On temperature corrections to Einsteinequations

G.E. VolovikLow Temperature Laboratory, Helsinki University of Technology

Box 2200, FIN-02015 HUT, Finlandand

L.D. Landau Institute for Theoretical Physics,117334 Moscow, Russia

andA.I. Zelnikov

Theoretical Physics Institute, University of Alberta

Edmonton, Alberta, Canada T6G 2J1and

P.N. Lebedev Physics InstituteLeninskii prospect 53, Moscow 119 991 Russia

September 16, 2003

Abstract

The temperature correction to the Einstein action is consideredwhich does not depend on the Planck energy physics. The leadingcorrection may be interpreted in terms of the temperature dependenteffective gravitational and cosmological constants. The correction tothe gravitational constant appears to be valid for all temperaturesT EPlanck. It is universal since it is determined only by the numberof fermionic and bosonic fields with masses m T . This universalmodification of the gravitational action can be used to study thermo-dynamics of quantum systems in condensed matter (such as quantum

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liquids superfluid 3He and 4He), where the effective gravity whichemerges for fermionic and/or bosonic quasiparticles is quite differentfrom the Einstein gravity.

1 Introduction

The corrections to the Einstein equations for the gravitational field typicallydepend on the underlying physics of the gravity. There are many ways toproduce the gravity: the gravity, of course, can be the fundamental field; butit also can be an effective field induced by quantum fluctuations of massivebosonic or fermionic fields according to the Sakharov scenario [1, 2, 3]; itcan gradually emerge in the low-energy corner of the fermionic system withmassless topologically protected fermions [4]; the metric/vierbein fields canemerge also as a result of the symmetry breaking as a vacuum expectationvalue of the bilinear combimations of spinor fields [5]; etc. In most cases thecorrections to the Einstein equations are non-universal since they dependon the trans-Planckian physics. However, there are special cases when thecorrections are completely determined by the infrared physics. These correc-tions are universal since they do not depend on the details of the underlyingphysics, and are the same irrespective of whether the gravity is fundamentalor effective. Here we consider such universal corrections which come from theinfrared fermionic and/or bosonic fields whose masses are small compared tothe temperature T .

The same universal terms in the action for the effective gravity arisein the condensed matter systems, in spite of the fact that the main actionwhich governs the effective metric does not resemble the Einstein acton evenremotely. We discuss such subdominating terms in the bosonic superfluid4He and fermionic superfluid 3He, where the effective metric field emerges forthe bosonic and fermionic quasiparticles respectively. Since the temperaturein these liquids is not very small compared to the corresponding ‘Planck’scale EPlanck (which is typically the Debye temperature or the temperature ofsuperfluid transition Tc), the effect of these terms is accesible for experiments.This allows us to experimentally verify many issues of the interplay of thegravity and thermodynamics.

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2 Contribution of massless fermions

2.1 Relativistic theory

Let us start with the modification of the action for gravity which comesfrom the massless fermions. Here we are interested in the contribution whichcomes from the thermal relativistic fermions in the background of the curvedspace. The free energy of the relativistic gas of fermions in the stationarygravitational field has the correction containing the Ricci curvature R of thegravitational field. If the gas is in the global equilibrium with temperatureT0 at infinity one obtains in the high-temperature limit T 2 h2R (butT EPlanck) [6, 7, 8]:

F = FT=0 − 7π2NF

360h3

∫d3x√−gT 4 +

NF

288h

∫d3x√−gT 2[R+ 6w2] . (1)

Here NF is the number of different species of massless chiral fermions; alter-natively this is valid for NF/2 species of Dirac fermions with masses m T .The temperature T is a local (red-shifted) temperature obeying the Tolman’slaw

T (r) =T0√|g00(r)|

, (2)

and T0 = Const. The quantity w2 = wµwµ , where wµ = 12∂µ ln g00(r) is

the 4-acceleration. The temperature dependent part of the free energy isinvariant under stationary conformal transformations [6]. This is becausethe thermodynamics is determined by the (quasi)particle spectrum, and inthe static spacetimes the energy spectrum E2 = pipkg

ik/g00 does not dependon the conformal factor. The 6w2 term in Eq.(1) provides the conformalinvariance of the free energy.

Adding the free energy of the stationary gravitational field

FT=0 = − 1

16πG

∫d3x√−gR , (3)

one obtains the free energy as a sum of matter and curvature terms

F = −7π2NF

360h3

∫d3x√−g

(T 4 − 15

14π2T 2w2

)− 1

16π

∫d3x

1

Geff

√−gR , (4)

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where the effective gravitational ‘constant’ Geff is renormalized due to thegravitational polarization of matter and becomes coordinate dependent dueto Tolman’s law (2):

G−1eff (r)−G−1(T = 0) = −πNF

18hT 2(r) . (5)

At high energy (temperature) the gravitational coupling increases because ofthe quantum effect of massless fermions onto the metric, which correspondsto the screening of gravity by the fermionic vacuum.

