Why Is the K/r^2 Formula Incompatible with Special Relativity?

[Sphinx]

Hello folks!
why a formula of type K/r^2 for the gravitationnal interaction is incompatible with the principle of special relativity ? ( the electric field is also defined with the same kind of relations)

 

[Nugatory]

The problem with $F=Gm_1m_2/r^2$ is that it predicts that if I move one of the masses, the force experienced by the other will immediately change, no matter how great the distance. Not only does the effect propagate faster than the speed of light, it propagates instantaneously.

The electrical force (you said "field" above, but you should have said "force" - in this context the distinction is crucial) described by Coulomb's $1/r^2$ law had the same problem until Maxwell discovered the laws describing how changes in the electrical field propagate. General relativity does something similar for gravitation.

 

[bcrowell]

Supposing that we fix the OP's mistake about how fields are defined, a different and deeper question is why it's possible to incorporate electromagnetic fields into special relativity, but it's not possible to do so with gravitational fields. One can, for example, describe gravity as a spin-2 field in flat spacetime, but the resulting theory is inconsistent unless you add corrections to it. Once you're done adding the infinite series of corrections, the original flat spacetime disappears and you have a theory that's equivalent to GR.

 

[atyy]

Supposing that we fix the OP's mistake about how fields are defined, a different and deeper question is why it's possible to incorporate electromagnetic fields into special relativity, but it's not possible to do so with gravitational fields. One can, for example, describe gravity as a spin-2 field in flat spacetime, but the resulting theory is inconsistent unless you add corrections to it. Once you're done adding the infinite series of corrections, the original flat spacetime disappears and you have a theory that's equivalent to GR.

That's not right. The resulting theory with corrections is still a theory in flat spacetime.

 

[PeterDonis]

The resulting theory with corrections is still a theory in flat spacetime.

No, it isn't. The "flat spacetime" that you started out with is unobservable; the actual metric is the curved metric including all the corrections, not the flat one you started with.

 

[Sphinx]

The problem with $F=Gm_1m_2/r^2$ is that it predicts that if I move one of the masses, the force experienced by the other will immediately change, no matter how great the distance. Not only does the effect propagate faster than the speed of light, it propagates instantaneously.

The electrical force (you said "field" above, but you should have said "force" - in this context the distinction is crucial) described by Coulomb's $1/r^2$ law had the same problem until Maxwell discovered the laws describing how changes in the electrical field propagate. General relativity does something similar for gravitation.

thanks a lot for the explanation

 

[bcrowell]

I was curious to understand how Einstein first realized that gravity was incompatible with SR. I found this paper:

Weinstein, "Einstein's Pathway to the Equivalence Principle 1905-1907," http://arxiv.org/abs/1208.5137

She gives a translation of a lecture Einstein gave in 1933.

"... I attempted to treat the law of gravity within the framework of the special theory of relativity.

"Like most writers at the time, I tried to establish a field-law for gravitation, since it was no longer possible to introduce direct action at a distance, at least in any natural way, because of the abolition of the notion of absolute simultaneity.

"The simplest thing was, of course, to retain the Laplacian scalar potential of gravity, and to complete the equation of Poisson in an obvious way by a term differentiated with respect to time in such a way, so that the special theory of relativity was satisfied. Also the law of motion of the mass point in a gravitational field had to be adapted to the special theory of relativity. The path here was less clearly marked out, since the inertial mass of a body could depend on the gravitational potential. In fact, this was to be expected on account of the inertia of energy.

"These investigations, however, led to a result which raised my strong suspicions. According to classical mechanics, the vertical acceleration of a body in the vertical gravitational field is independent of the horizontal component of its velocity ... But according to the theory I tried, the acceleration of a falling body was not independent of its horizontal velocity, or the internal energy of the system."

This led him to the equivalence principle and its implication of gravitational time dilation, which he published in 1907.

 

[pervect]

Hello folks!
why a formula of type K/r^2 for the gravitationnal interaction is incompatible with the principle of special relativity ? ( the electric field is also defined with the same kind of relations)

Lets discuss the electromagnetic case first, it's much easier. First, let's look at what the force law in electromagnetism between two charges actually is. You've noticed that it is not K q1 q2 / r^2.

