The Schwartzschild solution - why no Stress Tensor?

[Altabeh]

I'd be tempted to get more into how bad this is, but it's hard to find textbooks that discuss it (they focus on doing things right in the first place, not disucssing the consequences of incorrect assumptions), and the discussion isn't proceeding well for a less formal approach.

Textbooks do not go into such unipmortant things because it is of no concern in the models we have discussed to date about the structure of spacetime in a nearly flat spacetime! If you claim you got something interesting and instructive, then I'm ready to discuss it! But on a personal point, there you won't come up with anything appealing if you want to take the binding energy into account because the correction terms appearing in the equation are very tiny compared to terms up to the linear order! In the same book, on page 158, when Schutz proves a theorem on local flatness, you can see that the non-vanishing of terms like g_{ab,cd}, g_{ab,cde} in the geodesic coordinates are definitely not guaranteed so they appear in the Taylor expansion of g'_{ab} meaning that in the neighbourhood of a particular point, say, P at which a coordinate basis is chosen so that the metric is flat, metric may not really be "nearly" flat according to the fact that there is an infinite sum over all terms of higher order in the metric derivatives! But this is simply negligible because infinitesimally small as an "assumption" means x^a-x^a_p are supposed to be really small that all that sum is inconsiderable!

Such assumptions as the one mentioned above are generally true even if your spacetime is stupidly curved and the the contribution of metric-derivative factors go over the top compared to the factors involving multiples of coordinate differences! But no one has ever claimed the theorem of local flatness wouldn't work somewhere around the intrinsic singularity of Schwartzschild metric!

I'm always ready to hear new ideas and if you think this is something never been worked out, it is time to shine and let us know if it deservers to be published!

AB

 

[TerryW]

Having now got as far as Chapter 18 and the Reisner Nordstrom solution, it is interesting to see that you put in the Maxwell Tensor, work through to a solution and lo, the solution contains a term which we can identify as the charge sitting at the origin. It also contains the mass sitting at the origin, but this appears without having started out with a Mass/Energy Tensor!

I'm kind of resigned to a train of thought that goes along the line that the general solution worked out from the canonical form of the metric produces a general solution which says that a space will be flat if there is no mass at the origin and curved if there is, but I can't say I'm all that happy!