SR vs GR: Logical Necessity of Signal Propagation

[unparadoxical]

TL;DR Summary

Does relativity theory necessarily disallow all physical effects that may be termed "instant"?

I guess it is safe to say that SR is, mathematically speaking, a framework that correlates discrete events in spacetime by way of a universal constant (the speed of light) so that the "proper distances" that separate them may be determined. I think this means that every conceivable "instant effect" is disqualified under this framework. Every physical "happening" in SR, in other words, would always depend on the logic of mediating influences between isolated point-like events.

But I don't see that this is necessarily the case under the framework of GR. I always hear popularizers affirming the idea that the theory of relativity disallows every "action at a distance". When it comes to "actions" related to physical effects that require the introduction of traveling electromagnetic disturbances, that is quite sensible.

But this doesn't mean that every conceivable physical effect — considered under the framework of GR — absolutely requires a mediating signal, right?

An obvious example is the application of the mathematics of GR to the physics of gravitation. It should be safe to admit that not every gravitational model developed under GR is strictly required to include the concept of propagating ("EM-like") signals, right?

I think the technical way of asking this is: Does GR strictly require that every curvature tensor that constitutes a gravitational field be "caused" by the influence of a spatially distinct stress-energy tensor? And if the tensors aren't spatially distinct, what sense is there in the idea that the values in the curvature tensors are actually "caused"?

 

[Nugatory]

Does GR strictly require that every curvature tensor that constitutes a gravitational field be "caused" by the influence of a spatially distinct stress-energy tensor?

The term "spatially distinct tensor" has no generally accepted meaning, so there's no way of answering this question until we know what you mean by it. What do you mean? Can you present an example of a tensor that is not "spatially distinct" and another that is?

However, you may want to back up and clarify your understanding of why even SR disallows the notion of action at a distance. It's not because signal propagation is required (if it were, hypothetical signals propagating at superluminal but not infinite speeds would still be allowed by the theory) but because it has to be possible to formulate the laws of physics in such a way that causes happen before effects. GR also respects this principle, for any physically reasonable spacetime.
(This last qualification is needed because there are unphysical solutions to the EFE which require a much more carefully worded statement of what GR does and does not allow.)

 

[Dale]

TL;DR Summary: Does relativity theory necessarily disallow all physical effects that may be termed "instant"?

Does GR strictly require that every curvature tensor that constitutes a gravitational field be "caused" by the influence of a spatially distinct stress-energy tensor?

I think what you are asking is if every solution of the Einstein field equations can be formulated as a solution to a Cauchy problem.

 

[robphy]

Note that there is a hierarchy of causality conditions for Spacetimes:
https://en.m.wikipedia.org/wiki/Causality_conditions

At the bottom of this list is “global hyperbolicity”, which is associated with the Cauchy problem.

So, in this context, causality is more primitive than the Cauchy problem.

In addition, depending on the intent of the OP, note that one may wish to distinguish “solutions” from “solutions requiring (say) energy conditions”.

In short, more care might be needed in formulating a question.

 

[pervect]

I think the technical way of asking this is: Does GR strictly require that every curvature tensor that constitutes a gravitational field be "caused" by the influence of a spatially distinct stress-energy tensor? And if the tensors aren't spatially distinct, what sense is there in the idea that the values in the curvature tensors are actually "caused"?

I'm not quite sure what you're asking, but certainly GR requires that the Einstein curvature tensor, $G_{\mu\nu}$ is equal to some constant times the stress-energy tensor, $T_{\mu\nu}$. I've ignored the cosmological constant, which can be folded into the stress-energy tensor, though Wiki, for example, doesn't take this approach.

As far as other curvature tensors go, we start with the metric tensor which defines the space-time, compute the Riemann curvature tensor from the metric tensor - it is basically a non-linear computation from the second partial derivatives of the metric tensor. From the Riemann tensor, we compute the Ricci tensor via contraction, and we contract the Ricci again to form the Ricci scalar. Finally we can write the Einstein tensor as

$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R$$

Here $G_{\mu\nu}$ is the Einstein tensor, $R_{\mu\nu}$ is the Ricci tensor, and R is the Ricci scalar.

