Geodesics

Geodesics are the paths that free-falling objects follow through curved spacetime. They generalize the concept of "straight lines" to curved spaces.

What is a Geodesic?

A geodesic is a curve that extremizes the spacetime interval between two events. In flat spacetime, geodesics are straight lines. In curved spacetime, they're the "straightest possible" paths.

Think of a geodesic as the path a particle takes when no forces act on it - pure free fall through curved spacetime.

Geodesic Motion

Particles following geodesics around a massive object. Notice how closer orbits move faster (Kepler's law).

Show relativistic precession (like Mercury's orbit)

Mass: 2.0

The Geodesic Equation

The geodesic equation describes how coordinates change along a geodesic:

Geodesic Equation

\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\rho\sigma} \frac{dx^\rho}{d\tau} \frac{dx^\sigma}{d\tau} = 0

Here $\tau$ is proper time along the curve, and $\Gamma^\mu_{\rho\sigma}$ are the Christoffel symbols.

Equivalently, using the covariant derivative:

Covariant Form

u^\nu \nabla_\nu u^\mu = 0

where $u^\mu = dx^\mu/d\tau$ is the 4-velocity.

Types of Geodesics

Timelike Geodesics

$ds^2 < 0$ : Paths of massive particles. Parameterized by proper time.

g_{\mu\nu} u^\mu u^\nu = -c^2

Null Geodesics

$ds^2 = 0$ : Paths of light (photons). Cannot use proper time as parameter.

g_{\mu\nu} k^\mu k^\nu = 0

where $k^\mu$ is the wave 4-vector.

Spacelike Geodesics

$ds^2 > 0$ : Not physical trajectories, but useful for defining "distances".

Variational Principle

Geodesics can be derived by extremizing the action:

Proper Time Action

S = \int d\tau = \int \sqrt{-g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}} \, d\lambda

Applying the Euler-Lagrange equations gives the geodesic equation. This is directly analogous to the principle of least action in mechanics!

Example: Schwarzschild Geodesics

For the Schwarzschild spacetime, symmetries give conserved quantities:

Energy Conservation

E = \left(1 - \frac{r_s}{r}\right) \frac{dt}{d\tau} = \text{constant}

Angular Momentum Conservation

L = r^2 \frac{d\phi}{d\tau} = \text{constant}

These lead to the effective potential for radial motion:

Effective Potential

V_{\text{eff}}(r) = \left(1 - \frac{r_s}{r}\right)\left(1 + \frac{L^2}{r^2}\right)

Planetary Orbits

The geodesic equation in Schwarzschild spacetime predicts the famous precession of Mercury's perihelion:

Perihelion Precession

\Delta\phi = \frac{6\pi G M}{c^2 a(1-e^2)}

For Mercury, this gives about 43 arcseconds per century - exactly matching the observed anomaly that Newtonian gravity couldn't explain!

Light Bending

Null geodesics (light paths) near a massive object curve. The deflection angle for light passing a mass at distance $b$ :

Light Deflection

\Delta\phi = \frac{4GM}{c^2 b}

For light grazing the Sun, this gives 1.75 arcseconds - confirmed during the 1919 solar eclipse, making Einstein famous worldwide!

Gravitational Lensing

Light rays bending around a massive object. At high mass, notice the Einstein ring formation.

Mass: 1.5

Show light rays

Parallel Transport

A vector is parallel transported along a geodesic if:

Parallel Transport

\frac{D V^\mu}{d\tau} = u^\nu \nabla_\nu V^\mu = 0

The 4-velocity itself is parallel transported along a geodesic - this is why geodesics are "straight": the velocity doesn't change direction (in the curved-space sense).

The Equivalence Principle

The geodesic equation embodies the equivalence principle: all objects, regardless of their mass or composition, follow the same geodesics in a given gravitational field. This is because gravity is not a force - it's the geometry of spacetime itself. Free fall is the natural state; it's standing on Earth's surface that requires a force!

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