(because of length contraction). But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by and a spatial distance Then the spacetime interval between the two events that are separated by a distance in space and by in the -coordinate is: or for three space dimensions, The constant the speed of light, converts time units (like seconds) into space units (like meters). The squared interval is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects undergoing events, or because a single object in space is moving inertially between its events. The separation interval is derived by squaring the spatial distance separating event B from event A and subtracting it from the square of the spatial distance traveled by a light signal in that same time interval . If the event separation is due to a light signal, then this difference vanishes and . When the event considered is infinitesimally close to each other, then we may write In a different inertial frame, say with coordinates , the spacetime interval can be written in a same form as above. Because of the constancy of speed of light, the light events in all inertial frames belong to zero interval, . For any other infinitesimal event where , one can prove that which in turn upon integration leads to . The invariance of interval of any event between all intertial frames of reference is one of the fundamental results of special theory of relativity. Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, means , etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning. The equation above is similar to the Pythagorean theorem, except with a minus sign between the and the terms. The spacetime interval is the quantity not itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than