# Relativity

An observer's clock and ruler are not absolute. Special relativity starts
from two postulates — the laws of physics look the same in every inertial
frame, and light travels at the same speed _c_ for everyone — and
follows them without flinching to conclusions that overturn Newton's
separate space and time.


_Figure 001 — The light cone through an event splits spacetime into causal future, past,
and an unreachable elsewhere; a worldline threads the interior.
_

_Figure 002 — A moving light clock runs slow: the photon's diagonal path is longer, so
each tick takes longer.
_

Space and time merge into a single four-dimensional
**spacetime**. The one quantity every observer agrees on is
the interval between events, and the light cone it defines fixes what can
cause what — sorting every event into an absolute future, past, or the
spacelike elsewhere no signal can reach.


The bridge between frames is the **Lorentz transformation**.
It mixes space and time while leaving the speed of light invariant, and its
consequences — time dilation, length contraction, the relativity of
simultaneity — are geometry, not illusion.


_Figure 003 — A Lorentz boost tilts the moving frame's space and time axes toward the
light line — which is why simultaneity is relative.
_

Dynamics is rebuilt to match. Momentum and energy join into a single
four-vector, mass reveals itself as a form of energy through
_E = mc²_, and Maxwell's electromagnetism turns out to have been
relativistic all along.


Then gravity. The **equivalence principle** — that free fall
is indistinguishable from floating in empty space — says gravity is not a
force but the curvature of spacetime, and matter is what does the curving.


_Figure 004 — Mass curves the spacetime grid, and a free body orbits by following the
straightest available path through it.
_

The **Einstein field equation** ties that curvature to energy
and momentum. Its first exact solution, the Schwarzschild metric, describes
spacetime around a star and predicts the bending of light, the precession
of Mercury, and clocks that tick faster with altitude.


_Figure 005 — A gravitational wave stretches and squeezes a ring of free test masses as
it passes.
_

Push a mass small enough and an **event horizon** forms: a
black hole, a region from which not even light escapes. Ripples in the
curvature — **gravitational waves** — travel outward at
_c_ and were first caught stretching a detector in 2015.


Turn the same equations on the universe as a whole and they give an
expanding cosmos with a finite past. Relativity is the geometry underneath
all of it.


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## Contents

### 1. Foundations of Relativity

1. [The Postulates of Special Relativity](/relativity/foundations/special-relativity-postulates)
2. [The Lorentz Transformation and Spacetime](/relativity/foundations/lorentz-transformation-spacetime)
3. [Time Dilation, Length Contraction, and Paradoxes](/relativity/foundations/time-dilation-length-contraction)
4. [Relativistic Momentum and Energy](/relativity/foundations/relativistic-momentum-energy)
5. [A Taste of General Relativity](/relativity/foundations/general-relativity)

### 2. Spacetime and the Lorentz Group

1. [Minkowski Spacetime and the Interval](/relativity/spacetime-and-the-lorentz-group/minkowski-spacetime-and-the-interval)
2. [Four-Vectors and Index Notation](/relativity/spacetime-and-the-lorentz-group/four-vectors-and-index-notation)
3. [The Lorentz Group and Rapidity](/relativity/spacetime-and-the-lorentz-group/the-lorentz-group-and-rapidity)
4. [Doppler, Aberration, and Appearance](/relativity/spacetime-and-the-lorentz-group/doppler-aberration-and-appearance)

### 3. Relativistic Dynamics

1. [Four-Momentum, Four-Force, and Accelerated Motion](/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion)
2. [Particle Decays and Two-Body Kinematics](/relativity/relativistic-dynamics/particle-decays-and-two-body-kinematics)
3. [Relativistic Collisions and Threshold Energies](/relativity/relativistic-dynamics/collisions-thresholds-and-the-cm-frame)
4. [Mandelstam Variables and Lorentz Invariants](/relativity/relativistic-dynamics/mandelstam-variables-and-invariants)

### 4. Covariant Electromagnetism

1. [The Four-Current and Four-Potential](/relativity/covariant-electrodynamics/four-current-and-the-four-potential)
2. [The Electromagnetic Field Tensor](/relativity/covariant-electrodynamics/the-electromagnetic-field-tensor)
3. [How E and B Transform](/relativity/covariant-electrodynamics/transformation-of-electric-and-magnetic-fields)
4. [Covariant Maxwell and the Stress–Energy Tensor](/relativity/covariant-electrodynamics/covariant-maxwell-and-the-stress-energy-tensor)

### 5. Curved Spacetime

1. [The Equivalence Principle](/relativity/curved-spacetime/the-equivalence-principle-formalized)
2. [Manifolds, Vectors, and the Metric](/relativity/curved-spacetime/manifolds-vectors-and-the-metric)
3. [Parallel Transport and the Covariant Derivative](/relativity/curved-spacetime/covariant-derivative-and-christoffel-symbols)
4. [Geodesics and the Newtonian Limit](/relativity/curved-spacetime/geodesics-and-the-geodesic-equation)
5. [Curvature and the Riemann Tensor](/relativity/curved-spacetime/curvature-riemann-and-geodesic-deviation)
6. [The Einstein Field Equations](/relativity/curved-spacetime/the-einstein-field-equations)

### 6. The Schwarzschild Solution

1. [The Schwarzschild Metric](/relativity/the-schwarzschild-solution/the-schwarzschild-metric)
2. [Orbits in the Schwarzschild Geometry](/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
3. [Null Geodesics and the Photon Sphere](/relativity/the-schwarzschild-solution/light-bending-and-null-geodesics)

### 7. Tests of General Relativity

1. [The Perihelion Precession of Mercury](/relativity/tests-of-general-relativity/perihelion-precession-of-mercury)
2. [Light Deflection and Gravitational Lensing](/relativity/tests-of-general-relativity/deflection-of-light-and-gravitational-lensing)
3. [Gravitational Redshift and the Shapiro Delay](/relativity/tests-of-general-relativity/gravitational-redshift-and-shapiro-delay)
4. [Relativity and the Global Positioning System](/relativity/tests-of-general-relativity/relativity-in-technology-gps)

### 8. Black Holes

1. [Horizons and Coordinate Singularities](/relativity/black-holes/horizons-and-coordinate-singularities)
2. [Rotating and Charged Black Holes](/relativity/black-holes/rotating-and-charged-black-holes)
3. [Black-Hole Thermodynamics](/relativity/black-holes/black-hole-thermodynamics)

### 9. Gravitational Waves

1. [Linearized Gravity and Wave Solutions](/relativity/gravitational-waves/linearized-gravity-and-wave-solutions)
2. [The Quadrupole Formula](/relativity/gravitational-waves/generation-and-the-quadrupole-formula)
3. [LIGO and the First Detections](/relativity/gravitational-waves/detection-ligo-and-the-first-events)

### 10. A Bridge to Cosmology

1. [The Cosmological Principle and the FLRW Metric](/relativity/cosmological-bridge/the-cosmological-principle-and-flrw-metric)
2. [The Friedmann Equations and Cosmic Dynamics](/relativity/cosmological-bridge/friedmann-equations-and-cosmic-dynamics)