Energy momentum tensor - 3

This video shows how momentum and energy are not conserved in curved space-time as matter and energy respond to local curvature resulting in changes in the direction of their velocity vector which, in turn, results in changes in their momentum and energy.

[@meowwwww6350]

These videos are a 💎

[@TensorCalculusRobertDavie]

Hello Prabhakaran and thank you for that! Much appreciated.

[@بنوشواش]

شكرا THANKS PROF.

[@TensorCalculusRobertDavie]

You're welcome.

[@lionelronaldo1574]

Great video

[@TensorCalculusRobertDavie]

Thank you!

[@gunsdark2000]

Sir,if the divergence of EMT isn't in general zero,why, when construct the einstein's field equation , assume that it is zero?

[@TensorCalculusRobertDavie]

The Einstein tensor was constructed to satisfy local conservation of the EM tensor (divergence of both sides is zero). The only possible combination that did so was a combination of the Ricci tensor and curvature scalar multiplied by the metric. Why local? Because the Riemann tensor, from which are derived the Ricci tensor and scalar, gives the curvature at each point of the manifold (assigns a tensor field to each point). At each point and local to that point, the EM tensor gives us the energy density. But in a small region of space-time, the Laws of Special Relativity are a sufficiently good approximation for describing the system. We also know that energy and momentum are conserved in special relativity. In short, the Einstein tensor has the form it does (to give it zero divergence) so that it agrees with local conservation of energy and momentum on the right-hand side. Further reading: Relativity, Hans Stephani, 3rd Ed. page 237. Gravity, James B. Hartle, bottom of page 480 and then page 482. Among other sources as well. Hope that helps?

[@premkhurana3691]

Thanx sir and plz also provide one video on question asked in interview for phd if our specialisation in general relativity

[@بنوشواش]

Prof , can you make a video about Lyra geometry

[@TensorCalculusRobertDavie]

Sorry, but I have some other videos to make first.

[@MetaMaths]

Sir, do you give private lessons ?

[@TensorCalculusRobertDavie]

Sorry I'm not able to give private lessons for the foreseeable future.

[@lionelronaldo1574]

Can you make a video what happen to the stress energy tensor in schwarszchild or other metrics?

[@TensorCalculusRobertDavie]

I will touch on the Schwarzschild metric in the next video.

[@بنوشواش]

sir how are you ?

[@TensorCalculusRobertDavie]

I am doing well thank you. How are you going?

[@بنوشواش]

@@TensorCalculusRobertDavie 🙂 thanks for God ,i am fine 🙂

[@johnstroughair2816]

This seems a little problematic. We know D.T is everywhere zero, this is local energy and momentum conservation. So the integral of D.T over any spacetime volume must also be zero. But this doesn’t force the hypersurface integral over n.T to vanish, there is a term (Christoffel Symbol) T that seems to have been dropped. Zee gives a brief but illuminating discussion at the end of chapter VI.4 of his book.

[@robertdavie1221]

Globally, momentum and energy are not conserved but locally, in a small enough region of spacetime where the laws of special relativity apply, they are conserved. Have a look at the response to the question above this one.

[@johnstroughair2816]

I am not disagreeing with the conclusion that energy/momentum is only locally conserved. I am worried about the equations at about 4 mins. You seem to come to the conclusion that D.T is not zero. But we know that it is zero, it has to be because it is the right hand side of the field equations. My point is that in curved space D.T = 0 doesn’t force global energy/momentum conservation.

[@TensorCalculusRobertDavie]

@@johnstroughair2816 The points P and Q are at different locations in spacetime and so in a curved space vectors at each of these points cannot be added or subtracted to vectors at the other point because they belong to different vector spaces. That is why D.T does not equal zero. We are thinking globally here. This is not an issue in flat Minkowski space where there is only one vector space and all vectors belong to it. That means D.T = 0 in this case. The Einstein tensor was constructed to satisfy local conservation of the EM tensor (divergence of both sides is zero). The only possible combination that did so was a combination of the Ricci tensor and curvature scalar multiplied by the metric. Why local? Because the Riemann tensor, from which are derived the Ricci tensor and scalar, gives the curvature at each point of the manifold (assigns a tensor field to each point). At each point and local to that point, the EM tensor gives us the energy density. But in a small region of space-time, the Laws of Special Relativity are a sufficiently good approximation for describing the system. We also know that energy and momentum are conserved in special relativity. In short, the Einstein tensor has the form it does (to give it zero divergence) so that it agrees with local conservation of energy and momentum on the right-hand side. Further reading: Relativity, Hans Stephani, 3rd Ed. page 237. Gravity, James B. Hartle, bottom of page 480 and then page 482. Among other sources as well.

[@johnstroughair1972]

@@TensorCalculusRobertDavie It is entirely possible that I am missing something here I am just learning GR but let me make a couple of points. 1. It seems to me that if you were to apply the reasoning of this video to the EM 4-current we would get the conclusion that charge is not conserved. 2. I understand that the vectors at P and Q are in different tangent spaces and cannot be compared, but we are not doing that the surface integral is a map from the tangent space into R and that's where the comparison takes place. 3. I am not sure the form of the divergence theorem you are using can be applied to a symmetric (0,2) tensor. Appendix E of Carroll's book seems to say the theorem only holds for vectors and antisymmetric tensors like F. A symmetric tensor like T would pick up extra terms involving the connection and it is these terms that lead to nonconservation of energy - this is explained in VI.4 of Zee's book.