Scope of General Relativity: Assumptions & Implications

[ohwilleke]

TL;DR Summary

Does GR have content beyond gravity, special relativity and E=mc^2?

Assumptions

1. General Relativity is the modern and most complete widely accepted theory of gravitation, formulated in a background independent, geometric way.

2. General Relativity is formulated in a manner consistent with Special Relativity and I could imagine that it might be possible to derived Special Relativistic effects from it (whether or not this is possible is beyond the scope of my question).

3. General Relativity is often cited as the source of the equation E=mc² which can be generalized to:

(If these equations arguably has another source, this is beyond the scope of my question.)

Question

My question is whether there is any content or scientific law that General Relativity is the source of beyond (1), (2) and (3)?

Reason For Question

I ask this, in part, because it isn't usually called a "law of gravity".

My goal is to determine if I have some blind spot in my understanding of General Relativity and what it implies that I am not aware of, so that I know if there is something that I should learn more about to have a fuller understanding of General Relativity.

 

[PeroK]

Well, it's also the basis of cosmology - of which gravity is one aspect; universal expansion being another.

 

[Grasshopper]

$E=mc^2$ comes from special relativity and from reading some summaries of the papers from the early 1900s, it partially predates it (but if I understand it correctly, Poincaré didn’t have the right physical understanding of what the equation meant). Also the full equation you wrote. If I'm not mistaken, Minkowski gave us that (although it may have predated his work which came shortly after Einstein's special relativity papers; but I'm fairly sure that the idea of 4D spacetime came from him).

 

[PeterDonis]

1. General Relativity is the modern and most complete widely accepted theory of gravitation, formulated in a background independent, geometric way.

Ok.

2. General Relativity is formulated in a manner consistent with Special Relativity and I could imagine that it might be possible to derived Special Relativistic effects from it (whether or not this is possible is beyond the scope of my question).

Yes, it is; all you have to do is adopt flat Minkowski spacetime as your working solution of the Einstein Field Equation.

3. General Relativity is often cited as the source of the equation E=mc²

No, special relativity is often cited as the source of that equation.

 

[PeterDonis]

My question is whether there is any content or scientific law that General Relativity is the source of beyond (1), (2) and (3)?

I'm not sure what you're looking for. If you include all possible solutions of the Einstein Field Equation in "gravitation", then obviously (1) encompasses all of GR.

 

[Will Learn]

Hi Ohwilleke,

It's probably fair to say GR is a theory of gravity (as you have done). It doesn't sound too important. However, in simple models there are only four fundamental forces. So that makes gravity one-quarter of conventional Physics, if you want to look at it that way.

 

[pervect]

Wiki has a longish list of experimental tests of GR, which is probably not completely up to date (there was no mention of the detection of Gravitational waves by LIGO that I saw in the list - I could have missed it, though).

I think these may be what you may be asking about? It is not entirely clear to me what you're asking for though. I would think experimental predictions would be included in the category of "content", though.

There are a lot of relatively minor, but measurable, effects in the solar system that the wiki article discusses. These are perhaps most important for testing the fine details of GR, as opposed to what I would call "big picture" tests.

One thing that is not exactly a prediction of GR is the role of GR in our system of timekeeping. GR based corrections to atomic clocks are important to modern timekeeping standards, as the clock rates have to be adjusted for altitude to give the atomic time standard (TAI, and all the time standards that arederive from it, including universal coordinated time UTC).

Another particularly interesting contribution of GR is near the end of the wiki article:

n 1922, Alexander Friedmann found that the Einstein equations have non-stationary solutions (even in the presence of the cosmological constant). In 1927, Georges Lemaître showed that static solutions of the Einstein equations, which are possible in the presence of the cosmological constant, are unstable, and therefore the static universe envisioned by Einstein could not exist (it must either expand or contract).

Hubble's observations followed a couple of years after Lematire, according to another wiki. So GR was instrumental in predicting the expansion of the universe, one of the basic features of modern cosmology. Wiki paid rather short attention to the WMAP results on the cosmic microwave background, but those are also an important confirmation of modern cosmology.