The law we are need is the Lorenz force law, which is $f = q(E + \vec{v} \times \vec{B})$. This is called the Lorentz force law, see for instance https://en.wikipedia.org/w/index.php?title=Lorentz_force&oldid=678002831

E and B are the electric and magnetic fields. This law currently isn't in the form of the "force between charges". To apply the law to get the force between charges, we need to find out what the E and B fields are generated by a charge.

The exact expression of this is rather complex. I'm not sure of the simplest, most basic presentation. The one that comes to mind at the moment is the idea of the "retarded potential", https://en.wikipedia.org/w/index.php?title=Retarded_potential&oldid=667679850, which winds up yielding Jeffmenko's equations, https://en.wikipedia.org/w/index.php?title=Jefimenko's_equations&oldid=598903545. This is probably too advanced in detail, since this is an I level question :(.

Another approach that comes to mind is a discussion of "how the electromagnetic force transforms". This yields some good insights though it doesn't really answer the question of what the EM field of a moving charge actually is in numeric detail. Since it doesn't cover the point needed, I'll skip giving a link for that, if there's some interest, ask.

The motivation for all this is perhaps simpler, and perhaps it will satisfy you as to why we need both the E and B fields to define the force between charges. The motivation is that we want the laws of physics to work in any reference frame, so that we get the same result in a frame where the first charge is stationary (and the second is moving), a frame where the first charge is moving and the second is stationary, or an arbitrary frame where both charges are moving.

This principle is called the principle of covariance.

So we can say then, in general, that the observed behavior of the electromagnetic interaction, which involves both electric and magentic fields, is compatible with special relativity because the physical laws (including the force laws) are relativistically covariant. And we can note that the coulomb force law you ask about $F = k \, q1 \, q2 / r^2$ is NOT relativistically covariant.

I'm not sure how much further we can go in an I level thread, really. A textook like Griffiths, "Introduction to Electromagnetism", will go through the electromagnetic force in detail, both from a classical viewpoint (using Maxwell's equations), and a purely relativistic treatment. To follow the relativisitc treatment in full, though, you'll need to learn enough special relativity to understand relativisitic kinematics (the Lorentz transform) and relativisitc dynamics (the treatment of forces in special relativity). It's also helpful to realize that Maxwell's equations are fully compatible with special relativity, and that Maxwell's equations can be regarded as inspiring special relativity, though I wouldn't claim that this is historically accurate.

[add]
Let me just point out here that there is a bit of a change in thought here, from the idea of a force between charges, which in the coulomb case instantaneous, to a field concept, where the charges radiate fields, the fields propagate at some velocity, and then the fields interact with charges after they propagate.

Now let's say a few words of what additional things we need for gravity. If we consider only linearized gravity, there's a theory called GEM that is very like Maxwell's equations that gives us a gravitational equivalent of a "magnetic force" along with the usual "coulomb-like" force you're familiar with. If we want to go deeper than linearized theory, we start running into the limits of the idea of describing gravity as "just a force". Gravity causes effects such as time dilation and changes in the spatial geometry that simply cannot be put into the "mold" of a force. Thus any treatment of gravity that treats gravity as "only a force" doesn't even have the concepts to describe these additional effects.

The pop-sci version of this is to say that "gravity is curved space-time". But do yourself a favor and skip over the ubiquitous bowling-ball-on-a-sheet picture, which is rather likely to give you some false ideas :(.

 

[atyy]

No, it isn't. The "flat spacetime" that you started out with is unobservable; the actual metric is the curved metric including all the corrections, not the flat one you started with.

No, it is correct. Unobservable does not mean it is not there. The theory with flat spacetime is the same theory as GR and makes the same predictions.

 

[Sphinx]

Lets discuss the electromagnetic case first, it's much easier. First, let's look at what the force law in electromagnetism between two charges actually is. You've noticed that it is not K q1 q2 / r^2.

