Relativity Intermediate

Special relativity rests on two axioms:

1. Principle of relativity: the laws of physics take the same form in all inertial (non-accelerating) reference frames.
2. Constancy of c: the speed of light in vacuum is c ≈ 3×10⁸ m/s for all observers regardless of the motion of source or observer.

Derived consequences

• Simultaneity is relative — events simultaneous in one frame are not in another.
• Time dilation: moving clocks run slow by factor γ.
• Length contraction: moving objects shrink along the direction of motion by factor 1/γ.
• Relativistic velocity addition replaces v₁ + v₂.

The Lorentz factor

γ = 1 / √(1 − β²), where β = v/c

At low speeds γ ≈ 1 and Newtonian physics is recovered.

When two inertial frames S and S' are related by relative velocity v along the x-axis, coordinates transform as:

x' = γ(x − vt)
t' = γ(t − vx/c²)
y' = y, z' = z

The inverse just flips the sign of v.

Simultaneity

If two events are simultaneous in S (t₁ = t₂) but at different x, they are generally not simultaneous in S' because t' includes the term vx/c².

Recovering time dilation and length contraction

Set x' = 0 (the moving clock's position) in the Lorentz transform to derive Δt = γ Δτ, where τ is proper time. Set Δt' = 0 (simultaneous endpoints) to derive L = L₀/γ.

The Lorentz group

Lorentz transformations form a group (LT of LT is an LT). Compositions in the same direction add velocities relativistically, not simply.

A spacetime diagram (Minkowski diagram) plots time on the vertical axis and space on the horizontal. Events are points; worldlines are paths.

Light cone

Light from an event E spreads at 45° lines on the diagram (if c = 1). This divides spacetime into:

• Timelike future: inside the upper cone — events E can causally influence.
• Timelike past: inside the lower cone — events that could have caused E.
• Spacelike region: outside both cones — no causal connection to E (would require faster-than-light travel).

Spacetime interval

s² = −c²Δt² + Δx² + Δy² + Δz²

This is invariant — all observers agree on s². If s² < 0: timelike. If s² > 0: spacelike. If s² = 0: lightlike (null).

Velocities do not simply add in special relativity. If frame S' moves at v relative to S, and an object moves at u' in S', its velocity u in S is:

u = (u' + v) / (1 + u'v/c²)

Key properties

• If u' = c: u = (c + v)/(1 + v/c) = c. Light is still c in every frame.
• If u' and v ≪ c: u ≈ u' + v. Newtonian addition is recovered.
• u always stays below c when u' and v are each below c.

Aberration of light

The direction of light changes between frames. Stars appear shifted toward the direction of motion — an effect called aberration observed astronomically.

Classical momentum p = mv fails at high speeds. The relativistic versions:

p = γmv
E = γmc²

Rest energy E₀ = mc² when v = 0.

Energy–momentum relation

E² = (pc)² + (mc²)²

For massless particles (photons): E = pc, so p = E/c = hf/c.

Why you can't reach c

As v → c, γ → ∞, so momentum and energy diverge. An infinite amount of energy would be required.

Kinetic energy

KE = (γ − 1)mc². At low speeds, (γ−1) ≈ v²/(2c²) so KE ≈ ½mv² — Newton is recovered.

Relativity is elegantly expressed using four-vectors — objects with one time and three spatial components that transform together under Lorentz boosts.

Key four-vectors

• Position: x^μ = (ct, x, y, z)
• Four-momentum: p^μ = (E/c, pₓ, pᵧ, p_z)
• Four-velocity: u^μ = γ(c, vₓ, vᵧ, v_z)

The Minkowski metric

The invariant interval: ds² = η_{μν} dx^μ dx^ν = −c²dt² + dx² + dy² + dz², where η = diag(−1, +1, +1, +1).

Covariant equations

A physical law written as an equation between four-vectors automatically holds in every inertial frame. This is the principle of covariance.

In 1907 Einstein had his "happiest thought": a person in free fall does not feel gravity — they feel weightless, exactly like an astronaut in orbit.

The weak equivalence principle

All objects fall at the same rate regardless of mass (Galileo's experiment). Gravitational mass equals inertial mass.

The strong equivalence principle

A uniformly accelerating frame is locally indistinguishable from a gravitational field. The laws of physics (including those for light) look the same in both.

Predictions from the equivalence principle alone

• Gravitational redshift: light climbing out of a gravity well loses energy, redshifting.
• Light bending: if acceleration deflects light, gravity must too.
• Gravitational time dilation: clocks lower in a gravity well run slower.

These predictions were confirmed before Einstein had even finished general relativity.

General relativity (1915) recasts gravity as the geometry of four-dimensional space-time.

Flat vs curved

In flat (Minkowski) spacetime, free particles travel in straight lines at constant speed. In curved spacetime, the straightest possible paths — geodesics — appear curved, just as the shortest path between two points on a globe is a great circle, not a straight line on a map.

Metric tensor

The geometry of spacetime is encoded in the metric tensor g_{μν}. The invariant interval becomes ds² = g_{μν} dx^μ dx^ν. In flat spacetime g_{μν} = η_{μν}; near a mass it deviates.

Einstein's insight

Mass and energy curve space-time (encoded in the Einstein field equations). Curved space-time tells matter how to move (along geodesics). Gravity is geometry.