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)
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where
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)
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Rearranging terms we get
(15.5) |
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where
(15.6)
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The symbol
We now take a look at the 00 component of the dust tensor.
(15.7)
This can then be written
(15.8) |
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This equation is reminiscent of the relativistic mass relation where
(15.9) |
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An extra
We note that in a co-moving frame we would expect the following condition.
(15.10) |
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since
(15.11)
If a matter field of dust, with proper
density
(15.12) |
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(15.13) |
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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:
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Consider the case when the free index of
the divergence equation above has
(15.17)
This equation can be rewritten more succinctly as
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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) |
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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
(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
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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
(15.24) |
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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) |
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Using the product rule we can get the equivalent equation
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Contract this equation with
Since we are using proper time the 4-velocity is normalized such that
(15.28) |
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This automatically implies that
(15.29) |
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Thus the equation (15.27) reduces to
(15.30) |
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This is a continuity type of equation and when it is substituted back into (15.26) we arrive at the result
(15.31) |
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This equation implies that the vector
(15.32) |
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Perfect Fluid Case
A perfect fluid is characterized by two scalar quantities (energy density and pressure) and one vector quantity (4-velocity).
(15.33) |
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(15.34) |
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(15.35) |
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In the limit that
(15.36) |
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Note that in terms of units
(15.37) |
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and
(15.38) |
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Sticking to pure General Relativity it can be shown that the conservation equations
(15.39) |
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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) |
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In many cases the physical system is in
equilibrium and we can assume that
(15.41) |
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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.