Geodesic Equation

Equivalent forms

\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha \dot{x}^\beta = 0

Newton's 'F = ma with F = -m grad(phi)' emerges as the weak-field, low-speed limit of this single geometric statement that free particles extremize proper time.

Symbol	Name	SI unit	Dimension	Range
$a$	Coordinate Accelerationoutput d^2 x^mu/dtau^2 induced by curvature	m/s^2	LT^-2	-50 – 50
$Gamma$	Christoffel Symbol Connection coefficient (curvature of coordinates)	1/m	L^-1	-1 – 1
$v$	Coordinate Velocity dx/dtau along the path	m/s	LT^-1	0 – 10

Dimensional analysis

d^2x/dtau^2 -> [L T^-2]

Gamma v^2 ->

[L^-1][L^2\,\mathrm{T}^-2] = [L T^-2]

Both accelerations.

\checkmark

Discovery

Albert Einstein / Tullio Levi-Civita · 1915

Levi-Civita's parallel transport gave Einstein the mathematical notion of 'straightest path'; the geodesic equation became general relativity's replacement for Newton's second law under gravity.

Research status: stable

Real-world applications

Spacecraft trajectory and gravity-assist modelling in strong fields
Pulsar-timing and binary-inspiral orbits
Gravitational lensing ray-tracing

Common misconceptions

Christoffel symbols are not tensors — they can be zeroed at a point (local inertial frame), which is exactly the equivalence principle
Geodesics extremize (often maximize) proper time, not 'shortest distance'
A body in orbit is in free fall/weightless, following a geodesic — no force acts on it

Experimental verification

Planetary orbits (Mercury's precession), the deflection of starlight, the Pound-Rebka redshift, and the exquisitely tested equivalence principle (Eot-Wash, MICROSCOPE to 1e-15) all confirm test bodies follow geodesics independent of composition.

Derivation

Extremize proper time

\tau =

integral sqrt(-g_{mu nu} dx^mu dx^nu)/c between two events.

The Euler-Lagrange equations of this action yield d^2 x^mu/dtau^2 + Gamma^mu_{alpha beta} (dx^alpha/dtau)(dx^beta/dtau

) = 0

, where

\Gamma ^\mu _{\alpha \beta } = 1/2\,\mathrm{g}^{\mu \nu }

(partial_alpha g_{nu beta} + partial_beta g_{nu alpha} - partial_nu g_{alpha beta}).

Free fall = maximal aging.

Limiting cases

\Gamma = 0

(flat space)⟶ d^2 x/dtau^

2 = 0

Straight-line motion at constant velocity — Newton's first law.

weak field, slow motion⟶

d^2\,\mathrm{x/dt}^2 =

-grad(phi)Christoffel symbols reduce to the Newtonian gravitational field; you recover

g = 9.8\,\mathrm{m/s}^2

near Earth.

photon (null geodesic)⟶ same equation with affine parameterLight bending and gravitational lensing follow the null version.

What if…

What if two nearby geodesics are followed?

Their relative acceleration is the geodesic DEVIATION equation, R^mu_{alpha nu beta}-driven — this is what tidal forces and gravitational waves actually are.

What if you add a real force (a rocket)?

The right-hand side becomes f^mu/m; you leave the geodesic and feel weight.

1

Newtonian limit near Earth

Given ·

metric:: weak field $g_tt = -(1+2\varphi /c^2)$

Find · recover g

Steps

Keep dt/dtau $\sim 1$ , dx/dtau small
$\Gamma ^i_{tt} = c^-2$ partial_i phi
$d^2\,\mathrm{x}^i/dt^2 = -c^2 \Gamma ^i_{tt}$ = -partial_i phi

Result · For slow motion, only Gamma^i_{tt} survives:

\Gamma ^i_{tt} = (1/c^2)

partial_i phi. The spatial geodesic equation becomes

d^2\,\mathrm{x}^i/dt^2 =

-partial_

i \varphi = -g

, reproducing free fall at 9.8 m/s^2.

The Metric Tensor→christoffel symbolsSchwarzschild Metric→Einstein Equivalence Principle→