We consider a new expression for the dependence of mass on velocity, more general than the corresponding law of the special theory of relativity (STR). The deviations from the STR become large with increasing rest mass. One should therefore measure the dependence of mass on velocity for objects with a large rest mass. The theory predicts that particles with real mass can travel with hyperlight velocities. The space-time picture discussed here is close to Mach's conception: It is assumed that the dynamical behavior of a particle in uniform translational motion is due to the “action” of all the other masses in the universe. Space-time is eliminated as an active cause and, in contrast to the STR, is not absolute within the theory discussed here. It turns out that effects based on the new transformation formulas (from the coordinates and time in a “stationary” frame to the coordinates and time in a “moving” frame) are identical to those expected from the Lorentz transformations. For example, it is known that rapidly “moving”Μ mesons decay with a longer half-life than “stationary”Μ mesons and the STR describes this effect quantitatively. However, there is no strong evidence for the validity of the STR because the theory given in this paper predicts the same result.