Variation of the free energy over gµν gives the equations for the stationarymetric fields at fixed T0. However, because of the temperature corrections tothe gravitational ‘constant’, Geff itself depends on the metric element g00. Asa result the obtained equations deviate from the classical Einstein equationsGµν = 8πGeffTM

µν . Instead one has

Gµν = 8πG(TM

µν + TM+Gµν

), (6)

where Gµν is the Einstein tensor; TMµν is the energy–momentum tensor of

matter; and TM+Gµν is the additional mixed term which is proportional to

T 20 and depends on derivatives of the metric tensor. This means that the

gravity in a medium is different from the gravity in the vacuum: the matter(here the relativistic massless fermions at given T0) is not only the sourceof gravity, but it also renormalizes the gravitational ‘constant’ and modifiesthe gravitational interactions, which now depend on the reference frame inwhich the global equilibrium is achieved.

The temperature correction to the effective gravitational constant andthe resulting modification of Einstein equations are the properties of the low-energy physics of the massless fields in the classical gravitational background:the correction is universal and does not depend on the physics at Planckscale. It is determined only by the number and type of massless fields whichare present in the Universe at given temperature, and these fields may havenothing to do with the underlying quantum fields whose vacuum fluctuationscontribute to the gravitational constant G at T = 0 in induced gravity. Thismeans that the temperature correction does not know whether the gravity isfundamental or effective.

The effect of the polarization of matter in the gravitational field is ex-tremely small in normal conditions, since the temperature is extremely small

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compared to the Planck energy scale. The renormalization may become im-portant only when g00 → 0, i.e. when the system is close to the thresholdof formation of the horizon or ergoregion. But in condensed matter thetemperatures are typically not very low compared with the analogs of thePlanck energy – the Debye temperature TDebye ∼ 10 K in superfluid 4He; thesuperfluid transition temperature Tc ∼ 1 mK in 3He-A; and the transitiontemperature Tc ∼ 10−100 K in superconductors. That is why this effect canbe pronounced in the effective gravity arising in condensed matter.

Moreover, when the thermodynamics or kinetic properties of a condensedmatter system are measured, such as the specific heat, entropy or heat con-ductivity, the T = 0 contribution drops out and the temperature dependentterms become dominating. In the high temperature regime T 2 h2R the T 4

term is dominating in thermodynamics. However, the curvature simulatedby textures in quantum liquids can be made so strong that the subdominat-ing T 2R term under discussion becomes comparable to the T 4 term, and inprinciple even the opposite regime T 2 h2R can be reached which is calledthe texture-dominating regime in condensed matter.

Here we consider the analogs of the T 2R term on the example of twocondensed matter systems.

2.2 Chiral fermions in 3He-A

Let us start with 3He-A where the effective metric felt by chiral fermionicquasiparticles living in the vicinity of Fermi points (i.e. in the low-energycorner) is [4]

gij =1

c2‖li lj +

1

c2⊥

(δij − lilj) , g00 = −1 ,√−g =

1

c‖c2⊥

. (7)

Here l is the unit vector showing the axis of the uniaxial orbital anisotropy ofthe superfluid vacuum; the longitudinal and transverse ‘speeds of light’ are

c‖ =pF

m∗ , c⊥ =∆0

pF

c‖ , (8)

where pF and pF /m∗ are Fermi momentum and Fermi velocity correspond-ingly, and ∆0 is the gap amplitude. We omitted the non-static elementsg0i produced by the superfluid velocity field, which will be discussed later for

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the Bose superfluids, and consider the contribution of only the soft Goldstonedegrees of freedom related to the deformation of the unit vector l (called tex-tures in condensde matter). That is why the ‘speeds of light’ are consideredas constants in space, as well as g00, which may be chosen as g00 = −1. Thiscorrespponds to the ultrastatic spacetime in general relativity.

Omitting the total derivatives one can calculate the integral of the Ricciscalar of the metric (7)

∫d3x

√−gR = −1

2

∫d3x c‖c2

⊥

1

c2⊥− 1

c2‖

2

((l · (∇× l))2 + surface terms.

(9)Let us consider how the term like that enters the free energy of 3He-A.

