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redtree
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Given that no assumption is of a point energy is necessary to derive the vacuum (Schwarzschild) solution to the EFE, why is the solution assumed to apply to spacetime surrounding a point energy?
None is assumed. You can have M=0.redtree said:Sure, but point or otherwise, there is no need to assume ANY nearby energy to derive the vacuum solution.
redtree said:why is the solution assumed to apply to spacetime surrounding a point energy?
redtree said:there is no need to assume ANY nearby energy to derive the vacuum solution
More correctly that is called the stress energy tensor. All of its components are identically 0 in a vacuum solution, by definition.redtree said:On the RHS of the EFE, energy is rigorously defined as a rank 2 Tensor in 4D for all locations of the spacetime manifold
There isn't such a transformation. The stress energy tensor is 0 regardless of the value of S.redtree said:What is the relationship between the energy tensor and the energy scalar? What is the transformation?
It fails to capture it. There is no relationship between the two, they are completely different things.redtree said:Either there is a transformation that relates the stress-energy tensor to this energy scalar or the stress-energy tensor fails to capture the information embodied in this energy scalar term.
Nonsense. It just means that like any differential equation you have to include the boundary conditions. If you have ever used any differential equation then this should be a familiar concept.redtree said:If the latter is true, it implies an incompleteness to the stress-energy tensor and thus the EFE.
PAllen said:You can search these forums for many discussions of the following points:
- the total energy of universe (including gravitational) is only well defined for asymptotically flat spacetime. The Schwarzschild solution is asymptotically flat, so it has a well defined total energy that can be associated with one free parameter of the solution. Cosmological solutions are not asymptotically flat and it is accepted that (pick your formulation, I prefer the first) : total energy is not defined; total energy is not conserved.
- a tensorial quantity defining local energy including gravitational energy cannot be defined. This is actually mathematically proven in the sense that if you list the minimum properties you would expect for localization of energy, it can be proven that no such tensorial object can exist in GR. Some people define a non-tensorial, coordinate dependent object (pseudo-tensor) representing energy. The majority of GR practitioners don't find this useful, precisely because it is non-tensorial, and (as expected from this) has very odd properties except in special coordinates (e.g. harmonic coordinates).
Landau and Lifshitz, Classical Theory of Fields is a good starting point on why you must use a pseudo-tensor, and what their properties are.redtree said:Can you give a good reference on your second point?
You have been given the answer several times the the SET of the SC solution is identically zero everywhere. How do you claim that connects to the mass parameter of the solution (which is NOT curvature scalar of any type). When the SET is zero, the type of curvature allowed is Weyl curvature, and there are curvature scalars that are non-zero for Weyl curvature (e.g. the Kreschmann invariant). However, Weyl curvature is not constant in the SC solution, so the Kretshcmann invariant is a function of position, not a single number.redtree said:Given that both the stress-energy tensor (obviously via the EFE) and the energy scalar--as I call it--(via the vacuum solution as a constant of integration, 1/S) both associate with the same thing, namely, spacetime curvature, it seems problematic to call them totally different things. Rather, they seem highly similar.
A primary difference appears precisely in the boundary conditions of each implied by the assumptions of the vacuum solution. Specifically, the vacuum solution sets a boundary condition of zero for the stress-energy tensor and no boundary condition for the energy scalar.
In this context, it seems more accurate to say that the vacuum solution to the EFE associates with two, not one free parameter, namely r and 1/S (the energy scalar).
Again, all of this begs the question: given that both affect local spacetime curvature, what is the relationship between the stress-energy tensor and the energy scalar? Clearly, in the current formulation of GR, they are independent. However, on deeper consideration, I don't think the answer is at all obvious.
The main assumptions of the Schwarzschild solution to the EFE are:
The assumption of spherical symmetry is important because it simplifies the mathematical calculations and allows for a closed-form solution to the EFE. It also reflects the symmetry of many astrophysical objects such as stars and planets.
In the context of the Schwarzschild solution, a vacuum spacetime means that there is no matter or energy present, and therefore the stress-energy tensor is equal to zero. This simplifies the EFE and allows for a solution that describes the curvature of spacetime caused solely by the presence of a massive object.
The assumption of staticity means that the solution does not change with time, and all physical quantities are time-independent. This simplifies the EFE and allows for a solution that describes the spacetime geometry around a massive object at a specific point in time.
The asymptotic flatness assumption is important because it ensures that the solution approaches flat spacetime at infinity. This means that far away from the massive object, the spacetime is flat and the effects of gravity are negligible. This is necessary for the solution to be physically realistic and consistent with observations.