In classical physics, distances between points are measured using the Pythagorean theorem. But in special relativity, we deal with events in spacetime — each described by both spatial and temporal coordinates. The appropriate "distance" between two events is called the spacetime interval, and it reflects the geometry of Minkowski space.
Unlike classical distance, this interval combines space and time in a non-Euclidean way — and remains invariant under Lorentz transformations. This invariance forms the geometric foundation of all of special relativity.
To define a notion of "length" for four-vectors, we introduce the Minkowski metricημν, which determines how components are combined. In the (−,+,+,+) signature (commonly used in physics), the metric is:
ημν=⎣⎢⎢⎢⎡−10000+10000+10000+1⎦⎥⎥⎥⎤
This is a flat, symmetric metric that defines the geometry of Minkowski space — the spacetime of special relativity.
Using the metric, we can "lower" the index of a contravariant four-vector:
Xμ=ημνXν
So the covariant version of Xμ=(ct,x,y,z) becomes:
In special relativity, the distance between two events in spacetime isn't measured with the Pythagorean theorem, but rather with a modified expression that accounts for the geometry of Minkowski space. This distance is called the spacetime interval.
Let two events be represented by four-vectors:
Xμ=(ct,x,y,z),Yμ=(ct′,x′,y′,z′)
We define their separation as the displacement four-vector:
ΔXμ=Xμ−Yμ=(cΔt,Δx,Δy,Δz)
To measure the “length” of this displacement, we use the Minkowski metricημν to construct the invariant inner product, just as we would normally use the standard inner product to find the length of a vector:
s2=ημνΔXμΔXν.
This gives:
s2=−c2(Δt)2+(Δx)2+(Δy)2+(Δz)2,
which is known as the spacetime interval between the two events. It is a real scalar — meaning it has the same value in all inertial frames. This invariance makes it analogous to the squared length of a vector in Euclidean space, but with one critical difference: the minus sign in the time component reflects the non-Euclidean geometry of spacetime.
One of the foundational principles of special relativity is that the laws of physics take the same form in all inertial frames. This means that physically meaningful quantities — like the spacetime interval — must be the same for all observers, regardless of their relative motion.
Let Xμ=(ct,x,y,z) be the four-position of an event in one frame, and let X′μ be the same event as seen in another inertial frame. The Lorentz transformation connects these frames:
X′μ=ΛνμXν
Now consider the invariant spacetime interval:
s2=ημνXμXν
Under the Lorentz transformation, we substitute Xμ=(Λ−1)ρμX′ρ to compute s2 in the new frame:
s2=ημν(Λ−1)ρμX′ρ⋅(Λ−1)σνX′σ
Or, equivalently:
s2=ηρσX′ρX′σ
This result holds if and only if the Lorentz transformation satisfies:
ημν=ΛμρΛνσηρσ
This is the defining condition of a Lorentz transformation: it preserves the Minkowski metric.
Just as the Euclidean dot product between two spatial vectors is preserved under a rotation (because rotations preserve angles and lengths), the Minkowski inner product between two four-vectors is preserved under a Lorentz transformation.
The invariance of s2 is not a consequence of relativity — it's built into the structure of the Lorentz transformation itself.
In Euclidean space, the dot product of a vector with itself gives its squared length:
∣x∣2=x2+y2+z2
In Minkowski space, the spacetime intervals2 plays a similar role — but reflects the mixed nature of time and space. It's a scalar invariant under Lorentz transformations and encodes the causal structure of spacetime.
Instead of a sphere, the invariant surface of constant s2=0 defines a light cone.
The light cone separates events into the causally connected interior (timelike), unreachable exterior (spacelike), and the lightlike boundary.