# 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 ] .

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.

- General fluids
- Conservation of energy- momentum
- Conservation of particles
- Perfect fluids
- The conservation equations

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