This is a bit long, but basically it says that if we have the metric tensor, we can compute everything else we need from it.

This constant relating the linear relation between the Einstein tensor and the Stress-energy tensor is 8 pi in geometric units and 8 pi G / c^4 in non-geometric units.

But I think you're asking about cause and effect. I would say that GR itself does not address the issue of cause and effect specifically. That's the part where your question isn't clear. I would say that GR consists of Einstein's field equations, as summarized above. So you need to ask questions about something other than GR itself to ask about cause and effect.

Possibly of interest (as I said, I'm not sure what question you are asking) is https://resolver.caltech.edu/CaltechAUTHORS:ECHprd91, "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory", Echeveria et al.

You'll see some references to the Cauchy problem in this paper. If you're not familiar with the Cauchy problem offhand, it might be worth reading up on the defintion of the Cauchy problem in the wikipedia to see if it helps you phrase your question a bit differently as another poster has already suggeste. If you take the Cauchy problem as a definition of a framework to discuss cause and effect, you can see that some GR spacetimes with closed timelike curves, such as the wormhole example in the paper I cited, are problematical in this regard, as classically there can be an infinite number of solutions for some given initial conditions (configuration of billiard ball positions and initial states of motion).

 

[Fractal matter]

TL;DR Summary: Does relativity theory necessarily disallow all physical effects that may be termed "instant"?

I guess it is safe to say that SR is, mathematically speaking, a framework that correlates discrete events in spacetime by way of a universal constant (the speed of light) so that the "proper distances" that separate them may be determined. I think this means that every conceivable "instant effect" is disqualified under this framework. Every physical "happening" in SR, in other words, would always depend on the logic of mediating influences between isolated point-like events.

But I don't see that this is necessarily the case under the framework of GR. I always hear popularizers affirming the idea that the theory of relativity disallows every "action at a distance". When it comes to "actions" related to physical effects that require the introduction of traveling electromagnetic disturbances, that is quite sensible.

But this doesn't mean that every conceivable physical effect — considered under the framework of GR — absolutely requires a mediating signal, right?

An obvious example is the application of the mathematics of GR to the physics of gravitation. It should be safe to admit that not every gravitational model developed under GR is strictly required to include the concept of propagating ("EM-like") signals, right?

I think the technical way of asking this is: Does GR strictly require that every curvature tensor that constitutes a gravitational field be "caused" by the influence of a spatially distinct stress-energy tensor? And if the tensors aren't spatially distinct, what sense is there in the idea that the values in the curvature tensors are actually "caused"?

Wrong. GR absolutely require "propagating signals". It is a local theory. However, i myself, not quite sure what locality means. It somehow takes time for a cause to make an effect.

 

[pervect]

Wrong. GR absolutely require "propagating signals". It is a local theory. However, i myself, not quite sure what locality means. It somehow takes time for a cause to make an effect.

I would say that propagation indeed happens in GR, but it's a local result that is a logical consequence of Einstein's field equations. My recollection (which I haven't refreshed) is that you can locally show propagation via expressing Einstein's field equations in the de-Donder gauge, and applying some theorems about certain types of differential equations. My recollection is that Wald discusses this in his text on GR.

However, this result is only a local result - it doesn't addrfess non-local issues like wormhole space-times, where there might be a short-cut through the wormhole.

GR as a theory allows oddities like non time-orientable manifolds, which I haven't studied and have little intuition for, but seem to be possibly relevant to what the OP is trying to ask. Wiki has an article that discusses this that I haven't read in depth - https://en.wikipedia.org/wiki/Causal_structure.

 

[PeterDonis]

@unparadoxical, as others have already pointed out, your OP is too vague as it stands to be answerable. However, there is an obvious way to formulate what might be your question more precisely. It is this: in SR, all causal influences on a given event A must come from events in the past light cone of A. Your question would then be, is this still true in GR? And the answer is yes.

 

[unparadoxical]

in SR, all causal influences on a given event A must come from events in the past light cone of A. Your question would then be, is this still true in GR?

