Modern Physicsγ = 1/√(1 - v²/c²)

Special Relativity Simulator

Visualize time dilation and length contraction. Adjust the velocity as a fraction of c and see relativistic effects on clocks and rulers.

Parameters

β = v/c

Rest length L₀

Proper time t₀

Computed

Lorentz factor γ1.000000

Dilated time t′0.0000 s

Contracted L0.0000 m

K_rel (proton)0.000 MeV

β0.500

γ1.0000

t′0.000 s

L0.000 m

K (proton)0.000 MeV

Special Relativity

Einstein's 1905 Special Theory of Relativity rests on two postulates: (1) the laws of physics are the same in all inertial frames, and (2) the speed of light c is the same for all inertial observers. These deceptively simple postulates lead to profound consequences for measurements of time and length.

ℹRelativistic effects are only significant at velocities approaching c. At everyday speeds (v/c < 0.01), the Lorentz factor γ ≈ 1 and Newtonian mechanics is an excellent approximation.

The Lorentz Factor

The Lorentz factor γ is the key quantity in special relativity. It is always ≥ 1 and increases rapidly as β = v/c approaches 1. At β = 0.9, γ ≈ 2.29. At β = 0.99, γ ≈ 7.09. At β = 0.999, γ ≈ 22.4.

Time Dilation

A clock moving relative to an observer ticks more slowly. If the proper time (time measured by the moving clock) is t₀, the observer measures a longer time t' = γt₀. This is not an illusion — the moving clock genuinely runs slower. GPS satellites must correct for this effect.

Length Contraction

An object moving relative to an observer appears shorter along the direction of motion. If the proper length (rest length) is L₀, the observer measures L = L₀/γ. Only the dimension along the direction of motion contracts; transverse dimensions are unchanged.

💡Set β = 0.866 for γ = 2: the moving clock shows half the time elapsed, and the ruler appears half as long. This is the simplest benchmark for relativistic effects.

β = v/c	γ	Time dilation factor	Length contraction factor
0.1	1.005	1.005×	0.995
0.5	1.155	1.155×	0.866
0.866	2.00	2.00×	0.500
0.9	2.294	2.294×	0.436
0.99	7.089	7.089×	0.141
0.999	22.37	22.37×	0.0447

Special Relativity Formulas

Lorentz Factor

γ = \frac{1}{1 - β ^{2}}, β = \frac{v}{c}

Time Dilation

t^{'} = γ t_{0}

Length Contraction

L = \frac{L _{0}}{γ}

Relativistic Kinetic Energy

K = (γ - 1) m c^{2}

Symbol	Name	Unit
β	Velocity ratio v/c	dimensionless
γ	Lorentz factor	dimensionless (≥1)
t₀	Proper time (rest frame)	s
t'	Dilated time (observer frame)	s
L₀	Proper length (rest frame)	m
L	Contracted length (observer frame)	m
c	Speed of light	2.998 × 10⁸ m/s

Frequently Asked Questions

Is time dilation a real physical effect or just an optical illusion?

It is a real physical effect. Muons created by cosmic rays at 15 km altitude survive long enough to reach the Earth's surface because their internal 'clocks' run slow relative to the Earth frame — exactly as predicted by γ ≈ 9 for β ≈ 0.994. Atomic clocks on aircraft and GPS satellites confirm time dilation daily.

If time dilation is symmetric, why does the moving twin age less in the twin paradox?

The symmetry breaks because the travelling twin must accelerate (turn around). From the stay-at-home twin's inertial frame, the traveller's clock always runs slow. The asymmetry in acceleration resolves the apparent paradox — special relativity applies only to inertial frames, but the resolution requires accounting for the acceleration phase.

Does length contraction make a spaceship actually smaller?

An observer in the rest frame measures the ship as shorter. Observers on the ship measure it at its rest length L₀. Both measurements are correct in their respective frames — there is no single 'true' length. The ship does not physically compress; it is a measurement effect from simultaneity differences.

Why can't anything travel at exactly c?

As β → 1, γ → ∞. The relativistic kinetic energy K = (γ−1)mc² diverges, requiring infinite energy to reach c for any object with mass m > 0. Only massless particles (photons, gluons) travel at exactly c.

How does GPS correct for special relativity?

GPS satellites orbit at about 3.87 km/s. Special relativity makes their clocks run slow by about 7.2 μs/day relative to Earth clocks. General relativity (gravitational time dilation) makes them run fast by about 45.9 μs/day. The net effect is +38.4 μs/day, corrected in software so GPS remains accurate.