From: Simon Clark Subject: Re: Unification of Electromagnetism and Gravity Date: Thu, 8 Nov 2001 03:02:16 GMT Newsgroups: sci.physics.research Summary: To what extent have Electromagnetism and Gravity been unified into one theory? Grace Shellac wrote: > I've been under the impression that gravity and electromagnetism > actually hasn't been unified. I asked Jerrold Marsden down at CalTech > if this was true. > > He responded: "...there is a well defined set of equations called the > Einstein-Maxwell equations that already unifies E and M with > gravity. They are discussed in standard books on general relativity." This is true, but... > "Thus, I would say that most researchers believe that these theories > are already unified. ...this is -very- misleading. This is not what physicists mean when talking about unifying theory X with theory Y. Take the vacuum (say) Maxwell equations for the field-strength 2-form F in differential form notation: dF = 0 (usually considered an identity since F=dA and d^2=0) d*F = 0 (= j gives the Maxwell equations with a source.) which are already valid in curved spacetime (if * is then taken to be the Hodge map associated with the non-flat metric tensor). Now write the Einstein equation as: Ein = T[F] where Ein is the Einstein tensor and T[F] is the energy-momentum tensor of the EM field associated with F. (Look up the exact expression for T in a GR book.) These are the Einstein-Maxwell equations you mentioned above. But I don't think many physicists would describe this as a unification. (Have we unified EM with electric charge? Newtonian gravity with mass? etc...) Generally a field equation takes the form: [Differential operator] Field = Source (or D F = S for short) where Source is independent of Field. Now if we have two (say) sets of field equations: D_1 F_1 = S_1 D_2 F_2 = S_2 where S_1 might depend on F_2 and S_2 might depend on F_1, an (at least partial) unification of the two theories would require that we write: D_u F_u = S_u where F_u is constructed in some way from F_1 and F_2 and S_u is independent of F_u. A good example of this would be unifying electricity and magnetism into electro-magnetism. We combine the electric field E and magnetic field B into the field-strength F. It -is- possible to arrange this for GR and EM (google for Kaluza-Klein) though the is much doubt as to the usefulness of doing this. I'll leave this for someone else to discuss. -- Simon Clark ============================================================================== From: Chris Hillman Subject: Re: Unification of Electromagnetism and Gravity Date: 7 Nov 2001 19:21:44 GMT Newsgroups: sci.physics.research On Tue, 6 Nov 2001, Grace Shellac wrote: > I've been under the impression that gravity and electromagnetism > actually hasn't been unified. Right; they haven't. > I asked Jerrold Marsden down at CalTech if this was true. > > He responded: "...there is a well defined set of equations called the > Einstein-Maxwell equations that already unifies E and M with > gravity. They are discussed in standard books on general relativity." > > "Thus, I would say that most researchers believe that these theories > are already unified. The big question is how to unify quantum theory > with gravity." Well, Marsden is a mathematician, not a physicist ;-/ Or in other words, I fear that Homer hath nodded. > So, what is the deal here. Am I misinformed? No, you are right. > Or is Prof. Marsden mistaken? His statement was misleading. > Doing a google.com search didn't seem to help as there > doesn't seem to be a consensus that gravity and electromagnetism are or > have been satisfactorily unified. I see web pages by people like > Sweetster and others but no definitive comment that indicates that > anyone actually thinks gravity and electromagnetism have been unified. Since Sweetser sometimes posts here, you may get somewhat diverging answers. Let me try to sketch an answer which I believe fairly represents the -current mainstream consensus-. Marsden was referring to the notion of an "electrovacuum solution to the EFE". This is indeed treated in all the textbooks. An "electrovacuum solution" is simultaneously 1. a sourcefree solution to the curved spacetime Maxwell equations, 2. a solution to the EFE, in which T^(ab) = G^(ab)/8/Pi arises as the stress-momentum-energy tensor of the EM field in (1). These are also called "Einstein-Maxwell solutions" (some authors include in that phrase things like a charged fluid; this introduces various complications but doesn't affect the gist of what I am about to say). The best way to explain the concept is by briefly discussing a few simple but nontrivial examples. The best known example of an electrovacuum is the "Reissner-Nordstrom electrovacuum" (1918). This can be defined by giving the static polar spherical coordinate chart ds^2 = -(1-2m/r+q^2/r^2) dt^2 + dr^2/(1-2m/r+q^2/r^2) + r^2 (du^2 + sin(u)^2 dv^2), -infty < t < infty, 0 < r < infty, 0 < u < Pi, -Pi < v < Pi where q is a nonzero constant and m is a positive constant, m > |q|, together with EM vector potential A = -qr/(q^2-2mr+r^2) d/dt It is convenient to work with the following ONB (orthonormal basis) of vectors for the chart above: e_1 = 1/sqrt(1-2m/r+q^2/r^2) d/dt e_2 = sqrt(1-2m/r+q^2/r^2) d/dr e_3 = 1/r d/du e_4 = 1/r/sin(u) d/dv Then, the vector potential A yields a pure electric field (vanishing magnetic field) E = q/r^2 e_2 You can check that this solves the source-free curved spacetime Maxwell equations (that is, with four-current J^a vanishing identically) and gives rise to the stress-momentum-energy tensor (wrt the above ONB!) q^2 [ 1 0 0 0 ] T^(ab) = -------- [ 0 -1 0 0 ] 8 Pi r^4 [ 0 0 1 0 ] [ 0 0 0 1 ] Now, if you compute the Einstein tensor wrt the above ONB, you'll find that Einstein's equation G^(ab) = 8 Pi T^(ab) is -also- satisfied. (In doing these computations, it is best to work as far as possible with exterior forms. See for example Misner, Thorne, and Wheeler, Gravitation, Freeman, 1973, for examples of how to compute the curvature tensors using differential forms, and how to compute the EM field on a curved spacetime, given the vector potential. "MTW" is no doubt one of the standard textbooks which Marsden had in mind.) Thus, we have here an "exact Einstein-Maxwell solution" or "electrovacuum". More specifically, this is a "nonnull electrovacuum" because the EM field tensor F_(ab) has the principle Lorentz invariants F_(ab) F^(ab) = -2 q^2/r^4 F_(ab) (*F)^(ab) = 0 do not -both- vanish. (The fact that the second invariant vanishes shows that while the "generic observer"-- for example, an observer in a stable circular orbit around the charged static spherically symmetric massive object modeled by the RN solution-- will measure nonzero electric and magnetic fields, some observers--such as the static observers defined by e_1 above, or radially infalling observers-- will measure only an electric field.) Another example of an electrovacuum is a "uniform circularly polarized EM wave". This can be defined by the "NIL chart" (referring the "Thurson NIL geometry, aka the Riemannian geometry of the three-dimensional Lie group of three by three real upper triangular matrices with ones on the diagonal) ds^2 = [-dt^2/2 + dt dx + z dt dy]/m + dy^2/4/m^2 + dz^2, -infty < t,x,y,z < infty where m > 0 is a constant, together with the EM vector potential A = m d/dy Again, it is convenient to work with an ONB of vectors: e_1 = sqrt(2m) d/dt e_2 = sqrt(2m) [ d/dt + d/dx ] e_3 = 2 m [ z d/dx + d/dy ] e_4 = d/dz (It isn't supposed to obvious that this ONB does indeed yield the above metric tensor; in effect I found the ONB first, using the "Farnsworth-Kerr Ansatz", found the coordinate chart from that, and then figured out what the solution represents physically. The same day I found a whole bunch of nonsense "solutions", since the Farnsworth-Kerr trick is a rule of thumb, not a sure-fire method, this was to be expected.) Then, the EM field turns out to be E = sqrt(m/2) e_3 B = -sqrt(m/2) e_4 and this solves the source-free Maxwell equations on the above spacetime. Furthermore, the EM stress-momentum-energy tensor (wrt the above ONB!) is m [ 1 -1 0 0 ] T^(ab) = ---- [ -1 1 0 0 ] 8 Pi [ 0 0 0 0 ] [ 0 0 0 0 ] and if you compute the Einstein tensor, you'll find that once again G^(ab) = 8 Pi T^(ab) so we have an exact electrovacuum. This one is a "null electrovacuum" because the principle Lorentz invariants of the EM field tensor F^(ab) both vanish. As you know, this characterizes "radiative solutions" in Maxwell's theory. You might well wonder why I said this is a "uniform -circularly- polarized EM wave" rather than a "uniform -linearly- polarized EM wave". The reason is rather subtle: while the ONB I gave above has the property that the timelike vector field e_1 defines a timelike geodesic congruence (i.e. the world lines of a family of -inertial- observers), the spatial vectors e_3, e_4 turn out to be -rotating- with a constant angular frequency around the spatial vector e_2 wrt a -gryostabilized- ONB. When we switch to the gyrostabilized ONB, we find that now the E and B fields are -rotating- with a constant angular frequency, so we have a circularly polarized EM wave. Coming back to the question of why "the Einstein-Maxwell theory" is -not- a "unification" of the EM and gravitational interactions: as you know, neither Maxwell's theory of EM nor Einstein's theory of gravitation are quantum theories. Therefore, they are not regarded as theories of fundamental interactions. However, we know that Maxwell's theory of EM is the "effective field theory" or "classical limit" of a renormalizable quantum field theory, QED. The latter -is- regarded as a fundamental theory. As you may know, after much work in the last part of the previous century, it turned out that the QFT which would be the "most straightforward" (not really straightforward at all, as it turned out!) "gravitational analogue" of QED is -nonrenormalizable- and therefore unworkable. This doesn't mean that there is no viable quantum theory of gravity, just that whatever such a theory might look like, it must be quite different from the QFTs physicists grew familiar