No, that would not be my question, because it uses invalid assumptions.

GR is not covered by the simplistic visual metaphor of "light-cones", although still reduces to it in the trivial case of a spacetime with no curvature, so that it becomes materially indistinguishable from SR.

To fully appreciate my position, one must start with a careful consideration of the fact that SR should only be used in effective physical models, while GR can also be used in any physical model — up to perfectly ideal models that take into account global features of the entire spacetime. Further, the geometric aspects of SR should only be considered as superficial, while those of GR, as truly intrinsic.

Here are some consequences of the above claims. Much like the characters in animated celluloid cartoons, the matter in SR is simply "superimposed" on the background spacetime, and since it cannot be formally constructed by the theory, it has no claim to representations that are anything other than "point-like" abstractions. But in GR, while the matter is also typically represented in such a phenomenological way, it may also be the result of a formal construction, and therefore posses a definite "space-like" description.

So the simplistic "light-cone" metaphor applies only to the SR setting, where the physical elements can only, by rights, be considered as "point-like", and without external influences, can only move in straight-line geodesics. In GR, however, this metaphor may easily become useless: for example, light, as is well known, cannot escape from black holes due to the infinite curvature of the spacetime.

Succinctly stated, SR is built atop the algebraic ("point-like") paradigm of discrete events IN spacetime, while GR is built atop the topological ("space-like") paradigm of contiguous neighborhoods OF spacetime.

The "moral" of this story is just that the impulse to reduce the entirety of the cosmos to trivial "events" and to demand that every conceivable event — disregarding the hypothetical initial event — strictly requires a precedent, is to miss the entire point of the relativistic paradigm.

And the answer is yes.

The answer to the question as stated above, therefore, most emphatically cannot be "yes".

(I now have a reformulated version of the original question, but first I want assurance that this particular sidetrack has been resolved.)

 

[PeroK]

GR is not covered by the simplistic visual metaphor of "light-cones",

Nonsense.

Further, the geometric aspects of SR should only be considered as superficial, while those of GR, as truly intrinsic.

Nonsense.

Here are some consequences of the above claims. Much like the characters in animated celluloid cartoons, the matter in SR is simply "superimposed" on the background spacetime, and since it cannot be formally constructed by the theory, it has no claim to representations that are anything other than "point-like" abstractions. But in GR, while the matter is also typically represented in such a phenomenological way, it may also be the result of a formal construction, and therefore posses a definite "space-like" description.

Nonsense.

So the simplistic "light-cone" metaphor applies only to the SR setting, where the physical elements can only, by rights, be considered as "point-like", and without external influences, can only move in straight-line geodesics. In GR, however, this metaphor may easily become useless: for example, light, as is well known, cannot escape from black holes due to the infinite curvature of the spacetime.

Nonsense. The curvature at the event horizon is not infinite. In fact, it may be arbitraily small for a sufficiently massive black hole.

Ironically, it's the geometry of the light cones at the event horizon that determines the inability to escape.

Succinctly stated, SR is built atop the algebraic ("point-like") paradigm of discrete events IN spacetime, while GR is built atop the topological ("space-like") paradigm of contiguous neighborhoods OF spacetime.

Pseudo-mathematical nonsense.

The "moral" of this story is

... that you need to get a GR textbook and study it.

 

[Dale]

GR is not covered by the simplistic visual metaphor of "light-cones", although still reduces to it in the trivial case of a spacetime with no curvature, so that it becomes materially indistinguishable from SR.

Light cones are definitely part of GR. What do you mean here?

To fully appreciate my position, one must start with a careful consideration of the fact that SR should only be used in effective physical models, while GR can also be used in any physical model — up to perfectly ideal models that take into account global features of the entire spacetime. Further, the geometric aspects of SR should only be considered as superficial, while those of GR, as truly intrinsic.

I don't agree with any of this, particularly not with calling it a "fact". "Should only be used ..." is an opinion, and it seems like a personal opinion with no justification. Do you have a peer-reviewed source that states this same opinion and justifies it substantially?

