prelude
The hard part is the equation governing the response of spacetime curvature to the presence of matter and energy. We will eventually find what we want in the form of Einstein's equation \[ \begin{equation} R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=8 \pi G T_{\mu \nu} \end{equation} \]
explanation
| Symbol | Mathematical definition | Key physical role |
|---|---|---|
| \(g_{\mu\nu}\) | Metric tensor (rank 2, symmetric), used to measure spacetime intervals \(ds^{2}=g_{\mu\nu}\,dx^{\mu}dx^{\nu}\). | Encodes the gravitational “potential”. Once you know \(g_{\mu\nu}\) you know what free‑fall looks like. |
| \(\Gamma^{\rho}{}_{\mu\nu}\) | Levi‑Civita connection (Christoffel symbols): \(\Gamma^{\rho}{}_{\mu\nu}=\tfrac12 g^{\rho\sigma}\bigl(\partial_\mu g_{\nu\sigma}+\partial_\nu g_{\mu\sigma}-\partial_\sigma g_{\mu\nu}\bigr)\). | Tells you how to take covariant derivatives and therefore how vectors are “dragged” from point to point. |
| \(R^{\rho}{}_{\sigma\mu\nu}\) | Riemann curvature tensor: uses one antisymmetric pair of derivatives of \(\Gamma\). | Measures the failure of vectors to return unchanged after parallel transport around an infinitesimal loop. |
| \(R_{\mu\nu}=R^{\rho}{}_{\mu\rho\nu}\) | Ricci tensor (rank 2) obtained by contracting Riemann. | Summarises how geodesic families converge or diverge. |
| \(R=g^{\mu\nu}R_{\mu\nu}\) | Scalar curvature. | A single scalar telling whether the volume of a small geodesic ball differs from its Euclidean value. |
| \(G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R g_{\mu\nu}\) | Einstein tensor. | Packages curvature so that its divergence vanishes identically: \(\nabla_\mu G^{\mu\nu}=0\). |
| \(T_{\mu\nu}\) | Stress–energy tensor: energy–momentum flux through surfaces. | Encodes matter/energy content; also satisfies \(\nabla_\mu T^{\mu\nu}=0\) (local energy‑momentum conservation). |
The expression on the left-hand side is a measure of the curvature of spacetime, while the right-hand side measures the energy and momentum of matter, so this equation relates energy to curvature, as promised. We will discuss it latter.
Free particles move along paths of "shortest possible distance" or geodesics. That says, particles will move on straight line, but in a curved spacetime there might not be any straight lines. Therefore their parameterized path \(x^{\mu}(\lambda)\) obey the geodesic equation: \[ \begin{equation} \frac{d^2 x^\mu}{d \lambda^2}+\Gamma_{\rho \sigma}^\mu \frac{d x^\rho}{d \lambda} \frac{d x^\sigma}{d \lambda}=0 \end{equation} \] Special relativity is a model that invokes a particular kind of spacetime (one with no curvature, and hence no gravity ).
Spacetime is a four-dimensional set, with elements labeled by three dimensions of space and one of the time. An individual point in spacetime is called an event. The path of a particle is a curve through spacetime, a parameterized one-dimensional set of events, called the worldline.
At any event we define a light cone