My understanding is that gravitomagnetism is essentially the same relativistic effect as magnetism. If so, why is it that I've heard so much about magnetic monopoles, but never gravitomagnetic monopoles? Aren't each just as theoretically well-motivated as the other? In a similar vein, one always hears that "there is no independent existence of an electric or magnetic field, only the electromagnetic field" because E and B fields transform into each other under Lorentz boosts. Shouldn't the same statement apply to gravity and gravitomagnetism?
Gravitational monopoles are forbidden by the positive mass theorem--- any configuration of GR has positive mass, and therefore is an ordinary "pole" in the analogy with electromagnetism. The analogy is not very good, because the energy is always positive, unlike charge.
The reason magnetic monopoles make sense in EM is because of electric-magnetic duality, a parity violating symmetry between the electric and magnetic charge in the free Maxwell equations that is obviously broken by the fact that sources don't have magnetic charge. The gravitational field has many magnetic field analogs, the field is the Levi-Civita connection, which decomposes into lots of different nonrelativistic things in different components, but none of these can appear without an ordinary gravitational field at long distances.
But there are ways of embedding electromagnetism into GR, as Kaluza Klein showed, and then there are special solutions of GR that can be interpreted as magnetic monopoles of the electromagnetic reduction. There is also a whole industry of finding self-dual solutions to GR, where the duality property can be thought of as analogous to electric magnetic duality, because it involves the epsilon tensor. Self dual gravitational fields, and the decomposition of the SO(3,1) holonomy in GR into two SU(2)'s are both extremely important and fascinating topics which contain many exact solutions, and the inspiration for loop quantum gravity.
If you would like some insight into self-dual solutions, this is a better question than asking for gravitational analogs to magnetic monopoles, because these analogs are not there.
You don't hear about gravitomagnetic monopoles because they, unlike magnetic monopoles, cannot be well defined within the context of GR. Gravitomagnetism is only a weak field approximation of GR which isn't Lorentz-invariant, so it is not to be expected that everything from electromagnetism has an analogy in GR.
In electromagnetism, there are two ways one can have magnetic monopoles:
Gravitational analogue of Dirac string can be shown to be observable in classical GR.
Second approach would require introduction of an equivalent to an additional potential in GR. Equivalent of the potential in GR is the metric, and equivalent of electromagnetic field are the Christoffel symbols. So, in order to introduce the gravitational analogue of magnetic monopoles in GR, something besides the metric should be included in modified Christoffel symbols, but that wouldn't make much sense, because both metric and Christoffel symbols are properties of the spacetime geometry itself.
In addition to the other responses it's also important to point out that the Maxwell-esque gravitomagnetic equations are not Lorentz invariant, unlike Maxwell's equations. This is because energy-density and the three-vector momentum density do not form a four-vector. Rather, they are some of the components of the stress-energy tensor. This is different from EM because charge-density and current-density do form a four-vector.
Birkhoff's theorem in general relativity states that in the case of spherical symmetry, the vacuum field equations have a solution, the Schwarzschild spacetime, which is unique up to its mass $m$. (The maximally extended version of the Schwarzschild spacetime does include a white hole as well as a black hole, but the white hole is also an object of mass $m$.) This tells us that general relativity doesn't have point-magnetic-masses in addition to point-masses.
As other answers have pointed out, the analogy with E&M doesn't quite work here, because in E&M, Lorentz invariance says that if you only knew about electricity, you'd be required to invent magnetism. GR, on the other hand, already incorporates Lorentz invariance, and the picture of gravity-plus-gravitomagnetism is actually an approximation that violates Lorentz invariance.
To be honest, we were somewhat surprised when reading your posts.
You talk of "gravitomagnetism". This is the theory of Oliver Heaviside which he published in 1893 and it is the Maxwell-analogy for gravity. However, it is known that the general relativity theory is based upon a totally different set of premises than gravitomagnetism. Also, it is known that the relativistic effects from the general relativity theory are not applicable to the Heaviside gravity fields, in which only the retardation of the fields due to the finite speed of light has to be taken into account.
The general relativity theory uses a non-linear, intrinsic variable space and time with the velocity, while gravitomagnetism uses an Euclidean space without an intrinsic space and time variation with velocity. Many other fundamental issues differ as well.
Needless to say that we are quite disappointed about the mixing up of the different theories in your posts.
If the general relativity theory is correct, it should suffice to use it to demonstrate whatever issue has to be analyzed. If the Heaviside gravity fields theory is correct, the calculus of each issue should be done by it.
But what is too often seen, like in your paper, is that some scientist use, as it suits them best, the Maxwell-analogy, the Post-Newtonian formulation, or the relativity theory.
One claims about Oliver Heaviside that he wasn't able to prove the missing perihelion advance of Mercury. The relevance is poor, because Heaviside wasn't aware of the speed of the Sun in the Milky Way, and he had no data of our galaxy. In the mean time, it has been proven that the missing perihelion advance of Mercury is due to the magnetic field of the Sun in the Milky Way, causing an anisotropic extra attraction force of Mercury, accounting exactly for the missing value.
Moreover, Anatoli Vankov proved that Einstein's deduction of the missing perihelion advance of Mercury was altered by approximations that were too important to be omitted. Due to that, the correct deduction of the equations result in a quasi zero effect instead of the required missing perihelion advance.
We are glad that you recognize the Heaviside gravity fields theory (Maxwell-Analogy) as a valid way to handle gravity issues. However, this theory is fundamentally different than the general relativity theory. So, would urge you, if you use this theory, to recognize the validity of Heaviside's theory, not only when it suits you, but entirely.
From then on, it is however not possible of accepting simultaneously the general relativity theory and you should make the reader aware of that. We remain at your disposal to provide you the references of the claims we made in this e-mail, and we hope that your insights will evolve accordingly.
protected by Qmechanic♦ Aug 12 '14 at 20:14
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