Birkoff theorem
On Apr 26, 1:29 am, Tonico wrote:
On Apr 25, Koobee Wublee wrote:
Let’s go back to a solution to the field equations that
is static and spherically symmetric.
** ds^2 = c^2 (1 – R / u) dt^2 – du^2 / (1 – R / u) – u^2 dO^2
Where
** dO^2 = dLongitude^2 cos^2(Latitude) + dLatitude^2
** R = Integration constant
** u = ANY FUNCTION OF r
Allow Him to emphasize that any function of u(r) within the
content of the equation above is a solution to the field
equations. shrug
** When [u = r],
The geometry is the Schwarzschild metric which was derived by
Hilbert.
** When [u = r (1 – R^3 / r^3)^(1/3)],
The geometry becomes Schwarzschild’s original solution --- the
first solution ever derived. Schwarzschild’ original solution
does not manifest black holes and also degenerates into Newtonian
law of gravity.
** When [u = r – R],
The geometry manifests no black holes but also explains Newtonian
law of gravity.
** When [u = r / (1 + r^2 / R / S)] where (S = another constant),
The geometry also degenerates into Newtonian law of gravity at
relative short distances (galactic scale) but antigravity at very
large distances (cosmic scale).
** When [u = R^2 / r],
The geometry is not asymptotically flat which proves Birkhoff
Theorem wrong.
Just to toss another bone for you to play with, the following
is also a solution to the field equations.
** ds^2 = c^2 dt^2 / (1 + R / u) – (1 + R / u) (du/dr)^2 dr^2
– u^2 (1 + R / u)^2 dO^2
Where
** u = ANY FUNCTION OF r
Thanks for the demystification.
You are welcome. shrug
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