Theoretical Physics · Relativity
Spacetime is the four-dimensional manifold in which all physical events occur. Three spatial dimensions and one temporal dimension are unified into a single continuum. Mass and energy curve this manifold; curvature determines the motion of matter and light. This is the basis of Einstein's theories of special and general relativity.
In classical mechanics, space and time are independent. Special relativity, published by Einstein in 1905, demonstrated that they are not — they are linked by the speed of light c. Hermann Minkowski formalised this in 1908, combining them into a four-dimensional spacetime with coordinates (t, x, y, z).
The separation between two events in spacetime is measured by the spacetime interval s, which is invariant under Lorentz transformations — all inertial observers agree on its value, even if they disagree on individual distances and time intervals:
The sign convention above uses (−, +, +, +). When s² < 0, the interval is timelike — a causal connection exists between the events. When s² > 0, the interval is spacelike — no signal can connect them. When s² = 0, the interval is lightlike (null), and only light can connect them.
Special relativity rests on two postulates:
c = 299,792,458 m/s in all inertial frames, regardless of the motion of the source or observer. This is not an approximation — it is exact by definition of the metre.All relativistic effects — time dilation, length contraction, mass-energy equivalence — can be derived from a single dimensionless quantity, the Lorentz factor γ, which depends only on velocity v:
At everyday speeds, v ≪ c, so γ ≈ 1 and relativistic effects are negligible. As v → c, γ → ∞. No object with mass can reach or exceed c, as this would require infinite energy.
A moving clock runs slower than a stationary one by a factor of γ. This is not a mechanical effect — it is a feature of spacetime geometry. Adjust velocity and elapsed Earth time below:
Lorentz Time Dilation Calculator
The "twin paradox": if one twin travels at high velocity and returns, they have aged less. This is not a paradox — the asymmetry arises because the travelling twin must accelerate (and decelerate), breaking the symmetry between the frames.
The most famous result of special relativity relates rest mass m₀ to energy. The full relativistic energy-momentum relation is:
For a particle at rest (p = 0), this reduces to E = m₀c². The factor c² is extremely large (~9 × 10¹⁶ J/kg), meaning a small mass corresponds to an enormous amount of energy. One kilogram of matter, fully converted to energy, would yield approximately 9 × 10¹⁶ joules — equivalent to about 21 megatons of TNT.
Published in 1915, general relativity extends special relativity to include gravity and accelerating reference frames. Its central insight is that gravity is not a force (as Newton described) but the curvature of spacetime caused by mass and energy.
The relationship between spacetime curvature and the distribution of mass-energy is encoded in the Einstein field equations (EFE), a system of ten coupled, nonlinear partial differential equations:
Where Gμν is the Einstein tensor (describing spacetime curvature), Λ is the cosmological constant, gμν is the metric tensor, G is Newton's gravitational constant, and Tμν is the stress-energy tensor (describing the distribution of mass and energy).
In curved spacetime, freely falling objects follow geodesics — the generalisation of straight lines. What appears as the gravitational attraction of a massive body is actually an object following the straightest possible path through curved spacetime. The geodesic equation is:
Here Γμαβ are the Christoffel symbols, which encode how the coordinate basis vectors change from point to point — they capture the curvature of the path due to the spacetime geometry.
Clocks run slower in stronger gravitational fields. This is distinct from velocity-based time dilation in special relativity. For a clock at radius r in the Schwarzschild metric (spherical, non-rotating mass M):
rs is the Schwarzschild radius — the radius at which the escape velocity equals c. When r = rs, the factor goes to zero: a clock at the event horizon appears frozen to a distant observer. For Earth, rs ≈ 8.9 mm (far smaller than Earth's actual radius).
The linearised EFE admit wave solutions — ripples in spacetime curvature that propagate at c. These gravitational waves are generated by accelerating masses, particularly asymmetric systems such as binary black holes or neutron stars. Their amplitude (strain) is:
where Iμν is the quadrupole moment of the source and r is the distance. The strains detected by LIGO are of order h ~ 10⁻²¹ — a length change smaller than one-thousandth the diameter of a proton across a 4 km detector arm.
