Worldline

Your whole life is a single curve through spacetime. Stand still and you still move, straight up the time axis at one second per second. Drag the boost slider and watch the frame tilt: your sense of "now" rotates, simultaneity bends, but one shape refuses to move at all.

The concept

In special relativity there is no universal clock. Each observer slices spacetime into "space now" and "time" along their own tilt, and two observers in relative motion disagree about which events are simultaneous. What they cannot disagree about is the light cone: the set of events a flash from here could reach. It is the same for everyone, because the speed of light is the same for everyone.

The math

Boosting to a frame moving at velocity $v$ (in units where $c = 1$, with $\beta = v$ and $\gamma = 1/\sqrt{1-\beta^2}$) mixes time and space by the Lorentz transformation:

$t' = \gamma\,(t - \beta x), \qquad x' = \gamma\,(x - \beta t)$

This is a hyperbolic rotation, not a circular one. The quantity it leaves unchanged is the spacetime interval:

$s^2 = t^2 - x^2$

Circles of constant $x^2 + y^2$ become hyperbolae of constant $t^2 - x^2$. The light cone $s^2 = 0$ is the asymptote every boost leaves fixed, which is exactly why nothing crosses from outside your future cone into it.

Why it stays strange

Tilt the frame far enough and two events you called simultaneous now happen in a definite order, yet no boost ever flips the order of cause and effect, because that would mean crossing the cone. Relativity is often sold as "everything is relative." The truth is sharper: almost everything is relative precisely so that one thing, the causal structure, can be absolute.

Further reading

• Taylor and Wheeler, Spacetime Physics - the spacetime-diagram bible.
• Einstein (1905), "On the Electrodynamics of Moving Bodies."

@book{taylor1992spacetime,
  author    = {Taylor, Edwin F. and Wheeler, John Archibald},
  title     = {Spacetime Physics},
  edition   = {2nd},
  publisher = {W. H. Freeman},
  year      = {1992}
}