Geodesics Flashcards
(39 cards)
What is a geodesic?
The path that extremizes the interval (ds) between two points in spacetime.
What equation defines a geodesic?
d²x^μ/dλ² + Γ^μ_{νρ} dx^ν/dλ dx^ρ/dλ = 0.
What is the affine parameter?
A parameter λ that varies linearly with proper time or arc length.
What is the geodesic equation for null paths?
Same as for timelike, but ds² = 0.
What is the physical significance of a geodesic?
It is the trajectory of a freely falling particle or light ray.
What happens to a particle in free fall in GR?
It follows a geodesic of the spacetime.
What is the difference between geodesics and straight lines?
In curved spacetime, geodesics generalize straight lines from flat space.
What is the Euler-Lagrange equation applied to geodesics?
It yields the geodesic equation when extremizing the action S = ∫ds.
What is the Lagrangian for a geodesic?
L = g_{μν} dx^μ/dλ dx^ν/dλ.
How is the geodesic equation derived?
From the Euler-Lagrange equations applied to the line element or action.
What does parallel transport along a geodesic mean?
A vector remains constant in direction along the geodesic.
What is a timelike geodesic?
A geodesic where ds² < 0; represents a massive particle’s path.
What is a spacelike geodesic?
A geodesic where ds² > 0; not traversable by particles.
What is a null geodesic?
A path with ds² = 0; followed by light or massless particles.
What quantity is conserved along a geodesic with a Killing vector?
ξ^μ u_μ = constant (projection of velocity onto symmetry direction).
What does the geodesic equation simplify to in flat space?
d²x^μ/dλ² = 0 — straight line motion.
What is the conserved quantity in static spacetimes?
Energy: E = -g_{tt} dt/dλ (if g_{tt} is time-independent).
What is the geodesic deviation equation?
Describes how nearby geodesics diverge: D²ξ^μ/Dλ² = R^μ_{νρσ} u^ν ξ^ρ u^σ.
What is the meaning of geodesic deviation?
Tells how tidal forces act — nearby free-fall paths change due to curvature.
What is a proper parameterization of a timelike geodesic?
Proper time τ, such that u^μ = dx^μ/dτ and u^μ u_μ = -1.
What is u^μ?
The four-velocity: u^μ = dx^μ/dτ for a particle moving along a timelike path.
What does ‘affinely parameterized’ mean?
The parameter λ is such that the geodesic equation takes its standard form.
What is the geodesic Lagrangian for null paths?
L = g_{μν} dx^μ/dλ dx^ν/dλ = 0.
What is a ‘first integral’ of the geodesic equation?
A conserved quantity derived from a symmetry, like energy or angular momentum.
