Table Of ContentPotentials in Energy Momentum Tensor
and the Equation of State
Ying-Qiu Gu∗
School of Mathematical Science, Fudan University, Shanghai 200433, China
9
0 (Dated: 20th January 2009)
0
2
The energy momentum tensor of the perfect fluid is a simplified but successful
n
a model of wide practical use in astrophysics. In the previous researches, some im-
J
1 portant information such as the self-energy of the interaction of the particles and
2
influenceofthegravitational potentialareusuallyignoredorintroducedphenomeno-
]
h
p logically. In this letter, the analysis and calculation show that, the interactive po-
-
n tentials of the particles can bedescribed by introducingan extra potential W, which
e
g
acts like negative pressure. For the gases in a star, the state functions are strongly
.
s
c
influencedby thegravitational potential, andonly thetemperatureT is independent
i
s
y
variable, but the other state functions such as mass-energy density ρ and pressure
h
p
P can be expressed as simple functions of T. These relations can provide some new
[
3 insights into the dynamical behavior and structure of a star.
v
3
4
Key Words: energy momentum tensor, equation of state, gravitational potential,
2
1
. self-energy of interaction
1
1
7
PACS numbers: 95.30.Tg,95.30.Lz, 05.70.Ce, 64.30.-t
0
:
v
i
X Inastrophysics, theenergymomentum tensor oftheparticles inastarisusually described
r
a by the Einstein’s perfect fluid model
Tµν = (ρ+P)UµUν Pgµν. (1)
−
Theoretical analysis and experiments all show (1) is a successful approximation. Due to the
effort of several researchers such as Israel and Stewart[1, 2], Carter[3] and Lichnerowicz[4],
the energy momentum tensor Tµν has been developed into a treasure box for classical fluid
theory, which contains not only energy and momentum, but also heat flux, spatial stress
∗
Electronic address: [email protected]
2
and viscosity[5, 6]. Furthermore, the relativistic hydrodynamics with variational principle
and Cartan’s exterior algebra was elegantly established[7].
However, some important information is ignored inthe perfect fluid model (1). One is the
self-energy of the internal interactions of the particles such as electromagnetic potential and
nonlinear potentialofaspinor, theotherisinfluence ofthegravityonthemotionofparticles.
In this letter, we discuss the two realistic problems in detail. The discussion shows that the
self-energy of interaction can be merged into the mass-energy density, and an extra potential
term W should be introduced into Tµν which acts as negative pressure. The meaning of
mass-energy density ρ should be redefined to include the contribution of the self-energy. The
gases in thermal equilibrium inside a star is a state automatically formed, so the equation
of state can not be arbitrarily given, and only the temperature is the independent variable.
This property is much helpful to study the interior behave of a star, and leads to an elegant
and natural model for stellar structure[8]. Besides, the analysis also shows the consistence
and connection among thermodynamics, classical mechanics and field theory.
In conventional thermodynamics, some concepts are usually not explained definitely, e.g.
even the temperature T was not clearly defined for large energy range. This is inconve-
nient for theoretical analysis and easily leads to confusion. In the case of the monatomic
gases, considering that the thermal motion of particles in the comoving coordinate system
is fundamental, so we define the temperature T of the gases as follows
N
2 2 1
kT K¯ = K , (2)
n
≡ 3 3N
n
X
¯
where k is the Boltzmann constant, K the mean shifting kinetic energy of particles in a
micro comoving volume element V, N the particle number in V, K the kinetic energy of
n
n-th particle. (2) is the most popular description in the usual textbooks.
