## Friday, August 19, 2011

### Does General Relativity Minus Gravity Equal Special Relativity?

An issue of rigor here. Everybody knows that general relativity provides a more accurate version of how gravity works than Newtonian gravity in a way that respects the principles of special relativity relating to the non-linear relationship between velocity, mass and distance.

Gravity in general relativity differs from Newtonian gravity in multiple ways. General relativity gives us concepts like black holes, the Big Bang, and gravitational time dilation. Newtonian gravity is not path dependent, while general relativity is path dependent. In general relativity both mass and energy give rise to and are subject to gravitational effects (allowing for phenomena like gravitational lensing), while only mass matters for Newtonian gravity. And, Newtonian gravity has precisely the same effect in all directions on the Euclidian space sphere surrounding an object, while in general relativity the direction of an accelleration due to gravity depends upon the mix of mass-energy components in the stress-energy tensor – a rotating sphere for example, has a different gravitational effect than the same sphere at rest, even if the sphere at rest has a slightly greater mass equivalent to energy attributable to the angular momentem of the rotating sphere. In a generalized form of Newtonian gravity that looked at total mass-energy rather than merely mass, those two systems would be equivalent, exerting a spherically symmetric force directed at the center of the sphere, but in general relativity the gravitational effect of the rotating sphere will have a “frame dragging” effect that gives the gravitational effect a bit of twist rather than bering directed straight towards the center of the sphere, while the sphere that is not rotating will not.

All of this is by way of prologue. The issue I'm interested in today is whether general relativity is simply special relativity plus a rather complex kind of gravitational force expressed through the medium of space-time distortion, or whether adds something beyond gravity, or distinct from gravity but not independent of it, to physics beyond special relativity.

You don't, for example, need general relativity to come up with the equation E=mc^2. You can get that in the limit of special relativity and it is critical to the formulation of important parts of quantum mechanics like Dirac's equation.

I'm intuition to think that there is more to general relativity (perhaps tucked away in usually unstated assumption) than a quirky law of gravity. But, it is tricky to define it. The equivalence principle, for example, defined to mean that gravitational mass and interial mass are identical, is surely nonsensical in the absence of gravity. But, the notion that there is no preferred reference frame and that all forces are created equal whatever their origin might have deeper meaning. It might be possible to ask what, if anything, is left of general relativity if one were to rest the gravitational force constant to zero, but I'm not entirely convinced that this procedure wouldn't throw out additional innovations of general relativity that are embedded in the field equations of general relativity by zeroing out not just terms of gravitational signficiance, at least in conventional formulations of general relativity, but it might have other terms or concepts or implications that are distinguishable from gravity, per se, if not precisely independent of gravity.

I've heard people explaining a difference between special relativity and general relativity as including the latter's “background independence” which is not present in special relativity. I suppose one way to pose that issue would be, “could you formulate physics that are consistent with special relativity, but which lacks gravity, in a background independent way that would differ in some respect other than providing an exactly equivalent but different mathematical form of the equations, from the ordinary Minkowski space of special relativity and quantum mechanics, that would make it more akin to general relativity.

Similarly, general relativity, in its usual formulation, is stated in terms of a continuous stress-energy field, rather than in terms of point masses. I imagine that one could formulate other physical theories in those terms as well. Indeed, reifying the quantum mechanics particle propogator to think of it as an actual continous smeared out location and momentum of particles in a way proportionate to its amplitudes at each point, rather than as point particles that hop from point A to point B with a certain probability, in relation to forces other than gravity, might very well be a fruitful endeavor.

Honestly, while I know enough to ask the question and even to identify some of the issues inherent in trying to formulate it that might lead to different kinds of answers, I don't know general relativity well enough to answer it.