General Relativity

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.

• Geometrized Unit
  • References and Notes
• Basic Principles of General Relativity
  • Tidal Force and Equivalence Principle
  • Examples of Equivalence Principle
  • References and Notes
• Mathematics in General Relativity
  • Vectors and Tensors
  • Description of Space-time Manifold
• Curved Spacetime
  • Christoffel Symbol
  • Curvature
  • Parallel Transport
  • Geodesics
  • References and Notes
• Energy Momentum Tensor
• Gravitational Waves
  • Gauge
• General Relativity Revisited
  • Hyperthesis
  • Derivation of Field Equation
  • Vacuum Solutions
  • Perturbation Theory of General Relativity
  • What Frame Are We In
  • Experiments
  • Summary Table
  • Footnote
• Spherical Solutions to Stars
  • Static
  • Equation of Motion
  • Exterior Solutions
  • Interior Solutions
  • References and Notes
• Black Holes
  • Observations of Black Holes
  • Schwarzschild Metric
  • Kerr Metric
  • Photons Travelling on Equatorial Plane
  • Penrose Process

Fields and Particles

Energy-Momentum Tensor for Particles

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

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,

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

Theorems

Killing Vector Related

\(\xi^a\) is Killing vector field, \(T^a\) is the tangent vector of geodesic line. Then \(T^a\nabla_a(T^b\xi_b)=0\), that is \(T^b\xi_b\) is a constant on geodesics.

Specific Topics

Redshift

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 \(\omega=-K_aZ^a\).

Stationary vs Static

Stationay

“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.

Static

“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.