The theory makes precise, testable predictions. Every one of the following has been confirmed experimentally:
| Effect | Description | Confirmed |
|---|---|---|
| Gravitational lensing | Light bends around massive objects. During the 1919 solar eclipse, Eddington measured starlight deflected by 1.75 arcseconds — matching GR exactly. | 1919 |
| Perihelion precession | Mercury's orbit precesses by 43 arcseconds per century more than Newtonian gravity predicts. GR accounts for this precisely. | 1915 |
| Gravitational redshift | Photons lose energy climbing out of a gravitational well, shifting to longer (redder) wavelengths. Confirmed to 1 part in 10⁴ by Pound–Rebka (1959). | 1959 |
| Frame dragging | A rotating mass drags spacetime around it (Lense–Thirring effect). Measured by Gravity Probe B (2011) to within 0.28% of prediction. | 2011 |
| Gravitational waves | LIGO detected GW150914 on 14 September 2015 — a merger of two black holes ~1.3 billion light-years away, radiating ~3 solar masses as gravitational radiation. | 2015 |
| GPS correction | Satellite clocks run fast by ~38 µs/day (gravitational dilation +45.9 µs, velocity dilation −7.2 µs). Without correction, GPS position errors accumulate at ~10 km/day. | Ongoing |
| Black hole imaging | The Event Horizon Telescope resolved the shadow of M87* (mass ~6.5 × 10⁹ M☉) in 2019 and Sgr A* (mass ~4 × 10⁶ M☉) in 2022. Shadow sizes match GR predictions. | 2019–22 |
| Symbol | Constant | Value |
|---|---|---|
| c | Speed of light (exact) | 299,792,458 m/s |
| G | Gravitational constant | 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² |
| ℏ | Reduced Planck constant | 1.05457 × 10⁻³⁴ J·s |
| Λ | Cosmological constant | 1.089 × 10⁻⁵² m⁻² |
| lP | Planck length √(ℏG/c³) | 1.616 × 10⁻³⁵ m |
| tP | Planck time √(ℏG/c⁵) | 5.391 × 10⁻⁴⁴ s |
Newton formalises absolute space and time, with gravity as an instantaneous force at a distance. Accurate for low velocities and weak fields.
Maxwell unifies electricity, magnetism, and light. The equations predict a fixed speed for electromagnetic waves — inconsistent with Newtonian absolute time.
Finds no evidence of the luminiferous aether. The speed of light is the same in all directions, regardless of Earth's motion through space.
Einstein's "On the Electrodynamics of Moving Bodies" derives time dilation, length contraction, and E = mc² from two postulates. Minkowski reformulates it geometrically in 1908.
Einstein recognises that gravitational and inertial mass are equivalent. A person in a closed box cannot distinguish free fall from being weightless in deep space.
Einstein presents the field equations to the Prussian Academy. Within weeks, Karl Schwarzschild derives the first exact solution, describing the spacetime geometry around a spherical mass.
Measurements of starlight deflection during a solar eclipse confirm GR's prediction of 1.75 arcseconds — twice the Newtonian value. Einstein becomes internationally famous.
Penzias and Wilson detect the CMB, providing evidence for the Big Bang — a solution to Einstein's equations predicting an expanding universe from a hot, dense initial state.
Hulse and Taylor discover a binary pulsar whose orbital decay matches GR's prediction from gravitational wave emission exactly. Nobel Prize 1993.
LIGO observes GW150914 — the merger of two black holes of ~29 and ~36 solar masses. The signal lasted 0.2 seconds and swept from 35 to 150 Hz.
The Event Horizon Telescope produces radio images of M87* and Sgr A*, confirming the predicted shadow structure and light ring geometry to within measurement uncertainties.
General relativity is extraordinarily successful, yet it is known to be incomplete. Several fundamental tensions remain unresolved: