The presence of the factor of c^2 and the negative sign in the spacetime interval formula is a consequence of the geometry of spacetime as described by the theory of relativity.
In the theory of relativity, space and time are unified into a four-dimensional framework called spacetime. The metric used to describe the geometry of spacetime is known as the Minkowski metric, and it incorporates the factor of c, the speed of light in a vacuum.
The factor of c arises because the theory of relativity postulates that the speed of light is a fundamental constant of nature, and it sets an upper limit on the speed at which information and causal influences can propagate. The speed of light, c, is approximately 3 x 10^8 meters per second.
The negative sign in the spacetime interval formula is related to the distinction between timelike and spacelike intervals in spacetime. The spacetime interval is a measure of the separation between two events in spacetime, and it characterizes the overall "distance" between them. When the interval is positive, it indicates that the events are separated by a timelike interval, meaning that the time component dominates the separation. When the interval is negative, it signifies a spacelike separation, where the spatial components dominate.
The negative sign in the spacetime interval formula (-c^2t^2 + x^2 + y^2 + z^2) ensures that the interval is positive for timelike separations and negative for spacelike separations. It is a mathematical representation that aligns with the empirical observations and experimental results consistent with the theory of relativity.
In summary, the factor of c^2 and the negative sign in the spacetime interval formula stem from the fundamental properties of spacetime and the postulate of the constancy of the speed of light in the theory of relativity. These mathematical expressions have been validated by numerous experiments and observations, supporting the theory's predictions and its understanding of the geometry of spacetime.