On May 9, 12:32 pm, Koobee Wublee wrote:
On May 9, 1:11 am, JanPB wrote:

On May 8, 1:08 pm, Koobee Wublee wrote:
ds^2 = G c^2 dt^2 / (1 + K^2 / r^2) – 4 r^2 (1 + K^2 / r^2) dr^2 / K^2
– r^4 dO^2 / K^2

The Schwarzschild metric has the following form.

ds^2 = c^2 dt^2 (1 + K / r) – dr^2 / (1 + K / r) – r^2 dO^2

Then the first formula is not a solution. Calculate its Ricci tensor
and you'll see it's nonzero.

Oh, you are correct. It is my mistake --- a typo. Please allow me to
correct the first equation.

ds^2 = G c^2 dt^2 / (1 + K^2 / r^2) – 4 r^2 (1 + K^2 / r^2) dr^2 / K^2
– (r^2 + K^2)^2 dO^2 / K^2

Its Ricci curvature is still nonzero if r denotes the same coordinate
as in Schwarzschild. Try it. I recommend Cartan moving frame method as
it's much faster than Christoffel symbols.

The following is also a solution where the gravitational force follows
the inverse cubed law.

ds^2 = G c^2 dt^2 (1 + K^2 / r^2) – 4 r^2 dr^2 / K^2 / (1 + K^2 / r^2)
– r^4 dO^2 / K^2

Same thing: if r is the same then Ricci of the above is nonzero. OTOH
if r denotes sqrt(K * Schwarzschild-r) then Ricci of the above equals
0 but in that case it's the same solution as Schwarzschild - you
simply change the labels you attach to the spheres from "r" to "r^2/
K".

--
Jan Bielawski