I Is Gravity inertia, acceleration or curvature in GR?

[stevendaryl]

Then how would you interpret “Inertia and gravity are phenomena identical in nature.” This is much to clear to be a German/ English translation problem.

Well, that was not a clear way for Einstein to have said it. But the sense in which gravity is like a inertia has been explained already. Locally, there is no difference between being at rest in a gravitational field and being at "rest" in an accelerating reference frame. The statement that gravity is the same as inertia doesn't mean very much.

It's really more worth your time trying to understand General Relativity, rather than understanding specific things Einstein might have said about it. His words are not Holy Writ.

 

[stevendaryl]

Understood, but I think for some people the statement “the force you feel when you are at rest in a gravitational field is not 'the force of gravity'", can cause as much confusion as clarification.

Maybe so, but it's the kind of confusion that actually leads to a deeper understanding. You can't feel gravity--you only feel resistance to gravity. So you don't feel gravity pushing you down, but you feel the floor holding you up against gravity.

 

[A.T.]

Without getting too technical, we can largely equate tidal gravity with curvature. So it's common to say that Einstein's elevator has a uniform gravitational field (even though an object will weight slightly different amounts at the top and bottom of the elevator), and thus it has no tidal gravity and no curvature.

Also see this post by DrGreg for the difference between elevator frame (B) and intrinsic curvature with tidal forces (C):

This is my own non-animated way of looking at it:

• A. Two inertial particles, at rest relative to each other, in flat spacetime (i.e. no gravity), shown with inertial coordinates. Drawn as a red distance-time graph on a flat piece of paper with blue gridlines.
  
• B₁. The same particles in the same flat spacetime, but shown with non-inertial coordinates. Drawn as the same distance-time graph on an identical flat piece of paper except it has different gridlines.
  
  B₂. Take the flat piece of paper depicted in B₁, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B₂ shows exactly the same thing as B₁, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid.
  
• C. Two free-falling particles, initially at rest relative to each other, in curved spacetime (i.e. with gravity), shown with non-inertial coordinates. This cannot be drawn to scale on a flat piece of paper; you have to draw it on a curved surface instead. Note how C looks rather similar to B₂. This is the equivalence principle in action: if you zoomed in very close to B₂ and C, you wouldn't notice any difference between them.

Note the diagrams above aren't entirely accurate because they are drawn with a locally-Euclidean geometry, when really they ought to be drawn with a locally-Lorentzian geometry. I've drawn it this way as an analogy to help visualise the concepts.

 

[A.T.]

What his equivalence principle says is that the force of gravity is the same (or is the same sort of force) as the "inertial forces" that you feel when accelerating.

Even this isn't quite right. A better statement would be that the force you feel when you are at rest in a gravitational field is the same sort of force as the inertial force you feel when accelerating. But the force you feel when you are at rest in a gravitational field is not "the force of gravity".

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 would put it like this:

Coordinate acceleration towards a mass (in coordinates where the mass is at rest) is attributed to:
- An interaction force by Newton
- An inertial force by Einstein

 

[MikeGomez]

Also see this post by DrGreg for the difference between elevator frame (B) and intrinsic curvature with tidal forces (C)...

"B2. Take the flat piece of paper depicted in B1, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B2 shows exactly the same thing as B1, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid."

So there actually is curvature due to proper acceleration, correct? If so, it seems to me that would mean that in the case of Einstein’s elevator, there would curvature, even if the Weyl curvature tensor would (almost) completely vanish.

 

[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