The law we are need is the Lorenz force law, which is $f = q(E + \vec{v} \times \vec{B})$. This is called the Lorentz force law, see for instance https://en.wikipedia.org/w/index.php?title=Lorentz_force&oldid=678002831

E and B are the electric and magnetic fields. This law currently isn't in the form of the "force between charges". To apply the law to get the force between charges, we need to find out what the E and B fields are generated by a charge.

The exact expression of this is rather complex. I'm not sure of the simplest, most basic presentation. The one that comes to mind at the moment is the idea of the "retarded potential", https://en.wikipedia.org/w/index.php?title=Retarded_potential&oldid=667679850, which winds up yielding Jeffmenko's equations, https://en.wikipedia.org/w/index.php?title=Jefimenko's_equations&oldid=598903545. This is probably too advanced in detail, since this is an I level question :(.

Another approach that comes to mind is a discussion of "how the electromagnetic force transforms". This yields some good insights though it doesn't really answer the question of what the EM field of a moving charge actually is in numeric detail. Since it doesn't cover the point needed, I'll skip giving a link for that, if there's some interest, ask.

The motivation for all this is perhaps simpler, and perhaps it will satisfy you as to why we need both the E and B fields to define the force between charges. The motivation is that we want the laws of physics to work in any reference frame, so that we get the same result in a frame where the first charge is stationary (and the second is moving), a frame where the first charge is moving and the second is stationary, or an arbitrary frame where both charges are moving.

This principle is called the principle of covariance.

So we can say then, in general, that the observed behavior of the electromagnetic interaction, which involves both electric and magentic fields, is compatible with special relativity because the physical laws (including the force laws) are relativistically covariant. And we can note that the coulomb force law you ask about $F = k \, q1 \, q2 / r^2$ is NOT relativistically covariant.

I'm not sure how much further we can go in an I level thread, really. A textook like Griffiths, "Introduction to Electromagnetism", will go through the electromagnetic force in detail, both from a classical viewpoint (using Maxwell's equations), and a purely relativistic treatment. To follow the relativisitc treatment in full, though, you'll need to learn enough special relativity to understand relativisitic kinematics (the Lorentz transform) and relativisitc dynamics (the treatment of forces in special relativity). It's also helpful to realize that Maxwell's equations are fully compatible with special relativity, and that Maxwell's equations can be regarded as inspiring special relativity, though I wouldn't claim that this is historically accurate.

[add]
Let me just point out here that there is a bit of a change in thought here, from the idea of a force between charges, which in the coulomb case instantaneous, to a field concept, where the charges radiate fields, the fields propagate at some velocity, and then the fields interact with charges after they propagate.

Now let's say a few words of what additional things we need for gravity. If we consider only linearized gravity, there's a theory called GEM that is very like Maxwell's equations that gives us a gravitational equivalent of a "magnetic force" along with the usual "coulomb-like" force you're familiar with. If we want to go deeper than linearized theory, we start running into the limits of the idea of describing gravity as "just a force". Gravity causes effects such as time dilation and changes in the spatial geometry that simply cannot be put into the "mold" of a force. Thus any treatment of gravity that treats gravity as "only a force" doesn't even have the concepts to describe these additional effects.

The pop-sci version of this is to say that "gravity is curved space-time". But do yourself a favor and skip over the ubiquitous bowling-ball-on-a-sheet picture, which is rather likely to give you some false ideas :(.

Thanks a lot , this is very instructive

 

[atyy]

Supposing that we fix the OP's mistake about how fields are defined, a different and deeper question is why it's possible to incorporate electromagnetic fields into special relativity, but it's not possible to do so with gravitational fields. One can, for example, describe gravity as a spin-2 field in flat spacetime, but the resulting theory is inconsistent unless you add corrections to it. Once you're done adding the infinite series of corrections, the original flat spacetime disappears and you have a theory that's equivalent to GR.

