In Minkowski spacetime, which is the mathematical framework used in special relativity, the speed of light is a fundamental constant denoted by "c." Unlike in Euclidean space, where distances are measured with the Pythagorean theorem, in Minkowski spacetime, distances are calculated using a spacetime interval known as the Minkowski metric.
The Minkowski metric incorporates both space and time dimensions and is expressed as:
ds² = -c²dt² + dx² + dy² + dz²
Here, ds represents the infinitesimal spacetime interval, dt is the infinitesimal time interval, and dx, dy, and dz represent the infinitesimal spatial intervals in the x, y, and z directions, respectively. The factor of c² appears with the time interval to ensure that the units are consistent.
The Minkowski metric includes a negative sign in front of the c²dt² term, which distinguishes the spacetime interval from a spatial distance. The negative sign reflects the notion of spacetime being "pseudo-Euclidean" and accounts for the time component in the metric.
The important point to note is that in Minkowski spacetime, the speed of light is defined as the ratio between the spatial and temporal components of the spacetime interval. That is:
c = sqrt(dx² + dy² + dz²) / dt
This expression shows that the speed of light, c, is a constant that relates the spatial and temporal aspects of motion in Minkowski spacetime. It signifies the maximum speed at which information or causality can propagate through spacetime according to the theory of special relativity.