Sign conventions and notations
I use the following sign conventions (Misner, Thorne, and Wheeler 1973) and notations.
- partial derivative
\partial_{\mu} = \frac{\partial}{\partial x^{\mu}}
(- + + +) metric signature
For example, the Minkowski metric \eta_{\mu \nu} in the Cartesian coordinates is written as the following matrix form. [\eta_{\mu \nu}] = [\eta^{\mu \nu}] = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}
The general metric tensor is denoted by g_{\mu \nu}.
Christoffel symbol
\Gamma^{\sigma}{}_{\mu \nu} = \frac{1}{2} g^{\sigma \alpha}(\partial_{\mu} g_{\alpha \nu} + \partial_{\nu} g_{\alpha \mu} - \partial_{\alpha} g_{\mu \nu})
- Riemann curvature tensor
R^{\mu}{}_{\nu \alpha \beta} = \partial_{\alpha} \Gamma^{\mu}{}_{\nu \beta} - \partial_{\beta} \Gamma^{\mu}{}_{\nu \alpha} + \Gamma^{\mu}{}_{\sigma \alpha}\Gamma^{\sigma}{}_{\nu \beta} - \Gamma^{\mu}{}_{\sigma \beta}\Gamma^{\sigma}{}_{\nu \alpha}
- Ricci tensor
R_{\mu \nu} = R^{\alpha}{}_{\mu \alpha \nu}
- Ricci scalar
R = R^{\alpha}{}_{\alpha}
- Einstein tensor
G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu}
Four-velocity
In Minkowski space with the Cartesian coordinates, [U^{\mu}] = \begin{bmatrix} \gamma c \\ \gamma u_{x} \\ \gamma u_{y} \\ \gamma u_{z} \\ \end{bmatrix}
Here, the Lorentz factor is denoted by \displaystyle \gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} and u^2 = u_{x}^2 + u_{y}^2 + u_{z}^2.
Energy-momentum tensor
perfect fluid
T^{\mu \nu} = \left(\rho_{0} + \frac{p}{c^2}\right)U^{\mu}U^{\nu} + p g^{\mu \nu}
- Here, \rho_{0} is the rest mass denstiy and \rho = \gamma^2 \rho_{0} is the relativistic mass density.
dust (p=0)
T^{\mu \nu} = \rho_{0} U^{\mu}U^{\nu}
- In this case, T^{00} = \rho c^2.
Linearized gravity
The Einstein Field Equation (EFE) is
G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^{4}}T_{\mu \nu}
where \Lambda is the cosmological constant.
In the linearized gravity, we ignore the cosmological constant, i.e. \Lambda = 0.
Then, EFE is
G_{\mu \nu} = \frac{8 \pi G}{c^{4}}T_{\mu \nu}
Consider the small perturbation h_{\mu \nu} of the metric.
g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}, \qquad |h_{\mu \nu}| \ll 1
For convenience, we define the following two quantities.
h = \eta^{\mu \nu} h_{\mu \nu}
\bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \eta_{\mu \nu} h
Also, we use the Lorenz gauge condition \partial_{\mu} \bar{h}^{\mu}{}_{\nu} = 0.
After many calculations, we get the following linearized EFE in the Lorenz gauge.
\Box \bar{h}_{\mu \nu} = - \frac{16 \pi G}{c^4} T_{\mu \nu}
Here, \Box = \eta^{\mu \nu}\partial_{\mu}\partial_{\nu} = \partial^{\nu} \partial_{\nu} is the d’Alembertian.
This is the wave equation.
In electrodynamics, the Maxwell equation in the Lorenz gauge \partial_{\mu}A^{\mu}=0 in the Gaussian units is
\Box A^{\mu} = -\frac{4 \pi}{c} J^{\mu}
Here, A^{\mu} is the four-potential and J^{\mu} is the four-current.
As you can see, the form of the linearized EFE is very similar to that of the Maxwell equation which predicts electromagnetic waves. Therefore, we can expect the existence of gravitational waves. The theory of gravitational waves in linearized gravity is also very similar to that of electromagnetic waves. For these reasons, Landau and Lifshitz wrote the famous physics textbook “The Classical Theory of Fields” which covers both electrodynamics and relativity.