Theory — Motion Near the Speed of Light
Einstein's special relativity rests on two postulates: the laws of physics are the same in every inertial frame, and the speed of light c is the same for every observer regardless of how the source or observer moves. From these follow time dilation, length contraction, and a new relationship between energy, momentum, and mass. General relativity extends these ideas to gravity, treating it as the curvature of spacetime, where clocks deeper in a gravitational field also run slow.
The Lorentz factor
Almost every relativistic result is built from one quantity, the Lorentz factor γ, which depends only on the speed v expressed as a fraction of c. At low speed γ is essentially one and the familiar physics returns; as v approaches c, γ grows without limit.
Time dilation and length contraction
A moving clock ticks slowly: a time interval t₀ measured in the clock's own frame is observed to last a longer time t = γ t₀ in a frame the clock moves through. At the same time, a length L₀ measured at rest is observed to be shorter along the direction of motion, L = L₀ / γ.
Length contraction: L = L₀ / γ (moving lengths shrink)
Relativistic energy, kinetic energy, and momentum
A particle of rest mass m has a rest energy E₀ = mc² even when standing still. When it moves, its total energy is E = γmc², its kinetic energy is the difference KE = (γ − 1)mc², and its momentum is p = γmv. These reduce to the Newtonian forms at low speed but differ sharply as v approaches c.
Total energy: E = γ mc² = γ E₀
Kinetic energy: KE = (γ − 1) mc²
Momentum: p = γ mv, so pc = γ β E₀
Mass-energy equivalence
The rest energy E₀ = mc² says mass and energy are two forms of the same thing. A small amount of mass corresponds to an enormous amount of energy because c² is so large, which is why nuclear reactions release so much energy from tiny changes in mass.
One factor, many effects
The Lorentz factor γ sets the size of time dilation, length contraction, and the rise in energy and momentum.
Speed limit
As v approaches c, γ grows without bound, so it takes ever more energy to go faster and no massive object can reach c.
Mass is energy
Rest energy E₀ = mc² means even a particle at rest carries energy, and tiny mass changes release large energies.
| Quantity | Relationship | Notes |
|---|---|---|
| Lorentz factor | γ = 1 / √(1 − β²) | β = v/c |
| Time dilation | t = γ t₀ | moving clocks run slow |
| Length contraction | L = L₀ / γ | along the motion |
| Total energy | E = γ mc² | rest energy when γ = 1 |
| Kinetic energy | KE = (γ − 1) mc² | Newtonian at low speed |
| Momentum | pc = γ β E₀ | E² = (pc)² + (mc²)² |
Apparatus
The equipment a real special-relativity experiment uses. In the simulation these are modelled for you, but the readings correspond to what each instrument would measure.
Instructions — Running the Virtual Experiment
This is a record, calculate, and compare lab. In each part you set the speed, calculate the requested quantities by hand, enter your values, and only then does the simulation reveal its results so you can compare. Record every value in your worksheet.
Simulation — The Relativity Bench
| v/c | Your γ | γ actual | Your t (s) | Your L (m) | t, L actual |
|---|---|---|---|---|---|
| No rows yet — set the speed, calculate γ, t, and L, and check. | |||||
Moving frame
| Particle | v/c | Your γ | Your E (MeV) | Your KE (MeV) | E, KE actual |
|---|---|---|---|---|---|
| No rows yet — choose a particle and speed, calculate γ, E, and KE, and check. | |||||
Fast particle
| Particle | v/c | Your pc (MeV) | pc actual | Check E²=(pc)²+E₀² |
|---|---|---|---|---|
| No rows yet — calculate the momentum and check. | ||||
Momentum
Mass-energy: E = mc²
Team Questions
Example Lab Report
A worked example showing the expected format and the record, calculate, and compare workflow.
Special Relativity
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Objective
To calculate the Lorentz factor and use it to find the dilated time and contracted length, to determine the relativistic energy, kinetic energy, and momentum of a fast particle, and to confirm the mass-energy relation E = mc², comparing every calculated value with the simulation.
Part A — Time Dilation and Length Contraction (worked example)
At v = 0.80c: γ = 1 / square root of (1 − 0.80²) = 1 / square root of 0.36 = 1.667. For a proper time of 1.00 s, the dilated time is t = γ t₀ = 1.667 s. For a proper length of 1.00 m, the contracted length is L = L₀ / γ = 0.600 m. Both matched the simulation.
Part B — Relativistic Energy (worked example)
For an electron (E₀ = 0.511 MeV) at 0.80c: γ = 1.667, so the total energy is E = γ E₀ = 0.852 MeV and the kinetic energy is KE = (γ − 1) E₀ = 0.341 MeV. The relativistic kinetic energy is larger than the Newtonian estimate, as expected near c.
Part C — Momentum and Mass-Energy (worked example)
For the same electron, pc = γ β E₀ = 1.667 × 0.80 × 0.511 = 0.681 MeV. Checking, square root of ((0.681)² + (0.511)²) = 0.852 MeV, equal to the total energy, confirming E² = (pc)² + (mc²)². Converting 1.00 g of mass gives E = mc² = 0.00100 kg × (3.00 × 10⁸)² = 9.00 × 10¹³ J.
Discussion and Conclusion
Every calculated value agreed with the simulation. As the speed approached c the Lorentz factor grew quickly, time dilated and length contracted, and the energy and momentum rose far above their Newtonian values, while the rest energy E = mc² showed that a tiny mass corresponds to an enormous energy.