# What different approximations yield Gravitoelectromagnetism and Weak Field Einstein Equations?

This question is inspired by this answer, which cites Gravitoelectromagnetism (GEM) as a valid approximation to the Einstein Field Equations (EFE).

The wonted presentation of gravitational waves is either through Weak Field Einstein equations presented in, say, §8.3 of B. Schutz “A first course in General Relativity”, or through the exact wave solutions presented in, say §9.2 of B. Crowell “General Relativity” or §35.9 of Misner, Thorne and Wheeler.

In particular, the WFEE show their characteristic “quadrupolar polarization” which can be visualized as one way dilations in one transverse direction followed by one-way dilations in the orthogonal transverse direction. GEM on the other hand is wholly analogous to Maxwell’s equations, with the gravitational acceleration substituted for the $\mathbf{E}$ vector and with a $\mathbf{B}$ vector arising from propagation delays in the $\mathbf{E}$ field as the sources move.

My Questions:

1. The freespace “eigenmodes” of GEM, therefore, are circularly polarized plane waves of the gravitational $\mathbf{E}$ and $\mathbf{B}$. This does not seem to square exactly with the WFEE solution. So clearly GEM and WFEE are different approximations, probably holding in different approximation assumptions, although I can see that a spinning polarization vector could be interpreted as a time-varying eigenvector for a $2\times 2$ dilation matrix. What are the different assumptions that validate the use of the two theories, respectively?
2. The Wikipedia page on GEM tells us that GEM is written in non-inertial frames, without saying more. How does one describe these non-inertial frames? Are they, for example, stationary with respect to the centre of mass in the problem, like for Newtonian gravity? There would seem to be very few GR-co-ordinate independent ways to describe, when thinking of GEM as an approximation to the full EFE, a departure from an inertial frame. It’s not like you can say “sit on the inertial frame, then blast off North from there at some acceleration”.
3. Are there any experimental results that full GR explains that GEM as yet does not? I’m guessing that these will be in large scale movements of astronomical bodies.
4. Here I apologise for being ignorant of physics history and also because I am at the moment just trying to rehabilitate my GR after twenty years, so this may be a naïve one: if GR can in certain cases be reduced to analogues of Maxwell’s equations, what about the other way around: are there any theories that try to reverse the approximation from GR to GEM, but beginning with Maxwell’s equations instead and coming up with a GR description for EM? I know that Hermann Weyl did something like this – I never understood exactly what he was doing but is this essentially what he did?

I am currently researching this topic, through this paper and this one, so it is likely that I shall be able to answer my own questions 1. and 2. in the not too distant future. In the meanwhile, I thought it might be interesting if anyone who already knows this stuff can answer – this will help my own research, speed my own understanding and will also share around knowledge of an interesting topic.

• I'm no expert on GEM, but I'm sure the key to the answer to 1 and 2 at least has to do with the fact that the GEM field is not the full gravitational field. Rather, the GEM potential comes from the top row of the metric perturbation, something like $A_\mu = h_{\mu\nu} U^\nu$ where $U$ is the velocity of the local frame, and the "gauge transformations" are some restricted version of diffeomorphisms rather than the full group. It seems like GEM is some sort of "square root" of GR, which would explain the spin 1 vs spin 2 difference. – Michael Brown Aug 23 '13 at 8:52
• It would be really interesting to see if this has any relation to the correspondence between scattering amplitudes in gauge theories and gravity (the stuff Bern and Arkani-Hamed talk about a lot). I doubt it, but there are superficial similarities to a non-expert like myself... – Michael Brown Aug 23 '13 at 8:54
• I may understood your questions 3, 4 incorrectly, but, nevertheless, I will try to answer. Maxwell's equations can be derived by applying the Lorentz transformations to the Coulomb law. It is very boring, but it will lead you to the Lorentz force with relativistic expressions of the fields $\mathbf E , \mathbf B$. Then you can take the curl and divergence of these fields and get Maxwell equations. – user8817 Aug 29 '13 at 23:03
• Analogical action with Newton law will lead you to the gravitational analogue of the Lorentz force with $\mathbf E , \mathbf B$ fields, and to the equations which are very similar to GEM equations. But there is little problem: it is a missing factor 2 near the $\mathbf B$ field in the expression of force (in the Wikipedia's article this factor is hidden by redefinition of $\mathbf B$). – user8817 Aug 29 '13 at 23:03
• It creates some problems with observations. For example, in 1911 Einstein derived formula for the deflection of rays in a gravitational field from special relativity. The predicted value of deflection in a Sun field was two times less than it should. – user8817 Aug 29 '13 at 23:04

