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Tensors and Relativity: Chapter 4

# The energy- momentum tensor

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Consider a pressure- less distribution of non- interacting particles [ called dust  ], with rest mass  m and number density  n in the momentarily comoving reference frame [ MCRF ]  .

The density  in this frame is

In a general frame the number density will go up by a factor

and

so

Thus the density is not a component of a four- vector. We will see that it is a component of a 2/0 tensor.

We can introduce a number flux  four- vector :

where is the flux per unit area across a surface with normals in the x direction etc, and can be interpreted as the flux across a constant ct surface. Thus combines the flux and the number density in a single four- dimensional quantity. Note that

The most convenient definition of the energy- momentum tensor  is in terms of its components in some arbitrary frame.

where is the flux of - momentum across a surface of constant . By - momentum we mean the component of the four- momentum  .

Let us see how this definition fits in with the discussion above. Consider first . This is defined as the flux of 0- momentum [ energy divided by c ] across a surface of constant t. This is just the energy density .

Similarly, is the flux of energy divided by c across a surface of constant :

Then is the flux of i- momentum across a surface of constant t: the density of i- momentum multiplied by c:

Finally is the j- flux of i- momentum:

For any particular system, giving the components of in some frame, defines it completely.

For dust , the components of T in the MCRF are particularly simple. There is no motion of the particles, so all i- momenta are zero and all spatial fluxes are zero. Therefore:

It is easy to see that the tensor has exactly these components in the MCRF, where is the four- momentum of a particle. It follows that, for dust we have

From this we conclude that the components of are:

or in matrix form:

In a frame with , we therefore have

These are exactly what we would calculate from first principles, for the energy density , energy flux , momentum density  and momentum flux  respectively. Notice one important property of : it is symmetric:

This will turn out to be true in general, not just for dust.

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