So this law establishes conservation, actual conservation of this quantity.

There is no energy flux through the boundary.

This is conserved in this region.

If there is energy flux going out, energy is reduced.

If in, energy is increased.

But in case of the gravity.

What we have been calling as the energy momentum conservation

doesn't have this form.

It does have the following form.

It has a form like this with the covariant long derivative.

And as a result we have the following situation.

So 0 is equal to, we just write explicitly this covariant derivative.

It's T mu nu we express it in this form, sorry.

It was one up and this lower index, and this is upper index.

Plus gamma, mu beta,

mu T, beta nu.

Minus gamma beta nu mu.

T mu beta.

And as a result, this is equivalent 1 over square root of the modulus of

the determinant of the metric, d mu acting on T mu nu.

Square root of the determinant

of the metric, minus 1/2

of d mu g mu alpha T mu alpha.

To obtain this expression, we have used explicit form.

Explicit form for gamma mu nu alpha through the metric.

It's symmetry properties.

And also the fact that energy momentum tensor is symmetric.

T mu nu, T nu mu and finally we have

used that gamma, this gamma.

The trace of gamma, gamma mu nu mu with two indices.

And some summation of the indices is 1 over square root

of g d nu square root of modulus of g.

This is, it follows from the definition of the gamma matrices.

So now what we have is that there is no such conservation.

Well, we have something similar to that expression here, but

we have additional term.

This conservation law, although we call it conservation law,

it doesn't express any explicit conservation of some quantity.

And it should be expected, on general grounds,

that if we are dealing with a theory containing gravity,

the energy momentum tensor of matter shouldn't be simply conserved.

Because there can be energy transmission from matter to gravity and

from gravity to matter.

So the question is what is the quantity which specifies what

means energy, momentum for the gravity.

How to find it and that's the issue we going to address now.

So now we're going to specify what we mean by the energy momentum for the gravity.

And what is the total energy momentum of gravity plus matter?

So let me first write this relation that we have ended with.