SR Review & Tensors

General relativity (GR) is built on the mathematics of curved spacetime. Before tackling curvature, we review special relativity in tensor language: the metric η_μν, four-vectors, and the invariant spacetime interval.

Key Concepts

Spacetime interval: ds² = -c²dt² + dx² + dy² + dz²
Minkowski metric: η_μν = diag(-1,1,1,1)
Four-vector: x^μ = (ct, x, y, z)
Raising/lowering indices: x_μ = η_μν x^ν
Invariant: x_μ x^μ = -c²t² + r² = -c²τ²

Key Equations

Spacetime interval

ds^2 = \eta_{\mu\nu}dx^\mu dx^\nu = -c^2dt^2 + dx^2+dy^2+dz^2

Four-velocity

u^\mu = \frac{dx^\mu}{d\tau} = \gamma(c, \mathbf{v})

Geodesic equation (flat)

\frac{d^2x^\mu}{d\tau^2} = 0

Einstein's equivalence principle

g_{\mu\nu}|_{local} = \eta_{\mu\nu}

Worked Example

Example Problem

Problem

A particle travels at v=0.8c. Find its proper time elapsed when coordinate time Δt=10 s elapses.

Solution

γ = 1/√(1-0.64) = 1/0.6 = 5/3. Δτ = Δt/γ = 10/(5/3) = 6 s.

Practice

Exercises

7 problems

1 of 7

Find γ for v=0.6c.

Your answer

2 of 7

A spaceship travels at v=0.6c for coordinate time Δt=5 s. Find proper time Δτ in s.

Your answer

3 of 7

Compute the spacetime interval for two events separated by Δt=3 s, Δx=5 light-seconds (in units where c=1). Is the interval spacelike or timelike? Report |ds²| in s².