1. In this answer, we take the point of view that the GEM equations are not a first principle by themselves but can only be justified via an appropriate limit (to be determined) of the linearized EFE$$^1$$ in 3+1D \begin{align} \kappa T^{\mu\nu}~\stackrel{\text{EFE}}{=}~G^{\mu\nu}~&=~-\frac{1}{2}\left(\Box \bar{h}^{\mu\nu} + \eta^{\mu\nu} \partial_{\rho}\partial_{\sigma} \bar{h}^{\rho\sigma} - \partial^{\mu}\partial_{\rho} \bar{h}^{\rho\nu} - \partial^{\nu}\partial_{\rho} \bar{h}^{\rho\mu} \right) ,\cr \kappa~&\equiv~\frac{8\pi G}{c^4}, \end{align}\tag{1} where \begin{align}g_{\mu\nu}~&=~\eta_{\mu\nu}+h_{\mu\nu}, \cr \bar{h}_{\mu\nu}~:=~h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h \qquad&\Leftrightarrow\qquad h_{\mu\nu}~=~\bar{h}_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\bar{h}. \end{align} \tag{2}

2. There may be other approaches that we are unaware of, but reading Ref. 1, the pertinent GEM limit seems to be of E&M static nature, thereby seemingly excluding gravitational waves/radiation.

3. Concretely, the matter is assumed to be dust:$$^2$$ $$T^{\mu 0}~=~cj^{\mu}, \qquad j^{\mu}~=~\begin{bmatrix} c\rho \cr {\bf J} \end{bmatrix}, \qquad T^{ij}~=~{\cal O}(c^0). \tag{3}$$

4. The only way to systematically implement a dominanting temporal sector/static limit seems to be by going to the Lorenz gauge $$\partial_{\mu} \bar{h}^{\mu\nu} ~=~0. \tag{4}$$ Then the linearized EFE (1) simplifies to $$G^{\mu\nu}~=~-\frac{1}{2}\Box \bar{h}^{\mu\nu}~=~\kappa T^{\mu\nu}. \tag{5}$$

5. In our convention, the GEM ansatz reads$$^3$$ \begin{align}A^{\mu}~=~\begin{bmatrix} \phi/c \cr {\bf A} \end{bmatrix}, \qquad&\qquad\bar{h}^{ij}~=~{\cal O}(c^{-4}),\cr -\frac{1}{4}\bar{h}^{\mu\nu} ~&=~\begin{bmatrix} \phi/c^2 & {\bf A}^T /c\cr {\bf A}/c & {\cal O}(c^{-4})\end{bmatrix}_{4\times 4} \cr \qquad\Leftrightarrow\qquad -h^{\mu\nu} ~&=~\begin{bmatrix} 2\phi/c^2 & 4{\bf A}^T/c \cr 4{\bf A}/c & (2\phi/c^2){\bf 1}_{3\times 3}\end{bmatrix}_{4\times 4} \cr \qquad\Leftrightarrow\qquad g_{\mu\nu} ~&=~\begin{bmatrix} -1-2\phi/c^2 & 4{\bf A}^T/c \cr 4{\bf A}/c & (1-2\phi/c^2){\bf 1}_{3\times 3}\end{bmatrix}_{4\times 4} \end{align}. \tag{6}

6. The gravitational Lorenz gauge (4) corresponds to the Lorenz gauge condition $$c^{-2}\partial_t\phi + \nabla\cdot {\bf A}~\equiv~ \partial_{\mu}A^{\mu}~=~0 \tag{7}$$ and the "electrostatic limit"$$^3$$ $$\partial_t {\bf A}~=~{\cal O}(c^{-2}).\tag{8}$$

