GR vs SR: Is a Connection Necessary?

[greypilgrim]

Hi,

When I started learning about GR I wondered if it emerged from SR (which the name suggests) or if the connection between the two is mere technical. GR describes the behaviour of the metric of space-time, which is locally Minkowskian and therefore SR applies.

But is a curvature-based theory of gravity possible where the metric is locally Euclidean, i.e. the speed of light is not constant and space and time are essentially uncoupled?

Or is there a closer relation between GR and SR that I am missing?

 

[ShayanJ]

One example of the kind of theory you're looking for, is the Newton-Cartan theory.

 

[Markus Hanke]

When I started learning about GR I wondered if it emerged from SR (which the name suggests) or if the connection between the two is mere technical.

We need to locally arrive at a situation where the laws of physics are the same in all inertial frames, because that is what we empirically observe. This is not really possible if you uncouple space from time, and consider only a Euclidean metric; as such, the connection is more than merely technical, it is empirical.

 

[greypilgrim]

We need to locally arrive at a situation where the laws of physics are the same in all inertial frames, because that is what we empirically observe. This is not really possible if you uncouple space from time, and consider only a Euclidean metric; as such, the connection is more than merely technical, it is empirical.

True, but I was thinking about putting electrodynamics aside for the moment and just considering gravity. It's more of a hypothetical question.

 

[Dale]

But is a curvature-based theory of gravity possible where the metric is locally Euclidean, i.e. the speed of light is not constant and space and time are essentially uncoupled?

As @Shyan mentioned this is Newton Cartan theory.

I don't have a rigorous proof of this, but my impression is that any theory in which the inertial mass equals the gravitational mass can be geometrized.

 

[haushofer]

One example of the kind of theory you're looking for, is the Newton-Cartan theory.

Or, this excellent PF-insights article:

https://www.physicsforums.com/insights/revival-Newton-cartan-theory/

:P