# Special relativity treatment of gravity?

DrGreg
Gold Member
I have not heard anyone prove the interior Schwarzschild solution is wrong or even suggest that. Maybe your concern is that the interior solution does not appear to show that the gamma factor within the hollow shell is uniform?
I may have misused the terminology. When I referred to "the interior Schwarzschild solution", I meant inside the event horizon but outside the central mass, so the standard ("exterior") Schwarzschild metric still applies (albeit with spacelike t and timelike r). My concern was that you appeared to be using that metric to claim outward acceleration (by the way, outward acceleration need not imply outward velocity) but then you talk of a shell of matter which invalidates that metric inside the shell. I still believe spacetime is flat inside a uniform shell.

I'm not sure where all the equations you quote come from. Are they from a book or other reliable source, or have you derived them yourself? If the latter, I'd want to know how you got them, as there might be some flaw in your logic.

I can quote some other published equations (I can't pretend I fully understand their derivations):

For a radially free-falling particle of unit mass, in units where G = c = 1, the following quantities E (energy) and L (Lagrangian) are conserved (constant over time):

$$E = \left(1 - \frac{2M}{r} \right) \frac{dt}{d\tau}$$
$$2L = \left(1 - \frac{2M}{r} \right) \left( \frac{dt}{d\tau} \right)^2 - \frac{1}{1 - 2M/r} \left( \frac{dr}{d\tau} \right)^2$$​

References:

1. Woodhouse, N.M.J. (2007), General Relativity, Springer, London, ISBN 978-1-84628-486-1, pages 100, 107, 124.

2. Woodhouse, N.M.J. (2003), http://www2.maths.ox.ac.uk/~nwoodh/gr/index.html [Broken] on which the book (1) was closely based, pages 54-55 (Lecture 12), p59 (L13), p73 (L15).

These sources also confirm Hartle's formula for "the proper acceleration due to gravity" quoted in post #17 (1, p.99) (2, p.54, L12) (3, p.230)

3. Rindler, W. (2006 2nd ed), Relativity: Special, General and Cosmological, Oxford University Press, Oxford, ISBN 978-0-19-856732-5.

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