 

[ohwilleke]

I'm not sure what you're looking for. If you include all possible solutions of the Einstein Field Equation in "gravitation", then obviously (1) encompasses all of GR.

If that's it, cool.

It isn't manifestly obvious that there is not, for example: (1) a physics principle applicable outside of gravitation too, that doesn't show up except as a footnote or definition in the Einstein Field Equation (e.g. about the nature of time generally, or about CPT conservation, for example), or (2) some other equation in addition to the Einstein Field Equation, that nobody ever talks about because it's less important or only comes up in weird circumstances.

So I figured that I'd ask, rather than merely assuming that I actually know what all of the parts of GR are.

 

[strangerep]

It isn't manifestly obvious that there is not, for example: (1) a physics principle applicable outside of gravitation too, that doesn't show up except as a footnote or definition in the Einstein Field Equation (e.g. about the nature of time generally, or about CPT conservation, for example),

Well, there's the causality puzzle about how time seems only to evolve in one direction, although the equations are invariant under time-reversal. Related to this is the matter of positivity of energy, the Raychaudhuri equations and consequent singularity theorems. And let's not forget the cosmological constant which is in the "who ordered that?" Jeopardy category.

 

[PeterDonis]

there's the causality puzzle about how time seems only to evolve in one direction, although the equations are invariant under time-reversal.

This is simply an example of the fact that equations that are invariant under time reversal don't have to have solutions that are. All that needs to be the case is that for any solution that is not invariant under time reversal, its time reverse is also a solution. This is true of GR.

Related to this is the matter of positivity of energy

What specific issue are you referring to here?

the Raychaudhuri equations and consequent singularity theorems.

These are interesting consequences of the field equation given certain assumptions. I'm not sure how they are additional physical principles, though.

And let's not forget the cosmological constant which is in the "who ordered that?" Jeopardy category.

Actually, if you take Hilbert's route of deriving the Einstein Field Equation from a Lagrangian, the cosmological constant should be there, since its term in the Lagrangian satisfies the requirements of the derivation (no derivatives of the metric higher than second).

 

[strangerep]

This is simply an example of the fact that equations that are invariant under time reversal don't have to have solutions that are. All that needs to be the case is that for any solution that is not invariant under time reversal, its time reverse is also a solution.

... in which case there must be some additional physical principle, (a kind of superselection rule) that excludes those time-reversed solutions from our physical experience.

What specific issue are you referring to here?

I should perhaps have said "lower-boundedness" of energy.

These are interesting consequences of the field equation given certain assumptions. I'm not sure how they are additional physical principles, though.

I didn't mean the Raychaudhuri equations by themselves are additional physical principles, only that the combination of properties of energy-stress-momentum, combined with these equations, etc, etc, leads to interesting singularity theorems. Mathematical singularities show that our equations and principles are missing something.

Actually, if you take Hilbert's route of deriving the Einstein Field Equation from a Lagrangian, the cosmological constant should be there, since its term in the Lagrangian satisfies the requirements of the derivation (no derivatives of the metric higher than second).

... in which case you've already added another principle.

 

[PeterDonis]

in which case there must be some additional physical principle, (a kind of superselection rule) that excludes those time-reversed solutions from our physical experience.

No, just initial conditions. Unless you are going to take the position that, absent some other physical principle, we should expect every solution of whatever theory of physics we have to actually be realized.

I should perhaps have said "lower-boundedness" of energy.

Hm, I see. I take it you are thinking of something like the requirement in quantum theory that the Hamiltonian should be bounded below? Yes, that would be an additional principle, but then the question is whether it involves "gravitation". Which I guess depends on what position one takes on whether we should expect gravity to be quantized.

Mathematical singularities show that our equations and principles are missing something.

Ah, got it. Yes, in that sense I agree, the fact that solutions that appear to be physically relevant have singularities is a strong indication that GR is incomplete.

in which case you've already added another principle.

It was already there; Hilbert's derivation was published in 1915, the same year as the field equation. Perhaps that would be one response to the OP's question.

 

[strangerep]

No, just initial conditions. Unless you are going to take the position that, absent some other physical principle, we should expect every solution of whatever theory of physics we have to actually be realized.

