubensky, Principles of Con-

densed Matter Physics (C.U.P. 1995).

[11] H. Bondi, Eu. J. Phys. 7,106 (1986)

[12] W. G. Unruh, Phys. Rev. D 14, 870 (1976).

[13] L. Mandel and E. Wolf, Optical Coherence and Quantum

Optics (C.U.P. 1995) Sec. 15.8.1

[14] R. H. Price and K. S. Thorne, Phys. Rev. D 33,915

(1986).

[15] J. Makela and A. Peltola, arXiv: gr-qc/0612078

[16] R. M. Wald, General Relativity (University of Chicago

Press, 1984)

[17] S. Deser and B. Tekin, arXiv: gr-qc/0701140

[18] H. Bondi, Proc. R. Soc. London A 427, 249 (1990).

[19] M. Akbar and Rong-Gen Cai, Phys. Lett. B 648, 243

(2007); arXiv: gr-qc/0612089