The conflict between Maxwellian electrodynamics and Newtonian kinematics motivated a new kinematic framework. Experiments such as the Michelson Morley result challenged the ether hypothesis and suggested that the speed of light is the same in all inertial frames. Albert Einstein, working at the Swiss Patent Office, elevated the constancy of light speed to a postulate and, together with the relativity principle, derived the mathematical relations now called the Lorentz transformations. Hendrik Lorentz at Leiden University earlier developed similar formulas to preserve Maxwell's equations under frame change, showing continuity with prior work.
Linearity and symmetry
To reach the transformations one assumes inertial frames related by a uniform relative velocity and that space and time are homogeneous and isotropic. These assumptions imply a linear mapping between coordinates because nonlinearity would single out special events or positions. Demanding that two frames S and S prime moving at velocity v along a common axis yield identical physical laws constrains the form of the linear map. The crucial additional constraint is that a light pulse traveling at speed c in S must also travel at speed c in S prime. Mathematically, the worldline x = ct must map to x prime = c t prime.
From invariance to explicit form
Write the most general linear relations for coordinates along the motion direction as x prime = A x + B t and t prime = C x + D t. Enforcing that x = ct implies x prime = c t prime leads to (A c + B) = c (C c + D). Requiring inverse symmetry under velocity sign change and matching the Galilean limit at small velocities fixes relations among A B C D. Solving yields A = D = gamma and B = -gamma v and C = -gamma v / c squared, where gamma equals 1 divided by the square root of 1 minus v squared over c squared. The result is the standard form x prime = gamma times x minus v t and t prime = gamma times t minus v x over c squared.
These formulas produce immediate physical consequences: time dilation where moving clocks run slower by factor gamma, length contraction of moving rods, and the relativity of simultaneity whereby simultaneous events in one frame need not be simultaneous in another. Modern pedagogical expositions by David J. Griffiths Reed College and Sean Carroll University of California Davis reinforce the same derivation and emphasize that these results follow from symmetry and the experimentally established invariant speed c. Nuanced physical interpretation recognizes these as statements about ideal inertial frames; gravitational or noninertial contexts require general relativity.