General relativity
Dr. Gyula Bene
Department for Theoretical Physics, Loránd Eötvös University
Pázmány Péter sétány 1/A, 1117 Budapest
3. week
The laws of physics in curved spacetime
Motion of mass point in gravitational field
Principle of least action:
$$\begin{align}
\delta S=-mc \delta\;\int ds =0
\end{align}$$
Equation of motion:
$$\begin{align}
\frac{Du^i}{Ds}=0
\end{align}$$
( $u^i=\frac{dx^i}{ds}$ stands for the four velocity), i.e.
$$\begin{align}
\frac{d^2 x^i}{ds^2}+\Gamma^i_{\phantom{1}kl}\frac{d x^k}{ds}\frac{d x^l}{ds}=0
\end{align}$$
Hamilton-Jacobi equation
$$\begin{align}
p^i=mcu^i
\end{align}$$
$$\begin{align}
p_ip^i=m^2c^2
\end{align}$$
$$\begin{align}
g^{ik}\frac{\partial S}{\partial x^i}\frac{\partial S}{\partial x^k}-m^2c^2=0
\end{align}$$
Light propagation
$$\begin{align}
\frac{d k^i}{d\lambda}+\Gamma^i_{\phantom{1}kl}k^k k^l=0
\end{align}$$
Weak gravitational field
Non-relativistic Lagrangian:
$$\begin{align}
L=-mc^2+\frac{mv^2}{2}-m\varphi
\end{align}$$
$$\begin{align}
ds^2=(c^2+2\varphi)dt^2-d\bf{r}^2
\end{align}$$
$$\begin{align}
g_{00}=1+\frac{2\varphi}{c^2}
\end{align}$$
Static gravitational field, gravitational redshift
Frequency measured in proper time:
$$\begin{align}
\omega=\frac{\omega_0}{\sqrt{g_{00}}}\approx \omega_0\left(1-\frac{\varphi}{c^2}\right)
\end{align}$$
$$\begin{align}
\Delta \omega=\frac{\varphi_1-\varphi_2}{c^2}\omega\phantom{reds}
\end{align}$$
Maxwell's equations in gravitational field
Field tensor:
$$\begin{align}
F_{ik}=A_{k;i}-A_{i;k}=\frac{\partial A_k}{\partial x^i}-\frac{\partial A_i}{\partial x^k}
\end{align}$$
Four electric current density:
$$\begin{align}
j^i=\frac{\rho c}{\sqrt{g_{00}}}\frac{dx^i}{dx^0}
\end{align}$$
Maxwell's equations:
$$\begin{align}
\frac{\partial F_{ik}}{\partial x^l}+\frac{\partial F_{li}}{\partial x^k}+\frac{\partial F_{kl}}{\partial x^i}=0
\end{align}$$
$$\begin{align}
F^{ik}_{;k}=\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^k}\left(\sqrt{-g}F^{ik}\right)=-\frac{j^i}{\epsilon_0 c^2}
\end{align}$$
Motion of charged particle in electromagnetic and gravitational fields:
$$\begin{align}
m\left(\frac{d u^i}{ds}+\Gamma^i_{\phantom{1}kl}u^k u^l\right)=qF^{ik}u_k
\end{align}$$
bene@arpad.elte.hu