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Special Relativity

Einstein's special theory of relativity — where space and time merge into spacetime, clocks slow down near the speed of light, and simultaneity becomes relative. Interactive Minkowski diagrams and Lorentz transformations in your browser.

1+ simulations Canvas 2D Lorentz · Minkowski · Spacetime

Category Simulations

Each simulation runs fully in the browser — no server, no installation

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★★★ Advanced New
Lorentz Transform
Interactive Minkowski spacetime diagram with two inertial frames S and S′. Set velocity β = v/c and drag event A to watch coordinates transform: x′ = γ(x − βct), ct′ = γ(ct − βx). Visualise time dilation, length contraction, and the relativity of simultaneity.
Lorentz Transform Minkowski Diagram Spacetime Interval Canvas 2D
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★★★ Advanced New
Time Dilation & Length Contraction
Watch clocks tick slower and rulers shrink as velocity approaches c. Animated dual-clock comparison, length contraction ruler, Minkowski spacetime diagram with moving world lines, and live Lorentz factor γ.
Time Dilation Length Contraction Lorentz Factor Twin Paradox
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New ★★☆ Intermediate
Mass–Energy Equivalence E=mc²
Explore nuclear fission (U-235) and fusion (D-T) through E=mc². See the binding energy curve, mass defect and chain reactions.
E=mc² Nuclear Fission Binding Energy
New ★★☆ Intermediate
Minkowski Spacetime Diagram
Interactive spacetime diagram. Adjust β = v/c to Lorentz-boost the moving frame. See world lines, light cone and simultaneity.
Spacetime World Line Lorentz Boost
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New ★★☆ Intermediate
Twin Paradox
Animate the twin paradox: one twin rockets away and ages less. Lorentz time dilation with spacetime diagram and clock readouts.
Time Dilation Proper Time Gamma Factor
New ★★☆ Intermediate
Lorentz Contraction
Watch a moving ruler compress along the direction of motion at relativistic speed. Includes transverse comparison and preset velocities.
Length Contraction Lorentz Factor Relativistic

Lorentz Transformation (S → S′)

The four fundamental equations of special relativity for a boost along x

x′ = γ(x − βct)
Space coordinate
Objects at rest in S appear length-contracted by 1/γ in S′: L′ = L/γ.
ct′ = γ(ct − βx)
Time coordinate
Clocks moving at β tick more slowly by 1/γ: Δt′ = γΔt (time dilation).
γ = 1/√(1 − β²)
Lorentz factor
γ ≥ 1, diverging as β → 1. At β = 0.87 already γ ≈ 2; at β = 0.995, γ ≈ 10.
s² = c²t² − x²
Spacetime interval
Lorentz-invariant scalar. Timelike (s²>0), lightlike (s²=0), spacelike (s²<0).
E² = (pc)² + (m₀c²)²
Energy–momentum relation
For rest mass m₀: kinetic energy K = (γ−1)m₀c². At rest: E = m₀c².
u′ = (u−v)/(1−uv/c²)
Relativistic velocity addition
No matter how fast two objects move, combined speed in any frame never exceeds c.

About Special Relativity Simulations

From postulates to Minkowski geometry — explored interactively

Special relativity (Einstein, 1905) rests on two postulates: the laws of physics are the same in all inertial frames, and the speed of light c is constant in all frames regardless of the motion of source or observer. These simple postulates force a radical restructuring of space and time into a unified four-dimensional spacetime.

The Minkowski diagram is the most direct way to visualise this new geometry. Each point (event) in spacetime has coordinates (x, ct). Two inertial frames S and S′ — where S′ moves at velocity v = βc relative to S — are related by the Lorentz transformation. In the diagram, the S′ axes appear tilted toward the 45° light cone as β increases: both axes tilt symmetrically, which ensures c remains the same in both frames.

The most counter-intuitive consequence is the relativity of simultaneity: events that are simultaneous in S (same ct coordinate) generally have different ct′ values in S′. There is no absolute "now" — only the spacetime interval s² = c²t² − x² is invariant. Timelike separated events can be causally connected; spacelike ones cannot (and their time-ordering can be reversed by boosting).

All Categories

Key Concepts

Topics and physics you'll find in this category

Lorentz Factor γ γ = 1/√(1−β²); multiplies all relativistic effects; diverges as v→c
Time Dilation Moving clocks tick slower: Δt = γΔτ where Δτ is proper time
Length Contraction Moving rulers appear shorter: L = L₀/γ along the direction of motion
Relativity of Simultaneity Two events simultaneous in S need not be in S′; there is no universal "now"
Spacetime Interval s² = c²t²−x² is Lorentz-invariant; classifies events as timelike/lightlike/spacelike
Minkowski Geometry Spacetime with metric signature (−,+,+,+); hyperbolic geometry, not Euclidean
Light Cone The set of all events reachable at |v| ≤ c; defines causal past and future
E = mc² Rest-mass energy; total energy E = γm₀c²; kinetic energy K = (γ−1)m₀c²

Frequently Asked Questions

Common questions about special relativity

What does the Minkowski diagram simulation show?
The Lorentz Transform simulation draws a spacetime diagram with two frames S (blue) and S′ (orange). You can set the relative velocity β = v/c with a slider and drag event A anywhere in spacetime. The diagram shows the S and S′ coordinate grids, light cone, projections of the event onto both sets of axes, and the spacetime interval type (timelike, lightlike, or spacelike).
Why do the S′ axes tilt toward the light cone?
Both the x′ and ct′ axes tilt toward the 45° light cone as β increases. This is a direct consequence of the invariance of c: the light cone must look the same in both frames, so both axes tilt inward symmetrically. The space axis tilts up (slope = β) and the time axis tilts right (slope = 1/β in ct-vs-x terms). At β = 1 both axes would coincide with the light cone.
Why do S′ tick marks appear stretched?
The Minkowski diagram uses a pseudo-Euclidean (hyperbolic) metric, not ordinary Euclidean geometry. Equal intervals of proper time or proper length lie on calibration hyperbolas x²−c²t² = const. The tick marks on S′ axes are placed on these hyperbolas, which means they appear further from the origin in Euclidean distance than the S tick marks — a visual artefact of the non-Euclidean nature of spacetime.
Is time dilation a real physical effect?
Yes. GPS satellites run their clocks fast by ~38 µs/day to compensate for combined special (time dilation, −7 µs/day) and general (gravitational blueshift, +45 µs/day) relativistic effects. Without correction, GPS position errors would accumulate at ~10 km/day. Muons produced by cosmic rays at 15 km altitude survive long enough to reach Earth's surface only because of time dilation; in their rest frame, the atmosphere is length-contracted.