The Stress Tensor and Fluids

g∂µJ^µ+√

g(∂ν`n√ g)J^ν

= ∂_µ(√ gJ^µ)

= ∂µ(√

g^{µν1···νn−1}ων1···νn−1)

= E^ν1···νn∂ν₁ων₂···νn

= dω (4.10)

Here, E is the Levi–Civita symbol (section (14.6).

For the stress tensor, however, these manipulations do not go through and hence the divergence equation does not lead to a conservation law. How did one get the usual conservation laws for special relativity? Recall that in the special relativistic context, the stress tensor is a tensor only relative to Lorentz transformations. Hence the only changes of integration variables permitted are the (constant) Lorentz transformations. For these restricted change of variables, the integration is well defined. Furthermore the space-time is flat and so in the Minkowskian coordinates the connection is zero.

Covariant divergence is then same as the ordinary divergence.

A physical way of stating this lack of conservation law is to note that the connection term is like a gravitational force (since metric is analogous to the gravitational potential). Presence of these terms implies that tidal forces can always do work on the matter and thus one cannot expect a separate conservation for matter.

There are cases where the divergence equation does lead to conservation equation. If we have a space-time with a symmetry i.e. transformations gen-erated by a Killing vector which leaves the metric invariant, then one can define a conserved quantity. For instance, if ξµ is a Killing vector field, i.e.

satisfies ξµ;ν+ ξν;µ = 0, then one can define J^µ := T^µνξν. Its covariant di-vergence is zero and because of the argument presented above, the quantity Q(Σ) :=R

ΣJ^µξµ where Σ is a hypersurface orthogonal to the Killing vector, is conserved. However, generic space-times do not admit any Killing vectors.

4.3 The Stress Tensor and Fluids

Let us consider the right-hand side of the Einstein equation – the Stress Tensor, T_µν. Only two properties are stipulated, namely, it is a symmetric tensor and it is covariantly conserved. One more property is implicit: if vari-ables representing matter are ‘set to zero’, the stress tensor vanishes. Hence,

Dynamics of Space-Time 43 Tµν is more accurately called as the matter stress tensor. Coupling of matter to the metric is only through its ‘energy’ and its ‘strength’ is determined by appealing to the Newtonian limit and has the specific 8πG coefficient. How is the Tµν to be determined for a given type of matter?

It is postulated ab initio as in the case of a perfect fluid and is derived for matter which can be described by an action containing the metric. The precise coefficients of coupling are a matter of convention in the sense, it depends on the coefficients in front of the Einstein–Hilbert action. These coefficients are chosen so that the equations of motion derived give the Einstein equation. Let us note some examples.

Perfect fluid: A perfect fluid is characterized by a fluid velocity, v^µ(x) (time-like), and two functions, P (x) and ρ(x), representing pressure and en-ergy density respectively. Its stress tensor is given by,

T^µν := ρv^µv^ν+ P (g^µν+ v^µv^ν) (4.11) This is the same form as in special relativistic formulation.

For matter which can be described by an action, It is most naturally de-rived from an action functional including matter denoted generically by Φ, coupled to a metric, via the definition: T_µν(Φ, g) := −√²

|g|

δSmatter(Φ,gµν) δg^µν . The specific sign and the factor of 2 are chosen so that for the standard nor-malization for actions on Minkowski space-time, the definition matches with the special relativistic definitions¹. Thus, in using this definition, we assume standard normalization of the flat space-time actions².

Here are some examples.

Free scalar field: The action is taken as S[g, φ] =

2The action for metric and matter together will be written as: αgSGravity + αmSmatter(Φ, g), the constant coefficients are chosen so that Einstein equation results. The gravitational action is taken to be:R d⁴xp|g|R(g). This implies, αm= 16πGαg.

E₀^µ := u^µ denote an orthonormal basis, so that g^µν = η^IJE_I^µE_J^ν. Then E := Tµνu^µu^ν(=: T00) is the energy density measured by the observer in his/her rest frame. For the three examples above, we get,

E_scalar = (E₀· ∇φ)²+1

2−(E0· ∇φ)²+ (E_i· ∇φ)²+ m²φ²

= 1

2(E0· ∇φ)²+ (E_i· ∇φ)²+ m²φ²

(4.14) Eem = F0IF0Jη^IJ+1

4FIKFJ Lη^IJη^KL, FIJ:= E_I^αFαβE_J^β,

= 1

2

 ~E · ~E + ~B · ~B

, Ei:= F0i, Bi:= 1

2E_i^jkFjk (4.15)

Efluid = ρ (4.16)

The example of electrodynamics also shows that T0i:= Tµνu^µE_i^ν = E_i^jkEjBk

which is the Poynting vector representing the flux of electromagnetic field energy.

Not every stress tensor which is symmetric and covariantly conserved need represent physical matter. For instance, we expect all physical matter to have positive energy density as measured by any observer in his/her rest frame. In our experience with Newtonian gravity, it is an attractive force only among two positive masses and we have never seen it being repulsive. This suggests that in the physical world, masses are only positive.

