The geometry of boosting between reference frames
Switch to a moving observer and space and time get mixed together. The Lorentz transformations are the exact rule for that mixing.
Click to place events, then drag the velocity slider to boost. Watch the axes tilt toward the 45° light line, and notice that events on one horizontal line no longer share a time.
The Lorentz transformations are the mathematical engine of special relativity. They replace the Galilean transformations of Newtonian mechanics and describe how space and time coordinates change when you switch between inertial reference frames. The result: space and time are not separate — they mix together under boosts.
A Minkowski diagram plots time (vertical) against space (horizontal). Light travels along 45° lines. When you boost to a moving frame, the axes tilt toward each other, squeezing the light cone from both sides while keeping it at 45°.
Try it: Place events on the diagram, then drag the velocity slider to boost. Watch how the simultaneity surface (horizontal line in the rest frame) tilts — events that were simultaneous in one frame happen at different times in another.
Drag the speed slider and watch the rod shrink along its direction of motion. At 86.6% of c, a 1-meter rod measures 0.5 meters.
A moving object is contracted along its direction of motion by a factor of 1/γ. A 1-meter rod moving at 86.6% of c appears to be only 0.5 meters long to a stationary observer. This is not an optical illusion — it's a real consequence of spacetime geometry.
Press Play to strike both ends of the train with lightning, then switch between Station Frame and Train Frame. Strikes that are simultaneous on the platform happen at different times for the train.
Einstein's famous train thought experiment: two lightning bolts strike the ends of a moving train simultaneously in the platform frame. But the train observer, rushing toward one flash and away from the other, sees them at different times. Simultaneity is relative.
Key insight: This isn't about signal delays or observation tricks. The events genuinely happen at different times in different frames. The ordering of spacelike-separated events depends on your reference frame.
Click or drag along the curve to read γ at any speed. It stays near 1 for most of the range, then shoots toward infinity as v approaches c.
The factor γ = 1/√(1 - v²/c²) appears everywhere in special relativity. It controls time dilation, length contraction, and relativistic momentum. Near v = 0, γ ≈ 1 and effects are negligible. As v → c, γ → ∞ and relativistic effects dominate.
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