The two pillars of Modern Physics that began to be developed at the turn of the century from the 19th to the 20th are Quantum Mechanics and the Theory of Relativity. Classical Physics, represented by Newton's Mechanics, turned out to be inadequate to explain the pnenomena in microscopic scale or those involving fast-moving bodies whose speed is colse to the speed of light. The totally new concepts of Quantum Mechanics were needed to account for the microscopic scale pnenomena. On the other hand, the fact that the light travels with the same speed in all directions, regardless of the motion of the source or the observer, led A. Einstein (1879-1955) to develop the Special Theory of Relativity in 1905. The Galilean coordinate transformation between the reference frames moving with respect to each other cannot be compatible with the constancy of the speed of light and had to be replaced by the Lorentz transformation. Einstein further postulated that all laws of physics are the same in all inertial reference frames whose coordinates are related to each other by Lorentz transformations.
The first of Newton's laws of motion, that a body moves with a constant velocity unless acted on by forces, is really a statement about a particular reference frame, an inertial frame, in which the second law holds. The second law, that a body is accelerated in proportion to and in the direction of the force acted on it, cannot be justified without the specification of the reference frame provided by the first law. Is the inertial frame unique? Obviously not. Any frame that is moving uniformly with respect to a given inertial frame also satisfies the first law and is an inertial frame. In Classical Mechanics, the coordinates of the inertial frames are related to each other by the Galilean transformations. Newton's second law can be seen to be invariant under the Galilean transformation if one only considers those forces that depend on the relative positions of the interacting bodies. Newton's laws of motion appear the same in all inertial frames and therefore, as far as Newton's Mechanics is concerned, there can be no physical criteria for a perferred inertial frame. The idea of the 'rest frame' loses its absolute meaning, and word 'velocity' can only retain the relative meaning. This fact may be called the 'Newtonian Relativity'.
The theory of Electromagnetism, summarized and completed by J. Maxwell (1831-1879), appeared to alter the situation. Maxwell's equations allow a wave solution which represents electromagnetic waves propagating with the same speed in all directions. The Galilean transformation cannot retain the constancy of the speed of light and there could be only one inertial frame in which the light travels with the same speed in all directions. One could designate this frame the absolute rest frame. However, all experimental searches for this frame have failed. Einstein accepted the constancy of the speed of light in all inertial frames as a postulate and showed that the coordinate transformation between inertial frames has to be the Lorentz transformation to retain the invariance of the speed of light. All inertial frames are on the same footing again and the concept of relativity can remain. But the problem is that Newton's equation of motion, being invariant under the Galilean transformation, is not invariant under the Lorentz transformation. In the process of rescuing the concept of relativity in Electromagnetism, one in turn faces the possibility of ruining the concept of relativity in Mechanics. Einstein, postulating that all laws of physics are the same in all inertial reference frames, chose to modify Newton's equation of motion and invented the Relativistic Dynamics which is invariant under the Lorentz transformation.
After inventing the Relativistic Dynamics, Einstein started modifying Newton's theory of gravitation to invent a new theory of gravitation which is compatible with his Special Relativity. His path to a new theory took an unexpected turn, eventually leading to a completely new concept of gravitation. The final result which he published in 1916 is the General Theory of Relativity. In this theory, the effect of gravity is manifested as the spacetime curvature. Matter affects the geometry of spacetime and the spacetime curvature in turn affects the motion of matter. Spacetime is no longer just a background stage where physical phenomena occur as in Classical Physics. It has its own dynamics in General Relativity.
This note is intended to include the contents of Relativity that can be covered in one semester of a two-hour-a-week undergraduate course. The details of the kinematics and dynamics of Special Relativity are given in Chapter 2. The full coverage General Relativity is not intended, but only a brief introduction of it is given in Chapter 3.