Well, if not, then there are implicit constraints on the solution set. But such constraints beg for deeper explanations/motivations (though perhaps we are unlikely ever to find them in our lifetimes).

Hm, I see. I take it you are thinking of something like the requirement in quantum theory that the Hamiltonian should be bounded below?

...and even in classical non-gravitational physics: the Hamiltonian is the generator time evolution, so its lower-boundedness is intimately entwined with causality.

It was already there; Hilbert's derivation was published in 1915, the same year as the field equation. Perhaps that would be one response to the OP's question.

You don't even need Hilbert's derivation. Inferring the EFE from the geodesic deviation equation carefully (and appealing to Newtonian gravitation in an appropriate limit) also yields a CC (though most textbook derivations of that kind overlook this).

[Aside: Imho, variational principles are really only useful to check that one's postulated equations are robust in the following sense. If small variations in the fields could drastically change the field equations, this would be blindingly obvious experimentally. Therefore, field equations which are insensitive to 1st-order variations are simply more likely to be physically correct. Hence, deriving them via a vanishing 1st-order extremal principle is a good idea.]

 

[PeterDonis]

Inferring the EFE from the geodesic deviation equation carefully (and appealing to Newtonian gravitation in an appropriate limit) also yields a CC (though most textbook derivations of that kind overlook this).

Do you have a reference? I'm not familiar with this argument.

 

[strangerep]

Do you have a reference? I'm not familiar with this argument.

It's from some ancient (pre-internet) lecture notes. I'll try to find them, and maybe start a new thread with the derivation.

 

[vanhees71]

... in which case there must be some additional physical principle, (a kind of superselection rule) that excludes those time-reversed solutions from our physical experience.I should perhaps have said "lower-boundedness" of energy.I didn't mean the Raychaudhuri equations by themselves are additional physical principles, only that the combination of properties of energy-stress-momentum, combined with these equations, etc, etc, leads to interesting singularity theorems. Mathematical singularities show that our equations and principles are missing something.... in which case you've already added another principle.

The question about how the "arrow of time" we observe comes about, given that the fundamental laws are time-reversal invariant (except the weak interaction, which we can however neglect for this discussion, because due to its weakness it does not explain the usual "arrow of time" we experience in everyday life) is common to all physical theories, not only GR.

The answer lies of course in statistical physics and Boltzmann's derivation of the H theorem in kinetic theory. Indeed, if you look at sufficiently simple situations like the free fall of a ball the time-reversed solution is not very unusual. Instead of the free fall of a ball initially at rest the time-reversed solution is just a ball being thrown upwards with the velocity given as the negative of the velocity of the ball in free fall at some time after the initial time.

The other example is some glass falling from you table and breaking in many pieces hitting the floor. That's a not too uncommon situation, but we've never seen some shattered pieces of glass moving up and ending as a intact glass on the table. Though the latter motion is not "forbidden" by any fundamental law and is a possible motion given the precise initial conditions of all the pieces, but that's the point: It's extremely difficult (in practice impossible) to give all the pieces these precise initial conditions. That's what explains this socalled "kinetic/thermodynamical arrow of time".

Another similar example has nothing to do with statistical physics: Suppose you have the usual textbook solution of the Hertzian oscillator in classical electrodynamics, giving electric outward dipole radiation as a simple model for an antenna. Suppose you switch the dipole oscillations on for a finite time, and you get some outgoing em. wave. Now it's easy to see that taking the state of the dipole and em. field at this finite time. Then taking the time precise reversal of this state as initial conditions for the Maxwell equations, you get the precise time-reversed time evolution, i.e., some em. wave running towards the dipole and at one point being completely absorbed (the time-reversed state of the initial condition of the standard solution). It's of course extremely unlikely to observe something like this, because again it's extremely difficult (practically impossible) to prepare the precise time-reversed initial state described above, and that's the reason why we choose the retarded propagator to solve Maxwell's equations with sources rather than the advanced one, introducing a breaking of the time-reversal invariance by the choice of a realistic initial-value problem. In this sense you do not only have a "thermodynamical arrow of time", related to statistical arguments but also a "electromagnetic arrow of time".