To further show why this is wrong, there is in fact a flat spacetime argument as to why a reformulation as curved spacetime is possible. Unfortunately, i don't understand the argument, but I do know it exists. The reformulation as curved spacetime is possible because of the equivalence principle. Then the question becomes whether the equivalence principle can be derived. Weinberg argues that the equivalence principle can be derived starting from the theory in flat spacetime. His argument is outlined in http://arxiv.org/abs/1007.0435v3 Section 2.2.2 and Appendix A "Weinberg low-energy theorem".

 

[PeterDonis]

Unobservable does not mean it is not there.

If it's a metric, yes it does, because the definition of the metric is "the thing that determines spacetime intervals". That thing is the curved metric, not the flat metric, so the flat metric being unobservable--not determining any spacetime intervals--is the same as it not existing; "existing" for a metric means "determining spacetime intervals".

The theory with flat spacetime is the same theory as GR and makes the same predictions.

Locally, the spin-2 field theory Lagrangian is the Einstein-Hilbert Lagrangian (in the classical limit), so yes, it makes the same predictions as GR. But globally, the spin-2 field theory on a background Minkowski spacetime only allows solutions with the same topology as Minkowski spacetime. That rules out both the Kerr-Newman and the FRW families of spacetimes, i.e., the solutions which get the most use in GR.

 

[atyy]

If it's a metric, yes it does, because the definition of the metric is "the thing that determines spacetime intervals". That thing is the curved metric, not the flat metric, so the flat metric being unobservable--not determining any spacetime intervals--is the same as it not existing; "existing" for a metric means "determining spacetime intervals".

I agree. But that is different from saying whether it is compatible with special relativity and flat spacetime.

Locally, the spin-2 field theory Lagrangian is the Einstein-Hilbert Lagrangian (in the classical limit), so yes, it makes the same predictions as GR. But globally, the spin-2 field theory on a background Minkowski spacetime only allows solutions with the same topology as Minkowski spacetime. That rules out both the Kerr-Newman and the FRW families of spacetimes, i.e., the solutions which get the most use in GR.

Yes, but one cannot get there by corrections from flat spacetime either. Starting from flat spacetime yields a consistent theory, which when reformulated as curved spacetime allows a generalization to cosmological solutions.

So yes, I agree that GR postulated as curved spacetime is a more general theory. But the reasoning given by bcrowell is wrong.

 

[atyy]

I should add that even if one uses the definition of metric in the operational sense, there is a good argument that full GR is not a theory of a spacetime metric.

http://arxiv.org/abs/gr-qc/9912051
Does General Relativity Require a Metric
James L. Anderson

 

[PeterDonis]

one cannot get there by corrections from flat spacetime either. Starting from flat spacetime yields a consistent theory, which when reformulated as curved spacetime allows a generalization to cosmological solutions.

Ah, I see; you're saying that, if we take the "spin-2 field in flat spacetime" approach, even though we end up with the Einstein-Hilbert Lagrangian after applying corrections to all orders, we still have the underlying assumption that the topology is the same as Minkowski spacetime. In order to generalize to solutions with other topologies, we can't view the "spin-2 field in flat spacetime" approach as fundamental; we have to reformulate the theory as a theory of curved spacetime from the start. I agree with that; the fact that the spin-2 field approach leads to the GR Lagrangian is highly suggestive, but it can't be a fundamental theory by itself.

 

[martinbn]

I should add that even if one uses the definition of metric in the operational sense, there is a good argument that full GR is not a theory of a spacetime metric.

http://arxiv.org/abs/gr-qc/9912051
Does General Relativity Require a Metric
James L. Anderson

This seems like philosophy with no physical content. It seems like saying "let's call $g$ a field and not metric, then GR is a theory of that field, no space-time metric involved".

 

[atyy]

Ah, I see; you're saying that, if we take the "spin-2 field in flat spacetime" approach, even though we end up with the Einstein-Hilbert Lagrangian after applying corrections to all orders, we still have the underlying assumption that the topology is the same as Minkowski spacetime. In order to generalize to solutions with other topologies, we can't view the "spin-2 field in flat spacetime" approach as fundamental; we have to reformulate the theory as a theory of curved spacetime from the start. I agree with that; the fact that the spin-2 field approach leads to the GR Lagrangian is highly suggestive, but it can't be a fundamental theory by itself.