7. Next define the field strength $$F_{\mu\nu}~:=~\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}, \qquad -{\bf E} ~:=~ {\bf \nabla} \phi + \partial_t{\bf A}, \qquad {\bf B} ~:=~ {\bf \nabla} \times {\bf A}. \tag{9}$$ Then the tempotemporal & the spatiotemporal sectors of the linearized EFE (1) become the gravitational Maxwell equations with sources $$\partial_{\mu} F^{\mu\nu}~=~\frac{4\pi G}{c}j^{\mu}. \tag{10}$$ Note that the gravitational (electric) field $${\bf E}$$ should be inwards (outwards) for a positive mass (charge), respectively. For this reason, in this answer/Wikipedia, the GEM equations (10) and the Maxwell equations have opposite$$^4$$ signs.

8. Interestingly, a gravitational gauge transformation of the form $$\delta h_{\mu\nu}~=~\partial_{\mu}\varepsilon_{\nu}+(\mu\leftrightarrow\nu), \qquad \varepsilon_{\nu}~:=~c^{-1}\delta^0_{\nu}~\varepsilon, \tag{11}$$ leads to $$\delta h~=~-2c^{-1}\partial_0\varepsilon \tag{12}$$ and thereby to the usual gauge transformations $$\delta A_{\mu}~=~\partial_{\mu}\varepsilon.\tag{13}$$ Such gauge transformations (13) preserve the GEM eqs. (10) but violate the GEM ansatz $$\bar{h}^{ij}={\cal O}(c^{-4})$$ unless $$\partial_t\varepsilon~=~{\cal O}(c^{-2}).\tag{14}$$ In conclusion, the Lorenz gauge condition (7) is not necessary, but we seem to be stuck with the "electrostatic limit" (8).

References:

1. B. Mashhoon, Gravitoelectromagnetism: A Brief Review, arXiv:gr-qc/0311030.

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$$^1$$ In this answer we use Minkowski sign convention $$(-,+,+,+)$$ and work in the SI-system. Space-indices $$i,j,\ldots \in\{1,2,3\}$$ are Roman letters, while spacetime indices $$\mu,\nu,\ldots \in\{0,1,2,3\}$$ are Greek letters.

$$^2$$ Warning: The $$j^{\mu}$$ current (3) does not transform covariantly under Lorentz boosts. The non-inertial frames that Wikipedia mentions are presumably because the $$g_{\mu\nu}$$-metric (2) is non-Minkowskian.

$$^3$$ We unconventionally call eq. (8) the "electrostatic limit" since the term $$\partial_t{\bf A}$$ enters the definition (9) of $${\bf E}$$.

$$^4$$ Warning: In Mashhoon (Ref. 1) the GEM equations (10) and the Maxwell equations have the same sign. For comparison, in this Phys.SE answer $$\phi~=~-\phi^{\text{Mashhoon}}, \qquad {\bf E}~=~-{\bf E}^{\text{Mashhoon}},$$ $${\bf A}~=~-\frac{1}{2c}{\bf A}^{\text{Mashhoon}}, \qquad {\bf B}~=~-\frac{1}{2c}{\bf B}^{\text{Mashhoon}}.\tag{15}$$

• Great, thanks for the thorough, succinct explanation and also for the paper. Point 1 was EXACTLY the route I wanted to take; one can find many different "first principles" motivations of the GEM equations and indeed they are kind of obviously motivated if you assume you somehow approximate the source as a four current (i.e. a flux of a scalar) rather than taking account of the stress energy tensor and indeed your point number 3 is illuminating for me. I was well aware of footnote 2 and indeed I now realize this was the point that was most bothering me about the whole idea, and ..... – WetSavannaAnimal Jul 16 '18 at 10:54
• .... which the dust assumption together with the explicit assumption that one can only use frames wherein the $T^{ij}~=~{\cal O}(c^0)$ approximation holds up is what is needed to make it all work. – WetSavannaAnimal Jul 16 '18 at 10:56