Einstein's Relativity | Tale of Space

Albert Einstein (1879–1955)

Einstein's Theory of Relativity

When space and time become relative — a revolution that transformed our understanding of the Universe

Special Relativity — 1905 General Relativity — 1915

Albert Einstein revolutionized physics in the early 20th century with two theories that changed our view of the universe. Special relativity (1905) unifies space and time into a space-time continuum, while general relativity (1915) reveals that gravity is not a force, but a curvature of this space-time caused by mass.

1905

Year of Wonders

Special Relativity

Space and time are no longer absolute

E = mc²

The most famous equation in physics: energy equals mass multiplied by the speed of light squared.

Central idea: Time and space are not absolute, but depend on the speed at which one travels. The only absolute is the speed of light: 299,792,458 m/s.

Principle 1

Principle of Relativity

The laws of physics are identical for all observers in uniform linear motion (no acceleration). No experiment can detect absolute motion.

Principle 2

Constance de c

Light in a vacuum always travels at 299,792,458 m/s, regardless of the speed of the object emitting or receiving it. This is an insurmountable limit.

Strange consequences

Time dilation

The faster you go, the slower time slows down for you compared to someone who is stationary.

Length contraction

A rapidly moving object contracts in the direction of motion.

E = mc²

A small mass contains enormous energy (c² is gigantic: 9×10¹⁶!).

Concrete examples

Cosmic muons: These particles created in the upper atmosphere should decay before reaching the ground, but thanks to time dilation (they travel at ~0.99c), they survive until they reach the ground.

GPS: Satellites move quickly and are subject to less gravity → their clocks run differently. Without relativistic corrections, GPS would drift by 10 km per day!

Einstein's postulates

1. Principle of relativity (Galilean extension): The laws of mechanics AND electromagnetism are identical in all inertial reference frames.

2. Invariance of c: The speed of light in a vacuum, c = 299,792,458 m/s, is independent of the motion of the source and the observer.

Lorentz transformation

Replaces Galilean transformations to connect two reference frames moving relative to each other at velocity v:

x' = γ(x - vt) t' = γ(t - vx/c²) where γ = 1/√(1 - v²/c²) (Lorentz factor) If v → 0: γ → 1 (classical physics applies) If v → c: γ → ∞ (extreme relativistic effects)

Mathematical Consequences

Time dilation: Δt' = γ × Δt — If v → c, then γ → ∞: time slows down dramatically
Length contraction: L' = L / γ — If v → c, then L' → 0
Addition of velocities: u' = (u + v) / (1 + uv/c²) — Ensures that no velocity exceeds c
Total energy: E² = (mc²)² + (pc)² — If p = 0: E = mc²

Minkowski space-time

The three spatial dimensions plus one temporal dimension form a four-dimensional continuum. The space-time interval is invariant:

s² = c²t² - x² - y² - z² • If s² > 0: time-like interval (events can be causally related) • If s² < 0: space-like interval (no causal relation possible) • If s² = 0: light-like interval (trajectory of a photon)

Quantitative Applications

GPS: Satellites at v ≈ 3.9 km/s → time dilation of -7 μs/day (special relativity) + 45 μs/day (general relativity) = +38 μs/day net
Cosmic muons: Rest lifetime τ₀ = 2.2 μs, but observed on the ground thanks to γ ≈ 10-100
LHC: Protons at 0.999999991c → γ ≈ 7500, effective mass 7500× greater

1915

Masterpiece

General Relativity

Gravity as a curvature of space-time

Revolutionary idea: Gravity is not a force, but a distortion of space-time caused by the presence of mass and energy.

Mass distorts space-time like a ball on a stretched sheet.

Principle 1

Principle of Equivalence

Being in a gravitational field is equivalent to being in an accelerated reference frame. In a rocket accelerating at 9.8 m/s², you feel exactly the same as you do on Earth.

Principle 2

General Covariance

The laws of physics have the same form in all reference frames (accelerated or not). Physics does not depend on the chosen coordinate system.

