The energy- momentum tensor
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.
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
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 :
For any particular system, giving the components of in some frame, defines it completely.
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.
- General fluids
- Conservation of energy- momentum
- Conservation of particles
- Perfect fluids
- The conservation equations