Divergence of (covaraint) energymomentum tensor

[Stavros Kiri]

I will address just the first part (as I am currently ~ in bed and my Diff. Geometry won't be that walken up yet).
In college, to get ready for General Relativity, I used:
D. F. Lawden "Introduction to Tensor Calculus, Relativity and Cosmology" (e.g. Dover publ.)
https://www.amazon.com/dp/0486425401/?tag=pfamazon01-20
It's kind of educational and affordable.
I believe you can also find it on line (e.g. pdf)
https://www.google.gr/search?ei=lvB...67j0i7i30j0i22i10i30j33i160j0i13.ide/B7FO8fE=

 

[Torg]

Sorry, there a little mistake in the second expression. The first term in the RHS should be the ordinary partial derivative instead of the covariant derivative.

 

[Orodruin]

I struggled whole night with the second expression below, I couldn't figure out the positions of the indices

That would be because you raised the index of the covariant derivative and you simply cannot do that without involving the metric tensor in your expression.
$$
T_{ab}^{;b} = \nabla^{b} T_{ab} = g^{bc} \nabla_c T_{ab} = g^{bc} (T_{ab,c} - \Gamma_{ca}^d T_{db} - \Gamma_{cb}^d T_{ad})
= T_{ab}^{,c} - g^{bc} \Gamma_{ca}^d T_{db} - g^{bc} \Gamma_{cb}^d T_{ad}
$$

 

[Torg]

I thought the metric goes immediately to work into the tensor indices to give a mixed tensor not to the differentiation index itself. I am learning a lot.
Thank you very much Orodruin.

 

[Orodruin]

Of course you can absorb the metric tensor into the energy momentum tensor with lower indices and obtain a mixed tensor, but that did not seem to be what you were after as your template expression included $T_{\cdot\cdot}$.

Edit: Note that in the term on the form $\Gamma_{\cdot\cdot}^\cdot T_{\cdot\cdot}$ there is no way to place indices such that there is only one free covariant index left, since there are 4 covariant and only one contravariant index and the left-hand side has only one free covariant index. This should immediately tell you that it is impossible to write the term on that form.

 

[Torg]

Dr. Stavros Kiri helped me with this ebook
[PDF]http://elisa.ugm.ac.id/user/archive/download/66514/b818578e2246cb5c10d3547197b62209
the author used the wrong expression as i did in page (53) equation (21.20).
I think my lecturer at our department is using that reference.

 

[Orodruin]

At least in the sections you are looking at, that reference discusses special relativity with the Euclidean metric using an imaginary time component. It also introduces relativistic mass, which is an essentially deprecated concept. I would dare to say that you will not find any of this in a modern textbook and I suggest that you get a more modern reference (that text is over 50 years old).

 

[Torg]

Thank you very much. What about non conserved Energy-momentum in General Relativity because the energy of the gravitational filed in not included in it, and that gravity can be removed locally "free fall"?

 

[Torg]

Edit: Note that in the term on the form
\[\begin{array}{l}
{\Gamma _{..}}^{.}{T_{..}} \\
\end{array}\]
there is no way to place indices such that there is only one free covariant index left, since there are 4 covariant and only one contravariant index and the left-hand side has only one free covariant index. This should immediately tell you that it is impossible to write the term on that form.

Very clever! helping a lot in understanding and memorizing it. Thank you.

 

[Orodruin]

Thank you very much. What about non conserved Energy-momentum in General Relativity because the energy of the gravitational filed in not included in it, and that gravity can be removed locally "free fall"?

The stress energy tensor is locally conserved in GR $\nabla_a T^{ba} = 0$. There is generally no good way of defining global energy conservation.

 

[Torg]

I quote "no good way of defining global energy conservation", why is that?

 

[Orodruin]

I suggest reading the following text by John Baez: http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

 

[Torg]

It is very well explained except the paragraph
"We will not delve into definitions of energy in general relativity such as the hamiltonian (amusingly, the energy of a closed universe always works out to zero according to this definition), various kinds of energy one hopes to obtain by "deparametrizing" Einstein's equations, or "quasilocal energy". There's quite a bit to say about this sort of thing! Indeed, the issue of energy in general relativity has a lot to do with the notorious "problem of time" in quantum gravity... but that's another can of worms."

 

[Stavros Kiri]

(Already shared these references in a conversation with Torg yesterday - they may be useful to others too)

Take a look at this thread:
https://www.physicsforums.com/threads/book-on-general-relativity.874853/

Tensor calculus is generally part of differential geometry. Spivak's book is the one I was trying to recall. Try a google search for:
Spivak, "Comprehensive Introduction to Differential Geometry"

For tensors: J.L. Synge, A. Schild, Tensor Calculus (e.g. Dover publ.)
(Traditional)

See also
https://www.physicsforums.com/threads/book-recommendations-in-differential-geometry.917075/

And
https://www.physicsforums.com/threads/differential-geometry-book-with-tensor-calculus.880156/

Or
https://www.physicsforums.com/threads/a-good-book-on-tensors.914341/

I hope that helps.