General Relativity
General relativity is Albert Einstein's geometric theory of gravitation, published in 1915, which describes gravity not as a force but as the curvature of spacetime caused by mass and energy. Massive objects warp the fabric of spacetime, and other objects follow curved paths (geodesics) through this warped spacetime, which we perceive as gravitational attraction. The theory has been confirmed by numerous observations including gravitational lensing, gravitational redshift, the precession of Mercury's orbit, and the detection of gravitational waves.
Key Formula
G_μν + Λg_μν = (8πG/c⁴)T_μν
LaTeX: G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}
| Symbol | Meaning | Unit |
|---|---|---|
| G_μν | Einstein tensor (spacetime curvature) | dimensionless |
| Λ | Cosmological constant | m⁻² |
| g_μν | Metric tensor | dimensionless |
| G | Gravitational constant | N·m²/kg² |
| T_μν | Stress-energy tensor (matter/energy distribution) | J/m³ |
Worked Example
Problem
Estimate the gravitational time dilation on Earth's surface. How much does a clock at sea level slow relative to a clock far from Earth's gravity?
Solution
Step 1: Gravitational time dilation formula: Δt_surface/Δt_infinity = √(1 − 2GM/rc²). Step 2: For Earth: G = 6.674 × 10⁻¹¹ N·m²/kg², M = 5.972 × 10²⁴ kg, r = 6.371 × 10⁶ m, c = 3 × 10⁸ m/s. Step 3: 2GM/rc² = 2 × 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / (6.371 × 10⁶ × 9 × 10¹⁶) = 1.39 × 10⁻⁹. Step 4: Δt/Δt_∞ ≈ 1 − 6.95 × 10⁻¹⁰ using the approximation √(1−x) ≈ 1 − x/2.
Answer
Clocks at sea level run slower by about 6.95 × 10⁻¹⁰ (roughly 60 microseconds per day)
Key Predictions of General Relativity and Their Verification
| Prediction | Phenomenon | Observed Value | GR Prediction | Status |
|---|---|---|---|---|
| Mercury precession | Orbital mechanics | 43.1″/century | 43.0″/century | Confirmed 1915 |
| Gravitational lensing | Light bending near Sun | 1.75″ | 1.75″ | Confirmed 1919 |
| Gravitational redshift | Pound-Rebka experiment | z ≈ 2.46 × 10⁻¹⁵ | Matches theory | Confirmed 1959 |
| Frame dragging | Gravity Probe B | 37.2 mas/yr | 39.2 mas/yr | Confirmed 2011 |
| Gravitational waves | LIGO detection | GW150914 | Binary BH merger | Confirmed 2015 |
Interactive Tools
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Related Terms
Special Relativity
Special relativity is a physical theory proposed by Albert Einstein in 1905 that describes the relationship between space and time for objects moving at constant velocities, particularly near the speed of light. It is founded on two postulates: the laws of physics are identical in all inertial frames of reference, and the speed of light in a vacuum is constant for all observers regardless of their motion. The theory reveals that time, length, and mass are not absolute but depend on the relative motion between observer and object, unifying space and time into a single four-dimensional continuum called spacetime.
Gravitational Waves
Gravitational waves are ripples in the fabric of spacetime generated by accelerating massive objects, predicted by Einstein's general theory of relativity in 1916 and directly detected for the first time by the LIGO collaboration on September 14, 2015 (event GW150914). These distortions propagate at the speed of light as transverse waves, alternately stretching and squeezing spacetime in perpendicular directions. The detection of gravitational waves opened an entirely new observational window on the universe, allowing astronomers to study events such as merging black holes and neutron stars that are otherwise invisible to electromagnetic telescopes.
Black Hole
A black hole is a region of spacetime where gravity is so extreme that nothing — not even light or other electromagnetic radiation — can escape once past the event horizon, the point of no return. Black holes form when massive stars collapse at the end of their lives (stellar black holes), or may grow supermassive through accretion and mergers in galactic centres. The boundary of a black hole is described by the Schwarzschild radius (for non-rotating black holes), and their properties are encapsulated by the "no-hair theorem": a black hole is fully described by only three parameters — mass, charge, and spin.
The word "general" was chosen by Einstein to extend his 1905 "special" relativity to all reference frames, including accelerating ones. The German term was "Allgemeine Relativitätstheorie" (general theory of relativity), published in November 1915 in the Prussian Academy of Sciences proceedings.