Distinction between special and general relativity

[bcrowell]

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 ..."

 

[PAllen]

Perhaps add, as an advanced reference on this point, what is almost certainly the most thorough book on this point:

https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

 

[DrGreg]

SR deals with flat spacetime, GR with curved spacetime

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.

 

[bcrowell]

Thanks, PAllen and DrGreg, for the helpful comments -- I've made appropriate edits to #1.

 

[m4r35n357]

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.

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

 

[stevendaryl]

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.

 

[marcus]

Perhaps add, as an advanced reference on this point, what is almost certainly the most thorough book on this point:

http://www.walmart.com/ip/Special-Relativity-in-General-Frames-From-Particles-to-Astrophysics/23649503

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/dp/3642372759/?tag=pfamazon01-20

 

[robphy]

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

<snip>

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 ..."

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].

 

[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).

 

[bcrowell]

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].

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.

 

[robphy]

I see. So Wald wants to define flat spacetimes with nontrivial topologies as not belonging to GR.

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.)

 

[bcrowell]

@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.

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

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.

 

[robphy]

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.

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.

 

[bcrowell]

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.

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

 

[stevendaryl]

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

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).

 

[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?

 

[robphy]

Possibly interesting reading:
https://books.google.com/books?id=p...och+horowitz+"global+structure+of+spacetimes"

 

[bcrowell]

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).

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

 

[bcrowell]

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?

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

 

[stevendaryl]

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

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.)

 

[martinbn]

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

I suppose maximality may be part of the definition of a space-time, otherwise what happens to a light ray that reaches the "end"? And one can deem them unphysical.

 

[bcrowell]

I suppose maximality may be part of the definition of a space-time, otherwise what happens to a light ray that reaches the "end"? And one can deem them unphysical.

Or in terms of initial conditions, it would be sort of like the picture advocated by a creationist who says that dinosaurs never lived on Earth, and God just put the fossils in the rocks. That's why I say I don't think it's operationally observable. In fancier mathematical language, I think the example you gave is not globally hyperbolic. If you make it a closed ball, it's not a manifold.

 

[martinbn]

Actually what is the exact definition of globally hyperbolic? The example seems globally hyperbolic to me, its Cauchy development is going to be bigger, which is suggestive for it not being physical, but why is it not globally hyperbolic?

 

[bcrowell]

Actually what is the exact definition of globally hyperbolic? The example seems globally hyperbolic to me, its Cauchy development is going to be bigger, which is suggestive for it not being physical, but why is it not globally hyperbolic?

The definition I had in mind was (1) no CTCs, and (2) for any two events p and q, the intersection of p's causal future with q's causal past is compact. I think it fails #2, because the intersection could include an open neighborhood of the sphere's boundary.

 

[atyy]

@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.

Wald is just being informal there, which is allowed since he is talking about the heuristic version of the equivalence principle.

If you read the entire passage, they key difference is that GR is a theory in which if gravity is present, spacetime is curved.

If I am not wrong, Joshi's book on global aspects of general relativity does refer to special relativity as the case in which spacetime is flat and gravity is absent, but I'm not able to check that at the moment.

 

[robphy]

Joshi, Global Aspects in Gravitation and Cosmology
p.52

"The above principles effectively imply that it is the space-time metric, and the quantities derived from it, that must appear in the equations for physical quantities and that these equations must reduce to the flat spacetime case when the metric is Minkowskian. This is the basic content of the general theory of relativity where the space-time manifold is now allowed to have topologies other than R4 and the metric g_ij could be non-flat."

So, the metric in general relativity "could be non-flat" [but not "must be non-flat"] implies that it could be flat.

So, this is consistent with what I wrote earlier: in addition to curved-spacetimes, GR includes SR and spacetimes with flat-metrics.

 

[martinbn]

The definition I had in mind was (1) no CTCs, and (2) for any two events p and q, the intersection of p's causal future with q's causal past is compact. I think it fails #2, because the intersection could include an open neighborhood of the sphere's boundary.

Ah, I see, what about half Mikowski space-time, say $t>0$? The usual slices $t=const$ are Cauchy surfaces.

 

[bcrowell]

Ah, I see, what about half Mikowski space-time, say $t>0$? The usual slices $t=const$ are Cauchy surfaces.

Right, that's globally hyperbolic.

I think that example also shows that my fake-fossil analogy doesn't really correspond exactly to the notion of global hyperbolicity. Your example is very much like the fake-fossil situation, but it's globally hyperbolic.

 

[atyy]

Joshi, Global Aspects in Gravitation and Cosmology
p.52

"The above principles effectively imply that it is the space-time metric, and the quantities derived from it, that must appear in the equations for physical quantities and that these equations must reduce to the flat spacetime case when the metric is Minkowskian. This is the basic content of the general theory of relativity where the space-time manifold is now allowed to have topologies other than R4 and the metric g_ij could be non-flat."

