- #1
JustinLevy
- 895
- 1
Newton-Cartan theory has been brought up many times on this forum as an example of any theory being able to be written in "general covariance". I've even used the example myself after hearing about it here. However, when I actually went to look up the definition:
http://en.wikipedia.org/wiki/Newton–Cartan_theory
it was not what I expected. I was expecting literally Newtonian gravity, just written in tensor formalism somehow. But instead it is a metric theory ... ie. spacetime is curved.
So either I've misused the example, or I am misunderstanding the formulation. Either way I am wrong. I may have just misunderstood what people were casually referring to when using this as an example and took it too literally.
Although I've also seem people here make the argument that operationally defined coordinate systems are geometric objects and thus even all the Newtonian physics is technically, albeit implicitly, defined in geometrically invariant terms.
So what is everyone's take on this?
----------------
Now, that aside, I'd like to actually play with this theory a bit.
In Newtonian gravity, I've seen the escape velocity argument to discuss Newtonian black holes. So I'm curious if that remains in Newton-Cartan.
Using the somewhat bizarre (to me at least) notation in wiki
[tex]R_{\mu \nu} = 4 \pi G \rho {\delta^0}_\mu {\delta^0}_\nu[/tex]
Which I assume means this?
[tex]R_{\mu \nu} = 4 \pi G \rho \left (
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right)
[/tex]
But it seems to be saying this is only correct in inertial coordinate systems. Which due to the curvature can't be global anymore, so I'm not understanding now to actually interpret this as an equation for the metric to solve.
Any help? Are there some papers that work out some simple cases in Newton-Cartan (the equivalent of Schwartzschild and Kerr maybe?)
EDIT:
Still looking, but this is interesting: http://arxiv.org/abs/gr-qc/9604054 Newton-Cartan Cosmology
http://en.wikipedia.org/wiki/Newton–Cartan_theory
it was not what I expected. I was expecting literally Newtonian gravity, just written in tensor formalism somehow. But instead it is a metric theory ... ie. spacetime is curved.
So either I've misused the example, or I am misunderstanding the formulation. Either way I am wrong. I may have just misunderstood what people were casually referring to when using this as an example and took it too literally.
Although I've also seem people here make the argument that operationally defined coordinate systems are geometric objects and thus even all the Newtonian physics is technically, albeit implicitly, defined in geometrically invariant terms.
So what is everyone's take on this?
----------------
Now, that aside, I'd like to actually play with this theory a bit.
In Newtonian gravity, I've seen the escape velocity argument to discuss Newtonian black holes. So I'm curious if that remains in Newton-Cartan.
Using the somewhat bizarre (to me at least) notation in wiki
[tex]R_{\mu \nu} = 4 \pi G \rho {\delta^0}_\mu {\delta^0}_\nu[/tex]
Which I assume means this?
[tex]R_{\mu \nu} = 4 \pi G \rho \left (
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right)
[/tex]
But it seems to be saying this is only correct in inertial coordinate systems. Which due to the curvature can't be global anymore, so I'm not understanding now to actually interpret this as an equation for the metric to solve.
Any help? Are there some papers that work out some simple cases in Newton-Cartan (the equivalent of Schwartzschild and Kerr maybe?)
EDIT:
Still looking, but this is interesting: http://arxiv.org/abs/gr-qc/9604054 Newton-Cartan Cosmology