On May 10, 8:42*pm, Koobee Wublee wrote:
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.

Then evaluate surface area of a sphere in both coordinate systems.
You'll see something interesting if you actually do the computation.


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

Based on your years of not studying the subject?


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

Not if they use the Schwarzschild coordinate system.


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

That's because we [JanPB, me] understand that the r you use is
different from the r in Schwarzschild even if you don't.

You appear to think that the coordinates determine the physics. You
claim otherwise and you can't explain why Minkowski space in Cartesian
coordinates is different from Minkowski space in spherical
coordinates, but when presented with the Schwarzschild metric in two
different coordinates your GR myopia sets in and you spray ink to
avoid discussing the subject.