Is Gravity inertia, acceleration or curvature in GR?

[A.T.]

So there actually is curvature due to proper acceleration, correct?

None of the worldlines in Dr Gregs diagram has any proper acceleration. They are all geodesic worldlines of free falling obejcts. Extrinsic curvature of the worldlines with respect to spacetime (proper acceleration) is not present in any of the cases. Intrinsic curvature of space-time is just case C.

What Dr Greg means the quote is that if you would distort the coordinates in B (and the worldlines with them), to look like those in A, then the straight worldlines would look curved. Basically what happens in the video below 0:28 - 0.32, but in reverse:

 

[stevendaryl]

I find both formulations confusing, because both talk about "feeling inertial forces". Inertial forces can be used to explain coordinate acceleration in a non-inertial frame, but you can "feel" only the frame invariant proper acceleration from interaction forces.

I think that's what Peter Donis was saying, and I was agreeing with.

 

[A.T.]

I think that's what Peter Donis was saying, and I was agreeing with.

Yes, but he also talks about the "inertial force you feel", which I think should be avoided.

 

[pervect]

I think you mean "intrinsic", correct?

oops, yes - fixed

 

[pervect]

The smoothest-sounding thing I can think of to describe what gravity is is the connection coefficients. This is basically the same as saying that it's the Christoffel symbols, but it appears to be less technical on the surface - though I suppose it really isn't, when you get into all the details.

The point is this. If you start with the notion of inertial frames, you can describe flat space-time with an inertial frame of reference. An accelerated reference frame, like Einstein's elevator, is not an inertial frame. However, while the elevator is not itself an inertial frame of reference, at any given instant of time, we CAN create a co-moving inertial frame of reference that has the same velocity as the elevator does.

Both the inertial frame of reference and the non-inertial, accelerated elevator frame of reference can be regarded at any instant in time as having an instantaneous (and co-moving) spatial frame of reference. The difference between the accelerated frame and the inertial frame is subtle. The real difference between the inertial frame and the non-inertial elevator frame is not in how the frame is made up, but how the different instantaneous frames are "connected" to make a whole. They're both made up of the same thing (instantaneous spatial frames of reference), but they're "hooked up" or connected differently.

To get more detailed than this requires a small, but significant, amount of technical language. One needs the idea of basis vectors. We can regard vectors for our purposes as little arrows, such as we've seen on many of the diagrams used, that represent a displacement. We imagine a 2-dimensional surface, and pick a particular pair of vectors, called "basis" vectors, and then note that we can regard any vector we want as a weighted sum of these two basis vectors. If we imagine a 3-dimensional object, we need 3 basis vectors to represent it. We can't visualize it, but we can regard a 4 dimensional space as having 4 basis vectors, etc.

Then what the connection coefficients do is "connect" the basis vectors at some time $t_1$ to some other time $t_2$. An inertial frame connects the basis vectors in one way, and the non-inertial, accelerated frame, connects them differently. They have different connections. And that's how Einstien's elevator, which we regard as "having gravity" differs from an inertial frame of reference, which we regard as "not having gravity". It all lies in the connections, how we "hook together" the instantaneous co-moving frames.

I've skipped over a few fine points, which may result in confusion, but probably less confusion than would result if I tried to explain them at this point. The point is that we can regard gravity as this very abstract idea, the idea of a connection, which is a relationship (it happens to be a particularly simple relationship, a linear map) between basis vectors that ties them together into a unified framework.

 

[MikeGomez]

None of the worldlines in Dr Gregs diagram has any proper acceleration. They are all geodesic worldlines of free falling obejcts. Extrinsic curvature of the worldlines with respect to spacetime (proper acceleration) is not present in any of the cases. Intrinsic curvature of space-time is just case C.

What Dr Greg means the quote is that if you would distort the coordinates in B (and the worldlines with them), to look like those in A, then the straight worldlines would look curved. Basically what happens in the video below 0:28 - 0.32, but in reverse:

Sorry, I should have said coordinate acceleration, so my question applies to both types, both for coordinate acceleration (freefall either above the Earth or in the elevator) and proper acceleration (man at surface of Earth or on floor of elevator). Is it correct that acceleration is associated with curvature of some form?

 

[A.T.]

Is it correct that acceleration is associated with curvature of some form?

Proper acceleration = Worldline has extrinsic curvature with respect to space-time
Coordinate acceleration = Worldline has extrinsic curvature with respect to some arbitrary coordinates

 

[PeterDonis]

So there actually is curvature due to proper acceleration, correct?

Proper acceleration means path curvature of a worldline. It has nothing to do with the curvature, or lack thereof, of spacetime.

 

[MikeGomez]

Proper acceleration means path curvature of a worldline. It has nothing to do with the curvature, or lack thereof, of spacetime.

Ok, and does coordinate acceleration have to do with curvature of space-time?

 

[Nugatory]

Ok, and does coordinate acceleration have to do with curvature of space-time?

No, it has to do with your choice of coordinates. An object following a given worldline will have coordinate acceleration in some coordinate systems but not others, regardless of whether the worldline passes through curved or flat spacetime and regardless of whether the object is experiencing proper acceleration.

 

[MikeGomez]

No, it has to do with your choice of coordinates. An object following a given worldline will have coordinate acceleration in some coordinate systems but not others, regardless of whether the worldline passes through curved or flat spacetime and regardless of whether the object is experiencing proper acceleration.

I see. I was thinking that the curvature of space-time is the source of gravitational acceleration, not your choice of coordinate systems.

 

[cyrilus]

Gravity shows up when you operate some coordinate transformations from a free falling inertial referential to the non galilean laboratory referential (in this case considered as accelerating in the upper direction)

Then it is 'easy' to express the Christoffel symbol with respect to the metric tensor, which demonstrates the direct link between the spacetime itself and the gravitational force (the metric tensor describes the spacetime).

You could even easily demonstrate a direct relation between the metric tensor and the gravitational field from the geodesic equation in Newtonian limit.

Hope this can help