Change the observer.
Keep the physics.
Special relativity replaces one universal space and one universal time with spacetime. Different inertial observers can disagree about distances, durations and simultaneity—while agreeing on causal order for timelike events, the speed of light, and the spacetime interval.
Light cones classify what can influence what.
Place Event B relative to Event A. The interval classifies the separation as timelike, lightlike or spacelike.
Move event B
What the classification means
Timelike: a slower-than-light signal could connect the events. Lightlike: only light-speed propagation connects them. Spacelike: no causal influence can connect them without exceeding light speed.
Lorentz transforms change coordinates, not the interval.
Boost to another inertial frame and watch x and t change while s² remains fixed.
Choose observer velocity
Same events, different coordinates
The transformed coordinates are x′ = γ(x − βt) and t′ = γ(t − βx) when c = 1. The coordinate values depend on the frame; the interval does not.
Simultaneity depends on the inertial frame.
Two flashes occur at the same time in one frame but at different positions. Change frames and compare their transformed times.
Two simultaneous flashes
Important limit
Only spacelike-separated events can reverse temporal order between inertial frames. If two events are timelike-separated, all inertial observers agree on which happened first.
Proper time belongs to the clock that travels the worldline.
Set a coordinate duration and a constant velocity. The moving clock accumulates less proper time between the same two events.
Move the clock
What time dilation does—and does not—mean
For inertial motion at constant speed, Δτ = Δt/γ. Each inertial observer regards their own local clock as normal. Comparing elapsed proper times requires specifying the worldlines and the pair of events being connected.
Velocities do not add the Newtonian way near light speed.
Transform a signal velocity into another frame. Ordinary speeds change; a light signal remains at c.
Transform a signal
The causal speed limit survives frame changes
Relativistic velocity addition is u′ = (u − v)/(1 − uv/c²). If u = ±c, every inertial observer still measures ±c. That invariance is what keeps the light cone—and the causal structure—consistent.