Questions: The Geodesic Equation

Question 1

A massive particle in free fall follows a path that:

• A. Minimizes the spatial distance traveled between two events
• B. Maximizes the proper time elapsed between two events (among nearby paths)
• C. Moves along a path of zero proper time
• D. Follows a straight line in the coordinate system centered on the gravitating body

Answer: (B)

In Lorentzian geometry, timelike geodesics maximize proper time among nearby paths — this is the opposite of the Riemannian case where geodesics minimize distance. A freely falling clock between two events records more elapsed time than any nearby accelerated clock (the relativistic twin effect generalized to curved spacetime). Option C describes null geodesics (light). Option D is generally false in curved spacetime — geodesics appear curved in most coordinate systems.

Question 2

In the Newtonian limit (weak gravitational field, slow motion), the geodesic equation reduces to Newton's second law for gravity: d²x^i/dt² = -∂Φ/∂x^i.

• T. True
• F. False

Answer: True

In the weak-field (g_μν ≈ η_μν + h_μν with |h_μν| << 1), slow-motion (v << c) limit, the geodesic equation reduces to d²x^i/dτ² ≈ -(1/2)∂_i g_{00} c². With g_{00} ≈ -(1 + 2Φ/c²), this gives d²x^i/dt² ≈ -∂_i Φ, which is precisely Newton's gravitational acceleration. The Christoffel symbol Γ^i_{00} encodes the Newtonian gravitational force in this limit.

Question 3

Why must null geodesics (the paths of light) use an affine parameter other than proper time?

Model answer: Along a null geodesic, ds² = g_μν dx^μ dx^ν = 0 by definition, so the proper time dτ = √(-ds²/c²) is identically zero along the entire path. Proper time therefore cannot serve as a parameter to track progress along the curve. Instead, null geodesics are parameterized by an arbitrary affine parameter λ, defined by the requirement that the geodesic equation retains its standard form: d²x^μ/dλ² + Γ^μ_{αβ}(dx^α/dλ)(dx^β/dλ) = 0. Any reparameterization λ → aλ + b (affine transformation) preserves this form.

The affine parameter for null geodesics has no direct physical interpretation as elapsed time, but it is mathematically essential for writing the equation of motion. In practice, for light moving in a static spacetime, coordinate time t is often used as a convenient (though generally non-affine) parameter.

Question 4

Derive the geodesic equation from a variational principle by extremizing the proper time functional τ = ∫√(-g_μν dx^μ dx^ν) along a timelike path.

Model answer: Treat the path x^μ(λ) as the dynamical variable and form the action S = ∫√(-g_μν (dx^μ/dλ)(dx^ν/dλ)) dλ. Applying the Euler-Lagrange equations to the integrand L = √(-g_μν ẋ^μ ẋ^ν) yields (d/dλ)(∂L/∂ẋ^μ) - ∂L/∂x^μ = 0. Choosing proper time τ as the parameter (so L = 1) simplifies the Euler-Lagrange equations to d²x^μ/dτ² + Γ^μ_{αβ}(dx^α/dτ)(dx^β/dτ) = 0, with the Christoffel symbols arising from the metric derivatives in the Euler-Lagrange equations. Alternatively, extremizing ∫g_μν ẋ^μ ẋ^ν dλ (which avoids the square root) gives the same geodesic equation directly.

The variational derivation connects geodesics to the Lagrangian mechanics framework. The 'trick' of extremizing the squared integrand instead of the square root is standard in practice — it produces the same geodesic paths (with affine parameterization automatically enforced) and is much easier to compute.