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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

equation723

In a general frame the number density will go up by a factor tex2html_wrap_inline1014

equation725

and

equation727

so

equation729

Thus the density tex2html_wrap_inline1016 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 tex2html_wrap_inline1020 :

equation733

where tex2html_wrap_inline1022 is the flux per unit area across a surface with normals in the x direction etc, and tex2html_wrap_inline1026 can be interpreted as the flux across a constant ct surface. Thus tex2html_wrap_inline1020 combines the flux and the number density in a single four- dimensional quantity. Note that

equation744

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

equation750

where tex2html_wrap_inline1032 is the flux of tex2html_wrap_inline1034 - momentum across a surface of constant tex2html_wrap_inline1036 . By tex2html_wrap_inline1034 - momentum we mean the tex2html_wrap_inline1034 component of the four- momentum tex2html_wrap_inline1044  .

Let us see how this definition fits in with the discussion above. Consider first tex2html_wrap_inline1046 . 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 .

equation752

Similarly, tex2html_wrap_inline1054 is the flux of energy divided by c across a surface of constant tex2html_wrap_inline1058 :

equation754

Then tex2html_wrap_inline1060 is the flux of i- momentum across a surface of constant t: the density of i- momentum multiplied by c: 

equation756

Finally tex2html_wrap_inline1070 is the j- flux of i- momentum: 

equation758

For any particular system, giving the components of tex2html_wrap_inline1076 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:

eqnarray760

It is easy to see that the tensor tex2html_wrap_inline1082 has exactly these components in the MCRF, where tex2html_wrap_inline1084 is the four- momentum of a particle. It follows that, for dust we have

equation770

From this we conclude that the components of tex2html_wrap_inline1076 are:

equation784

or in matrix form:

eqnarray786

In a frame tex2html_wrap_inline1088 with tex2html_wrap_inline1090 , we therefore have

eqnarray794

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 tex2html_wrap_inline1076 : it is symmetric: 

equation808

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



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