The gradient expansion of the free energy in terms of the gradients of l-fieldhas been elaborated by Cross in the limit c⊥ c‖ [9]. At that time it wasnot known that the effective gravity for quasiparticles emerges at low energy.But now when we look at the gradient energy derived by Cross we find thatit does contain the twist term which quadratically depends on T [4]:

Ftwist(T )− Ftwist(T = 0) = − 1

288h

∫d3x

c‖c2⊥

T 2(l · (∇× l))2 (10)

≡ 1

144h

∫d3x√−gT 2R . (11)

Here T = T0 since g00 = −1. This term exactly coincides with the tem-perature correction in Eq.(1) in the limit c⊥ c‖, since the number ofthe massless chiral fermions living in the low-energy corner of 3He-A isNF = 2. The thermal part of the free energy does not contain micro-scopic (trans-Planckian) parameters, and is completely determined by thePlanck constant h and by the parameters of the spectrum of fermionic quasi-particles in the low-energy limit when the spectrum becomes ‘relativistic’:E2 = gikpipk = c2

‖(p · l)2 +c2⊥(p× l)2 (here the momentum p of quasiparticles

is counted from the Fermi point). This demonstrates the universality of theT 2R-term, i.e. its independence of the trans-Planckian physics.

We discussed the contribution to the effective action which comes frommassless fermions. In 3He-A there are also analogs of massless photons andgauge bosons, moreover in the logarithmic approximation the effective elec-trodynamics obeys the same metric gµν as fermionic quasiparticles, and thus

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one could expect the similar contribution of the gauge bosons to G−1eff . How-

ever, this contribution has not been found. The reason for that is thatin the effective electrodynamics emerging in 3He-A the non-renormalizablenon-logarithmic terms are comparable with the covariant terms containingthe logarithmically divergent running coupling [4]. As result the ‘speed oflight’ for photons differs from the ‘speed of light’ for chiral fermions. Whilec‖(gauge bosons) = c‖(chiral fermions), the transverse speed of light is es-sentially bigger: c⊥(gauge bosons) c⊥(chiral fermions). Since accordingto Eq.(10) c⊥ is in the denominator, the contribution of the gauge bosons tothe T 2R term is negliegible.

The contribution of the scalar fields with their own (acoustic) metric willbe discussed further in the paper.

3 Contribution of scalar fields

3.1 Bosonic field

The origin of the effective gravity in superfluid 4He is essentially differentfrom that in 3He-A. The superfluid 3He-A belongs to the same universal-ity class of the fermionic systems as the quantum vacuum of the StandardModel, and one can expect that these systems have a common origin of theemerging gravity represented by the collective modes related to the defor-mation of Fermi points. The superfluid 4He is the Bose-liquid where bosonicquasiparticles – phonons – play the role of a scalar field, propagating in thebackground of the effective metric gµν provided by the moving superfluidcondensate (superfluid quantum vacuum). The gravity field gµν experiencedby phonons is simulated mainly by the superfluid velocity field vs (velocityof the superfluid quantum vacuum). The dynamics of the velocity field is de-termined by the hydrodynamic equations instead of the Einstein equations.Nevertheless, the universal T 2R term must enter the effective action for gµν ,and this term must be universal, such as in Eq.(5) but with the prefactordetermined now by the number of bosonic fields. The free energy of the rel-ativistic gas of Ns scalar fields with masses m T in the gravitational fieldis [6, 10]:

F = −π2Ns

90h3

∫d3x√−gT 4 − Ns

144h

∫d3x√−gT 2[R+ 6w2] . (12)

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This means that the contribution of scalar fields to the effective gravitationalconstant is

G−1eff −G−1(T = 0) =

π

9hNsT

2 . (13)

At high energy the gravitational coupling decreases, which reflects the anti-screening of gravity by the vacuum of the massless scalar field.

3.2 Phonon contribution in superfluid 4He.

Phonons obey the following effective metric (which is called the acousticmetric [11]):

√−g =1

c3, g00 = −

(1− v2

s

c2

), gij =

1

c2δij , g0i = −vsi

c2, (14)

where c is the speed of sound; vs is the superfluid velocity; and we neglectedthe conformal factor which depends on the mass density ρ of the liquid. Letus consider the simplest case when the velocity field is stationary. This meansthat there is a preferred reference frame, in which the velocity field does notdepend on time, and vs(r) is the velocity with respect to this frame. Themetric is thus stationary, but not static. Let us also assume for simplicitythat c and ρ are space independent. Then the expression for curvature interms of the velocity field is [12]:

R =1

2(∇× vs)

2 +∇ · (vs(∇ · vs)) +∇ · ((vs · ∇)vs). (15)

For pure rotation vs = Ω × r the curvature is zero R = 0, since this cor-responds to Minkovski space-time in the rotating frame. That is why letus consider the case when the velocity of the superfluid vacuum is curl-free(∇ × vs = 0), which is the case for superfluid 4He when quantized vorticesare absent. Then one has

R = ∇i∇k(vsivsk) . (16)

If in addition the superfluid vacuum is incompressible (∇ · vs = 0), thecurvature transforms to