Assuming the distribution of kinetic energy of the particles is given by
K dK
d = f , (3)
P kT kT
(cid:18) (cid:19)
which satisfies
∞ ∞ 3 ∞ 3
d = 1, Kd = kT, K2d = (kT)2, (4)
P P 2 P 2σ
Z0 Z0 Z0
whereσ isaconstantrelatedtotheconcretedistributionfunctionforgivenparticles. Inusual
statistical mechanics, the distribution functions are usually defined in the phase space of
3
coordinate and momentum of the particles, but such selection is sometimes inconvenient for
calculation of state functions, especially in the case of relativistic gases. Since the following
discussions have not any explicit relation with distribution function f, and at most use the
second order moment, so the kinetic energy distribution (3) is much convenient and the
quantities defined by Eq(4) are enough. The following examples show how σ varies with
distribution f,
K 4K dK 2
d = exp , σ = , (Maxwell distribution), (5)
P −kT πkT kT 5
(cid:18) (cid:19)r
−1
40 K dK 15
d = 1+ exp , σ=˙ , (6)
P 177 kT ln(217/40)kT 38
(cid:18) (cid:18) (cid:19)(cid:19)
−n
2K 2(n 1)dK n 3
d = 1+ − , σ = − , (7)
P 3(n 2)kT 3(n 2)kT 3(n 2)
(cid:18) − (cid:19) − −
K 3 dK 2
d = δ , σ = , (the upper bound of σ). (8)
P kT − 2 kT 3
(cid:18) (cid:19)
For the quantized nonlinear fields such as spinors, in microscopic view, the energy mo-
mentum tensor can be expressed by the following classical approximation[9, 10, 11]
Tµν = (m uµuν +w gµν)δ3(~x X~ ) 1 v2, (9)
k k k k − k − k
Xk q
where m is the proper mass of the k-th particle, w is the proper potential, uµ 4-vector
k k k
~
velocity, v the speed, X (t) the central coordinate. In the case of ideal gas, we have
k k
w = 0,( k). The physical meaning of w can be explained by the following example.
k k
∀
Consider the following Lagrangian of self-interaction
1
= , = (∂ A ∂µAν s2AαA )+e qµA , (10)
LA LA|k LA|k 2 µ ν|k k − k α|k k k kµ
k
X
µ
where A stands for the self potential of the k-th spinor, s = 0 corresponds to the electro-
k
magnetic field. Making classical approximation, we generally have[11, 12, 13]
Tµν = ∂µA ∂νAα gµν
|A α|k k − LA
k
X
[(m′ +p′)uµuν p′gµν] 1 v2δ3(~x X~ ), (11)
→ k k k k − k − k − k
k q
X
where X~ is the central coordinate of the particle, m′ stands for the proper energy of Aµ
k k k
in mean sense, and p′ for the internal pressure. By (11), the parameters in the central
k
4
coordinate system are calculated by
1
m′ = dx¯ = e qµA d3x¯ > 0, (12)
k − LA|k 2 k k kµ
ZR3 ZR3
1 1
p′ = m′ + (¯Aα¯A +4 )d3x¯ = m′ + s2AαA d3x¯ , (13)
k 3 k ∇ k∇ α|k LA|k −3 k k α|k
(cid:18) ZR3 (cid:19) (cid:18) ZR3 (cid:19)
in which we have used ∂¯ Aα = 0 and ∆Aα+s2Aα = eqα in the central coordinate system.
0 − k k k
From (13) we learn p′ < 0. So different from the photons which have p = 1ρ , the static
k ph 3 ph
field Aα provides negative pressure. By (12) and (13), we find (9) is of general meaning, in
k
which m can be explained as the total mechanical mass of the k-th particle including the
k
self-energy of interaction, p′ < 0 is the total potential acting as negative pressure, which
k
leads to w > 0 in (9).
k
In order to calculate the statistical expectation value of Eq(9), denote
1 m
P (Tx +Ty +Tz ), K = k m , (14)
≡ −3 x y z k 1 v2 − k
− k
then we have e p
T0 = K +m +w 1 v2 δ3(~x X~ ), (15)
0 k k k − k − k
k (cid:18) q (cid:19)
X
1
P = K +m (m +3w ) 1 v2 δ3(~x X~ ). (16)
3 k k − k k − k − k
Xk (cid:18) q (cid:19)
e
In Eq(15) and (16), we can directly solve the mean value ofK ,m as the function of kT and
k k
m¯ by (4), but the rest terms include the factor 1 v2, which is difficult for estimation.
− k
We resolve the problem by the following trick, whpich naturally includes the influence of the
gravitational potential.
To research the thermodynamical properties of gas, we use piston and cylinder to drive
gas. In astrophysics, we have more ideal piston and cylinder, that is the spacetime with
Friedmann-Robertson-Walker(FRW) metric, which is absolutely adiabatic and reversible.
The FRW metric drives the gases homogeneously expanding and contracting as the scale
factor a varies, and the results have generally meanings according to the principle of equiv-
alence.
