Minkowski Spacetime and the Interval

The Lorentz transformation mixes space and time coordinates so that the speed $c$ is the same in every inertial frame. Read algebraically, it is a rule for converting one observer's numbers into another's. Read geometrically, it is a symmetry of a single four-dimensional object — spacetime — that leaves one quantity fixed. That quantity is the interval, and building the rest of the theory on it turns coordinate bookkeeping into geometry. This lesson sets up the arena; the tensor language that follows in the next lessons is the calculus that runs on it.

Events, coordinates, and worldlines

The primitive object is the event: a point in space at an instant of time, a single here-now with no extent and no duration. An inertial observer labels each event with four numbers, one time and three space,

The time coordinate is carried as $x^0=ct$ so that all four components have the dimension of length. The index sits upstairs for a reason that the next lesson makes precise; for now it is a label running over the four coordinates. The collection of all events is Minkowski spacetime, written ${\mathbb{R}}^{1,3}$ to record that one coordinate plays a different role from the other three.

A worldline is the complete history of a particle, not a trajectory swept out in time. Nothing moves along it — the whole curve exists at once, and different observers slice it into space-at-an-instant differently. A light pulse from the origin traces the surface $c^2{t}^2=x^2+y^2+z^2$, which in a diagram with one space axis suppressed is a pair of lines at $45^{\circ }$, and in two space dimensions is a cone.

The invariant interval

Two nearby events separated by coordinate differences $\mathbf{d}{x}^{\mu }=(c\mathbf{d}t,\mathbf{d}x,\mathbf{d}y,\mathbf{d}z)$ have a spacetime interval built from a specific quadratic combination. Adopting the signature convention used throughout the general-theory modules,

The content of the Lorentz transformation is that $\mathbf{d}{s}^2$ takes the same value in every inertial frame. Substituting the boost $x^{\prime }=\gamma (x-vt)$, $t^{\prime }=\gamma (t-vx/c^2)$ into $-c^2\mathbf{d}{t}^{\prime 2}+\mathbf{d}{x}^{\prime 2}$ and expanding, the cross terms cancel and the ${\gamma }^2(1-{\beta }^2)=1$ identity collapses the result to $-c^2\mathbf{d}{t}^2+\mathbf{d}{x}^2$. The transverse terms are untouched. Hence

Individual coordinate differences are frame-dependent — this is time dilation and length contraction — but this one combination is not. The interval is to spacetime what squared distance is to Euclidean space, with the single sign difference in the time slot carrying all of relativity.

Causal structure

The sign of the interval between two events is invariant, so it sorts pairs of events into three classes that all observers agree on. For two events separated by $\Delta t$ in time and $\Delta r=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}$ in space, with $\Delta s^2=-c^2\Delta t^2+\Delta r^2$:

• Timelike ($\Delta s^2<0$): $c\left|\Delta t\right|>\Delta r$. A signal slower than light can travel between the two events, and every observer agrees on their time order. A massive particle can have both events on its worldline.
• Null or lightlike ($\Delta s^2=0$): $c\left|\Delta t\right|=\Delta r$. Only a light signal connects them. The two events lie on each other's light cone.
• Spacelike ($\Delta s^2>0$): $c\left|\Delta t\right|<\Delta r$. No signal at or below $c$ connects them. Their time order is frame-dependent, so neither can be a cause of the other.

The set of events null-separated from a given event $O$ forms $O$'s light cone, and it partitions spacetime into the absolute future (timelike, later), the absolute past (timelike, earlier), and the elsewhere (spacelike). Because the partition is built from the invariant $\Delta s^2$, it is the same partition for every inertial observer.

That the causal partition is invariant is the reason relativity never lets an effect precede its cause even while it lets observers disagree about simultaneity. If $B$ lies in the future light cone of $A$, no boost can carry $B$ outside it; the ordering $A$ before $B$ is absolute exactly for the pairs that could be cause and effect.

Proper time as the length of a worldline

Along a timelike worldline $\mathbf{d}{s}^2<0$, and the natural positive quantity is the proper time, the time read by a clock carried along the worldline. Define it by

Dividing through by $\mathbf{d}{t}^2$ and writing the coordinate speed as $v^2=(\mathbf{d}{x}^2+\mathbf{d}{y}^2+\mathbf{d}{z}^2)/\mathbf{d}{t}^2$,

This reproduces time dilation, now read off the geometry: the proper time elapsed on a worldline is its Minkowski arc length (up to the factor $c$), and because $\gamma \ge 1$, a moving clock's proper time advances more slowly than coordinate time. Integrating along a worldline from event $\mathcal{A}$ to event $\mathcal{B}$,

Proper time is a property of the worldline itself, independent of the coordinates used to compute the integral, because $\mathbf{d}\tau$ is built from the invariant $\mathbf{d}{s}^2$. Two worldlines connecting the same pair of events generally accumulate different proper times.

The twin paradox as extremal proper time

The twin paradox has a one-line geometric statement. Take two timelike-separated events, departure $\mathcal{A}$ and reunion $\mathcal{B}$. One twin stays inertial: a straight worldline from $\mathcal{A}$ to $\mathcal{B}$. The other travels out and back: a bent worldline through the same endpoints. The proper times differ because the worldlines differ, and the inertial one is longer.

The traveling twin, whose worldline bends at the turnaround, returns younger. No paradox survives once the two histories are recognized as different curves between the same endpoints; asking which twin really moved is asking which curve is straight, and only one of them is. The asymmetry is the turnaround, where the traveling twin's frame is non-inertial.

The straight worldline is the flat-spacetime instance of a geodesic, the extremal-proper-time curve between two events. In flat spacetime the geodesic is a straight line and it maximizes proper time; the curved-spacetime modules generalize the extremal principle, and free fall in a gravitational field turns out to be exactly geodesic motion — the longest-proper-time worldline through a curved geometry.

The metric as a measuring convention

The interval assigns a number to every pair of nearby events, and that assignment is the metric of Minkowski spacetime. In the coordinates $x^{\mu }=(ct,x,y,z)$ it is the diagonal array

with the summation over repeated indices anticipated. The metric is what converts coordinate differences into the physical, frame-independent interval; it is simultaneously the clock (for timelike directions, via $\mathbf{d}\tau$) and the ruler (for spacelike directions, via proper length). On the light cone it reads zero — light-cone directions have no proper time and no proper length, which is why one cannot ride alongside a light beam and watch it stand still.

The array ${\eta }_{\mu \nu }$ is the object that the next lesson promotes to a tensor, giving the rules for raising and lowering indices, forming invariant scalar products, and defining the four-velocity and four-acceleration that carry relativistic dynamics. Everything downstream is bookkeeping organized so that the invariance of $\mathbf{d}{s}^2$ is automatic.