So the simplistic "light-cone" metaphor applies only to the SR setting, where the physical elements can only, by rights, be considered as "point-like", and without external influences, can only move in straight-line geodesics. In GR, however, this metaphor may easily become useless: for example, light, as is well known, cannot escape from black holes due to the infinite curvature of the spacetime.

This is incorrect. The event horizon occurs in a region with arbitrarily low curvature. Light not escaping is not due to infinite curvature. Also, the light cones pointing inward at the horizon is not a useless metaphor but a useful geometrical fact that explains why nothing material can escape the horizon.

IMO, your assertions here and above are unjustified. Please post a professional scientific source that makes these claims.

Succinctly stated, SR is built atop the algebraic ("point-like") paradigm of discrete events IN spacetime, while GR is built atop the topological ("space-like") paradigm of contiguous neighborhoods OF spacetime.

The "moral" of this story is just that the impulse to reduce the entirety of the cosmos to trivial "events" and to demand that every conceivable event — disregarding the hypothetical initial event — strictly requires a precedent, is to miss the entire point of the relativistic paradigm.

Events are just as much a part of GR as they are of SR. And the spacetime of SR is every bit as much a topological space as GR's manifold. They are both pseudo-Riemannian manifolds and inherit all of the topological properties of topological manifolds, but SR is also an affine space whereas general curved manifolds are not.

 

[Orodruin]

Further, the geometric aspects of SR should only be considered as superficial, while those of GR, as truly intrinsic.

Just to pile on: This is the Lorentzian manifold equivalent of saying that geometrical aspects of Euclidean space are only superficial and only those of Riemannian manifolds are truly intrinsic.

On the contrary, Euclidean space certainly holds fundamental and intrinsic geometrical aspects.

 

[Fractal matter]

The answer to the question as stated above, therefore, most emphatically cannot be "yes".

ADM formulation is a reformulation of GR. In its decomposition of spacetime, spacelike surfaces parametrized by time are "nonlocal". They are simultaneous. If this is what you're looking for.

 

[PeterDonis]

No, that would not be my question, because it uses invalid assumptions.

GR is not covered by the simplistic visual metaphor of "light-cones"\

You are wrong. There is most certainly a valid concept of light cones in GR. See any GR textbook. For the full gory details, see Hawking & Ellis. Wald also has a more detailed discussion than most.

To fully appreciate my position, one must start with a careful consideration of the fact that SR should only be used in effective physical models, while GR can also be used in any physical model — up to perfectly ideal models that take into account global features of the entire spacetime. Further, the geometric aspects of SR should only be considered as superficial, while those of GR, as truly intrinsic.

I have no idea where you are getting any of this from. Personal theories are off limits here.

the matter in SR is simply "superimposed" on the background spacetime, and since it cannot be formally constructed by the theory, it has no claim to representations that are anything other than "point-like" abstractions.

This is wrong. The only necessary assumption for adding matter to an SR model is that the spacetime curvature produced by the matter must be negligible to a good enough approximation for the model to make accurate predictions. There are many, many situations in which this is a very, very good approximation. You can model matter with a stress-energy tensor in SR just as you do in GR.

light, as is well known, cannot escape from black holes due to the infinite curvature of the spacetime.

The curvature of spacetime at a black hole's horizon is not infinite.

Succinctly stated, SR is built atop the algebraic ("point-like") paradigm of discrete events IN spacetime, while GR is built atop the topological ("space-like") paradigm of contiguous neighborhoods OF spacetime.

I have no idea where you are getting this from either. It's wrong. Both SR and GR model spacetime as a continuum. "Events" are points in the continuum in both SR and GR, and have open neighborhoods in both SR and GR.

(I now have a reformulated version of the original question, but first I want assurance that this particular sidetrack has been resolved.)

You will get no such thing. Your understanding of relativity is so off base that I seriously doubt we will be able to have a productive discussion unless you forget everything you think you know about relativity and start from scratch learning what it actually says.

 

[Fractal matter]

A also wonder, what does locality really mean? I think this is what
unparadoxical presume: SR local, GR not.

 

[PeterDonis]

SR local, GR not.

This is not true on any definition of "locality" that I am aware of.

