Energy-Momentum Tensor

Incoherent Matter Case

One of the simplest energy-momentum tensors is the dust energy momentum tensor. This type of matter field consists of noninteracting incoherent matter. The matter field depends on one scalar quantity and one vector quantity. These two quantities are as follows:

(15.1)

This density is the density that would be measured by an observer moving with the flow of the dust fluid. Such an observer is called a co-moving observer.

(15.2)

where  is the proper time along the path that the dust particle is taking. The simplest energy-momentum tensor that can be constructed from these two dust quantities is the following:

Note that this pattern of physical terms is similar to the pattern for kinetic energy in classical mechanics.

We will now look at the behaviour of this energy-momentum tensor by seeing how it works in special relativity. The relation between the proper time and normal coordinate time is found by considering the Minkowski line element (coordinates given by  ).

(15.4)

Rearranging terms we get

 (15.5)

where  is the usual special relativity speed factor

(15.6)

 .

The symbol  is the 3-D velocity magnitude of the dust fluid particle.

We now take a look at the 00 component of the dust tensor.

(15.7)

This can then be written

 (15.8)

This equation is reminiscent of the relativistic mass relation where

 (15.9)

An extra  factor is picked up in the case of  because we are dealing with a density with volume involved. We need to transform the volume along the direction of motion of the dust particle and this brings in the  factor to make .

We note that in a co-moving frame we would expect the following condition.

 (15.10)

since  by definition in such a frame. Thus we have

(15.11)

If a matter field of dust, with proper density  and 3-D velocity , flows past a fixed observer, that observer measures the density as

 (15.12) .

 (15.13)

We are therefore led to a physical interpretation of the 00 component of the dust energy-momentum tensor as follows.

(15.14)

The other components of the dust energy momentum tensor (15.3) can be written

The divergence of the energy momentum tensor for dust leads to two very important equations involving energy and momentum. We shall now investigate how these two equations arise. The divergence equation is written as follows:

 .

Consider the case when the free index of the divergence equation above has . Substituting the matrix equation (15.15) into the above equation then gives (exercise)

(15.17)

This equation can be rewritten more succinctly as

This is just the equation of continuity of a fluid. It links up the change in time of the density with a corresponding change in space. In special relativity conservation of mass density is essentially the same as conservation of energy density we can interpret the 0 component of the divergence equation

 (15.19)

as a relativistic conservation of energy equation for the dust fluid. We now consider the other important equation that comes out of the divergence equation (15.16) by letting . We have

(15.20)

.

When combined these three equation gives the following result (exercise).

(15.21)

Using the equation of continuity (15.18) we see that this equation can be succinctly written as

In classical fluid mechanics the equivalent of Newton's 2nd Law equation of motion is called the Navier-Stokes Equation. It is written as

(15.23)

where  is the pressure in the fluid and  is called the external body force per unit mass. We see that the dust equation of motion (15.22) is just a special case of the Navier-Stokes equation when there's no pressure and no external forces acting on the body. From the two derived equations (15.18) and (15.22) we can conclude that the divergence equation (15.16) in special relativity is a statement of conservation of energy and conservation of linear momentum in the matter field. This explains where this tensor gets its name: energy-momentum tensor. Making the simplest generalization we see that the equation of energy momentum conservation in General Relativity should be given as

 (15.24) .

This equation in General Relativity actually contains the geodesic equation of motion. We will now derive this fact. With dust the divergence equation becomes

 (15.25)

Using the product rule we can get the equivalent equation

Contract this equation with  to give

(15.27)

Since we are using proper time the 4-velocity is normalized such that

 (15.28) .

This automatically implies that

 (15.29) .

Thus the equation (15.27) reduces to

 (15.30)

This is a continuity type of equation and when it is substituted back into (15.26) we arrive at the result

 (15.31)

This equation implies that the vector  is tangent to a geodesic. Thus geodesic motion must apply for dust fluid particles. This is made explicit by expanding this equation in terms of the proper time derivatives to give:

 (15.32)

Perfect Fluid Case

A perfect fluid is characterized by two scalar quantities (energy density and pressure) and one vector quantity (4-velocity).

 (15.33)

 (15.34)

 (15.35)

In the limit that , the perfect fluid tensor should reduce to the dust tensor. It can be shown that in order to have this reduction and also give rise to the equivalent of the equation of continuity and the Navier-Stokes in Special Relativity the perfect fluid tensor must have the following form:

 (15.36)

Note that in terms of units

 (15.37)

and

 (15.38)

Sticking to pure General Relativity it can be shown that the conservation equations

 (15.39) ,

when applied to the perfect fluid tensor, also give (1) an equation-of-continuity-type equation and (2) an equation-of-motion equation that looks like a geodesic equation with a forcing function. This forcing function arises from the derivative of the pressure. This pressure derivative makes the fluid deviate from pure geodesic motion.

In general when we model some matter-energy distribution in General Relativity we need another equation that tells us how the matter reacts with pressure and absolute temperature. This equation is called the equation of state of the matter-energy system. It can be generically written as

 (15.40)

In many cases the physical system is in equilibrium and we can assume that . Then the equation of state for an equilibrium situation is generically written as

 (15.41)

If you are modelling a star you will need an equation of state that depends on the type of star. If you are modelling the universe in a cosmological calculation you will need an equation of state depending on what stage of the universe after the Big Bang you are considering.