Acceleration and Curvature: Understanding the Relationship

[Naty1]

From another thread:

... Presumably transforming to a frame which is accelerating with respect to the flat frame would result in curvature...

No, no! Acceleration is different from gravity. Take simple Classical Mechanics and choose an accelerating RF. Such a change of variables introduces additional forces but the space remains flat, that's for sure.

(Bob for short's reply)

I thought acceleration DID curve space...
I'm coming from this perspective:

say in the rotating "rigid" disc...and via Einstein's equivalence principle...

for example, Brian Greene in THE ELEGANT UNIVERSE says:

And so Einstein realized that the familiar geometrical relationships..that pertain to flat space of the Greeks do not hold from the perspective of an accelerating observer. ..In fact accelerated motion not only results in a warping of space, it also results in an analogous warping of time."

any clarifications appreciated.

 

[Bob_for_short]

If you start from a flat space-time, no variable change can alter the invariant R=0.

If you start from a Riemann space-time, you always stay within the same curvature R.

Einstein used a Riemann space-time, not that of Minkowsky's.

 

[Fredrik]

I thought acceleration DID curve space...

What he and I said is that acceleration doesn't curve spacetime. If you define "space" as a hypersurface (in Minkowski space) of constant time coordinate, then the curvature of space will of course depend on what coordinate system you're using.

The coordinate system I would associate with the motion of the point at the center of a rotating disc has flat hypersurfaces of constant time. To get a curved "space" you would have to do something like pick a point on the edge and define a local coordinate system by taking a segment of the world line of that point to be the time axis, and then use the standard synchronization convention to define the rest of it.

I don't like Greene's explanation of these things.

 

[Naty1]

I could have equally well said I thought acceleration could be thought to curve spacetime...as well as space...

I doubt I really understand yet...I thought Einstein's equivalence said that acceleration and gravitational potential had the "same" effects...


Greene also uses a three dimensional disgram of an object moving thru spacetime (FABRIC OF THE COSMOS, page 61) ...say x,y are space and z is time, t...so a particle with constant velocity will trace out a straight line along, say, the t axis; rotational motion motion appears as a corkscrew along the t direction, and uniform accelerated motion as a curved trajectory...I believe I also saw this in another author's text but can't find it...

you guy's don't like such a representation??

 

[madness]

I'm not an expert, but I believe the equivalence principle states that they have the same effect locally. However tidal effects occur due to the second derivative in the Riemann tensor, which cause neighbouring geodesics to diverge.

 

[A.T.]

Greene also uses a three dimensional disgram of an object moving thru spacetime (FABRIC OF THE COSMOS, page 61) ...say x,y are space and z is time, t...so a particle with constant velocity will trace out a straight line along, say, the t axis; rotational motion motion appears as a corkscrew along the t direction, and uniform accelerated motion as a curved trajectory...I believe I also saw this in another author's text but can't find it...

you guy's don't like such a representation??

I like them. Epstein uses a similar diagram to show an alternative way to visualize curvature. In the case of rotational motion (orbiting a bigger mass) the corkscrew is caused by a "density gradient" of the 2+1 spacetime, that bends the worldline around a column of higher density, because it advances slower in the denser region. This slower advance also models gravitational time dilation. Analogy: light rays are bent in a media with varying optical density / variable propagation speed.

 

[Fredrik]

I could have equally well said I thought acceleration could be thought to curve spacetime...as well as space...

I doubt I really understand yet...I thought Einstein's equivalence said that acceleration and gravitational potential had the "same" effects...

What it says is that an experiment that measures some of those effects won't be able to distinguish between gravity and acceleration if the region of spacetime in which the experiment is performed is small enough.

Greene also uses a three dimensional disgram of an object moving thru spacetime (FABRIC OF THE COSMOS, page 61) ...say x,y are space and z is time, t...so a particle with constant velocity will trace out a straight line along, say, the t axis; rotational motion motion appears as a corkscrew along the t direction, and uniform accelerated motion as a curved trajectory...I believe I also saw this in another author's text but can't find it...

you guy's don't like such a representation??

I don't dislike spacetime diagrams at all. In fact, I think they are by far the best way to explain almost anything in special relativity.