Yes. So I'm saying two things:

1) Starting from spin 2, we can get a consistent theory that is equivalent to a restricted regime of GR. So this method cannot be used to argue that gravity is inconsistent with special relativity - it is consistent and spin 2 in flat spacetime and the restricted regime of GR are equivalent formulations of the same theory.

2) However, the form of the equations we get from the restricted regime suggests a more general theory, where we allow cosmological solutions. It is not a theoretical inconsistency with flat spacetime that is the problem, rather it is the observation of the expanding universe (and the acceleration for the positive cosmological constant) that makes the curved spacetime form the more fundamental postulation.

And a third point, which maybe is the most important for bcrowell's deeper question.

3) Although I disagree with the technicalities of bcrowell's phrasing, I do agree that there is a deeper question, which I would rephrase as: can the equivalence principle be derived from other "reasonable" principles? I reformulate it this way because in the presence of matter, it is the universality of the minimal coupling that allows the conception of the gravitational field as a metric in some limit. Then, the best answer I know (I don't understand it, but I do believe this is what the literature suggests) is that if we take the quantum spin 2 theory in flat spacetime, we can "derive" (informally, at the level of Weinberg's QFT) the equivalence principle. So in a sense to answer bcrowell's deeper question, we cannot (yet) reject the formulation of GR (in a restricted regime) as spin 2 in flat spacetime.

I would love to know whether it is possible to extend Weinberg's argument to spin 2 in curved spacetime.

 

[PeterDonis]

the form of the equations we get from the restricted regime suggests a more general theory, where we allow cosmological solutions.

Not just cosmological solutions; also black hole solutions.

 

[atyy]

Not just cosmological solutions; also black hole solutions.

Yes. The one I'm not sure about is the Schwarzschild vacuum black hole solution. Can that be covered by a single set of harmonic coordinates?

I googled a bit and found http://arxiv.org/abs/gr-qc/0503018v1 and http://relativity.livingreviews.org/Articles/lrr-2000-5/ (section 3.3) which discusses horizon-penetrating harmonic-like coordinates.

 

[PeterDonis]

The one I'm not sure about is the Schwarzschild vacuum black hole solution. Can that be covered by a single set of harmonic coordinates?

The topology of the maximally extended Schwarzschild solution is #R^2 x S^2$, not$R^4##. So it has a different topology from Minkowski spacetime.

If you're just talking about an open neighborhood of a Schwarzschild solution joined to some other non-vacuum neighborhood describing gravitating matter (either a static gravitating body, which precludes a horizon being present, or a collapsing matter region as in the Oppenheimer-Snyder solution), the vacuum region can be covered by a single chart. I'm not sure about the topology of the entire solution (including the matter region) in the latter case, though. (In the static case the entire spacetime can obviously be viewed as a perturbation on Minkowski spacetime, because there can't be a horizon present.)

 

[atyy]

The topology of the maximally extended Schwarzschild solution is #R^2 x S^2$, not$R^4##. So it has a different topology from Minkowski spacetime.

If you're just talking about an open neighborhood of a Schwarzschild solution joined to some other non-vacuum neighborhood describing gravitating matter (either a static gravitating body, which precludes a horizon being present, or a collapsing matter region as in the Oppenheimer-Snyder solution), the vacuum region can be covered by a single chart. I'm not sure about the topology of the entire solution (including the matter region) in the latter case, though. (In the static case the entire spacetime can obviously be viewed as a perturbation on Minkowski spacetime, because there can't be a horizon present.)

Yes, I'm talking about the maximally extended Schwarzschild solution. Can one fit a Minkowski metric on $R^2$ X $S^2$? For example, the Minkowski metric can fit on the cylinder $R^3$ X $S^1$, so that would still be a special relativistic spacetime even though the topology is different from $R^4$.

 

[PeterDonis]

Can one fit a Minkowski metric on ##R^2 X S^2##?

