Spacetime Equation: A Geometrization of Special Relativity Based on "Temporal Flow Rate"

Starting from the time dilation effect of special relativity, this paper introduces a new physical quantity with the dimension of velocity — "time flow rate", defined as the product of the rate of change of the proper time of a moving observer relative to the coordinate time of the laboratory frame and the speed of light. Based on this definition, we derive a concise geometric equation:v2+t2=c2 where v is the relative velocity of an object, t its "time flow rate", and c the speed of light. Mathematically equivalent to the Lorentz transformation, this equation provides a novel and intuitive geometric picture for understanding spacetime: in the velocity–time flow rate plane, all physically allowed motion states form a quartercircle of radius c. Velocity and time passage act as two orthogonal components, whose resultant vector has a constant magnitude equal to the speed of light. This formulation not only clearly reveals the conserved tradeoff between velocity and time dilation, but also naturally connects to the Minkowski metric of fourvelocity, possessing significant pedagogical and interpretive value.