Speed of light in relativistic physics
RAD4921 said:
It seem that if gravitational waves travel at C, that this is a hint that there is unification between gravity and electromagnetism.
Symbreak said:
There is a way we can conduct an experiement to test this hypothesis:
observe gravitational waves (GR) passing through a variable electromagnetic field (emf/v).
(the idea being, the displacement in the field caused by GR leads to the emf/v emmiting electromagnetic waves. The GR energy is transferred into light.)
... so somewhere along the line, is it possible to 'convert' GR into light?
RAD4921: it seems to me that this doesn't really concern UNIFICATION at all. Better to say that in relativistic physics, it is more or less inevitable that physical fields will be "updated" THERE and THEN to take account of changes HERE and NOW in the source of the field by signals traveling at (at most) the speed of light in vacuo. So Maxwell's theory of EM predicts "EM radiation" which travels at c, gtr (and related theories of gravitation) predicts "gravitational radiation" exists and travels at c, toy theories of scalar fields "coupled" in various ways to curvature predict "scalar radiation" which travels at the speed of c, and so on and on.
Symbreak: a "metric theory" of gravitation such as gtr more or less automatically "incorporates" other theories such as Maxwell's theory of EM or a scalar field theory, by allowing the energy of the EM field, or of the scalar field, to gravitate and thus to affect the spacetime geometry. Going the other way, it is more or less inevitable that gravitational radiation, which changes the spacetime geometry, will slightly change a distribution of charge, and thus can affect electromagnetic fields.
Similarly, a passing gravitational wave can affect the propagation of an EM wave. Indeed, without loss of generality, assuming they are not propagating parallel to each other, they can be treated as the collision of a gravitational and an EM wave, and the study of exact solutions answering to this description often have some initially surprising features! (See Griffiths, Colliding Plane Waves in General Relativty, Clarendon Press, 1991, for a very readable survey of this fascinating subject, unfortunately now 15 years out of date.)
Indeed, according to gtr, such effects can be literally "seen", in principle, in the presence of a single gravitational planar "sandwich wave". Imagine an inertial observer in a region of flat spacetime. Along comes a gravitational wave with planar wavefronts and some roughly sinusoidal oscillations as in MTW section 35.9. If the observer looks through approaching wavefronts at more distant objects (e.g. "constellations"), he sees no optical distortions--- of course not, or he would have advance warning of a signal approaching him at the speed of light! But if after the wave passes his location, he turns and looks through departing wavefronts at more distant objects, he will see optical distortions, as revealed by the nonzero expansion and shear scalars of the appropriate null geodesic congruence.
(In mathematical terms, this analysis involves an exact gravitational plane wave solution and a null geodesic congruence, "opposing" the wave vector field of the gravitational plane wave. In physical terms, this scenario involves a strong gravitational wave interacting with weak EM radiation, corresponding to the light from distant constellations or whatever. Needless to say, such a scenario, while fun and instructive to explore as an exercise in gtr, is probably rather hard to arrange in the real universe!)
So it is true that according to gtr, gravitational waves can affect EM waves and vice versa, and even that electromagnetic and gravitational radiation can be "partially interconverted". It turns out that these are very tiny effects--- but you had the right idea!
But again, this kind of mutual influence and partial interconversion is not the same thing at all as unification. In gtr, EM fields and hypothetical scalar fields are treated quite differently from the gravitational field. In a unification, these should somehow stand revealed as different aspects of the same thing, but that doesn't happen in gtr, where various sharp contrasts (in terms of physically observable behavior and mathematical treatment) between gravitational radiation and EM radiation (and between these and scalar radiation) tend to be even more striking than their similarities.
About "incorporation": if one accepts the principle that all forms of mass-energy gravitate, then it is more or less inevitable any good theory of gravitation will be rather generous about what it accepts as admissable forms of mass-energy. It can sometimes be helpful to think of relativistic gravitation theories such as gtr as analogous to thermodynamics, where we seek a theory telling us how energy transformations and transport works in very general terms, without needing to specify a theory of matter. In the same way, a gravitation theory should specify how gravitation works, without needing to specify a theory of matter, or a theory of some field which might also be contributing to the ambient mass-energy present in some location. And indeed, in gtr, to make specific predictions we might need to know something about the stress-energy tensor, but the theory itself happily accepts (almost too happily!) pretty much anything as a term contributing to the stress-energy, if you let it. This isn't really a defect, but a more or less inevitable feature of gravitation theories. My point is that any good gravitation theory will have to "incorporate" EM and other theories, even toy theories made up in school one day. Again, this is not the same thing at all as "unification".
(By the way, the very rough analogy between thermodynamics and gravitation may remind some of a rather specific analogy which turns out to be, apparently, much more than a mere analogy! C.f. "black hole thermodynamics", which has led to such intriguing concepts as optical and acoustical black holes, which may some day afford experimental verification of the existence of Hawking radiation, albeit associated with optical or acoustical holes, rather than the gravitational kind. Presumably the Nobel Prize committee will not quibble if this comes to pass, however! Certainly the optical Hawking radiation would be no less real than gravitational Hawking radiation, if either does indeed exist.)
Hurkyl: you already know about "ascent", in which one tries (roughly speaking) to obtain solutions of some kind (e.g. vacuum solutions) in four-dimensional gtr (or some comparable theory), with specified properties, by trying to extract such solutions from solutions of some kind, with specified properties, in higher dimensions (possibly in some theory other than n-D gtr). But you will probably be intrigued to learn (if you don't already know) that there is a kind of "dual" notion of"descent", in which, for example, one represents vacuum solutions having particular (rather restrictive) symmetry properties as solutions to the 3-D Einstein equations for an artificial type of scalar field. Since 3-D gtr is mathematically and physically quite different from 4-D gtr, the very possibility of making such a representation immediately implies that the originally sought symmetrical vacuum solutions can't exhibit (for example) gravitational radiation.
My point is that in addition to "unification" versus "incorporation" we find a third important theme: "representation" (parse that "re-presentation"), e.g. of 4-D special vacuum solutions as certain nonvacuum solutions in 3-D, and so on.
Chris Hillman