The line element of the FRW spacetime is given by
ds2 = dt2 a2(t) dr2 +h2(r)dθ2 +h2(r)sin2θdϕ2 , (17)
−
(cid:0) (cid:1)
5
where the specific form of h(r) is not important. Solving the geodesic we have the following
relations[10, 14, 15]
b a
v = k , 1 v2 = . (18)
k a2 +b2 − k a2 +b2
k q k
σm¯b σm¯b2
p p
a = , kT = , (19)
kT(kT +2σm¯) a(a+√a2 +b2)
where b ,b are determinepd by initial velocity. Since we firstly solve the micro motion of
k
particles and then solving the expectation value of a variable, the results should be the same
as that we firstly do statistical average for the variable and then solve its macro changing
value, so we have the following calculation. By (18), we have relation
d a 1 d
1 v2 = = [a(K +m )]. (20)
− k da 1 v2 mk da k k
q − k
Then by (19), we get p
1 d a 3σm¯kT
m 1 v2d = (K +m )d = m¯ , (21)
N k − k P da N k k P − 2(σm¯ +kT)
!
Xk Z q Xk Z
1 d a w 3σµ¯m¯kT
w 1 v2d = k (K +m )d = w¯ ,(22)
N k − k P da N m k k P − 2(σm¯ +kT)
k !
k Z q k Z
X X
where the mean parameters are defined by
N N
1 1 w
k
w¯ = w , µ¯ = . (23)
k
N N m
k
k k
X X
In the case w > 0, by Eq(22) we have
k
2w¯
w 1 v2d > 0, σµ¯ < . (24)
k − k P 3m¯
k Z q
X
For the same kind particles, by w¯ = µ¯m¯, we get σ < 2.
3
Similar to thermodynamical laws have not relation with piston and cylinder, Eq(21)
and (22) also have not any relation with the ‘piston’ a(t), so they are independent of any
dynamical process, andaregenerally validforidealgases inthe comoving coordinatesystem.
We define the mass-energy density ρ and pressure P of the particles in comoving coordi-
nate system as usual[9]
m 1 m v2
ρ k δ3(~x X~ ), P k k δ3(~x X~ ), (25)
k k
≡ 1 v2 − ≡ 3 1 v2 −
Xk − k Xk − k
p p
6
Then, in mean sense we get
1 3kT
ρ = d (K +m )δ3(~x X~ )dV = ̺ 1+ , (26)
k k k
V P − 2m¯
Z ZV k (cid:18) (cid:19)
X
where ̺ = Nm¯ is the static mass density. By Eq(15), (16) and (21), (22), we find the
V
expectation value of Eq(9) should take the following form
Tµν = (ρ+P)UµUν +(W P)gµν, (27)
−
where Uµ is the mean velocity satisfying U Uµ = 1, and W is an ignored term corresponding
µ
to potentials w , which acts as negative pressure.
k
Then according to (21) and (22), Eq(26) and (27) give
1 w¯ 3µ¯σkT
W = d T0dV ρ = ̺ , (28)
V P 0 − m¯ − 2(σm¯ +kT)
Z ZV (cid:18) (cid:19)
1 kT kT
P = d PdV +W = ̺ 1 . (29)
V P m¯ − 2(σm¯ +kT)
Z ZV (cid:18) (cid:19)
e
In case of ideal gases or spinors, Eq(29) can be also derived according to the principle of
action[14]. In the ultra-relativistic case kT m¯, we have P 1ρ , which is obviously
≫ → 3
reasonable in physics.
W can be explained as potentials. If w = 0, we get W = 0, and the other results reduce
k
to the relations for ideal gas[14, 15]. So the introduction of W provides an entry to deal
with the self-energy of internal interaction potentials of the particles.
The equations (26), (28) and (29) are generally valid relations for any ideal gases in local
equilibrium. By (19), noticing ̺ = Nm¯ and V a3 [kT(kT + 2σm¯)]−32, we get the
V ∝ ∝
following state functions in dimensionless form
3
̺ = ̺0[J(J +2σ)]2, (30)
3 3
3
ρ = ̺(1+ J) = ̺0[J(J +2σ)]2(1+ J), (31)
2 2
J(J +2σ) 1
5
P = ̺ = ̺0[J(J +2σ)]2 , (32)
2(J +σ) 2(J +σ)
3µ¯σJ 3µ¯σJ
3
W = ̺ µ = ̺0[J(J +2σ)]2 µ , (33)
− 2(J +σ) − 2(J +σ)
(cid:18) (cid:19) (cid:18) (cid:19)
where ̺ is a constant depending on parameters (m¯,b,σ,µ,µ¯), and
0
kT w¯
J = , µ = . (34)
m¯ m¯
7
Surprisingly, under theconstraint ofgravity, amongthe statefunctions onlythetemperature
is independent.
