# Special Relativity Calculator - Time Dilation, Twin Paradox & Minkowski Diagram

> Lorentz factor, time dilation and length contraction with a rocket animation, relativistic energy, velocity addition, twin paradox and a Minkowski diagram.

- URL: https://qilingtools.com/en/math/relativity
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

One speed slider (or type β, km/s or γ directly) shows the main effects of special relativity side by side: the Lorentz factor, moving clocks running slow and length contracting along the motion, with an animated rocket that shrinks and a ship clock that falls behind. The energy tab compares relativistic and classical kinetic energy for an electron, a proton or a 1 kg object, and can solve for the speed from a kinetic energy. Velocity addition and twin-paradox trips, at constant speed or a steady 1 g, each get their own calculator and world-line chart. In the spacetime diagram you click to place events and drag β to watch simultaneity change.

## How to use

1. Drag the speed in the first tab or type β, km/s or γ, or pick a preset such as a GPS satellite, a cosmic-ray muon or an LHC proton
2. Read off γ, time dilation and length contraction, and enter a proper time or proper length for concrete numbers
3. Use the energy, velocity-addition and twin-paradox tabs for those calculations
4. In the spacetime diagram, click to add events, drag β or switch frames, and watch the lines of simultaneity and the order of events change

## Features

- Speed as β, km/s or γ, with γ − 1 computed accurately even at everyday speeds
- Animated length-contracted rocket and two ticking clocks
- Rest, total and kinetic energy (relativistic vs classical) and momentum, auto-scaled in eV; speed from kinetic energy
- Relativistic velocity addition and additive rapidities
- Twin paradox at constant speed or 1 g constant acceleration, with proper-time ticks on the world lines
- Interactive Minkowski diagram: light cone, skewed axes, simultaneity lines, interval type and event order

## FAQ

### Do GPS satellites really need relativity?

Yes. A GPS satellite moves at about 3.87 km/s, and special-relativistic time dilation makes its clock lose about 7.2 μs per day (enter that speed here to check). It also sits higher in Earth's gravity well, and the general-relativistic effect makes it gain about 45.7 μs per day, for a net gain of about 38 μs per day. Left uncorrected, positions would drift by more than 10 km a day, so the onboard atomic clocks are deliberately set slow before launch.

### If motion is relative, why is the travelling twin younger?

The situations aren't symmetric. The twin on Earth stays in one inertial frame, while the traveller turns around and switches frames. At the turnaround the traveller's line of simultaneity tilts, and “now on Earth” jumps forward by years. A spacetime diagram makes it clear: of two world lines joining the same start and end events, the straight (inertial) one has the most proper time and the bent one less. Acceleration itself isn't the cause; changing frames is.

### Why can't anything be accelerated to the speed of light?

Kinetic energy is (γ − 1)mc², and γ → ∞ as v → c, so the required energy grows without limit. Momentum γmv diverges as well. The LHC brings protons to 6.8 TeV, γ ≈ 7250 and 0.99999999c; adding more energy raises γ but barely changes the speed. The addition rule w = (u + v)/(1 + uv/c²) likewise guarantees that combining two sub-light speeds always gives a sub-light speed.

### What is a line of simultaneity, and why does “at the same time” depend on the observer?

In the rest frame S, simultaneous events lie on a horizontal line. A frame S′ moving at β has its time axis along x = β·ct, and its lines of simultaneity, parallel to the x′ axis, along ct = β·x, tilted upward. So two events simultaneous in S happen one after the other in S′. Only spacelike-separated events (Δs² < 0) can swap order, and they can't be causally connected. Timelike events keep their order in every frame, so causality survives.

### How does E = mc² relate to relativistic kinetic energy?

Even at rest a body has rest energy E₀ = mc². In motion its total energy is E = γmc², and the extra (γ − 1)mc² is the kinetic energy. At low speed (γ − 1)mc² ≈ ½mv², the classical formula. For an electron at 0.9c the relativistic kinetic energy is already more than three times the classical value. Energy and momentum obey E² = (pc)² + (mc²)², so for a massless photon E = pc.

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