 

[stevendaryl]

... in which case there must be some additional physical principle, (a kind of superselection rule) that excludes those time-reversed solutions from our physical experience.

This is a little bit subtle to reason about. If absolutely EVERYTHING were time-reversed, we wouldn't even notice any difference. We would just relabel the future and the past.

So what our physical experience tells us is that all sections of the universe appear to have the same direction of the arrow of time. Since all of the observable universe presumably came from the same Big Bang, then the observation boils down to: The direction of the thermodynamic arrow of time (toward increasing entropy) is observed to be the same as the direction away from the Big Bang.

 

[stevendaryl]

Another similar example has nothing to do with statistical physics: Suppose you have the usual textbook solution of the Hertzian oscillator in classical electrodynamics, giving electric outward dipole radiation as a simple model for an antenna. Suppose you switch the dipole oscillations on for a finite time, and you get some outgoing em. wave. Now it's easy to see that taking the state of the dipole and em. field at this finite time. Then taking the time precise reversal of this state as initial conditions for the Maxwell equations, you get the precise time-reversed time evolution, i.e., some em. wave running towards the dipole and at one point being completely absorbed (the time-reversed state of the initial condition of the standard solution). It's of course extremely unlikely to observe something like this, because again it's extremely difficult (practically impossible) to prepare the precise time-reversed initial state described above, and that's the reason why we choose the retarded propagator to solve Maxwell's equations with sources rather than the advanced one, introducing a breaking of the time-reversal invariance by the choice of a realistic initial-value problem. In this sense you do not only have a "thermodynamical arrow of time", related to statistical arguments but also a "electromagnetic arrow of time".

Well, as I said in another post (in another thread, maybe?) the electrodynamic arrow of time in this example follows from the fact that we are assuming that there is no electromagnetic radiation prior to the jiggling of the dipole. If we had instead specified that there was no electromagnetic radiation AFTER the jiggling of the dipole, then we would have gotten the advanced solution.

So it's connected with the fact that we know initial conditions, not final conditions.

 

[dextercioby]

Isn't the Poincaré's recurrence theorem a severe setback for all entropy increase / arrow of time theory?

 

[atyy]

Isn't the Poincaré's recurrence theorem a severe setback for all entropy increase / arrow of time theory?

The recurrence generally takes too long to matter for current observations. Wikipedia says "likely longer than the life of the universe".
https://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Poincaré_recurrence_theorem

There are issues for speculative cyclic models of cosmology.
https://en.wikipedia.org/wiki/Cyclic_model

 

[dextercioby]

The recurrence generally takes too long to matter for current observations. Wikipedia says "likely longer than the life of the universe".
https://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Poincaré_recurrence_theorem
[...]

Yes, I knew that, it is just that people put Boltzmann's work and the whole 2nd law -> thermodynamics arrow of time on a pedestal, when it is clear that the same mechanistic foundation of statistical mechanics allows at a theoretical level that all atoms from a shattered glass would after probably 10^10^100 years turn back to form an identical glass (actually, this is oversimplified, the atoms in glass fragments would still be tied into the semi-crystalline structure and the macroscopic fragments couldn't overcome gravity to climb back on the table).

Anyway, don't mind me, as this is off-topic to GR, carry on.

 

[strangerep]

Resuming this thread... I've been pondering something @PeterDonis said in post #10 about time-reversed solutions...

[...] there's the causality puzzle about how time seems only to evolve in one direction, although the equations are invariant under time-reversal.

This is simply an example of the fact that equations that are invariant under time reversal don't have to have solutions that are. All that needs to be the case is that for any solution that is not invariant under time reversal, its time reverse is also a solution. This is true of GR.

Which particular solutions in GR did you have in mind?

 

[PeterDonis]

Which particular solutions in GR did you have in mind?

Any solution that is not time reversal invariant. For example, an expanding critical density FRW universe (with an initial singularity) is not time reversal invariant; its time reverse is a collapsing critical density FRW universe (with a final singularity).