Consider a pressure- less distribution of non- interacting particles [ called dust ], with rest massm and number densityn 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
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.