with in the last century, in particular from the three QFTs for the EM, weak, and strong interactions. As you no doubt know, in the last century, these three QFTs were incorporated into a single QFT usually called the "Standard Model", which predicts that at very high energies, the three interactions become indistinguishable, but at the kind of energies we can readily observe, they appear to be different interactions with -very- different properties! Another way of understanding why the Standard Model can be said to "unify" the EM, weak, strong interactions is by observing that its gauge group or fundamental symmetry group includes as subgroups the symmetry groups of the three specialized QFTs, e.g. the abelian circle group U(1) for QED (the other two are "nonabelian gauge theories"). Furthermore, the way in which the symmetry of the big gauge group has been "broken" by a kind of "phase transformation" which occurs at "low energies" corresponds exactly to the way in which the three subgroups fit into the big gauge group. It is universally believed that at still higher energies (near the "Planck energy", if not before), the gravitational interaction must somehow treated in a "quantum" manner. It is also possible that at these energies (or maybe at still higher energies), the gravitational interaction may become unified with the other three. Superstring theory attempts to provide a -unified- quantum theory of all four fundamental interactions; that is, it aims to become a TOE. Its most popular competitor, loop quantum gravity, "merely" attempts to provide a quantum theory of gravity. In both cases, it is widely believed that gtr must be the effective field theory of the gravitational interaction at low energies, but it is possible that some more complicated classical relativistic field theory will turn out to be the effective field theory for one or both of these competitors. If this is the case, this more complicated theory must be indistinguishable from gtr at all scales/energies thus far investigated by experimenters or astronomers. Going back to the issue of what it means to say that the EFE can be solved -simultaneously- with other classical field equations, I think it is helpful to think of gtr as being analogous to the most general of all theories of classical physics, namely classical thermodynamics. Recall that the latter is a very general theory about (roughly) "dissipation" and "transportation" of "energy". It applies to any reasonable "theory of matter". In particular, if you regard ordinary matter as being made up of atoms, you can try to show how thermodynamic properties emerge from the atomic theory as statistical phenomena--- that's the basic idea "statistical mechanics", a theory which was initiated by Maxwell and Kelvin but which is still very much under development. In much the same way, you can "feed" any reasonable effective field theory to the RHS of the EFE (Einstein Field Equation), by adding suitable terms to the stress-momentum-energy tensor T_(ab). For example: some quantum fields can be modeled in the classical limit as a "massless scalar field", which is the simplest kind of effective field theory. Thus, you can search for something which is simultaneously: 1. a scalar field which satisfies the curved spacetime wave equation, 2. a solution to the EFE in which T^(ab) arises as the stress-momentum-energy tensor of the massless scalar field in (1). In particular, you can search for a static spherically symmetric solution of this type, which is analogous to the Reissner-Nordstrom electrovacuum. The resulting solution is called the "Janis-Newman-Winacour" solution. Again, the existence of such solutions doesn't mean that QFTs which can be modeled in the classical limit as a massless scalar field have been -unified- with gtr. It just reflects that fact that gtr is a very general theory of gravitation which is automatically compatible with any reasonable classical relativistic effective field theory. (By the way, it may sound like finding such simultaneous solutions is hard--- in fact, it is really quite easy to find interesting solutions by an elementary "Ansatz method" in which we stipulate a "geometric Ansatz" wherein we write down an ONB in terms of two undetermined functions of one or two coordinates, and then systematically use the field equations to determine first one, then the other function. When you actually carry out this process, it often does seem as though "miracles" keep occuring--- but there is, of course, an underlying explanation: symmetry!) A third example: one of the most popular types of T^(ab) to place on the RHS of the EFE is the stress-momentum-energy tensor of a "perfect fluid". In this case, wrt an appropriate ONB, T^(ab) has the simple form T^(ab) = diag(rho,p,p,p) where rho is the mass-energy density and p is the pressure. In particular, the well known FRW cosmological models are perfect fluid solutions (isotropic pressure; no viscosity or heat flux) to the EFE. Or we can contemplate more complicated things like charged fluids (in which T^(ab) would have two