 

[unparadoxical]

Okay, let's just see if I can refocus this thread by formulating a non-controversial distinction between SR and GR, regarding the concept of curvature and its theoretical consequences.

SR: no curvature means no geometric models of matter. All of the matter must be introduced as "material points", which means that all of the density calculations of non-zero quantities would become infinite. So, all of the "physically meaningful" material bodies under SR would be in the form of singularities.

GR: the introduction of curvature means that the matter can attain a definite geometric description [1], which just means that it would become space-filling, and the corresponding density calculations would be finite.

Am I assuming too much here? Can we all agree on this much?

[1] My reference for an example of singularity-free matter under GR is the Einstein-Rosen paper from 1935, "The Particle Problem in the General Theory of Relativity": https://www.nevis.columbia.edu/~zajc/acad/W3072/EinsteinWormhole.pdf

 

[PeroK]

SR: no curvature means no geometric models of matter. All of the matter must be introduced as "material points", which means that all of the density calculations of non-zero quantities would become infinite. So, all of the "physically meaningful" material bodies under SR would be in the form of singularities.

There is nothing problematic about having particles in SR, per se. And certainly no more so than in Newtonian physics, which is the low-speed approximation of SR.

Moreover, the standard model of particle physics is built on SR. There is no issue that would be solved directly by considering curved spacetime.

GR: the introduction of curvature means that the matter can attain a definite geometric description [1], which just means that it would become space-filling, and the corresponding density calculations would be finite.

The Einstein paper predates Quantum Field Theory, so I'm not sure how relevant it is to modern particle physics.

 

[Orodruin]

All of the matter must be introduced as "material points", which means that all of the density calculations of non-zero quantities would become infinite.

Wrong. Nothing prohibits continuum mechanics in special relativity. Electromagnetism is also a perfectly fine example of a classical field theory with continuous energy and momentum densities.

It is not matter that has a geometry in relativity, it is the spacetime itself. The difference between GR and SR is that SR assumes that geometry to be flat whereas in GR the spacetime geometry is determined by the stress energy tensor.

Edit: As PeroK also mentioned, building understanding on papers that are almost 90 years old is not really going to be productive.

 

[Ibix]

Can we all agree on this much?

SR is just a special case of GR, the case where curvature is zero. I don't think there's any problem with having pointlike or non-pointlike objects in SR. The only restriction is that gravity must be negligible.

Pointlike objects are actually a problem in GR, because curvature grows without bound and implies all pointlike particles ought to behave like black holes (which aren't points, although that's another story). Since GR is a classical theory we simply ignore this and treat matter as a region with a smoothly varying stress-energy tensor (i.e. continuous, rather than discrete, matter). We don't expect the theory to be correct on atomic scales anyway (we can put atoms in superposed states but can't describe superposed gravitational fields, so something must be wrong somewhere). So if you're worried about the inaccuracies here you're misusing the theory anyway.

 

[PeterDonis]

SR: no curvature means no geometric models of matter.

I'm not sure what you mean by this, but:

All of the matter must be introduced as "material points"

This is wrong. As I have already said, you can model matter continuously using a stress-energy tensor in SR, in fact this is common when treating fluids or electromagnetic fields. What you cannot do is connect the matter to any spacetime curvature; as I have said, you must assume that the matter's stress-energy does not produce any non-negligible spacetime curvature, which for many applications is a very good assumption.

Am I assuming too much here? Can we all agree on this much?

Yes. No. See above.

My reference for an example of singularity-free matter under GR is the Einstein-Rosen paper from 1935, "The Particle Problem in the General Theory of Relativity"

This is a very bad choice of reference, since it is an attempt to formulate a classical unified field theory of gravitation and electromagnetism that would include a viable model of matter. It was one of many such attempts that Einstein made in the latter part of his life, all of which failed.

Our current model of matter is based on quantum field theory, which, in almost all of its applications, is formulated in flat spacetime and ignores any gravitational effects. This works just fine for most applications. In the classical limit, the quantum field theory just gives the familiar classical field theory from which a classical stress-energy tensor can be derived (for example, the classical limit of QED is just Maxwell's Equations and the classical EM stress-energy tensor derived from them).

