Understanding Special & General Relativity: Inertial vs. Gravitational Frames

[nomadreid]

I know this is a naive question that has almost certainly been brought up numerous times before, but my search abilities seem not to be sufficient for finding a good answer, so if anyone just refers me, that would be fine. The question: Special relativity concerns comparisons between pairs of inertial reference frames, whereas general relativity deals with accelerated reference frames (equivalent to the presence of mass). Acceleration can be seen as an infinite number of velocities, each over an infinitesimal piece of spacetime. So why cannot we derive the laws for an accelerated reference frame by treating it as an infinite number of inertial reference frames over an infinitesimal section of spacetime? That is, why is general relativity not derivable from special?

 

[Dale]

Special relativity concerns comparisons between pairs of inertial reference frames, whereas general relativity deals with accelerated reference frames (equivalent to the presence of mass)

You can certainly handle accelerated reference frames in special relativity. What you cannot handle is curved spacetime (tidal gravity).

 

[mfb]

Special relativity works fine with accelerated reference frames, it just makes equations more complicated.
Only the introduction of gravity makes general relativity necessary.

 

[nomadreid]

Thanks, Dale and mfb. Then, to modify the question:
(1) to Dale, why can't curved spacetime be treated as a infinite collection of infinitesimal sections of flat spacetime, analogous to treating the surface of a sphere as an infinite collection of infinitesimal planes?
(2) If gravity is equivalent to an accelerated frame, then if SR can handle accelerated reference frames, why can't it handle gravity?

 

[mfb]

If gravity is equivalent to an accelerated frame

Only locally, not globally. Locally, you can always use special relativity, but you can't use it for the whole system.

 

[nomadreid]

Thanks, mfb. Gives me food for thought and impulse for more reading.

 

[Dale]

why can't curved spacetime be treated as a infinite collection of infinitesimal sections of flat spacetime, analogous to treating the surface of a sphere as an infinite collection of infinitesimal planes?

You can, but you don’t gain anything by doing so. At the junction between each plane you would have to do the equivalent of parallel transport anyway, and that is the thing that makes curved spacetime strange. Nothing in SR would tell you how to do that.

 

[PeterDonis]

analogous to treating the surface of a sphere as an infinite collection of infinitesimal planes?

In order to do this, you need to know the connection between the planes at different points on the sphere--in other words, how to map vectors in the plane at one point to vectors in the plane at another point. Nothing in the geometry of the planes themselves tells you how to do that.

This is the same thing @Dale said, just in different words.

 

[nomadreid]

Thanks very much, Dale and PeterDonis. I had not thought about parallel transport, aka the role of the curvature in the infinite sum. Enlightening.

 

[Ibix]

In order to do this, you need to know the connection between the planes at different points

And this is why the connection coefficients are so named, right? This is what they do (well, strictly they operate on the tangent spaces).

 

[PeterDonis]

this is why the connection coefficients are so named, right?

Yes.

This is what they do (well, strictly they operate on the tangent spaces).

The "planes" being referred to are the tangent spaces.

 

[Ibix]

The "planes" being referred to are the tangent spaces.

Hm. I'd thought of the tangent spaces as some flat vector space associated with a point on the manifold. But what the OP seemed to me to be proposing was breaking the manifold into an infinite number of small flat planes - and I didn't think that was quite the same thing as the tangent spaces.

Are we disagreeing on what the tangent spaces are, or on what the OP meant?

 

[PeterDonis]

I'd thought of the tangent spaces as some flat vector space associated with a point on the manifold.

That's correct.

what the OP seemed to me to be proposing was breaking the manifold into an infinite number of small flat planes

If the OP meant actually trying to "break up" the manifold itself into flat planes, that's impossible. The manifold isn't flat, it's curved, and the curvature is continuous.

I had assumed the OP was (clumsily) trying to get at the concept of tangent spaces. At any rate, that's the only well-defined concept I'm aware of that corresponds to "flat planes" at each point of the manifold.

 

[Ibix]

If the OP meant actually trying to "break up" the manifold itself into flat planes, that's impossible. The manifold isn't flat, it's curved, and the curvature is continuous.

Well, perhaps rather than "break up" I should say I think he was trying to regard the manifold as the limiting case of a set of small flat vector spaces. Like regarding a sphere as some kind of limiting case of an n-faced polyhedron (see #4).

Not sure, then, if there's a contradiction between what you said in #8 and @Dale said in #7 (which I read as based on the interpretation I made in my previous paragraph), or if we're all trapped in a maze of misinterpreting each other.

 

[PeterDonis]

I should say I think he was trying to regard the manifold as the limiting case of a set of small flat vector spaces.

If that's the case, then the tangent spaces are what you get in the limit, so he's basically referring to the tangent spaces anyway.

Not sure, then, if there's a contradiction between what you said in #8 and @Dale said in #7

I don't think so; in the limit, the "junction" between each of the adjacent small flat planes just becomes the continuous curvature of the manifold, and the "equivalent of the connection" that you would need to do at each junction just becomes the connection coefficients in the continuous curved manifold.

 

[nomadreid]

Thanks to Ibix and PeterDonis for that helpful discussion. I think that PeterDonis used the correct term in describing my attempts when he wrote "clumsily" ; yes, what I was trying to get at was the connections on the tangent spaces, but my background in differential geometry is very weak. It is obvious that I need to do some more reading in the topic, but as the topic is huge, this gives me a direction to aim for. Thanks again!