R =1

2∆(v2

s ) =1

2c2∆g00 = −∆Φ , (17)

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where Φ is the analog of the gravitational potential: g00 = −(1 + 2Φ/c2).Let us apply Eq.(12) to the superfluid 4He where Ns = 1. For simplic-

ity we consider the static effective spacetimes, i.e. those which satisfy thecondition ∇× (vs/(c2 − v2

s )) = 0. Then integrating by parts one obtains:

F − FT=0 =∫

d3x

− π2T 4

90h3c3− T 2

144hc5

∇(v2s ) · ∇(v2

s )

1− v2s

c2

. (18)

Let us remind that T (r) = T0/√

1− v2s (r)/c

2 is a red-shifted temperaturemeasured by the local observer who lives in the liquid and uses phononsfor communication, while T0 is the real temperature of the liquid, which isconstant across the container in a global equilibrium.

Note that the action for the gravity itself (the hydrodynamic energy ofsuperfluid 4He in terms of vs and ρ) is very different from the Einstein action,and it certainly cannot be represented only in terms of the curvature term√−gR. This is because the hydrodynamic equations being determined bythe ultra-violet ‘Planck-scale’ physics are not obeying the effective acousticmetric (14). The same actually occurs in 3He-A, though the effective gravitythere is essentially improved as compared to that in superfluid 4He, and onecan even identify some of the components of the Einstein action. However,in both liquids the temperature correction to the curvature term in Eqs.(18)and (11) is within the responsibility of the emerging infra-red relativisticphysics and is independent of the microscopic (Planck) physics.

3.3 Fields with different effective metrics.

In the fermionic liquid, the velocity-dependent T 2R term (such as the secondterm in Eq.(18)) is also present. Actually there are even several such terms:one comes from NF = 2 chiral fermions with their metric gF

µν in Eq.(7)modified by the superfluid velocity field (see Eq.(9.13) in [4]); another onecomes from the massless scalar field of the sound with its acoustic metricgs

µν in Eq.(14); and there is also two massless scalar fields of spin waves withtheir own ‘spin-acoustic’ metric. These contributions to the T 2 correction

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can be combined in the following way

F2 = − T 20

288h

∫d3x

∑

F

√−gF

|gF00|

(RF + 6w2

F

)− 2

∑s

√−gs

|gs00|

(Rs + 6w2

s

) .

(19)Here T0 is the real temperature in the liquid, which is constant in a globalequilibrium, while the local red-shifted temperature is different for differentquasiparticles, since it depends on the effective metric experienced by a givenexcitation.

4 Discussion

There are many condensed matter systems where the low-energy quasipar-ticles obey the effective Lorentzian metric. The dynamics of this effectivemetric is, typically, essentially different from the dynamics of the gravita-tional field in Einstein theory. However, the thermodynamics appears to beidentical, since it is determined solely by the infra-red physics, which does notdepend on whether the gravity is fundamental or effective. This similaritycan be useful both for condensed matter and for gravity.

In particular this will be useful for the consideration of the low-temperaturelimit. The procedure of the gradient expansion in quantum liquids assumesthat the gradients are smaller than all other quantities, in particular, thecorrection to the action for the metric field in Eq.(11) is valid only if theeffective curvature is small enough: h2R T 2 E2

Planck ≡ ∆20, i.e. when

the temperture is relatively high. However, in quantum liquids or super-conductors the opposite (texture-dominating) limit T 2 h2R E2

Planck isalso achievable. Moreover the thermodynamic effects related to the textures(curvature) become dominating in this limit. An example is provided by the−T 2

√B term in the free energy of the quasi-two-dimensional d-wave super-conductor in applied magnetic field B [13]. This term comes from the gaplessfermions in the background of the U(1) field, whose curvature B substitutesthe Riemann curvature R of the gravitational field. It determines the ther-modynamics of d-wave superconductors in the low-T limit T 2 B, and inparticular the specific heat C ∝ T

√B has been measured in cuprate su-perconductors [14]. Whether the corresponding gravitational term −T 2

√Rappears at T 2 h2R in the free energy of quasi-two-dimensional systems

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with effective Lorentzian metric is the problem for future investigations.Since these effects are determined solely by the infra-red physics, one can

use for their calculations the methods developed in the relativistic theory.Since these terms are not sensitive to the dynamics of the effective gravityfield, one can assume that this dynamics is governed by the Einstein equa-tions. Then the powerful theorems derived for the Einstein gravity can beused, including the connection between the curvature terms in the free en-ergy and the entropy of the horizon. Thus the condensed matter can serveas the arena where the connection between the gravity and thermodynamicscan be exploited and developed.

This work was supported by ESF COSLAB Programme and by the Rus-sian Foundations for Fundamental Research. AZ is grateful to the KillamTrust for its financial support.

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