Equations (31)-(33) are based on the assumption that the particles move along geodesic.
But the analysis in [11] shows that the central trajectory of a spinor with self interactions,
such as nonlinear potential and electromagnetic field, will slightly depart from the exact
geodesic. So in strict sense, we should solve the exact trajectories to replace the equations of
geodesics (18) and (19). This is a complicated problem. However, in the case 0 w m ,
k k
≤ ≪
the above results are good approximations to high precision. Moreover these results can be
modified by the following treatments.
In (31)-(33), we treat density ̺ or equivalently the volume V as an independent state
function, and then derive the function ̺(J) according to the continuity equation. In the
comoving coordinate system with the following Gaussian type metric[9]
g = diag(g , g ), (a,b) 1,2,3 , (35)
µν 00 ab
− ∈ { }
where g is the spatial metric, for eneregy-momentum tensor (27), we check the following
ab
Gibbs-Duhem’s law
e
δQ = d[(ρ+W)V]+(P W)dV, (36)
−
where δQ denotes the heat received by the N particles, and
V = g∆x∆y∆z, (g = det(g )) (37)
ab
p
is the micro spatial volume occupeied by N given pearticles.e
We trace the motion of these particles. In the adiabatic process, we have δQ = 0. Then
by U Tµν = 0, we get the continuity equation for Eq(27) as
µ ;ν
Uµ∂ (ρ+W)+(ρ+P)Uµ = 0. (38)
µ ;µ
Denoting the proper time by dτ, then d = Uµ∂ , Eq(38) becomes
dτ µ
d d
0 = g (ρ+W)+(ρ+P) g ∂ Uµ + g ,
µ
| |dτ | | dτ | |
(cid:18) (cid:19)
p p p
d d
= √g (ρ+W) g +(P W) g (39)
00
dτ − dτ
(cid:18) (cid:19)
(cid:16) p (cid:17) p
d
+(ρ+P) g √g ∂ Ueµ + √g , e
00 µ 00
dτ
(cid:18) (cid:19)
p
e
8
where g = det(g ) = g g. In the comoving system, we have Uµ = ( g00,0,0,0) and
µν 00
dτ = √g dt, then we get p
00
e
d 1 1
√g ∂ Uµ + √g = √g ∂ + ∂ √g = 0. (40)
00 µ 00 00 t t 00
dτ √g √g
00 00
Multiplying Eq(39) by the comoving volume element ∆x∆y∆z for the N given particles,
(39) gives (36) in the case δQ = 0.
One may oppugn the generality of Eq(36) due to the collision of particles. As shown
in [14], Eq(36) holds in mean sense for any process with elastic collisions, which can be
understood asrelay raceof particles. (36) include the gravitationalpotentialof the particles,
which is different from but consistent with the energy-momentum conservation law Tµν = 0.
;ν
For example, Tµν = 0 holds for a little solid ball, in which the molecules are all fixed at
;ν
definite position by electromagnetic interaction, and then the influence of the gravity on the
thermal movement vanishes. However, in astrophysics the gravity is dominant inside a star,
where the gravity drives the particles moving rapidly, and such motion should be exchanged
andbalanced withthermalmotionoftheparticles. Thisistheunderlying andsubtlerelation
between gravity and equation of state, which is ignored before. If the gravitational potential
is taken into account, we get elegant and singularity-free models for the stellar structure[8].
Inthecaseof δQ = 0, (36)provides anentry to dealwiththeradiationandheat produced
6
by nuclear reaction inside a star. Obviously, δQ takes the form δQ = (̺q(J) E )dτ.
radiation
−
Then in the equilibrium process, we also have δQ = 0.
Now we derive the relation ̺(J) from (36) for the equilibrium process with δQ = 0. For
clearness, we use the the static mass density ̺ = Nm¯ to replace V. Substituting Eq(26),
V
(28) and (29) into Eq(36), we get dimensionless differential equation
1 d̺ 3[(J +σ)2 µ¯σ2]
= − , (41)
̺dJ (J +σ)[J2 +BJ 2µσ]
−
and the solution
α
̺ = ̺0(J +σ)3µ¯3+µ¯1 J2 +JB 2σµ 23(1−3µ¯µ¯+1) 2J +B −A 2A , (42)
− 2J +B +A
(cid:18) (cid:19)
(cid:0) (cid:1)
where parameters are defined by
B = (2+3µ¯)σ 2µ, A = 8σµ+B2, α = 3(1+4µ¯)[2µ 3µ¯σ]. (43)
− −
p
(42) shows how the internal potentials w influence the mass density. (42) reduces to (31)
k
if µ = µ¯ = 0.
