Theory of relativity

Minkowski spacetime

Space and time form a four-dimensional spacetime; events are separated by an interval invariant under Lorentz transformations.

An event is a point in spacetime with coordinates (ct,x,y,z). Its geometry assigns opposite signs to time and space components, so it is not ordinary four-dimensional Euclidean distance.

s^2=c^2\Delta t^2−\Delta x^2−\Delta y^2−\Delta z^2

Definition: Definition

For two events, $s^2>0$ is timelike, $s^2=0$ lightlike, and $s^2<0$ spacelike, using the sign convention shown. Spacelike-separated events cannot be causally connected at or below speed c.

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Physical meaning

The worldline of a massive object lies inside the light cone; a light ray lies on it. The cone separates events that can influence, or be influenced by, a given event.

Example: Worked example

For Δt=5 μs and Δx=900 m with no other spatial displacement, take c=300 m/μs. Find s² and classify the interval.

Solution

$s²=(300·5)²−900²=1,440,000$ m²>0. The separation is timelike; a frame exists in which the events occur at the same position.

Each inertial observer chooses one time axis and three spatial axes, but a boost changes only how spacetime is decomposed into those coordinates. The light cone divides events into future, past, and regions unreachable by subluminal signals. This causal structure is observer-independent.

Example: Classifying a spacetime separation

Two events are separated by $\Delta t=4\,\mu s$ and $\Delta x=1{,}500\,m$ along one axis. With $c=300\,m/\mu s$ and $s^2=c^2\Delta t^2-\Delta x^2$ , can the events be causally connected?

Solution

$c\Delta t=1{,}200\,m$ , so $s^2=1{,}200^2-1{,}500^2=-810{,}000\,m^2<0$ . The separation is spacelike; no signal traveling at or below $c$ can connect the events.

The coordinate $ct$ has units of length, giving the interval $ds^2=-c^2dt^2+dx^2+dy^2+dz^2$ in this signature. A Lorentz transformation is a hyperbolic rotation preserving this quadratic form, just as a Euclidean rotation preserves $x^2+y^2$ . The opposite sign creates the light cone and distinguishes timelike, spacelike, and null separations; time is not simply another Euclidean spatial direction.

Quick check

If s²=0 for two events, how are they separated?

Which quantity is invariant under Lorentz transformations?

References

Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime Physics
Robert Resnick (1968). Introduction to Special Relativity