I don't think so, but whether it can or not, I think there are complications here that we haven't discussed. See below.

the Minkowski metric can fit on the cylinder $R^3 X S$

This is true in 1 + 1 dimensions, i.e., a 2-d Minkowski metric can fit on $S^1 \times R$. But is it true for 3 + 1 dimensions?

Even if it isn't true for $R^3 \times S$, though, AFAIK a flat Euclidean metric can be put on a spatial 3-torus, so we can have a Minkowski metric on $S^3 \times R$. And that raises a question: I had said in an earlier post that assuming flat spacetime means assuming a global $R^4$ topology. However, I see now that that's not necessarily true, mathematically speaking. Any manifold that globally admits a Minkowski metric would be a valid candidate for the underlying manifold of a spin-2 field theory in flat spacetime.

Of course that raises the question: would such a nontrivial global topology with a flat metric make sense, physically speaking? The usual (unstated) assumption seems to be that some sort of source of gravity is required to have a nontrivial global topology. But if that's true, it would mean that the only valid "ground state" solution of the spin-2 field theory--i.e., the only valid state with no sources present--would be Minkowski spacetime on $R^4$. That raises the question of how solutions with any other topology could arise; adding sources would somehow have to change the global topology that is used to define the field theory in the first place.

 

[bcrowell]

Yes, but one cannot get there by corrections from flat spacetime either. Starting from flat spacetime yields a consistent theory, which when reformulated as curved spacetime allows a generalization to cosmological solutions.

This is intellectually stimulating -- it would be very interesting if we could use cosmological data to infer the laws of physics. Normally we work the other way around, by using the laws of physics to construct cosmological models, and then seeing which of those models seem to fit the cosmological data.

But I think there are at least two flaws in what you're proposing.

(1) Not all cosmological solutions have a topology that is incompatible with a flat metric. All currently available cosmological data are consistent with the possibility that the universe has the topology of R^4. (AFAIK we have absolutely zero data -- concerning black holes, cosmology, large extra dimensions, or anything else -- that would be incompatible with the hypothesis that spacetime has any topology other than that of R^4.)

(2) As Peter Donis has noted, a flat Lorentzian metric is compatible with more than one topology.

A more general point is that although Newtonian mechanics uses nonlocal laws of physics, that's not how anyone has put together their proposed laws of physics for the last hundred years. Today our entire notion of a physical law is that it's a local thing (typically expressed in a differential equation). Therefore if we have two versions of the laws of physics, and they make the same predictions locally, then I would say they're the same laws of physics.

Of course that raises the question: would such a nontrivial global topology with a flat metric make sense, physically speaking? The usual (unstated) assumption seems to be that some sort of source of gravity is required to have a nontrivial global topology. But if that's true, it would mean that the only valid "ground state" solution of the spin-2 field theory--i.e., the only valid state with no sources present--would be Minkowski spacetime on $R^4$. That raises the question of how solutions with any other topology could arise; adding sources would somehow have to change the global topology that is used to define the field theory in the first place.

This doesn't sound right to me. You're talking about cause and effect, but how would you establish here what is the cause and what is the effect? GR in its standard formulation doesn't allow topology change, so in general it is not possible for us to start with matter fields in some initial state and have that lead, at some later time, to a certain topology. No topology is assumed in stating SR, except in the sense that some manifolds don't admit a flat metric.

 

[PeterDonis]

GR in its standard formulation doesn't allow topology change, so in general it is not possible for us to start with matter fields in some initial state and have that lead, at some later time, to a certain topology.

I agree; in GR the topology has to be put in "by hand", you can't derive it from the equations. See below.

No topology is assumed in stating SR, except in the sense that some manifolds don't admit a flat metric.

Not in SR, no. But SR does not have a field equation; it just assumes that spacetime is flat, without explaining why it is that way. So it's incomplete as a theory.

GR attempts to be more complete in this respect, by insisting that whatever spacetime geometry exists is explained by the distribution of matter and energy, via the Einstein Field Equation. But, since the EFE is local, there could be multiple possible global solutions with the same local metric, corresponding to different possible topologies for the underlying manifold. So GR isn't really complete in this respect either: it can tell you the local metric, but not the global topology--as above, the latter has to be put in "by hand".

