# Distinction between special and general relativity

1. Aug 14, 2015

### bcrowell

Staff Emeritus
FAQ: Can special relativity handle accelerated frames of reference? What is the distinction between special and general relativity?

Einstein believed, erroneously, that the crucial difference between special and general relativity was that general relativity allowed accelerated frames of reference. In fact, it has been known for many decades that special relativity is capable of handling accelerated frames of reference.[Gourgoulhon] This can be done, for example, using coordinates popularized by Rindler,[Rindler] which were apparently known as far back as 1936.[Einstein]

GR describes gravity as the curvature of spacetime. The correct modern definition of the distinction between SR and GR, universally agreed upon by modern relativists, is that SR deals with flat spacetime, GR with curved spacetime.[Carroll],[Wald]

References

Sean Carroll, Lecture Notes on General Relativity, ch. 1, "Special relativity and flat spacetime," http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html

Einstein and Rosen, "A Particle Problem in the General Theory of Relativity," Physical Review 48 (1936) 73

Gourgoulhon, Special Relativity in General Frames: From Particles to Astrophysics, 2013

Rindler, Essential Relativity: Special, General, and Cosmological

Wald, General Relativity, p. 60: "...the special theory of relativity asserts that spacetime is the manifold R^4 with a flat metric of Lorentz signature defined on it. Conversely, the entire content of special relativity ... is contained in this statement ..."

Last edited: Aug 14, 2015
2. Aug 14, 2015

### PAllen

3. Aug 14, 2015

### DrGreg

For this to be understood by the widest possible audience, SR ignores gravity (or, more precisely, the tidal effects of gravity), whereas GR does not.

4. Aug 14, 2015

### bcrowell

Staff Emeritus

5. Aug 16, 2015

### m4r35n357

I'd go further from a beginner's perspective.
Special relativity: Speed of light (including e = mc2)
General relativity: Gravity

6. Aug 16, 2015

### stevendaryl

Staff Emeritus
I think that there are several stepping stones on the way from Special Relativity to General Relativity:
1. Special Relativity in Cartesian, inertial coordinates.
2. Special Relativity in noninertial, curvilinear coordinates.
3. Using the equivalence principle to relate physics in an approximately constant gravitational field to physics in an accelerated coordinate system.
4. The mathematics of curved spacetime (geodesics, parallel transport, curvature tensors, etc.)
5. The field equations giving stress-energy as the source of spacetime curvature.
Some people only consider step 1 to be SR, and all the rest to be GR, but I think step 2 is still SR. Step 3 isn't really GR, either; it's just SR plus the hypothesis of the equivalence principle. Einstein got to that point, which allowed him to predict gravitational time dilation, in 1907, which was a full 8 years before GR was finished. So most of the really hard parts of GR are yet to come after the EP.

7. Sep 26, 2015

### marcus

The amazon page for Gourgoulhon's book has "Look inside" so you can for example browse the table of contents. I didn't see that feature on the walmart page. It also has a bit more information about the book and some reviews. Folks may want to to give amazon some competition but still consult the amazon page.
https://www.amazon.com/Special-Relativity-General-Frames-Astrophysics/dp/3642372759

8. Sep 26, 2015

### robphy

Strictly speaking,
I think the statements should be
1. (as you quote what Wald says) "the special theory of relativity asserts that spacetime is the manifold R^4 with a flat metric of Lorentz signature defined on it."
2. General relativity should then be (as Wald says on page 68) "In general relativity, we do not assert that spacetime is the manifold R4 with a flat metric g defined on it...
Spacetime is a manifold M on which is defined a Lorentz metric g."

So, GR includes SR (with its flat Minkowski spacetime),
as well as any other spacetime with Lorentz metric, as defined above--
certainly including curved spacetimes
but also including flat spacetimes whose manifold is not R^4 (like a cylinder, Mobius band, torus, or even a punctured R^4 [Wald, p 191]) [which cannot be fully treated by SR].

9. Sep 27, 2015

### vanhees71

In standard GR it's a torsion free pseudo-Riemannian manifold with a pseudometric of signature (1,3) (for west-coastlers or (3,1) for east-coastlers).

10. Sep 27, 2015

### bcrowell

Staff Emeritus
I see. So Wald wants to define flat spacetimes with nontrivial topologies as not belonging to GR. I would be interested in knowing whether this is something that is widely agreed on as a matter of terminology. I would be inclined to go the other way, since the generalization of SR to nontrivial topologies is pretty straightforward.

I also see no strong reason to define any of this specifically as requiring four dimensions. It's often nicer pedagogically to talk about SR in 1+1 dimensions. Generalizing GR from 4 dimensions to 5 or more is trivial. I suppose we have problems in lower-dimensional GR because certain phenomena, such as propagating degrees of freedom, may not exist at all. In any case, I don't think the number of dimensions is relevant to the distinction between SR and GR.

Last edited: Sep 27, 2015
11. Sep 27, 2015

### robphy

My reading of Wald (and my opinion of the situation) is that
"flat spacetimes with nontrivial topologies" belong to GR, but not SR.

A "flat spacetime with a nontrivial topology" is no longer a vector space.

(I'm not considering issues with dimensionality.)

12. Sep 27, 2015

### bcrowell

Staff Emeritus
@robphy: Yes, I agree with your reading of Wald. I'm just wondering whether what Wald does is a widely accepted standard, and stating as a matter of preference that I dislike what he does.

That's true, but I don't see it as very relevant. It's not difficult to cleanse SR of the assumption that spacetime is a vector space.

13. Sep 27, 2015

### robphy

With nontrivial topologies, one can have various causality violations or problems with initial-value-problems
that one doesn't get in SR.

Maybe in a nice enough and small enough subset, you can do much of SR just fine....
However, globally there is a difference.

Observers in such spacetimes will eventually discover that their universe is not Minkowski spacetime.

14. Sep 28, 2015

### bcrowell

Staff Emeritus
That all makes sense, but it doesn't require any reformulation of the basic theory in order to handle it.

15. Sep 28, 2015

### stevendaryl

Staff Emeritus
Well, there is a sense in which it's not just nontrivial topologies, but curvature as well. If you know the laws of physics in Minkowsky space, then you can generalize them to curved spacetime by replacing regular derivatives by covariant derivatives.

So there is a sense in which knowing SR gets you almost to GR with very few additional assumptions. The one thing that you don't get is the field equations (which took Einstein 10 years after stating the equivalence principle linking SR to GR).

16. Sep 28, 2015

### martinbn

What about flat, trivial topology in the sense above (simply connected), but not maximal, say an open ball of $\mathbb R^4$, is that SR or GR?

17. Sep 28, 2015

### robphy

18. Sep 28, 2015

### bcrowell

Staff Emeritus
If it took Albert Einstein 10 years to figure out, I don't think it's a straightforward generalization :-)

19. Sep 28, 2015

### bcrowell

Staff Emeritus
This is an interesting possibility to think about. However, it seems to me that such a universe would be operationally indistinguishable from Minkowski space.

20. Sep 28, 2015

### stevendaryl

Staff Emeritus
Definitely not. As Misner, Thorne and Wheeler put it (paraphrased):

Spacetime tells matter how to move; matter tells spacetime how to curve.​

The first part is sort of a straightforward generalization of SR, once you have the idea of spacetime curvature. Einstein was most of the way there when he stated the equvialence principle and used it to predict gravitational time dilation. The second part is a lot more difficult, and wasn't really a generalization of anything that already existed. (In hindsight, it's possible to formulate Newton's theory of gravity so that it looks something like the field equations, but of course, Newton didn't think of it as describing curvature.)