On May 10, 3:09 am, JanPB wrote:
On May 9, 12:32 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^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.

Its Ricci and Einstein tensors both vanish. In doing so, it is also
sharing the same coordinate system with the Schwarzschild metric.

I recommend Cartan moving frame method as
it's much faster than Christoffel symbols.

There is no short cut in such type of mathematics. shrug

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.

But mathematically, its Ricci and Einstein tensors also vanish.
shrug

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".

Perhaps, there is a flaw in your Cartan whatever. As I said, anyone
who possesses mathematical software like Eric Gisse can easily verify
it, and I do not see any complaint from that multi-year super-senior.
shrug