Look up linearized Einstein field equations anywhere and the first thing you'll see will be a discussion of gravitational waves. Using the linearized EFE's is pretty handy when studying gravitational waves, but it doesn't seem like they are used anywhere else! Is this true? If not, what are the other applications?
[Physics] Applications of the Linearized Einstein Field Equations (EFE)
general-relativitygravitational-waveslinearized-theory
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I won't do it with the messy formulas, because this is not the right way to check signs.
There is a right way to check signs--- by examples. Then you know for sure what the term means. The term whose sign you don't know is the only term without a derivative. So if you make a perturbation to g which is constant in the coordinate system you are using, it will be the only term.
The quickest way to check is to start in Riemann normal coordinates in the neighborhood of a single point. This is very useful for every local calculation--- it's a freefalling frame with the best possible conventional local form for the bending of the coordinate axes. In such a frame the metric near a point gives a simple and canonical expression for the curvature,
$$ g = \delta_{\mu\nu} + {1\over 6} R_{\mu\alpha\nu\beta} x^\alpha x^\beta $$
Now you can add a constant perturbation to g, say in the diagonal component:
$$ g_{\mu\nu} = \delta_{\mu\nu} + h_{\mu\nu} - {1\over 6} R_{\mu\alpha\nu\beta} x^\alpha x^\beta$$
Were $h_\mu\nu$ is a diagonal perturbation, which can be reconverted back to Riemann normal coordinates by rescaling each x by the corresponding factor of $(1+h/2)$, and this has the effect of multiplying R by (1-h) for each x it contracts. The first order change in R and then in G is with a minus sign, as you said it should be.
But I am sure you didn't really screw up. The two obvious places where a sign difference can creep in between your calculation in your conventions and your source:
- A different convention for h: $g' = g+h$ or $g= g'+h$. Both look natural.
- a different order convention for the index order of the Riemann tensor, or its overall sign (there are good conventions for both).
I am sure that your disagreement is due to one of the two points above. What is most important for the sign issue is having enough quick checks (like Riemann normal coordinates, an explicit sphere metric, AdS metric, 2d metrics, etc) and only then will you stop getting confused over signs.
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 $$ \begin{align} \kappa T^{\mu\nu}~\stackrel{\text{EFE}}{=}~& G^{\mu\nu}\cr ~=~&-\frac{1}{2}\left(\Box \bar{h}^{\mu\nu} + \eta^{\mu\nu} \partial_{\rho}\partial_{\sigma} \bar{h}^{\rho\sigma} - \partial^{\mu}\partial_{\rho} \bar{h}^{\rho\nu} - \partial^{\nu}\partial_{\rho} \bar{h}^{\rho\mu} \right) ,\cr \kappa~\equiv~&\frac{8\pi G}{c^4}, \end{align}\tag{1}$$ where $$ \begin{align}g_{\mu\nu}~&=~\eta_{\mu\nu}+h_{\mu\nu}, \cr \bar{h}_{\mu\nu}~:=~h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h \qquad&\Leftrightarrow\qquad h_{\mu\nu}~=~\bar{h}_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\bar{h}. \end{align} \tag{2}$$
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.
Concretely, the matter is assumed to be dust:$^2$ $$ T^{\mu 0}~=~cj^{\mu}, \qquad j^{\mu}~=~\begin{bmatrix} c\rho \cr {\bf J} \end{bmatrix}, \qquad T^{ij}~=~{\cal O}(c^0). \tag{3}$$
The only way to systematically implement a dominanting temporal sector/static limit seems to be by going to the Lorenz gauge$^3$ $$\partial_{\mu} \bar{h}^{\mu\nu} ~=~0. \tag{4}$$ Then the linearized EFE (1) simplifies to $$ G^{\mu\nu}~=~-\frac{1}{2}\Box \bar{h}^{\mu\nu}~=~\kappa T^{\mu\nu}. \tag{5}$$
In our convention, the GEM ansatz reads$^5$ $$\begin{align} A^{\mu}~=~&\begin{bmatrix} \phi/c \cr {\bf A} \end{bmatrix}, \qquad\bar{h}^{ij}~=~{\cal O}(c^{-4}),\cr -\frac{1}{4}\bar{h}^{\mu\nu} ~=~&\begin{bmatrix} \phi/c^2 & {\bf A}^T /c\cr {\bf A}/c & {\cal O}(c^{-4})\end{bmatrix}_{4\times 4}\cr ~\Updownarrow~& \cr -h^{\mu\nu} ~=~&\begin{bmatrix} 2\phi/c^2 & 4{\bf A}^T/c \cr 4{\bf A}/c & (2\phi/c^2){\bf 1}_{3\times 3}\end{bmatrix}_{4\times 4} \cr ~\Updownarrow~& \cr g_{\mu\nu} ~=~&\begin{bmatrix} -1-2\phi/c^2 & 4{\bf A}^T/c \cr 4{\bf A}/c & (1-2\phi/c^2){\bf 1}_{3\times 3}\end{bmatrix}_{4\times 4}. \end{align}\tag{6}$$
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"$^4$ $$ \partial_t {\bf A}~=~{\cal O}(c^{-2}).\tag{8}$$
Next define the field strength $$\begin{align} F_{\mu\nu}~:=~&\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}, \cr -{\bf E}~:=~&{\bf \nabla} \phi+\partial_t{\bf A}, \cr {\bf B}~:=~&{\bf \nabla}\times {\bf A}.\end{align} \tag{9} $$ 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$^5$ signs.
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 $$\delta A_{\mu}~=~\partial_{\mu}\varepsilon.\tag{13}$$ 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:
- 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.
$^3$ The Lorenz gauge (4) is the linearized de Donder/harmonic gauge $$ \partial_{\mu}(\sqrt{|g|} g^{\mu\nu})~=~0.\tag{15}$$
$^4$ We unconventionally call eq. (8) the "electrostatic limit" since the term $\partial_t{\bf A}$ enters the definition (9) of ${\bf E}$.
$^5$ Warning: In Mashhoon (Ref. 1) the GEM equations (10) and the Maxwell equations have the same sign. For comparison, in this Phys.SE answer $$\phi~=~-\phi^{\text{Mashhoon}}, \qquad {\bf E}~=~-{\bf E}^{\text{Mashhoon}}, $$ $${\bf A}~=~-\frac{1}{2c}{\bf A}^{\text{Mashhoon}}, \qquad {\bf B}~=~-\frac{1}{2c}{\bf B}^{\text{Mashhoon}}.\tag{16}$$
Best Answer
Calculating the relativistic precession of Mercury, for one. This post-diction was one of the key things that helped with the rapid acceptance of general relativity.
Modeling GPS, and calculating the orbits of LAGEOS and Gravity Probe B, for another. A full-blown general relativistic formulation works quite nicely on (and is absolutely essential for) black holes and neutron stars precisely because gravity about those extremely massive objects is simple. Earth's gravity field isn't so nice and simple. It's rather lumpy compared to a neutron star. One of the more recent models of the Earth's gravity field, Earth Gravity Model 2008 (EGM2008), is a 2159x2159 spherical harmonics model. How are you going to handle that with general relativity? The answer is to linearize the field equations.
Modeling the behavior of the solar system, for yet another. All three of the leading models of planetary ephemerides use a first order post-Newtonian approximation of gravity. (But apparently they're starting to wonder if they need to step beyond that. To second order.)
One last use: "weigh" the Earth. See my answer to the question "How is the mass of the Earth determined?" at the earth science stackexchange sister site.