General relativity is a theory of gravity. The idea is to find a set of “proper” coordinate system to describe physics on a curved space and make connection between these “proper” coordinate systems.

\[S_p \equiv -m c \int \int \mathrm d s\mathrm d\tau \sqrt{-\dot x ^\mu g_{\mu\nu} \dot x^\nu} \delta^4(x^\mu - x^\mu (s)) ,\]

in which \(x^\mu(s)\) is the trajectory of the particle. Then the energy density \(\rho\) corresponds to \(m\delta^4(x^\mu- x^\mu(s))\).

The Largrange density

\[\mathcal L = -\int\mathrm ds mc \sqrt{-\dot x^\mu g_{\mu\nu}\dot x^\nu}\delta^4(x^\mu - x^\mu(s))\]

Energy-momentum density is \(\mathcal T^{\mu\nu} = \sqrt{-g}T^{\mu\nu}\) is

\[\mathcal T^{\mu\nu} = -2 \frac{\partial \mathcal L}{\partial g_{\mu\nu}}\]

Finally,

\[\begin{split}\mathcal T^{\mu\nu} &= \int \mathrm ds \frac{mc\dot x^\mu \dot x^\nu}{\sqrt{-\dot x^\mu g_{\mu\nu} \dot x^\nu}} \delta(t-t(s))\delta^3(\vec x - \vec x(t)) \\
&= m\dot x^\mu \dot x^\nu \frac{\mathrm d s}{\mathrm d t} \delta^3(\vec x - \vec x(s(t)))\end{split}\]

In geometrical optics limit, the angular frequency \(\omega\) of a photon with a 4-vector \(K^a\), measured by a observer with a 4-velocity \(Z^a\), is \(\omega=-K_aZ^a\).

“A stationary spacetime admits a timelike Killing vector field. That a stationary spacetime is one in which you can find a family of observers who observe no changes in the gravitational field (or sources such as matter or electromagnetic fields) over time.”

When we say a field is stationary, we only mean the field is time-independent.

“A static spacetime is a stationary spacetime in which the timelike Killing vector field has vanishing vorticity, or equivalently (by the Frobenius theorem) is hypersurface orthogonal. A static spacetime is one which admits a slicing into spacelike hypersurfaces which are everywhere orthogonal to the world lines of our ‘bored observers’”

When we say a field is static, the field is both time-independent and symmetric in a time reversal process.

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