Are the simple energy momentum tensor and the one derived from the action the same?
Answers
Answered by
0
This leads directly to the way to give meaning to the components like for example T00T00 being the energy density, T0iT0i being the momentum density and so on.
It seems that when Einstein derived his GR theory the energy momentum tensor he had in mind was exactly that.
Now, if one picks the Einstein-Hilbert action and derive the equations of motion coupling to a matter action one is lead to the equation
116πG(Rμν−12Rgμν)+1−g−−−√δSMδgμν=0116πG(Rμν−12Rgμν)+1−gδSMδgμν=0
Which upon comparison to Einstein's equations suggest that one defines
Tμν=−21−g−−−√δSMδgμνTμν=−21−gδSMδgμν
which would give rise to a different energy momentum tensor. One usually then considers this last equation the correct definition of the energy momentum tensor for GR.
I find this quite hard to swallow. I mean, I do understand that this comes from the requirement that the Einstein-Hilbert action yields the same equations of motion than Einstein's equations, but this last definition is just one abstract mathematical object defined as a derivative of the action, which, at first sight, doesn't carry much physical meaning.
Einstein's idea is: turn matter into the source of spacetime curvature. For this t here's nothing better than that naive definition I posted. With hit, Gμν=κTμνGμν=κTμν actually carries the meaning that "matter distribution curves spacetime".
Now, writing Gμν=κTμνGμν=κTμν with the other TμνTμν we are just coupling curvature to one strange and abstract tensor derived from one action, which doesn't seem to tell at first that matter is the responsible for spacetime curvature.
Finally, energy and momentum should be, as we know from virtually all physics the generators of time and spatial translations. This is, as far as I know, the most general way to understand energy and momentum. It is not clear at all how this is built in into the "action-based" TμνTμν while it is obvious how is this built in into the traditional TμνTμν since pμpμ is defined as the generator of spacetime translations.
Anyway: we have two energy momentum tensors to use with EFE, one closely related to energy and matter, and which seems to be the one used by Einstein's original approach. The other which is just abstract and comes out of mathematical manipulations, and doesn't seem to relate to energy and matter at all. How do these two relate and how can they be the same?
I mean, they ought to be the same, because if they are not, the idea that "matter curves spacetime" is certainly captured just by the first one.
It seems that when Einstein derived his GR theory the energy momentum tensor he had in mind was exactly that.
Now, if one picks the Einstein-Hilbert action and derive the equations of motion coupling to a matter action one is lead to the equation
116πG(Rμν−12Rgμν)+1−g−−−√δSMδgμν=0116πG(Rμν−12Rgμν)+1−gδSMδgμν=0
Which upon comparison to Einstein's equations suggest that one defines
Tμν=−21−g−−−√δSMδgμνTμν=−21−gδSMδgμν
which would give rise to a different energy momentum tensor. One usually then considers this last equation the correct definition of the energy momentum tensor for GR.
I find this quite hard to swallow. I mean, I do understand that this comes from the requirement that the Einstein-Hilbert action yields the same equations of motion than Einstein's equations, but this last definition is just one abstract mathematical object defined as a derivative of the action, which, at first sight, doesn't carry much physical meaning.
Einstein's idea is: turn matter into the source of spacetime curvature. For this t here's nothing better than that naive definition I posted. With hit, Gμν=κTμνGμν=κTμν actually carries the meaning that "matter distribution curves spacetime".
Now, writing Gμν=κTμνGμν=κTμν with the other TμνTμν we are just coupling curvature to one strange and abstract tensor derived from one action, which doesn't seem to tell at first that matter is the responsible for spacetime curvature.
Finally, energy and momentum should be, as we know from virtually all physics the generators of time and spatial translations. This is, as far as I know, the most general way to understand energy and momentum. It is not clear at all how this is built in into the "action-based" TμνTμν while it is obvious how is this built in into the traditional TμνTμν since pμpμ is defined as the generator of spacetime translations.
Anyway: we have two energy momentum tensors to use with EFE, one closely related to energy and matter, and which seems to be the one used by Einstein's original approach. The other which is just abstract and comes out of mathematical manipulations, and doesn't seem to relate to energy and matter at all. How do these two relate and how can they be the same?
I mean, they ought to be the same, because if they are not, the idea that "matter curves spacetime" is certainly captured just by the first one.
Answered by
4
Explanation:
Are the simple energy momentum tensor and the one derived from the action the same? In the most basic treatments one defines the energy momentum tensor in the following way: The component Tμν of the energy momentum tensor is the flux of four-momentum pμ
Similar questions