Spacetime: The Geometry of the Universe
Special relativity revealed that space and time aren’t separate. An event that’s “here and now” for you might be “there and then” for someone moving relative to you. The stage on which physics happens isn’t 3D space plus a separate clock — it’s a single 4D structure: spacetime.
Minkowski spacetime
Hermann Minkowski (Einstein’s former teacher) formalised this in 1908. Spacetime is a 4-dimensional space with coordinates $(ct, x, y, z)$ — we multiply time by $c$ so all four dimensions have units of length.
The key structure is the spacetime interval between two events:
$$\Delta s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$
This is like the Pythagorean distance formula, but with a minus sign on the time component. That sign changes everything.
Three types of separation
The sign of $\Delta s^2$ classifies every pair of events:
| $\Delta s^2$ | Type | Meaning |
|---|---|---|
| $< 0$ | Timelike | One event can influence the other. A massive particle can travel between them. |
| $= 0$ | Lightlike (null) | Only light can connect them — they’re on each other’s light cone. |
| $> 0$ | Spacelike | No signal can connect them. They’re “simultaneous” in some frame. |
For timelike-separated events, the proper time $\Delta\tau = \sqrt{-\Delta s^2}/c$ is what a clock travelling between them would actually measure. This is frame-independent — all observers agree on it.
Light cones
A spacetime diagram with one space dimension (horizontal) and time (vertical). The dashed 45° lines are the light cone — the boundary of causal influence.
Drag the yellow event. Events inside the light cone (shaded) are causally connected to the origin. Events outside are spacelike-separated — unreachable by any signal.
At each event, the light cone divides spacetime into three regions:
- Future light cone: all events this event can influence (above, inside the cone)
- Past light cone: all events that could have influenced this one (below, inside the cone)
- Elsewhere: spacelike-separated events that are causally disconnected
Nothing with mass can cross the light cone. Worldlines of massive objects are always inside the cone — always timelike.
Four-vectors
In 3D, a vector has three components $(v_x, v_y, v_z)$. In spacetime, a four-vector has four: $(v^0, v^1, v^2, v^3)$, where component 0 is the time part.
The four-position: $x^\mu = (ct, x, y, z)$
The four-velocity: $u^\mu = \gamma(c, v_x, v_y, v_z)$ — the rate of change of position with respect to proper time. Its “length” (computed with the spacetime metric) is always $-c^2$, regardless of speed.
The four-momentum: $p^\mu = m \, u^\mu = (\gamma mc, \gamma m v_x, \gamma m v_y, \gamma m v_z) = (E/c, p_x, p_y, p_z)$
The time component of four-momentum is energy (divided by $c$). The space components are ordinary momentum. Energy and momentum are unified — they’re components of the same four-vector, seen from different frames.
The invariant “length” of the four-momentum gives:
$$E^2 = (pc)^2 + (mc^2)^2$$
For a particle at rest ($p = 0$): $E = mc^2$.
Why this matters for general relativity
Minkowski spacetime is flat — the interval formula uses constant coefficients $(-1, +1, +1, +1)$ everywhere. This is fine when gravity is absent.
General relativity says: in the presence of mass and energy, these coefficients become functions of position. The geometry of spacetime itself becomes curved. The constant Minkowski metric becomes the dynamic metric tensor — and the story of gravity becomes a story of geometry.