The nature of space and time in current relativity physics is defined by Einstein’s special relativity theory (SRT), with length contraction and time dilation in moving inertial systems as fundamental elements in SRT’s structure of space and time. Both are essentially ad hoc concepts, supported by a wealth of complex and exacting experiments. This paper instead starts by considering the simplest possible relativistic experiment — the propagation of a beam of light, as seen from the principle of relativity point of view. It is found that the principle of relativity by itself transforms space and time coordinates of an optical wave between inertial systems, and secures the covariance of the Maxwell equations — both otherwise the prerogative of the Lorentz transformation. When analyzed by those coordinate transformations defined by the principle of relativity, crucial relativistic experiments that provide the experimental support of SRT’s concepts of space and time are explained without recourse to length contraction or time dilation. Consequently, time dilation and length contraction of rigid bodies inside moving inertial systems do not occur in a real world ruled by the principle of relativity. In particular, experiments show that time proceeds equally fast in a moving inertial system as in a stationary system. Concepts of space and time in current special relativity theory therefore are not compatible with the principle of relativity. A consistent special theory of relativity would have its basic structure of space and time defined by the principle of relativity, not by the physical interpretation of the Lorentz transformation. That shall simplify relativistic concepts, to make relativity theory more transparent and true to relativity physics, possibly opening up for further advances directly from the basis of relativity.