How are Christoffel symbols different in rotating reference frames?

[Jonnyb42]

A very important idea in General Relativity is, same laws in all reference frames.
How does that work in rotating reference frames?

Jonny

 

[bcrowell]

Good question!

I would say that a very important idea in *special* relativity is that you have the same laws of physics in all frames of reference.

General relativity doesn't really have frames of reference. (Well, it has them, but only locally.) What general relativity really has is coordinate systems, and the laws of physics are the same regardless of what coordinate system you pick. You can pick a certain coordinate system, then transform to a second coordinate system that you might be able to interpret as rotating relative to the first. The laws of physics are the same in both coordinate systems.

What this does *not* mean is that rotation is purely relative in GR. Rotation is absolute in GR. For example, if you're locked in a sealed lab with no windows, you can determine whether the lab is rotating by observing a gyroscope.

 

[Jonnyb42]

but that bothers me

you said: "The laws of physics are the same in both coordinate systems."

Then how are centrifugal forces explained in the "rotating" frame?

 

[Jonnyb42]

Also, is there a connection between Frame-Dragging and inertia?

 

[xienohp]

The meaning of "The laws of physics are the same in both coordinate systems." actually means, you won't get F=ma^2 in a rotating system. If you measure things in a rotating frame, and calculate it inside the frame, then all the things follows what you have learnt. However, if you need to transform your law in rotating frame to steady frame, then you need to add some "frame transformation" term. Just as Feynman said: Goldfish may have its own rule under water, for example light bends as the water density under sea is increasing, but the physics is the same. So for us, remember to use Snell's law to understand.

Actually I don't know what is inertia, even after years of study. If someone knows, explicitly, tell me.

 

[Garth]

A very important idea in General Relativity is, same laws in all reference frames.
How does that work in rotating reference frames?

Jonny

That should read "same laws in all inertial reference frames."

Rotating frames are not inertial.

Garth

 

[Bill_K]

A very important idea in General Relativity is, same laws in all reference frames.
How does that work in rotating reference frames?

This is a time-worn question with a simple answer. Special relativity is not restricted to inertial reference frames. The equations of electromagnetism, relativistic mechanics, etc, can just as well be written in a curvilinear coordinate system, and you are still doing special relativity. You haven't changed the physics merely by using curvilinear coordinates!

The laws of physics do look the same in all coordinate systems. They are covariant, and have covariant derivatives and Christoffel symbols in them. The inertial forces appear in the Christoffel symbols. But spacetime is still a flat Minkowski space.

In special relativity, geodesics are straight lines and gravity is an applied force which makes particles follow curved trajectories. General relativity only comes in when you change the physics and recast gravity as the curvature of space with a Riemann tensor.

 

[bcrowell]

but that bothers me

you said: "The laws of physics are the same in both coordinate systems."

Then how are centrifugal forces explained in the "rotating" frame?

The laws of physics in GR are the Einstein field equations, which are coordinate-independent.

GR doesn't really talk about forces. The same geodesic can be described in either the rotating frame or the nonrotating frame. The geodesics in both frames obey the geodesic equation. The geodesic equation involves Christoffel symbols, which are different in different coordinates.