Since the Ricci tensor is determined by the matter stress tensor through the Einstein equation, the properties of stress tensor have a direct bearing on the geometrical properties manifested through the Ricci tensor. Even though we can have very complicated physical stress tensor arising out of different species of matter, we expect certain general properties to be satisfied by any such tensor. These stipulations go under the name of energy conditions. Below are commonly used energy conditions [17].

Weak Tµνv^µv^ν ≥ 0 ∀ v · v < 0

Strong T_µνv^µv^ν ≥ −¹₂T_αβg^αβ ∀ v · v = −1

Dominant −T^µ_νv^ν is future directed causal ∀ v future directed causal.

There is also the Null energy condition which replaces the time-like v^µ by a light-like k^µ.

To appreciate their implications, it is useful to have an algebraic classifi-cation of the stress tensor, which allows us to put the stress tensor in some canonical forms. The classification holds point-wise in the space-time. Math-ematically, one considers the eigenvalue equation: T^µ_νX^ν = λX^µ or equiva-lently, T_µνX^ν= λg_µνX^ν. Note that T^µ_ν = g^µαT_αν is not a symmetric matrix, only T_µν = T_νµ. Hence, diagonalizability of T^µ_ν is not assured. Secondly, the metric is Lorentzian which means that eigenvectors X^µ can be time-like and light-like as well. Nevertheless, symmetric nature of the stress tensor implies

Dynamics of Space-Time 45 that eigenvectors corresponding to distinct eigenvalues are necessarily orthog-onal. This implies that among the eigenvectors, there can be at the most only one time-like eigenvector. Likewise, since no two distinct light-like vectors can be orthogonal, if they are eigenvectors, they must have the same eigenvalue.

The classification is now arranged according to number of distinct eigenval-ues [18].

Type I: Four distinct eigenvalues. This implies 4 orthogonal eigenvectors.

If one of these is light-like, then the remaining three must be space-like and this is impossible. Hence the eigenvectors must form a (pseudo-)orthonormal basis, E_µ^a, a = 0, 1, 2, 3, E^a· E^b= η^ab. The T^µ_ν is diagonalizable and can be expressed as,

T_µν = −ρE_µ⁰E_ν⁰+

3

X

i=1

p_iE_µⁱE_νⁱ.

The p_i are called principle pressures and ρ is called the rest energy density.

Stress tensors of all observed massive and massless fields are of type-I, except for a very special case below.

Type II: Three distinct eigenvalues. So one eigenvalue is repeated once.

The eigenspace of the repeated eigenvalue is two-dimensional. If every vector in this space is space-like or if one of these vectors is time-like, then we are back to the previous case. So a new case arises when the eigenvector for the repeated eigenvalue is light-like and with no other eigenvector in the subspace.

This case thus has a double light-like eigenvector and two space-like ones.

Two get a canonical form, let k^µ be the double light-like eigenvector with eigenvalue λ and let E_µ², E_µ³ be the remaining two eigenvectors. Introduce another light-like vector `µ which is orthogonal to E<sup>2</sup>, E<sup>3</sup> and k · ` = 2. Then the stress tensor can be written as,

T_µν = σk_µk_ν+λ

2(k_µ`<sub>ν</sub>+ k<sub>ν</sub>`_µ) + p₂E²_µE²_ν+ p₃E³_µE_ν³, σ 6= 0 is arbitrary.

Only known situation is a massless field with radiation going along kµ (all eigenvalues are zero i.e. λ, p2, p3= 0) This is said to represent ‘null fluids’.

Type III: Two distinct eigenvalues. By similar argument as above, a new case arises when the repeated eigenvalue has a triple, light-like eigenvector and of course one space-like one. Let k^µ be the triple light-like eigenvector with eigenvalue λ. Introduce ` as before and another orthogonal vector eµwhich is space-like and is also orthogonal to E_µ³. The canonical form is then expressed as,

Tµν =λ

2(kµ`ν+ kν`µ) + (kµeν+ kνeµ) + λe²_µe²_ν+ p3E_µ³E_ν³. This ensures that only kµ and E_µ³ are the eigenvectors.

There are no physical examples of this type.

Type IV: There is no time-like or light-like eigenvector at all. There are no physical examples of this type either. This happens when there is a pair of

complex conjugate eigenvalues. This cannot happen in the space-like subspaces hence it suffices to have the canonical form as,

Tµν = (σ −ρ)kµkν−(σ +ρ)`µ`ν−ρ(kµ`ν+kν`µ)+p2E_µ²E_ν²+p3E_µ³E_ν³, ρ²< σ² Note that if all eigenvalues are equal, then T_µν = Λg_µν and this falls in the type I of diagonalizable stress tensors.

For diagonalizable stress tensor, the energy conditions take the form:

Weak ρ ≥ 0, ρ + pi ≥ 0 ;

Strong ρ +P pi ≥ 0, ρ + p_i ≥ 0 ;

Dominant ρ ≥ |pi|

These energy conditions play a role in the discussion of singularity theorems.