However, even if the topology has to be put in "by hand", we can still ask whether a given topology makes sense, physically, given other features of a given solution. For example, if you show me a solution with a flat metric but a nontrivial global topology like $S^3 \times R$, I think it's fair of me to ask what, physically, is your justification for giving the solution that topology instead of $R^4$, which would seem to be the "natural" topology for a flat metric.

Of course this is all heuristic and handwaving; mathematically speaking any solution with a manifold topology compatible with a flat metric is valid. But there are lots of mathematically valid solutions that nobody thinks are physically reasonable. My question is whether a flat metric with any global topology other than $R^4$ is in that category. If it is, then it seems to me that a spin-2 field theory on flat spacetime has a problem, since the ground state of the theory (flat spacetime with no sources) would have one topology ($R^4$), but states with sources present might have other topologies.

Of course, if it turns out that we can account for all observations using solutions with $R^4$ topology (or patches of such solutions stitched together), then this is a non-issue.

 

[bcrowell]

If it is, then it seems to me that a spin-2 field theory on flat spacetime has a problem, since the ground state of the theory (flat spacetime with no sources) would have one topology ($R^4$), but states with sources present might have other topologies.

Spacetime in SR doesn't have a ground state. Spacetime in SR doesn't have different states with different topologies. Spacetime isn't dynamical in SR, so it doesn't make sense to describe the geometry of that spacetime as a "state." It's not a state, it's just a background. SR makes no claims about the topology of spacetime (except that the topology must be compatible with flatness). The R^4 topology has no special status in SR.

It seems to me that you want to take the assumption that SR describes a spacetime with the topology of R^4, and dress up that assumption in some kind of fancy language to make it more than what it is: simply an arbitrary assumption that could turn out to be wrong.

For example, if you show me a solution with a flat metric but a nontrivial global topology like $S^3 \times R$, I think it's fair of me to ask what, physically, is your justification for giving the solution that topology instead of $R^4$, which would seem to be the "natural" topology for a flat metric.

I disagree that this is fair. We're not even discussing the actual geometry of the universe, just a fictitious geometry that represents the starting point for a process of iteration in the spin-2 picture. If we were discussing the actual geometry of the universe, then it would still be unfair to expect someone else to be able to come up with some reason for the universe to have a certain topology. If multiple topologies are compatible with the observed matter fields in the universe, then we have no physical principle that would allow us to pick one of them a priori. It's purely a matter of observation.

 

[PeterDonis]

Spacetime in SR doesn't have a ground state.

But a spin-2 field theory in flat spacetime does, just like any field theory. That's the ground state I was referring to. Since the spin-2 field determines the actual, physical metric, specifying the ground state also specifies the metric in that state.

It seems to me that you want to take the assumption that SR describes a spacetime with the topology of R^4, and dress up that assumption in some kind of fancy language to make it more than what it is: simply an arbitrary assumption that could turn out to be wrong.

The theory under discussion is not SR by itself. It's spin-2 field theory on a flat spacetime background. SR is part of that theory, but not all of it.

We're not even discussing the actual geometry of the universe, just a fictitious geometry that represents the starting point for a process of iteration in the spin-2 picture.

But if this fictitious geometry turns out to be physically unreasonable--if it turns out to be impossible, or at least highly implausible, to get to a geometry that can describe what we actually observe, from such a starting point--then the theory isn't viable, even if it's perfectly consistent mathematically. I'm not disputing that, mathematically speaking, the procedure is perfectly consistent and it works with any topology that is compatible with a flat metric. Everything I'm saying is, as I've already admitted, just heuristic and handwaving. (Which means we're probably getting to the point where this subthread is off topic.)

