# Gravitation vs relativity

1. Aug 29, 2015

### 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)

2. Aug 29, 2015

### Staff: Mentor

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.

3. Aug 29, 2015

### bcrowell

Staff Emeritus
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.

4. Aug 29, 2015

### atyy

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

5. Aug 30, 2015

### Staff: Mentor

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.

6. Aug 30, 2015

### Sphinx

thanks a lot for the explanation

7. Aug 30, 2015

### bcrowell

Staff Emeritus
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.

8. Aug 30, 2015

### pervect

Staff Emeritus
Lets discuss the electromagnetic case first, it's much easier. First, lets 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.

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 lets 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 :(.

Last edited: Aug 30, 2015
9. Aug 31, 2015

### atyy

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.

10. Aug 31, 2015

### Sphinx

Thanks a lot , this is very instructive

11. Aug 31, 2015

### atyy

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".

12. Aug 31, 2015

### Staff: Mentor

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".

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.

13. Aug 31, 2015

### atyy

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

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.

14. Aug 31, 2015

### 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

15. Aug 31, 2015

### Staff: Mentor

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.

16. Aug 31, 2015

### martinbn

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".

17. Aug 31, 2015

### atyy

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.

Last edited: Aug 31, 2015
18. Aug 31, 2015

### Staff: Mentor

Not just cosmological solutions; also black hole solutions.

19. Aug 31, 2015

### atyy

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/ [Broken] (section 3.3) which discusses horizon-penetrating harmonic-like coordinates.

Last edited by a moderator: May 7, 2017
20. Aug 31, 2015

### Staff: Mentor

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.)