- #1

tom.stoer

Science Advisor

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## Main Question or Discussion Point

I have some questions regarding Newtonian spacetime; reference is MTW chap. 12.

MTW translate the Lagrange e.o.m. for Newtonian mechanics (with a potential phi derived from a mass density rho via Poisson eq.) into a geodesic equation in 4-dim. spacetime. They explicitly construct the connection Gamma, Riemann curvature R and Ricci tensor Ric.

They present an exercise to prove that the connection cannot be derived from a metric on spacetime. So the first consequence is that this Newtonian spacetime 4-manifold is not a Riemann manifold.

They show that the Ricci tensor contains the mass density only. That means that in vacuum the manifold is Ricci-flat Ric = 0.

1st question: what type of manifold is this? not Riemann due to the missing metric; not affine b/c it's not globally flat (see 3rd question)

2nd question: how can I further study the curvature? I would proceed with the Weyl tensor C, but I was not able to find a definition how to extract C from R w/o using the metric

3rd question: what does it mean that a manifold is flat? vanishing of R is too strong b/c it does contain coordinate effects; vanishing of Ric is too weak b/c it misses the Weyl curvature

Thanks

Tom

MTW translate the Lagrange e.o.m. for Newtonian mechanics (with a potential phi derived from a mass density rho via Poisson eq.) into a geodesic equation in 4-dim. spacetime. They explicitly construct the connection Gamma, Riemann curvature R and Ricci tensor Ric.

They present an exercise to prove that the connection cannot be derived from a metric on spacetime. So the first consequence is that this Newtonian spacetime 4-manifold is not a Riemann manifold.

They show that the Ricci tensor contains the mass density only. That means that in vacuum the manifold is Ricci-flat Ric = 0.

1st question: what type of manifold is this? not Riemann due to the missing metric; not affine b/c it's not globally flat (see 3rd question)

2nd question: how can I further study the curvature? I would proceed with the Weyl tensor C, but I was not able to find a definition how to extract C from R w/o using the metric

3rd question: what does it mean that a manifold is flat? vanishing of R is too strong b/c it does contain coordinate effects; vanishing of Ric is too weak b/c it misses the Weyl curvature

Thanks

Tom