Simple analogy

Imagine a sheet stretched out representing space-time. Place a bowling ball (a star) on it: it distorts the sheet. A marble (planet) rolling nearby will follow the curve created → this is the orbit! Planets do not "fall" toward the Sun because of a force; they simply follow the most natural path in curved space-time.

Einstein's Equation

G_μν + Λg_μν = (8πG/c⁴) × T_μν Where: • G_μν: Einstein tensor (describes the curvature of space-time) • Λ: cosmological constant (dark energy) • g_μν: metric tensor (geometry of space-time) • T_μν: energy-momentum tensor (matter-energy distribution)

Meaning: "The geometry of space-time (left side) is determined by the distribution of matter and energy (right side)."

Founding Principles

Principle of equivalence (weak): Inertial mass = gravitational mass. m_inertia (F=ma) ≡ m_gravity (F=GMm/r²)
Equivalence principle (strong): Locally (small region), it is impossible to distinguish between acceleration and gravity through any physical experiment.
General covariance principle: Physical laws are expressed by tensor equations that are valid in any reference frame.

Schwarzschild metric (static black hole)

ds² = -(1 - r_s/r)c²dt² + dr²/(1 - r_s/r) + r²(dθ² + sin²θ dφ²) Schwarzschild radius: r_s = 2GM/c² Predictions: • Event horizon at r = r_s • Central singularity r = 0 • For the Sun: r_s ≈ 3 km • For the Earth: r_s ≈ 9 mm

Experimental Tests

Mercury's perihelion precession: +43 arc seconds per century (unexplained by Newton)
Light deflection by the Sun: 1.75 arc seconds (measured during the 1919 eclipse by Eddington)
Gravitational redshift: Δν/ν = -GM/(rc²), verified with atomic clocks (Pound-Rebka, 1959)
Gravitational waves: Direct detection 2015 (GW150914, merger of black holes 36+29 M☉)
Image of black hole M87*: Event Horizon Telescope (2019)

Friedmann equations (Cosmology)

H² = (8πG/3)ρ - k/a² + Λ/3 H = Hubble expansion rate ρ = energy density k = spatial curvature a = scale factor of the Universe Λ = cosmological constant → Age of the Universe: 13.8 billion years (ΛCDM model)

The Equation That Describes the Universe

G_μν + Λg_μν = (8πG/c⁴) T_μν

This tensor equation relates the curvature of space-time (left side) to the distribution of matter and energy (right side). It predicts black holes, gravitational waves, the expansion of the Universe, and the Big Bang. It is one of the most beautiful and powerful equations in all of physics.

Verified Predictions

A century of spectacular experimental confirmations

Light Deflection

Verified in 1919

Starlight is bent as it passes close to the Sun during an eclipse.

Black Holes

Image of M87* in 2019

Objects so dense that even light cannot escape

Gravitational Waves

Detected in 2015

"Wrinkles" in space-time caused by cataclysmic events

Gravitational Redshift

Verified in 1959

Light loses energy as it "climbs" in a gravitational field.

Expansion of the Universe

Discovered in 1929

Space-time itself expands (Hubble's discovery)

Gravitational Lenses

Observed since 1979

Massive galaxies bend light like cosmic magnifying glasses.

Modern Applications

Relativity is not just an abstract theory—it is essential to the functioning of everyday technologies.

GPS

Without relativistic corrections (restricted + general), GPS would drift by 10 km per day. Every smartphone uses relativity!

Particle Accelerators

At the LHC, protons reach γ ≈ 7500. Their effective mass is 7500 times greater than at rest.

Nuclear Energy

E = mc² explains the energy released by nuclear fission and fusion. 1 kg of matter = 90 petajoules.

Cosmology

Big Bang, age of the Universe (13.8 billion years), dark energy, dark matter—everything is based on general relativity.

Detection of Exoplanets

Gravitational microlensing allows planets to be detected by observing the bending of light.

LIGO/Virgo

Gravitational wave detectors capable of measuring deformations of 10⁻²¹ meters.