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Geodesics: How Things Fall

Replace 'force of gravity' with a single principle — free bodies follow the straightest paths through curved spacetime — and derive the equation of motion that recovers Newton.

The straightest possible path

In flat spacetime a free particle moves in a straight line at constant speed — Newton's first law. The natural generalisation to a curved manifold is a geodesic: the locally straightest possible curve, and for a massive particle the worldline that maximises the proper time between two events. Einstein's boldest move was to declare this the entire law of motion under gravity: free fall is geodesic motion.

On a curved surface the 'straight line' between two points is a geodesic — a great circle on a sphere. In spacetime, that straightest worldline is the trajectory of a freely falling body.

The geodesic equation

Extremising the proper time \tau=\int d\tau with the calculus of variations yields the geodesic equation — the equation of motion of general relativity:

\frac{d^2 x^\mu}{d\tau^2} + \Gamma^{\mu}{}_{\alpha\beta}\,\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0

The four-acceleration is zero — but 'zero acceleration' now includes the Gamma term that accounts for the bending of the coordinate grid.

The first term is the naive coordinate acceleration; if it were alone, we would be back to a straight line. The second term, built from the Christoffel symbols \Gamma^{\mu}{}_{\alpha\beta}, is the whole content of gravity: it says how much the notion of 'straight' twists as you move through a curved (or merely curvilinear) geometry. What Newton called the gravitational force is repackaged entirely into these \Gamma's.

Christoffel symbols from the metric

Crucially, you do not guess the \Gamma's — they are computed directly from the metric and its first derivatives. This is the machinery that turns geometry (g_{\mu\nu}) into motion:

\Gamma^{\lambda}{}_{\mu\nu} = \tfrac{1}{2}\,g^{\lambda\sigma}\!\left(\partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}\right)

Give me the metric and I can grind out every Christoffel symbol, then the geodesics. This is how you actually predict orbits in GR.

Covariant derivative and parallel transport

Behind the geodesic equation lies a deeper tool. In curved space the ordinary derivative of a vector field is not itself a proper (tensorial) object, because the basis vectors change from point to point. The fix is the covariant derivative \nabla_\mu, which corrects the plain derivative with a Christoffel term:

\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^{\nu}{}_{\mu\lambda}\,V^\lambda

The covariant derivative is the 'true' rate of change that subtracts off the mere twisting of the coordinate basis.

This gives the cleanest picture of a geodesic. Parallel transport means carrying a vector along a curve while keeping it 'as parallel as possible' (\nabla along the curve equals zero). A geodesic is then simply a curve that parallel-transports its own tangent vector — it keeps pointing straight ahead. Curvature reveals itself dramatically here: carry a vector around a closed loop on a curved manifold and it returns rotated. The size of that rotation is the fingerprint of curvature — the subject of Guide 4.

Sanity check: recovering Newton

A new theory must reproduce the old one where the old one worked. Take the geodesic equation in the limit that made Newton succeed: speeds much less than c, a weak field, and a static geometry. Then the motion collapses back to exactly Newtonian free fall.

  1. Slow motion means the space components of the four-velocity are tiny next to the time component, so only \tfrac{dx^0}{d\tau}\approx c\tfrac{dt}{d\tau} survives in the sum. The geodesic equation reduces to \tfrac{d^2 x^i}{d\tau^2} \approx -\Gamma^{i}{}_{00}\left(\tfrac{dx^0}{d\tau}\right)^2.
  2. For a static, weak field the only surviving Christoffel symbol is \Gamma^{i}{}_{00} \approx \tfrac{1}{c^2}\,\partial_i \Phi, obtained by feeding g_{tt}\approx-(1+2\Phi/c^2)c^2 into the Christoffel formula.
  3. Since the field is weak and slow, proper time \tau and coordinate time t nearly coincide, so we may replace d\tau\to dt. The two c's cancel.
  4. The result is Newton's equation of motion, with -\nabla\Phi playing the role of the gravitational acceleration \mathbf g.
\frac{d^2 x^i}{dt^2} = -\,\partial_i \Phi \;=\; g^i

Curved-spacetime geodesic motion contains Newtonian gravity as its slow, weak-field limit — exactly as required.