Table of Contents
- Introduction
- Flat vs Curved Spacetime
- Curvature in General Relativity
- The Metric Tensor and Line Element
- Geodesics: The Straightest Paths
- Deriving the Geodesic Equation
- Christoffel Symbols and Covariant Derivatives
- Geodesics in Schwarzschild Geometry
- Types of Geodesics: Timelike, Null, Spacelike
- Gravitational Lensing and Geodesic Bending
- Free Fall as Geodesic Motion
- Geodesic Deviation and Tidal Forces
- The Twin Paradox in Curved Spacetime
- Applications and Observations
- Conclusion
1. Introduction
In general relativity, spacetime is not flat. Instead, it bends in response to mass and energy. Objects under the influence of gravity are not pulled by a force but follow paths called geodesics, which are the closest analogs to straight lines in a curved geometry.
Understanding geodesics allows us to describe planetary orbits, black hole accretion, and even light bending due to gravity.
2. Flat vs Curved Spacetime
In special relativity, spacetime is flat and described by the Minkowski metric:
\[
ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2
\]
In general relativity, the presence of mass-energy curves spacetime, and distances are measured via:
\[
ds^2 = g_{\mu\nu} dx^\mu dx^\nu
\]
Where \( g_{\mu\nu} \) is the metric tensor of spacetime.
3. Curvature in General Relativity
Spacetime curvature is described using:
- Riemann curvature tensor \( R^\rho_{\ \sigma\mu\nu} \)
- Ricci tensor \( R_{\mu\nu} \)
- Ricci scalar \( R \)
These quantities appear in Einstein’s field equations:
\[
R_{\mu\nu} – \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}
\]
Curvature determines how geodesics behave and deviate.
4. The Metric Tensor and Line Element
The metric tensor \( g_{\mu\nu} \) encodes the geometry of spacetime.
The line element:
\[
ds^2 = g_{\mu\nu} dx^\mu dx^\nu
\]
This is used to compute intervals, proper time, and distances in curved spacetime. It generalizes the Pythagorean theorem to curved, 4D manifolds.
5. Geodesics: The Straightest Paths
A geodesic is the generalization of a straight line to curved spacetime. It is defined as the path of extremal proper time (for timelike curves) or extremal path length (for spacelike curves).
Freely falling particles follow timelike geodesics. Light rays follow null geodesics.
6. Deriving the Geodesic Equation
We derive the geodesic equation by extremizing the action:
\[
S = \int ds = \int \sqrt{-g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}} \, d\lambda
\]
Applying the Euler–Lagrange equation leads to:
\[
\frac{d^2 x^\rho}{d\lambda^2} + \Gamma^\rho_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda} = 0
\]
Where \( \Gamma^\rho_{\mu\nu} \) are Christoffel symbols.
7. Christoffel Symbols and Covariant Derivatives
Christoffel symbols describe how coordinates and vectors change in curved space:
\[
\Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} – \partial_\sigma g_{\mu\nu} \right)
\]
They appear in covariant derivatives, which replace ordinary derivatives in curved spacetime.
8. Geodesics in Schwarzschild Geometry
The Schwarzschild metric (outside a static spherical mass):
\[
ds^2 = -\left(1 – \frac{2GM}{rc^2}\right) c^2 dt^2 + \left(1 – \frac{2GM}{rc^2}\right)^{-1} dr^2 + r^2 d\Omega^2
\]
This describes black holes and stars. Geodesics in this geometry yield:
- Planetary orbits (including perihelion precession)
- Light bending (gravitational lensing)
9. Types of Geodesics: Timelike, Null, Spacelike
- Timelike: \( ds^2 < 0 \) — paths of massive particles
- Null: \( ds^2 = 0 \) — paths of light
- Spacelike: \( ds^2 > 0 \) — hypothetical non-causal paths
These determine causal structure in spacetime.
10. Gravitational Lensing and Geodesic Bending
Massive objects bend light by curving spacetime. Light follows null geodesics, which curve around stars and galaxies.
This leads to:
- Einstein rings
- Multiple images of quasars
- Maps of dark matter via lensing
11. Free Fall as Geodesic Motion
Objects in free fall are not accelerating in their local frame — they move along geodesics.
This resolves the apparent paradox of why astronauts feel weightless — they are simply following spacetime’s geometry.
12. Geodesic Deviation and Tidal Forces
Geodesic deviation equation:
\[
\frac{D^2 \xi^\mu}{d\tau^2} = -R^\mu_{\ \nu\alpha\beta} u^\nu \xi^\alpha u^\beta
\]
Where \( \xi^\mu \) is the separation vector. This explains:
- Tidal forces
- Stretching and compression in gravitational fields
13. The Twin Paradox in Curved Spacetime
In GR, proper time depends on the entire path in spacetime.
The twin who travels and returns follows a geodesic with less proper time due to acceleration and gravity — explaining the paradox without contradiction.
14. Applications and Observations
Geodesics play central roles in:
- GPS satellite calibration
- Black hole simulations
- Cosmology (light paths from distant galaxies)
- Predicting gravitational wave signatures
15. Conclusion
Geodesics and curved spacetime lie at the heart of general relativity. They explain how matter and light move in the presence of gravity — not as a force, but as a manifestation of geometry.
From black holes to cosmological expansion, understanding geodesics gives us a profound map of how the universe bends, twists, and evolves.
