Modern Physics · Upper Division

General Relativity

Einstein's general theory of relativity identifies gravity with the curvature of spacetime. Mass and energy curve spacetime; curved spacetime tells matter how to move. The theory has passed every experimental test and underlies GPS, black holes, and cosmology.

Prerequisites

Special relativity (Ch. 19) \cdot Vectors and tensors \cdot Differential equations

Learning Goals

State the equivalence principle and derive gravitational time dilation and light deflection from it.
Interpret the metric tensor and the geodesic equation as replacements for Newtonian gravity.
Identify the key terms in Einstein's field equations and explain the physical meaning of each side.
Use the Schwarzschild metric to calculate the event horizon radius, ISCO, and gravitational redshift.
Describe the observational tests of general relativity: Mercury's precession, lensing, LIGO, and GPS.

GR.1 The Equivalence Principle

Definition GR.1 — The Equivalence Principle

Weak Equivalence Principle (WEP): The trajector

y of a freely falling test body is independent of its composition or mass (inertial mass =

mass). Known to 10^{-15} precision (Eötvös experiments, MICROSCOPE satellite

Einstein Equivalence Principle (EEP): In a small, freely falling laboratory, all non-gravitational physics is identical to what it would be in a flat spacetime inertial frame. Consequence: gravity can always be locally transformed away by going to a freely falling frame — gravity is not a force but a geometric effect.

Immediate consequences of EEP: (1) Light deflects in a gravitational field — photons follow geodesics. (2) Clocks run slow in a gravitational potential (gravitational time dilation):

\Delta\tau/\Deltat = \sqrt(1 - 2GM/rc^{2}) \approx 1 - GM/(rc^{2}) \qquad (gravitational redshift)

(GR.1)

GR.2 Spacetime and the Metric

General relativity describes spacetime as a 4D Lorentzian manifold with a metric tensor gμν. The spacetime interval between nearby events is:

ds^{2} = g\mu\nu dx\mu dx\nu \qquad (summation over repeated indices)

(GR.2)

In flat spacetime: gμν = ημν = diag(−1, +1, +1, +1) (Minkowski metric). The metric encodes the gravitational field — it determines distances, angles, and the paths of freely falling objects (geodesics).

Freely falling particles follow geodesics — the straightest possible paths in curved spacetime — determined by the geodesic equation:

d^{2}x\mu/d\tau^{2} + Γ\mu_\alpha\beta (dx\alpha/d\tau)(dx\beta/d\tau) = 0

(GR.3)

where Γμ_αβ are the Christoffel symbols, computed from derivatives of gμν. These play the role of the gravitational force in Newtonian mechanics — but they are coordinate-dependent (fictitious forces), not true forces.

GR.3 Einstein's Field Equations

The field equations relate the geometry of spacetime (left side) to the energy-momentum content (right side):

G\mu\nu \equiv R\mu\nu - \frac{1}{2} g\mu\nu R = 8\piG/c^{4} \times T\mu\nu \qquad (Einstein's equations)

(GR.4)

Here: Rμν is the Ricci tensor (curvature from Christoffel symbols), R = gμν Rμν is the Ricci scalar, Gμν is the Einstein tensor, and Tμν is the stress-energy tensor (energy density, momentum flux, pressure). In vacuum (Tμν = 0): Rμν = 0. The field equations are 10 nonlinear second-order PDEs — exact solutions are rare and precious.

Example GR.1 — The Schwarzschild Metric

The unique spherically symmetric vacuum solution (Birkhoff's theorem) is the Schwarzschild metric:

Metric:

ds^{2} = -(1-r_{s}/r)c^{2}dt^{2} + (1-r_{s}/r)^{-1}dr^{2} + r^{2}d\Omega^{2} \qquad (r_{s} = 2GM/c^{2}

Schwarzschild radius:

r_{s} = 2GM/c^{2}. For the Sun: r_{s} = 2\times6.67\times10^{-11}\times2\times10^{30}/(3\times10^{8})^{2} = 2.95 km (Sun's radius is 6

96,000 km — safe).

