Relativity & Spacetime — Einstein's Framework — Antariksham

Special Relativity (1905)

Einstein's special theory of relativity rests on two postulates: the laws of physics are identical in all inertial (non-accelerating) frames, and the speed of light $c$ is constant for all observers regardless of the motion of source or observer.

These simple postulates have profound consequences.

Time Dilation:

A moving clock runs slower than a stationary one. If a clock moves at velocity $v$ relative to an observer, it ticks at a rate:

t' = \frac{t}{\gamma}, \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

where $\gamma \geq 1$ is the Lorentz factor. At $v = 0.9c$ , $\gamma \approx 2.29$ — the moving clock runs at less than half the rate of the stationary one.

Length Contraction:

Objects in motion appear contracted along the direction of travel:

L' = \frac{L}{\gamma}

Mass-Energy Equivalence:

The most famous equation in physics:

E = mc^2

More completely, for a moving particle: $E^2 = (pc)^2 + (mc^2)^2$ , where $p$ is momentum.

General Relativity (1915)

Special relativity handles inertial frames. General relativity extends this to accelerating frames and gravity. Einstein's key insight was the equivalence principle: there is no local experiment that can distinguish free fall in a gravitational field from inertial motion in empty space.

Gravity is not a force — it is the curvature of spacetime caused by mass and energy. The Einstein field equations relate spacetime curvature to the distribution of matter and energy:

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

Where $G_{\mu\nu}$ is the Einstein tensor (encoding curvature), $g_{\mu\nu}$ is the metric tensor, $\Lambda$ is the cosmological constant, and $T_{\mu\nu}$ is the stress-energy tensor (encoding matter and energy).

Gravitational Time Dilation:

Clocks in stronger gravitational fields run slower. At height $h$ above a massive body of radius $R$ :

\frac{\Delta t_h}{\Delta t_0} = \sqrt{1 - \frac{2GM}{c^2(R+h)}} \Bigg/ \sqrt{1 - \frac{2GM}{c^2 R}}

For small $h$ , this approximates to:

\frac{\Delta t_h}{\Delta t_0} \approx 1 + \frac{gh}{c^2}

GPS: Relativity in Engineering

The Global Positioning System provides the clearest real-world demonstration that relativistic corrections are essential engineering, not abstract theory.

GPS satellites orbit at ~20,200 km altitude, moving at ~3.87 km/s. Two effects operate simultaneously:

Special relativistic effect (time dilation): The satellite's velocity causes its clock to run slower than Earth-surface clocks by approximately $-7.2$ microseconds per day.

General relativistic effect (gravitational time dilation): The weaker gravitational field at altitude causes satellite clocks to run faster than surface clocks by approximately $+45.9$ microseconds per day.

The net effect is approximately $+38.4$ microseconds per day — satellite clocks gain 38 microseconds per day relative to Earth clocks. Since GPS position accuracy depends on time measurements accurate to nanoseconds, this ~38,000 nanosecond daily error would accumulate to position errors of roughly 10 km per day if uncorrected.

GPS satellite clocks are therefore pre-compensated: they are set to tick slightly slower before launch, so that in orbit they run at the correct rate from Earth's perspective. Every GPS fix you take is a practical application of both special and general relativity.