Bishal,
I suggest you post these queries on Quora or Stack Exchange and await for informed answers from experts in the field. Sorry, I lack the time to dig into your model and understand the context of your questions.
Note that the relationship between the stress energy tensor and the matter Lagrangian is also covered in Caroll's book:
https://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Carroll_SG.pdf
In addition, these links may be of help: https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action#Derivation_of_Einstein.27s_field_equations https://www.physicsforums.com/threads/understanding-how-to-derive-the-stress-energy-tensor-formula.988498/
Hi Bishal
Apologies for the delay in replying. In the last two months, I have started three new jobs and have been away on holiday, so I've been very busy.
I'm glad to see you're taking an interest in such fascinating topics. We discuss relativity and other topics in physics and mathematics every Tuesday at the meetings of the CAMP group at the Ronin Institute. It would be lovely if you could come to some of these meetings. Next week is a special meeting, as we have a guest speaker. I will send you an email at your Ronin Institute address very soon.
I'm afraid I don't think I can answer your question. I know in principle that the energy-momentum density tensor for matter is proportional to the the Euler-Lagrange derivative of the matter Lagrangian. (I prefer this term to stress energy tensor, though they mean the same thing - I'll abbreviate it to EMDT.) But I don't recall ever doing this in practice for a particular matter system. And I certainly don't know how to do the reverse in any detail: going from the EMDT to the matter Lagrangian it can be obtained from.
Furthermore, I'm not entirely clear what system of matter you're looking at and how you derived the EMDT. My impression is that the interval you state is for the exterior solution of a Schwarzschild-type system, where M represents a mass and Phi is the scalar gravitational potentional resulting from it. If this is the case, it would seem odd to me to have Phi contained in the EMDT. The EMDT usually describes properties of the matter, and wouldn't usually describe the gravitational field, as this is described by curvature and therefore falls in the Einstein(-Hilbert) part of the Lagrangian (density), not the matter part.
If you'd like to come to one of the CAMP meetings, we could discuss it there. (Although not this coming Tuesday, as we have a guest speaker.)
Best wishes
Tom
Tom Lawrence Conventionally, your are right. But the line element I have stated is symmetrical, static but non-vacuum. I could assure the validity of this line element although, it is derived on the basis of positive gravitational potential (which has authentic derivation) with all classical tests has been verified as well as it reduces to same form as Schwarzschild solution for lower mass case. The non-vacuum here represents the results that the non-zero EMDT follows outside the mass M (as you mentioned). So, I believe that the EMDT splits in two ways for interior (fluid idealization) and exterior (representing the gravitational field behavior) portions that the exterior part is conventionally assumed to have non-existence believing that Christoffel's get vanished at outer local regions. But what the problems this kind of assumption leads is well explained by Penrose in his book "The Road to Reality", one among is the Energy localization problem, whose solution would be fulfilled from our EMDT. I would appreciate if you have any queries further. For your kind information, I have solved those issues I mentioned in the question.
Good to hear you've now answered the question that you posed at the start. That's often the way, I find - I ask others a question, then end up solving it myself.
But if that is the line element for the interior, I'm still intrigued by the form you have for the EMDT. Is curly M a mass? If so, how do you end up with a mass - which presumably is spread over a region - appearing in the EMDT field for each point? I'd expect some kind of derivative of that to appear in the expression - a local density.
Tom Lawrence Thank you for your question! The line element I had posted is the exterior one, field originated. The complete one including the interior, fluid idealized is not mentioned there. I had already mentioned that the total EMDT splits into two portions. Yes, the local density should coupled for interior solution in EMDT.
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