Classical Physics is often misunderstood as Newtonian Physics. The theory of Electromagnetism developed by Maxwell in 1856 governs the properties of light as an electromagnetic wave, and is also contained within the domain of classical physics. Newton was vehemently interested in matter, energy and the dynamics of particles involved, within the classical scope of length, mass and time. And so it is wise to identify Newtonian Physics as a field involving the mechanics of particles to within the domain of classical physics.
The remarkable success of Newtonian Mechanics (compared to Electromagnetism) made Newtonian mechanics a rigid foundation for the rest of physics, meaning that even in the likelihood of a future better theory, Newtonian Mechanics must be encompassed within it.
Einstein realized that the measure of mass, length and time depend upon the use of light in an inherent way. And in the cases of high velocity (when the velocity of an object approaches the velocity of light), it was obvious that the Newtonian Mechanics failed badly. As for example; when an electron accelerated by a potential difference of 10 million volts and moving with velocity 0.998c is further accelerated four times, its velocity isn’t equal to 1.98c as it should be according to the Newtonian mechanical relation (K.E.=half. m. v squared). It was found experimentally to be equal to 0.999c. For some reason, the electron was not obeying the Newtonian dynamics. It was a paradox. Ironically, paradoxes are the most important elements of Science. If it wasn’t for paradoxes and our unquenchable thirst to understand and solve these ironies, we would still be at the forests as hunter-gatherers.
Thomas Kuhn uses the word ‘paradigm shift’ in his book ‘The Structures of Scientific Revolutions’ to precisely denote such moment of intellectual urgency. And yes, it was a time for paradigm shift. Einstein realized a need for the complete revision of entities like length, mass and time by reformulating the idea of space and time to space-time. With that revision in our conception, he was confident that such paradoxes would be resolved while still preserving the Newtonian Mechanics.
The mathematical background for Newtonian Mechanics is the Galilean transformation. Under Galilean transformation, the acceleration of a body is invariant though the quantities like velocity, momentum, K.E are not. What this means is that the laws of mechanics (Beware, not laws of Physics) are invariant in all inertial systems (Even if the magnitude of Kinetic energy is different, if conservation of K.E. holds in one inertial frame, it holds in another inertial frame too). An inertial system is the one which is at rest or at motion with constant velocity with respect to the fixed system of distant stars. (But for convenience, we can take any frame of reference as an inertial one if it is at rest or at motion with constant velocity with respect to Earth).
The invariance of laws of mechanics under a Galilean transformation explains why it is impossible to identify the state of rest or motion (with constant velocity) of a spaceship by observing the behavior of an apple within the system. The state of rest or motion can only be known by visually comparing the position of the spaceship with respect to the outward objects. Now, let us assume, a supernova explosion took place within the viewing range of the spaceship and the Earth. The passengers on the spaceship and we (at the Earth) as different inertial observers in different inertial systems, will certainly not agree with our measurements (Clearly, the equations of Galilean transformation are here at play). But, the laws of physics will be the same in both the systems. Both the observers will conclude the same regarding whether conservation of momentum took place or not. No observation is preferred over another since all inertial observers are equivalent. This law that every observers are equivalent and there is no such thing called absolute measurement is often called Newtonian Relativity.
The only flaw with the Galilean transformation system and hence with the Newtonian Relativity is the bold assumption that ‘time for all inertial observers is the same’ i.e. ‘Time is an Absolute quantity’. The motion of an inertial observer would certainly not be a pain in the ass for time. If that would be the case then, any event occurring at a given duration of time would be the same for all inertial observers. This implies that the distance is also an invariant quantity because, distance is basically the ‘dynamics of light happening at an instant of time’ (the length of a fish is the distance between the point of strike of a pulse of light at its head and tail at an instant). So if, Galilean transformation is taken for granted, the length of a fish would also be an absolute quantity for all inertial observers. This showed that simultaneity is an immediate absolute property of the universe.
But when Einstein realized the failure of Newtonian Mechanics as its inability to incorporate the velocity of light, he came up with an ingenious idea that the velocity of light is same in all inertial systems as given by ‘c’. He further tried to set up the equations of motion so that the constant velocity of light is preserved for all inertial observers. He used Lorentz Transformation system (developed decades before him) as the background mathematical structure. In doing so, not just mechanics but all laws of physics (laws of mechanics including electromagnetism) were conserved for all inertial observers. This is obvious since he used light, an electromagnetic wave, for the foundation of his theory.
This new theory of Einstein showed that the length, mass and time aren’t absolute quantities as conceived earlier but variational properties of space-time depending upon the motion of inertial observers. Now, simultaneity also became a relative concept. Two events that were simultaneous in one frame mightn’t be so in another inertial frame, thereby the new term ‘relativity of simultaneity’.
The equations of motion under a Lorentz Transformation yields back the Galilean transformation equation for low velocity cases. The classical theory of Electromagnetism developed by Maxwell was found to be consistent with Einstein’s theory of Relativity. What Einstein’s Relativity essentially brought up is the modification of Newtonian laws to incorporate high velocity cases.