terms, one from the fluid and one from the EM fields). And so forth. The analogy between gtr and thermodynamics is more than just an analogy: as you probably know, Hawking, Press, and Bardeen proved around 1972 that when two black holes merge, the area of the event horizon new hole is no smaller than the sum of the areas of the event horizon of the original pair. In fact, they proved three laws of "black hole mechanics" which were clearly formally analogous to the three laws of thermodynamics. This immediately led Bekenstein to suggest that analogy was not a mere formality and that the area of the event horizon of a black hole should be identified (up to some scale factor) with its physical entropy. Since this would require a black hole to have a "uniform temperature", and since classically, this doesn't make any sense, this proposal was generally dismissed as being based upon a seductive but faulty analogy. Then, in 1974, Hawking showed that if you take quantum effects into account ("semiclassical approximation" for QFTs on a curved spacetime), black holes emit black-body radiation, with a temperature corresponding to the "surface gravity" (this doesn't mean what Newtonian intuition might suggest; the important point is that it is well-defined and [subject to some generous conditions] -uniform- over the event horizon) of the hole. This confirmed that Bekenstein was right: the area of a black hole -can- be identified with its physical entropy. Much more recently, superstring theory and loop quantum gravity have each yielded computations showing that according to these theories, black holes do indeed have the entropy which is predicted by the semiclassical approximation. In both cases, this is roughly analogous from passing from classical thermodynamics to statistical mechanics, since both computations compute the number of (two different notions of) "microscopic configurations" for the hole which are consistent with "the macroscopic data". More precisely, superstring theory gets exactly the right entropy, but only for near-extremal holes! LQG, on the other hand, gets the entropy right for -any- hole, but only up to a undetermined scale factor! So at present, both theories fall a bit short, but in slightly different ways. Right now, they are pretty much racing neck and neck toward the goal of a viable quantum theory of gravity, although as I said, superstring theory has a more ambitious ultimate goal. Meanwhile, Visser and others have turned this around by suggesting that none of this implies that thermodynamics and -gtr- are fundamentally related; rather, it may be that essentially any reasonable gravitation theory (including the zillions of relativistic classical theories which are self-consistent but happen not to agree with all observations/experiments) should obey a generalized thermodynamics including black holes with entropy and all that. IOW, thermodynamics may be more fundamental than either quantum theory alone or gravitation alone. Indeed, they have proposed that Hawking radiation should have physical analogues in other types of "quantum systems". At least two research groups are very actively working toward creating a quantum system (using a "Bose-Einstein condensate") which has an optical (photon) or acoustical (phonon) analogue of an event horizon, in hope of measuring the expected blackbody radiation in their lab. If they succeed, this would surely lead to a Nobel Prize for Hawking and for successful experimenters. (Ted Jacobson has also made a very interesting proposal regarding the relationship between generalized thermodynamics and the EFE, but since he often reads this group, I'll let him describe his suggestion.) As always, I would welcome correction from the experts here if I have gotten any of the above wrong. Quite a few regulars here know MUCH more about black hole thermodynamics, the semiclassical approximation, QFTs, and the Standard Model, than I do--- so I am sure someone will catch any errors I may have commited. Indeed, the only reason I tried to talk here about things I don't know much about is that there is no better way to learn than to try to explain the stuff you don't know much about, and to hope that an expert uncovers a misconception in what you said! So keep your eyes peeled for followups :-/ Chris Hillman Home page: http://www.math.washington.edu/~hillman/personal.html ============================================================================== From: Chris Hillman Subject: Re: Unification of Electromagnetism and Gravity Date: 8 Nov 2001 07:59:45 GMT Newsgroups: sci.physics.research On 7 Nov 2001, I wrote: > The best known example of an electrovacuum is the "Reissner-Nordstrom > electrovacuum" (1918). This can be defined by giving the static polar > spherical coordinate chart > > ds^2 = -(1-2m/r+q^2/r^2) dt^2 + dr^2/(1-2m/r+q^2/r^2) > > + r^2 (du^2 + sin(u)^2 dv^2), > > -infty < t < infty, 0 < r < infty, 0 < u < Pi, -Pi < v < Pi ^^^^^^^^^^^^^ m + sqrt(m^2-q^2) < r < infty Sorry for the goof. Chris Hillman Home page: http://www.math.washington.edu/~hillman/personal.html