 

[bcrowell]

By the way, when we say that a black hole solution has a particular topology, that's a statement that cannot be tested empirically against reality -- not even in principle. The topology of such a solution has a topological "hole" cut out of it where the singularity is, analogous to the situation where you remove one point from the plane and turn its topology into that of a cylinder. To verify the existence of that topological hole at the singularity of a black hole, we would have to observe it by receiving signals from it, which we can't do. (And in any case, I don't know any physicists who believe that GR's prediction of black hole singularities is literally correct.)

But if this fictitious geometry turns out to be physically unreasonable--if it turns out to be impossible, or at least highly implausible, to get to a geometry that can describe what we actually observe, from such a starting point

But I don't see any reason to believe such a thing.

Everything I'm saying is, as I've already admitted, just heuristic and handwaving. (Which means we're probably getting to the point where this subthread is off topic.)

It's been an interesting discussion, but I would tend to agree that we're probably not going to make any further progress without wandering off into la-la land.

 

[atyy]

This is intellectually stimulating -- it would be very interesting if we could use cosmological data to infer the laws of physics. Normally we work the other way around, by using the laws of physics to construct cosmological models, and then seeing which of those models seem to fit the cosmological data.

But I think there are at least two flaws in what you're proposing.

(1) Not all cosmological solutions have a topology that is incompatible with a flat metric. All currently available cosmological data are consistent with the possibility that the universe has the topology of R^4. (AFAIK we have absolutely zero data -- concerning black holes, cosmology, large extra dimensions, or anything else -- that would be incompatible with the hypothesis that spacetime has any topology other than that of R^4.)

(2) As Peter Donis has noted, a flat Lorentzian metric is compatible with more than one topology.

A more general point is that although Newtonian mechanics uses nonlocal laws of physics, that's not how anyone has put together their proposed laws of physics for the last hundred years. Today our entire notion of a physical law is that it's a local thing (typically expressed in a differential equation). Therefore if we have two versions of the laws of physics, and they make the same predictions locally, then I would say they're the same laws of physics.

Yes, I was too hasty to agree with everything Peter Donis said. Anyway, I don't think this is backwards from the usual way. It's just that there are at least 2 ways to get to GR equations (1) as spin 2 on flat spacetime compatible with special relativity and reformulate as curved spacetime (2) directly postulate the curved spacetime so that we are not constrained by the theory having an equivalent formulation as spin 2 in flat spacetime. Both yield consistent theories, but the second is more general since it doesn't have the constraint of also being spin 2 in flat spacetime. If observations require solutions that can be reformulated as spin 2 on flat spacetime, then we can say observations don't force us to abandon special relativity. If we need a solution that cannot be reformulated as spin 2 on flat spacetime, then we have to abandon global special relativity as a fundamental postulate.

So why do at least some cosmological solutions fail to be reformulatable as spin 2 on flat spacetime? I think in the case of the non-zero cosmological constant, it is because the boundary conditions are not asymptotically flat, not because the topology is not $R^4$. So this would apply to the current cosmological model.

But how about FRW with zero cosmological constant? My impression was that this is also not asymptotically flat, so it cannot be written as spin 2 on flat spacetime, but I am not sure whether this is the right reason. The other possible reason for a failure is, I believe, one needs some condition like being able to use harmonic coordinates to cover the whole spacetime. I think FRW with zero cosmological constant can be covered by a single chart, but that chart cannot be harmonic, so it cannot be formulated as spin 2 on flat spacetime. The reason I'm not sure this is correct is that while harmonic coordinates are sufficient to guarantee a formulation as spin 2 on flat spacetime, I don't know whether it is necessary.

 

[atyy]

Thinking about it a bit more, I think it is the harmonic coordinate condition that is more important, since I don't see how one would have asymptotically flat boundary conditions on a torus. However, I still don't know whether the harmonic condition is necessary, or whether there are other similar conditions needed to formulate GR as a fields in flat spacetime. A reference is

http://relativity.livingreviews.org/Articles/lrr-2014-4/articlese5.html#x8-440005 : "Eq. (http://relativity.livingreviews.org/Articles/lrr-2014-4/articlese5.html#x8-50001r77 ) is exact, and depends only on the assumption that the relevant parts of spacetime can be covered by harmonic coordinates."