- #36

Pyter

- 157

- 16

My main problems boil down to these:

1) how can you build the stress-energy tensor if you don't have a complete Riemannian manifold yet?

2) If you have a complete Riemannian manifold, you don't need to solve the EFE because you can compute the metric tensor directly from the manifold chart(s).

I'll try to explain it in other words.

(In my previous posts, "map" should be read as "charts", which is the proper terminology).

In relativity, our continuum is modeled after a 4D Riemannian manifold (with Minkowskian metric). The latter is - simplifying - a N-dimensional surface, immersed in a N+1 dimension space, equipped with one or more invertible charts which map a point on the surface to a linear N-space (not necessarily Cartesian, they could be polar or spherical or cylindrical coordinates, but they are linear) and back.

In SR, there are no gravity sources and the motion is inertial, so the chart coincides with the identity chart. We know that ##g_{\mu \nu} = \eta_{\mu\nu}## everywhere.

So far so good.

In GR, instead, the charts depend on the gravity sources (or, equivalently, the accelerated/rotatory motion of the observer).

To determine the position of a point in the manifold our continuum is modeled with, you need its charts, just like when you want to determine the position of a point on a 2D spherical surface with fixed radius, you may use the linear coordinates ##\theta, \phi## and the sine and cosine charts mapping the linear coordinates to the 3D coordinates of the surface points.

To build the SET, you need to specify the position and times of the masses (or of a single extended mass, like a star) in the 4D manifold, and in order to do so you need its charts.

But those charts depends on the manifold's curvature and you don't know it because you haven't solved the EFE yet.

But you can't solve the EFE until you build the SET.

And if you somehow find the right charts to build the SET, why should you need to solve EFE, since you can express the metric tensor analytically by taking the partial derivatives of the chart functions?

And even if you solve the EFE for the ##g_{\mu\nu}## (as a function of the linear coordinates), is it mathematically guaranteed that you can univocally find a chart expressing that metric tensor?

Can you see my conundrum?

1) how can you build the stress-energy tensor if you don't have a complete Riemannian manifold yet?

2) If you have a complete Riemannian manifold, you don't need to solve the EFE because you can compute the metric tensor directly from the manifold chart(s).

I'll try to explain it in other words.

(In my previous posts, "map" should be read as "charts", which is the proper terminology).

In relativity, our continuum is modeled after a 4D Riemannian manifold (with Minkowskian metric). The latter is - simplifying - a N-dimensional surface, immersed in a N+1 dimension space, equipped with one or more invertible charts which map a point on the surface to a linear N-space (not necessarily Cartesian, they could be polar or spherical or cylindrical coordinates, but they are linear) and back.

In SR, there are no gravity sources and the motion is inertial, so the chart coincides with the identity chart. We know that ##g_{\mu \nu} = \eta_{\mu\nu}## everywhere.

So far so good.

In GR, instead, the charts depend on the gravity sources (or, equivalently, the accelerated/rotatory motion of the observer).

To determine the position of a point in the manifold our continuum is modeled with, you need its charts, just like when you want to determine the position of a point on a 2D spherical surface with fixed radius, you may use the linear coordinates ##\theta, \phi## and the sine and cosine charts mapping the linear coordinates to the 3D coordinates of the surface points.

To build the SET, you need to specify the position and times of the masses (or of a single extended mass, like a star) in the 4D manifold, and in order to do so you need its charts.

But those charts depends on the manifold's curvature and you don't know it because you haven't solved the EFE yet.

But you can't solve the EFE until you build the SET.

And if you somehow find the right charts to build the SET, why should you need to solve EFE, since you can express the metric tensor analytically by taking the partial derivatives of the chart functions?

And even if you solve the EFE for the ##g_{\mu\nu}## (as a function of the linear coordinates), is it mathematically guaranteed that you can univocally find a chart expressing that metric tensor?

Can you see my conundrum?

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