Distinction between special and general relativity

[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.

 

[martinbn]

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.

If it is flat and has matter, the matter consists of only test objects, so it can be consistant with SR and with GR.

 

[atyy]

If it is flat and has matter, the matter consists of only test objects, so it can be consistant with SR and with GR.

Yes, one can define test objects in GR or not, depending on taste. I personally prefer not to consider test objects to be fundamental, at least in classical GR.

 

[martinbn]

Yes, one can define test objects in GR or not, depending on taste. I personally prefer not to consider test objects to be fundamental, at least in classical GR.

Then how do you answer any question about test obejcts? Say Mercury's perihelion presesion?

 

[atyy]

Then how do you answer any question about test obejcts? Say Mercury's perihelion presesion?

(1) Using intuition to postulate geodesic motion as a good approximation and no proper justification

(2) With great difficulty, eg. deriving geodesic motion as a good approximation for the motion of a small body, eg. http://arxiv.org/abs/1506.06245. The philosophy is similar to that given by Bill K for spinning bodies in https://www.physicsforums.com/threads/fermi-walker-transport-and-gyroscopes.704108/#post-4463372.

 

[martinbn]

(1) Using intuition to postulate geodesic motion as a good approximation and no proper justification

(2) With great difficulty, eg. deriving geodesic motion as a good approximation for the motion of a small body, eg. http://arxiv.org/abs/1506.06245. The philosophy is similar to that given by Bill K for spinning bodies in https://www.physicsforums.com/threads/fermi-walker-transport-and-gyroscopes.704108/#post-4463372.

What is the difference with assuming that Mercury is a test object!?

 

[atyy]

What is the difference with assuming that Mercury is a test object!?

The presence of small correction terms to the geodesic motion. The correction terms are absent if mercury is a test object, since it will follow a geodesic exactly.

 

[martinbn]

The presence of small correction terms to the geodesic motion. The correction terms are absent if mercury is a test object, since it will follow a geodesic exactly.

But you ignore these corrections, which is the same as having a test object!

 

[PAllen]

My take:

SR is encompasses flat 4-D spacetime (pseudo-Riemannian metric) with topology of R4, including any matter/energy theories expressed on this basis (classical EM, relativistic mechanics, QED, QCD). Gravity is totally excluded. The spacetime is an axiom, not subject to boundary or initial conditions.

GR encompasses all other 4-d spacetimes (pseudo-Riemannian metric) . Topological features are governed mostly by boundary conditions. Given any such manifold with metric, one may derive the stress-energy tensor, via the EFE. Classically, physical plausibility is defined by the dominant energy condition plus global hyperbolicity. In the context of quantum mechanics, I make no claim.

 

[atyy]

But you ignore these corrections, which is the same as having a test object!

No, it is not the same, since we don't have to say that mercury is made of matter that does not cause spacetime curvature.

 

[martinbn]

No, it is not the same, since we don't have to say that mercury is made of matter that does not cause spacetime curvature.

But you never have to say that, including when you consider the planet a test body. You say that the contribution to the curvature is negligible for the problem and therefore we consider the planet a test object. It is exactly the same.

 

[atyy]

But you never have to say that, including when you consider the planet a test body. You say that the contribution to the curvature is negligible for the problem and therefore we consider the planet a test object. It is exactly the same.

Negligible does not mean zero. They two views are not in opposition. One has a hierarchy of theories, each of which is internally mathematically consistent (ie. within that theory certain things are not approximations), but one theory is a good approximation in some limited regime from the view of the more general theory.

Example 1:

(1A) SR - flat spacetime with mass-energy, no gravity

(1B) GR - curved spacetime in the presence of mass-energy

Here within SR, spacetime is exactly globally flat. However, from the point of view of GR, SR is a good approximation when the spacetime curvature is small enough.

Example 2:

(2A) test matter or field theory on curved spacetime, but the field does not contribute to spacetime curvature

(2B) GR - curved spacetime in the presence of mass-energy

Here 2A has matter which exactly has no gravitational field - that is not an approximation from the view of the theory in 2A. But from the point of view of 2B, 2A is a good approximation in some regime (eg. mercury perihelion precession, Hawking radiation).

To be clear, as I said earlier, I don't object to postulating the existence of test matter. I was just saying from the most general viewpoint, I prefer not to have postulate the existence of test matter. I prefer to derive the geodesic motion as a superb approximation. There is no problem mathematically with postulating the existence of test matter, but we will have a conceptual problem if we cannot also derive it as an approximation.

 

[martinbn]

I still don't get the Freudian distinction that you make. One one hand we say that the object's influence is negligible and we treat it as a test object. On the other we don't use the words test objects, we say that the influence of the object is negligible and we ignore the corrections.

 

[atyy]

I still don't get the Freudian distinction that you make. One one hand we say that the object's influence is negligible and we treat it as a test object. On the other we don't use the words test objects, we say that the influence of the object is negligible and we ignore the corrections.

The difference is in whether test particles are postulated as fundamental or not.

 

[martinbn]

The difference is in whether test particles are postulated as fundamental or not.

Ok, going back to the beginning.

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.

Fundamental or not, as long as they are there, test objects on flat space-times with non R4 topology are going to be GR.

p.s. I side comment is that SR in that convention means that the space-time is Minkowski, not just flat and R4 topology.

 

[atyy]

Ok, going back to the beginning.
Fundamental or not, as long as they are there, test objects on flat space-times with non R4 topology are going to be GR.

p.s. I side comment is that SR in that convention means that the space-time is Minkowski, not just flat and R4 topology.

Well, they are not there if they are not postulated.

 

[martinbn]

Well, they are not there if they are not postulated.

But you said that your preference is that they are derived, so their are not postulated but derived, they are still their. Or are you saying that what you wrote above about Mercury is not GR.

I really don't know what we are discussing and where this is going! Anyway...

 

[atyy]

But you said that your preference is that they are derived, so their are not postulated but derived, they are still their. Or are you saying that what you wrote above about Mercury is not GR.

I really don't know what we are discussing and where this is going! Anyway...

Derived as an approximation, not exactly. In exact terms, the don't exist, because of the corrections. So conceptually we don't ignore the correction terms, it is just that the corrections are smaller than the experimental error.

Another way to say it is that if the test particles are postulated, then when experiment is fine enough to detect the deviations from geodesic motion, there is no theory that will systematically give you the corrections to match experiment.

However, if one uses the Einstein field equations without postulating test particles, then when experiment is fine enough to detect the deviations from geodesic motion, there is a theory that will systematically give you the corrections to match experiment.