Gravitational time dilation:

A clock at r ticks at rate \sqrt(1-r_{s}/r) relative to one at infinity. At r = 2r_s: rate = 1/\sqrt

2 \approx 71% — significant slowing

Black hole:

At r = r_{s}: g_{tt} = 0, g_{rr} \to \infty. The event horizon — an infalling observer crosses it in fi

nite prop

er time but a distant observer never sees them cross (infinite coordinate time). Inside r_

t swap roles; everything must fall toward r=0 (future singularity

Orbits:

Innermost stable circular orbit (ISCO): r_{ISCO} = 3r_s = 6GM/c^{2}. Below this, there are no s

table circular orbits — matter accreting in a black hole spirals rapidly to the ISCO then plunges in.

GR.4 Observational Tests and Applications

GR has passed every precision test:

Perihelion precession of Mercury: GR predicts 42.98″/century extra precession (Newtonian mechanics gives 532″, observations give 575″ — the discrepancy of 43″ is explained exactly by GR). Einstein famously said his heart trembled when he computed this in 1915.

Gravitational lensing:Light deflection angle θ = 4GM/(rc²) at impact parameter r. For the Sun: θ = 1.75″ (confirmed by Eddington's 1919 eclipse expedition). Strong lensing creates Einstein rings; weak lensing maps dark matter.

Gravitational waves: Linearized GR predicts waves propagating at c in the metric perturbation hμν. First direct detection: LIGO 2015 (GW150914), from two ~30 M☉ black holes merging 1.3 billion light-years away. Peak luminosity ~50 M☉c²/s — briefly outshining all stars in the observable universe.

GPS: Combines both GR effects: gravitational time dilation (clocks on satellites run 45 μs/day fast at altitude) and special relativistic time dilation (clocks run 7 μs/day slow from velocity). Net: +38 μs/day — uncorrected, GPS would accumulate 10 km position error per day.

Definition GR.2 — Common Traps

Gravity is geometry in GR: free-falling objects follow geodesics, not force paths in flat space.
Coordinates can mislead: physical conclusions should use invariants or observable intervals.
Local inertial frames still exist: curvature appears through tidal effects across finite regions.
Schwarzschild radius is not a material surface: the event horizon is a causal boundary.

Exercises — GR.1–GR.4 General Relativity

Calculate the Schwarzschild radius of the Sun.

Straightforward

What is the net daily clock correction required for GPS satellites combining both GR and SR effects?

μs/day

Straightforward

Calculate the gravitational redshift of light from the surface of a white dwarf (M = 1M\odot

R = 0.0085 R\odot). Compare to the historical measurement from Sirius B

Straightforward

4.Using the Schwarzschild effective potential, find the innermost stable circular orbit (ISCO). What is the binding energy of a particle there, and why does this matter for astrophysics?

Intermediate

5.Estimate the gravitational wave strain from a binary black hole merger at 1 Gpc. What determines the frequency at merger?

Intermediate

6.Derive the Friedmann equation governing cosmological expansion. Describe the history of the universe: radiation-dominated, matter-dominated, and dark-energy-dominated eras.

Challenging

Key Takeaways

EEP: gravity is locally indistinguishable from acceleration. Gravity curves spacetime.
$Metric ds^{2} = g\mu\nu dx\mu dx\nu encodes geometry. Geodesic equation gives free-fall trajectories$ .
$Einstein equations: G\mu\nu = 8\piG/c^{4} \times T\mu\nu. Mass-energy tells spacetime how to curve$ .
$Schwarzschild metric: r_{s} = 2GM/c^{2} is the event horizon. ISCO at 3r_s$ .
$Tested: Mercury perihelion, gravitational lensing (4GM/rc^{2}), LIGO gravitational waves (201$ 5).
$GPS: +45\mus/day GR time dilation, -7\mus/day SR time dilation, net +38\mus/day must be correcte$ d.