How much is Special Relativity a needed foundation of General Relativity

lalbatros

If one had to built an invariant theory for gravitation, applicable in any system of coordinate, could it not be possible to create one without knowing about SR (constancy of c, EM, ...).

Could such an off-road journey teach us something, and couldn't SR pop up in some other way?

Thanks for your ideas,

Michel

robphy

Some approaches to Quantum Gravity are trying to obtain GR in an appropriate continuum limit. One question that comes up... without invoking SR, how could a Lorentzian structure arise?

SR pops up in the tangent space of an event in a GR spacetime manifold.

actionintegral

What does the word "invariant" mean if you don't know about SR?

Stingray

It is always possible to write down a theory in invariant form (though it might be very difficult to do). The most important example is of Newtonian gravity. There is a very elegant formulation of it due mainly to Cartan which is completely covariant.

It turns out that GR actually pops out of this as a very direct generalization. It would actually be unnatural to write down SR as an intermediate step. But of course, it's still in there. The principles of SR are all embedded within GR, so there's no way of completely avoiding it.

actionintegral

I don't understand the meaning of the word invariant. Invariant with respect to what?

Daverz

This might be of interest:

http://adsabs.harvard.edu/abs/1974PhRvD..10.2330W

Aether

The 'constancy of c' is not inherent within special relativity, but rather it is a convention used in its standard formulation.The 'constancy of c' is not a principle of SR per se, and can be easily avoided if desired.

Stingray

Invariant with respect to coordinate transformations. The laws of physics can be made to look the same in any coordinate system. Of course doing this requires defining what coordinate transformations mean, etc., which is usually done by giving the theory a geometric structure of some sort. The structure of Newtonian gravity turns out to be more complicated than the structure of general relativity, though it does involve one less parameter (c).

robphy

...of course, as long as it is replaced by some similar condition, e.g. finite maximum signal speed.

Stingray

I'm not sure what you're trying to say. I meant that SR is just a special case of general relativity, so everything in SR is contained in GR.

Within both special and general relativity, there is an unavoidable constant we call c. Of course it isn't necessary that that parameter has anything to do with electromagnetic phenomena, but experimentally, it does.

Aether

If you limit this statement to 'round trip average c', then it is a part of the nonconventional content of SR. However, the standard formulation of SR extends this statement to include the one-way istotropy of c and that is not a part of the nonconventional part of SR per se; e.g., that is the conventional part that can be easily avoided if desired.

lalbatros

actionintegral,

"I don't understand the meaning of the word invariant. Invariant with respect to what?"

I just mean that the laws are the same in any coordinate system.

Michel

actionintegral

The reason I said that was I think "laws" is a synonym for "speed of light".
So SR would be implied in "laws".

Thrice

Apparently you can also get GR by looking for a field that describes massless spin-2 particles.

Daverz

On a background Lorentz spacetime.

Hurkyl

There are precisely three local geometries we can have on a 4-dimensional pseudo-Riemannian manifold. (the thing we use for space-time in GR)

One is that of 4-d Euclidean space.
One is that of Minkowski space. (the thing we use for space-time in SR)
One corresponds to a signature of (2, 2). (so it's kind of like 2 spatial and 2 temporal dimensions)

pess5

Well, to ask whether we needed SR is one thing. But to ask whether we needed an invariant c is another thing altogether. Without invariant c, we are stuck with aether, absolute space, and an independent time. Basically, we would have Newton, where gravity works by magic from afar instantaneously. We'd have no concept of the mechanism which creates the mechanics. Having the correct mechanism then leads to further valid extension of the physics, and unification then becomes more probable.

Maxwell changed it all. His theory had symmetry in it, which required EM to exist at only one rate in vacu. We either ignore this or we don't. If we ignore it, we remain with Newton & Gallileo. But these things cannot be ignored, because it is not in the nature of mankind.

Einstein's Special Theory lead to a number of things, all of which gave the insight to Einstein for his geometric model of space/time & matter/energy. SR lead to Minkowki's notion of a fused spacetime continuum. This allowed Einstein to think in terms of a single spacetime fabric entity. Add the equivalecy principle, providing the insight that the continuum might be warped. Einstein's own E=mc^2 lends support to this notion since it showed that matter is just energy of another form, and the gravity field goes everywhere the mass goes. So gravity wells and rest mass must be mutually coexistent, the rest mass forming at the expense of surrounding medium uniformity. The genius of assuming the medium to support only a speed c change within itself, required gravity to setup, break down, and quake at c ... and so all the limitations of Newton's model are then surpassed as no instantaneous force from afar is required.

Hard to imagine GR in the absence of SR, personally. It's like asking whether SR would have been developed had Maxwell never developed his theory of electromagnetism. Noone would then have believed that Michelson/Morley's null result was anything but a bad test setup. Or if Maxwell could have never developed his theory had Faraday and Gauss never made their contributions first. Or Newton's mechanics in the absence of Gallileo's inertia, gravity, and kinematics.

Had Einstein not existed, we'd have been stuck with Lorentz's aether theory. As close as he was, he fell short. Einstein wasn't stuck on the aether, nor an absolute space. It is possible that Lorentz and Poincare might have eventually got it right, but it may have taken a long time. I doubt anyone would have taken on gravitation though. Einstein was gifted, had keen insight and knew it, was confident as could be, likely spent most his entire life just thinking about these things, and sacroficed his family to do what a group of geniuses were unlikely to even attempt.

That said, my guess is that although many folks would produce many models, none would likely be right had SR not been developed first. If anything close to GR had eventually arisen, I'd bet it would have taken a very very long time with dozens of gifted theoretical physicists to produce a much lesser model, if we were lucky. But then, stranger things have happened

pess