General relativity

Idea

The easiest way to start conceptualizing gravity as a geometric effect is to ponder the simplest toy model: the convergence of great circles on a sphere. Two nearby meridians are “as close to parallel as they can be” at the equator, and ships/planes that follow them will certainly not be accelerating sideways, but they will nevertheless draw closer together.

That takes the geometry for granted not explaining why (or in precisely what manner) space-time geometry is curved by the presence of mass, but it captures the essence of the kinematics. The worldline through spacetime of a free-falling body is a geodesic, and the failure to follow a geodesic, which involves what’s known as a “proper acceleration”, requires a force, and is perceived as weight.

The weight you feel as you stand, motionless, with respect to the Earth is due to the fact that your worldline is curved compared to the spacetime geodesic that would take you towards the centre of the Earth. But in principle, that’s no stranger than the sideways force required to keep a ship/plane at a constant (non-equatorial) latitude, rather than following a great circle.

Surprisingly, if you are on Earth and we let a stone free fall and it hits your head, it is not the case that the stone is accelerating, it is following a geodesic: you are accelerating towards the stone!

Keywords

According to the development by Schuller GR lectures

1. relativistic spacetime
2. worldlines
3. observers
4. vielbein
5. matter in GR
6. energy-momentum tensor
7. Einstein field equations
8. Einstein--Hilbert action

Schematic view: practical use of GR

In GR, spacetime and matter evolve together, and neither evolves independently:

1. Matter → Geometry:
   You compute the metric $g_{\mu \nu }$ by solving the Einstein field equations using the current state of matter (via energy-momentum tensor $T_{\mu \nu }$). Example: Schwarzschild solution.

2. Geometry → Matter Motion:
   Then, using this metric, you compute geodesics, fluid evolution, etc., to update the matter fields (density $\rho$, velocity, etc.).

3. Back to Step 1:
   Now that matter has moved, $T_{\mu \nu }$ has changed → you must recompute the metric.

This is a continuous and fully coupled system. That’s why the Einstein equations are nonlinear and challenging.

Everything comes from the total action (see Einstein--Hilbert action)

1. Einstein Field Equations (Geometry from Matter). Vary the total action with respect to the metric $g_{\mu \nu }$:

• $S_{\text{EH}}$ gives the Einstein tensor $G_{\mu \nu }$
• $S_{\text{matter}}$ gives the stress-energy tensor $T_{\mu \nu }$ via:$$T_{\mu \nu }=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{matter}}}{\delta g^{\mu \nu }}$$

1. Matter Field Equations (Motion from Geometry). Now vary the action with respect to the matter fields $\psi$:

This gives the equations of motion for the matter (e.g., for a scalar field, fluid, electromagnetic field), which describe how matter evolves in curved spacetime. Because the matter Lagrangian depends on $g_{\mu \nu }$, these equations contain information about the curved geometry, like covariant derivatives or curvature terms.