9
Insummary, the abovecalculations work outallrelations intheenergy-momentum tensor
of the monatomic particles, which include the classical approximation for the self-energy of
interaction and the influence of the gravitational potential. Although the realistic process
inside astariscomplicated, theabovetreatment includes thedominant partsofthese effects.
Noticing the effectiveness of the assumptions, the results will provide some new insights into
the interior behavior and structure of the stars. In addition, we list some interesting prop-
erties of the state functions derived above, which show the profound relationship between
the empirical laws and the theoretical concepts:
1. In the case of ideal gas with w¯ = 0 at low temperature, (31)-(32) gives the equation
of state for the adiabatic monatomic gas
5
P=˙ρJ=˙P0ρ3, (J σ, or kT m¯), (44)
≪ ≪
which is identical to the empirical law in thermodynamics. When J 1, we have
≫
P 1ρ, so the adiabatic index is not a constant for large range of temperature due to
→ 3
the relativistic effect. These results show the validity of (31)-(33) and the consistence
with normal thermodynamics.
2. By (31), letting J or m¯ 0, we get the Stefan-Boltzmann’s law ρ T4.
→ ∞ → ∝
This means that the above results automatically include photons, and the Stefan-
Boltzmann’s law is also valid for the ultra-relativistic particles.
3. In general relativity, all processes occur automatically, and ̺ is independent of any
0
practical process, so it is a constant representing the strength of gravity constraining
free particles. Of course, ̺ is related to the property of particles. Furthermore,
0
equation of state (31)-(32) provides a self consistent theory for stellar structure in
thermal equilibrium[8].
4. In the case w¯ > 0, the motion of the particles with internal interaction will slightly
deviate from the geodesic[11], so consequently the above relations have a very tiny
violation. By (42) we find, ̺ = 0 leads to J = 2σµ µ, which means the zero
B ≈
temperature can not reach. This result may be caused by the deviation of motion
from geodesic Eq(18).
10
Acknowledgments
The author is grateful to his supervisor Prof. Ta-Tsien Li for his encouragement and
guidance. Thanks to Prof. Tie-Hu Qin and Prof. Ji-Zong Li for meaningful discussions.
[1] J. M. Stewart, On Transient Relativistic Thermodynamics and Kinetic Theory, Proc. Roy.
Soc. London A, 357, 59-75(1977).
[2] W. Israel, J. M. Stewart, Transient Relativistic Thermodynamics and Kinetic Theory, Ann.
Phys., 118, 341-372(1979).
[3] B. Carter, Convective Variational Approach to Relativistic Thermodynamics of Dissipative
Fluids, Proc. Roy. Soc. London A, 433, 45(1991).
[4] A. Lichnerowicz, Relativistic hydrodynamics and magnetohydrodynamics, Benjamin, New
York, 1967
[5] N. Andersson, G. L. Comer, Relativistic fluid dynamics: physics for many different scales,
gr-qc/0605010.
[6] W. A. Hiscock, L. Lindblom, Stability and Causality in Dissipative Relativistic Fluids, Ann.
Phys., 151, 466-496(1983).
[7] E. Gourgoulhon, AN INTRODUCTION TO RELATIVISTIC HYDRODYNAMICS, EAS
Publ.Ser. 21, 43(2006), gr-qc/0603009
[8] Y. Q. Gu, Structure of the Star with Ideal Gases, arXiv:0712.0219
[9] S. L. Weinberg, Gravitation and Cosmology(Ch.2.8, Ch.11), Wiley, New York, 1972
[10] Y.Q.Gu,A Cosmological Model with Dark Spinor Source, toappearinIJMPA,gr-qc/0610147
[11] Y. Q. Gu, Local Lorentz Transformation and “Lorentz Violation”, arXiv:0708.1643
[12] Y. Q. Gu, Mass-Energy Relation of the Nonlinear Spinor, hep-th/0701030
[13] Y. Q. Gu, New Approach to N-body Relativistic Quantum Mechanics, IJMPA Vol.22(No.11)
2007-2019(2007), hep-th/0610153
[14] Y. Q. Gu, Thermodynamics of Ideal Gas in Cosmology, arXiv:0708.2962
[15] Y.Q.Gu,G.J.Ni,TheRelation betweenCosmicTemperature andScale Factor,gr-qc/0609098