For the rare cases where we need to include gravitational effects, since we do not have a working quantum field theory of gravity, we either use the classical limit of our quantum field theory of matter in order to derive a classical stress-energy tensor as above, but in curved spacetime instead of flat, or we do quantum field theory in curved spacetime and use the expectation value of whatever stress-energy tensor operator applies to the quantum field theory of matter we are using as the source in the classical Einstein Field Equation. (In fact, in most applications of quantum field theory in curved spacetime, we don't even do that, we take the curved spacetime as fixed and not sourced by the quantum fields we are studying, but just treat the quantum fields the same way as we do in the flat spacetime case, as having negligible gravitational effects.)

None of these approaches are anything like what Einstein was trying to do.

 

[vanhees71]

There is nothing problematic about having particles in SR, per se. And certainly no more so than in Newtonian physics, which is the low-speed approximation of SR.

I think there are big problems having (classical) point particles in SR. What works of course are non-interacting point particles. It's just described by time-like straight lines within an arbitrary inertial frame of reference, and even the massless limit is no problem for them.

The next more complicated case is the motion of a single point particle in some external field. In practice that's used for charged particles in an electromagnetic field. This works to the extent that you neglect the radiation-reaction issue, and the latter can effectively quite satisfactorially be described by the Landau-Lifshitz approximation of the Lorentz-Abraham-Dirac equation. This can also be justified as an approximation to a quantum-Langevin approach, which however leads to a non-Markovian description, which has not the deficiencies (causality, run-away solutions) as the LAD equation.

Trying to formulate a theory of closed interacting point-particle systems does not work because of the famous no-go theorem by Leutwyler et al, which shows that any consistent Hamiltonian relativistic set of equations must describe non-interacting particles.

What works much better are continuum-mechanical descriptions of various kinds, starting from relativistic perfect fluid dynamics to viscous fluid dynamics (most recently also in terms of Navier-Stokes-like first-order-in-gradients equations without the deficiencies of the naive Navier-Stokes prescriptions) or relativistic kinetic transport equations, which can be derived from many-body quantum field theory (Kadanoff-Baym equations).

Moreover, the standard model of particle physics is built on SR. There is no issue that would be solved directly by considering curved spacetime.

The Einstein paper predates Quantum Field Theory, so I'm not sure how relevant it is to modern particle physics.

 

[Dale]

I think there are big problems having (classical) point particles in SR.

There are big problems with classical point particles in Newtonian mechanics too. I think the problem is more with the idea of a classical point particle than with SR.

 

[Ibix]

There are big problems with classical point particles in Newtonian mechanics too. I think the problem is more with the idea of a classical point particle than with SR.

And adding curvature doesn't fix it.

 

[Dale]

And adding curvature doesn't fix it.

Indeed, and not having curvature doesn’t necessitate using those problematic point particles

 

[vanhees71]

There are big problems with classical point particles in Newtonian mechanics too. I think the problem is more with the idea of a classical point particle than with SR.

Hm, but in Newtonian mechanics you have a consistent theory with quite some success using the point-particle model. You can describe a lot within the model of a closed system of particles interacting via central pair potentials, most notably the entire edifice of celestial mechanics with the gravitational interaction with its $1/r$ potential.

Which specific problems within Newtonian theory are you referring too?

 

[Dale]

Which specific problems within Newtonian theory are you referring too?

You can get an infinite velocity and infinite separation in a finite time using 5 point particles acting under Newtonian gravity.

Z. Xia, “The Existence of Noncollision Singularities in Newtonian Systems,” Annals Math. 135, 411-468, 1992

 

[vanhees71]

Here's the DOI to the paper:

https://doi.org/10.2307/2946572

The first formula of this paper should have a - sign on the right-hand side, right?

It should be
$$m_i \ddot{\vec{q}_i} = -\sum_{j \neq i} m_i m_j \frac{\vec{q}_i - \vec{q}_j}{|\vec{q}_i-\vec{q}_j|^3}.$$
Gravitation is attractive!