What different approximations yield Gravitoelectromagnetism and Weak Field Einstein Equations?

Question

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.

Answer 1

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

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:$^2$ $$\begin{align} T^{0\mu}~\equiv~&T^{\mu 0}~=~cj^{\mu}, \cr j^{\mu}~=~&\begin{bmatrix} c\rho \cr {\bf J} \end{bmatrix}, \cr T^{ij}~=~&{\cal O}(c^0), \end{align}\tag{3}$$ and we assume for simplicity that the cosmological constant $\Lambda=0$ is zero$^3$.

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

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

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

7. Next define the field strength $$\begin{align} \bar{F}_{\mu\nu}~:=~&\partial_{\mu} \bar{A}_{\nu} -\partial_{\nu} \bar{A}_{\mu}, \cr -\bar{\bf E}~:=~&{\bf \nabla} \bar{\phi} +\partial_t\bar{\bf A}, \cr \bar{B}_{ij}~:=~&\bar{F}_{ij}.\end{align} \tag{9} $$ Then the tempotemporal & the spatiotemporal sectors of the linearized EFE (1) become the gravitational Maxwell equations with sources $$ \partial_{\mu} \bar{F}^{\mu\nu} ~\stackrel{(7)}{=}~\Box\bar{A}^{\nu} ~\stackrel{(5)+(6)}{=}~\frac{c^2\kappa}{2}j^{\nu} ~\equiv~\frac{4\pi G}{c^2}j^{\nu} . \tag{10} $$ 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$^5$ signs.

8. Interestingly, a gravitational gauge transformation of the form $$\begin{align}\delta h_{\mu\nu}~=~&\partial_{\mu}\varepsilon_{\nu}+(\mu\leftrightarrow\nu), \cr \varepsilon_{\nu}~:=~&c^{-1}\delta^0_{\nu}~\varepsilon, \end{align}\tag{11} $$ leads to $$\delta h~=~-2c^{-1}\partial_0\varepsilon \tag{12}$$ and thereby to the usual gauge transformations $$\delta \bar{A}_{\mu}~=~\partial_{\mu}\varepsilon.\tag{13}$$ 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 $(-,+,\ldots,+)$ and work in the SI-system. Space-indices $i,j,\ldots \in\{1,2,\ldots, d\!-\!1\}$ are Roman letters, while spacetime indices $\mu,\nu,\ldots \in\{0,1,2,\ldots, d\!-\!1\}$ 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.

$^3$ To include a small cosmological constant $\Lambda$, let us introduce the notation that $\bar{\bar{h}}_{\mu\nu}$ solves the linearized EFE with $\Lambda=0$ in the Lorenz gauge $$\quad\partial_{\mu}\bar{\bar{h}}^{\mu\nu}~=~0.\tag{15}$$ The solution for the linearized EFE (1) with a small cosmological constant $\Lambda$ then takes the following form.

• Lorentz-invariant ansatz: $$\begin{align} \bar{h}_{\mu\nu}~=~&\bar{\bar{h}}_{\mu\nu} +\frac{\Lambda x^2}{2(d-1)}\eta_{\mu\nu} \cr ~\Updownarrow~& \cr h_{\mu\nu}~=~&\bar{\bar{h}}_{\mu\nu}-\frac{1}{d-2}\eta_{\mu\nu}\bar{\bar{h}} -\frac{\Lambda x^2}{(d-1)(d-2)}\eta_{\mu\nu} .\end{align}\tag{16}$$

• Stationary spherically symmetric ansatz: $$\begin{align} \bar{h}_{\mu\nu}~=~& \bar{\bar{h}}_{\mu\nu}\cr &+ \frac{\Lambda r^2}{2(d-2)} \begin{bmatrix} -\frac{d-3}{d-1} & {\bf 0}^T \cr {\bf 0} & {\bf 1}_{(d-1)\times (d-1)} \end{bmatrix}_{d\times d} \cr ~\Updownarrow~& \cr h_{\mu\nu}~=~&\bar{\bar{h}}_{\mu\nu}-\frac{1}{d-2}\eta_{\mu\nu}\bar{\bar{h}}\cr &+ \frac{\Lambda r^2}{(d-1)(d-2)} \begin{bmatrix} 2 & {\bf 0}^T \cr {\bf 0} & -{\bf 1}_{(d-1)\times (d-1)} \end{bmatrix}_{d\times d} .\end{align}\tag{17}$$

$^4$ The Lorenz gauge (4) is the linearized de Donder/harmonic gauge $$ \partial_{\mu}(\sqrt{|g|} g^{\mu\nu})~=~0.\tag{18}$$

$^5$ We unconventionally call eq. (8) the "electrostatic limit" since the term $\partial_t{\bf A}$ enters the definition (9) of ${\bf E}$.

$^6$ Warning: In Mashhoon (Ref. 1) the GEM equations (10) and the Maxwell equations have the same sign. For comparison, in this Phys.SE answer with $d=4$ $$\bar{\phi}~=~-\phi^{\text{Mashhoon}}, \qquad \bar{\bf E}~=~-{\bf E}^{\text{Mashhoon}}, $$ $$\bar{\bf A}~=~-\frac{1}{2c}{\bf A}^{\text{Mashhoon}}, \qquad \bar{\bf B}~=~-\frac{1}{2c}{\bf B}^{\text{Mashhoon}}.\tag{19}$$