How does one calculate the metric tensor?

[Alan Grayson]

Wiki states it's a linear function of two variables whose domain is the vector space of velocity vectors on the tangent plane at some point P, say, on a spacetime manifold. But assuming we know the function, given by its 4x4 matrix representation defined on some coordinate system, which pair of vectors do we chose to calculate its real value? TY, AG

[Alan Grayson]

On Wednesday, October 2, 2024 at 1:47:48 AM UTC-6 Alan Grayson wrote:

Wiki states it's a linear function of two variables whose domain is the vector space of velocity vectors on the tangent plane at some point P, say, on a spacetime manifold. But assuming we know the function, given by its 4x4 matrix representation defined on some coordinate system, which pair of vectors do we chose to calculate its real value? TY, AG

I haven't mentioned the stress-energy tensor in the above question or any of the other tensors in Einstein's Field Equation because I am doubtful that it matters, since any tangent space on which the metric tensor is defined, will always contain an infinite set of vectors in the vector space defining that tangent space, so the same problem arises; namely, which pair of vectors must one chose, to calculate the metric tensor?  TY, AG 

[Alan Grayson]

I'm guessing that the metric tensor's value on the spacetime manifold depends on the Stress-Energy tensor. OTOH, I've seen videos by the usual suspects (e.g., Sean Carroll), that one needs to first determine the metric tensor, and presumably its value on the spacetime manifold, to calculate the SE tensor and therefore the curvature of spacetime. Surely, someone on this MB knows how this issue is resolved and can, at least, point me in the right direction. TY, AG