So, the metric in general relativity "could be non-flat" [but not "must be non-flat"] implies that it could be flat.

So, this is consistent with what I wrote earlier: in addition to curved-spacetimes, GR includes SR and spacetimes with flat-metrics.

Technically, it doesn't say that the topology in SR must be R4. It only says that Minkowski spacetime is on R4 (ie. Minkowski spacetime is not completely synonymous with SR).

Also, it allows that GR includes flat metrics. But it doesn't say that GR include flat metrics with matter, where matter is defined as things with local stress-energy.

Does Joshi not have an example of SR on S1 X R?

 

[robphy]

Technically, it doesn't say that the topology in SR must be R4. It only says that Minkowski spacetime is on R4 (ie. Minkowski spacetime is not completely synonymous with SR).

[snip]

Does Joshi not have an example of SR on S1 X R?

Elsewhere (p.6), Joshi writes
"For the Minkowski space-time, which is the space-time of special relativity, the manifold is globally Euclidean with the topology of R4...

[snip]

However, if we allow for arbitrary topologies for the space-time, globally the future light cone of p may bend to enter the past of p, thus giving rise to causality violations. For example, consider the two-dimensional Minkowski space defined by the metric $ds^2 = -dt^2 + dx^2$ and where we identify the lines t =-1 and t = L. This space is topologically $S^1 \times R$ and contains closed timelike curves through every point."

So, Joshi starts with 1+1 Minkowski spacetime then makes an identification to obtain $S^1 \times R$, which is no longer Minkowski spacetime since it doesn't have the topology of $R^2$.

 

[atyy]

Elsewhere (p.6), Joshi writes
"For the Minkowski space-time, which is the space-time of special relativity, the manifold is globally Euclidean with the topology of R4...

[snip]

However, if we allow for arbitrary topologies for the space-time, globally the future light cone of p may bend to enter the past of p, thus giving rise to causality violations. For example, consider the two-dimensional Minkowski space defined by the metric $ds^2 = -dt^2 + dx^2$ and where we identify the lines t =-1 and t = L. This space is topologically $S^1 \times R$ and contains closed timelike curves through every point."

So, Joshi starts with 1+1 Minkowski spacetime then makes an identification to obtain $S^1 \times R$, which is no longer Minkowski spacetime since it doesn't have the topology of $R^2$.

I see - so if we keep to Joshi's terminology strictly, flat spacetimes with matter on topologies other than R4 would be neither SR nor GR.

 

[robphy]

I see - so if we keep to Joshi's terminology strictly, flat spacetimes with matter on topologies other than R4 would be neither SR nor GR.

GR has a not-necessarily-R4-manifold and has not-necessarily-flat torsion-free Lorentz-signature metric.
SR has a necessarily-R4-manifold and has necessarily-flat Lorentz-signature metric, and is covered as special case of GR.

Matter is described by a tensor field on spacetime...
that is to say, the spacetime-manifold and metric has already been established before a specification of a tensor field for matter.
So, matter or no-matter is already under covered by GR.

[edit: torsion-free refers to the metric-compatible derivative operator]

 

[atyy]

GR has a not-necessarily-R4-manifold and has not-necessarily-flat torsion-free Lorentz-signature metric.
SR has a necessarily-R4-manifold and has necessarily-flat Lorentz-signature metric, and is covered as special case of GR.

Matter is described by a tensor field on spacetime...
that is to say, the spacetime-manifold and metric has already been established before a specification of a tensor field for matter.
So, matter or no-matter is already under covered by GR.

Flat spacetime with matter cannot be a special case of GR, because it is inconsistent with the Einstein field equations.

 

[robphy]

Flat spacetime with matter cannot be a special case of GR, because it is inconsistent with the Einstein field equations.

That's fine. I agree. I'm not even talking about the field equations or any matter models that satisfy various energy conditions.
I will attach "spacetime-structure of" henceforth...

I'm referring to distinction that the "spacetime-structure (manifold and metric) of GR" includes that of SR and that of all flat-nonR4 as well as curved spacetimes. When I first entered this discussion, I disagreed with the idea that the "spacetime-structure-of-GR" had to be curved.
My comments and quotes try to say that "spacetime-structure-of-GR" need not be curved--so it can be flat and that SR is a proper subset of GR. If the spacetime-structure is curved, it's certainly not the "spacetime-structure of SR"

 

[bcrowell]

Flat spacetime with matter cannot be a special case of GR, because it is inconsistent with the Einstein field equations.

That would make SR a completely useless theory. SR applies when the curvature is negligible. We don't need the curvature to vanish identically.