Lorentz Transformations | Special Relativity Ch. 3

Special relativity’s core move is replacing ordinary “sliding” of spacetime diagrams with a specific kind of geometric transformation that keeps the speed of light the same for every observer. On a spacetime diagram, an observer’s own worldline is always a straight vertical line at x = 0. Changing to a moving observer’s perspective therefore can’t be done by merely shifting the diagram; the worldline of the moving object must rotate to become vertical, meaning the transformation must change angles—i.e., it must be a “boost.”

The first candidate is a shear-like transformation: slide each constant-time snapshot by an amount proportional to time so that the moving object’s worldline becomes vertical while the time coordinate of each event stays the same. This works mathematically for ordinary velocities: if a cat moves at speed v relative to one observer, the observer can be made to appear to move at the same speed v relative to the cat, matching the symmetry of relative motion. In the cat’s frame, the other observer’s velocity changes exactly as expected from the sliding rule.

But shear transformations fail a decisive experimental test: the speed of light in vacuum. Everyday objects obey velocity addition—light does not. Experiments show that light rays have the same measured speed no matter how fast the source or observer is moving. A shear transformation changes all velocities in a way tied to time-snapshot sliding, so even if it makes one worldline vertical, it would also alter the slope of light-ray worldlines. Since light’s speed must remain unchanged, shear transformations are ruled out.

That leaves the other geometric possibilities for turning an angled worldline vertical while preserving the angle between worldlines: transformations that behave like a “squeeze rotation,” where points are mapped to earlier or later times as part of the rotation-like adjustment. These squeeze-rotation boosts can keep one particular speed unchanged—and, crucially, can keep the speed of light unchanged in every direction. In the example with a slow-moving sheep and faster cats, sliding alone would spoil the cats’ speed when switching frames. A squeeze-rotation transformation instead preserves the cats’ worldline angles (their speeds) while also producing the correct new perspective for the sheep.

With the standard spacetime-diagram scaling—where one second is one tick vertically and 299,792,458 meters is one tick horizontally—the speed of light appears as 45° lines. Under a Lorentz transformation, spacetime is “squeezed” along one 45° direction and “stretched” along the other in a precise proportional way. The angles of non-light worldlines change, reflecting different perceived speeds, yet the 45° light rays stay fixed.

These squeeze-rotation boosts are the Lorentz transformations, named after early derivations by Hendrik Lorentz. They form the mathematical backbone of special relativity and set up the later results—time dilation, length contraction, relativity of simultaneity, and the twins paradox—by dictating how motion and causality look across different inertial frames. The episode also highlights a physical “time globe” device that implements Lorentz transformations mechanically, letting learners explore frame changes without starting from heavy algebra.