24 Special Theory of Relativity

Ashok Goyal

1. Introduction

In Newtonian world view space and time are independent absolute entities. The Newton’s Laws of motion are invariant under Galilean transformations when we go from one inertial frame to another which is moving with a uniform velocity with respect to the one another. The Galilean transformations are given by .  and  are the position vectors measured in the unprimed and primed initial frames which are moving with a relative velocity  with respect to each other. In Newtonian frame work time is independent of the reference frame. Maxwell’s equations of electro–magnetism however, do not satisfy the principle of Galilean relativity. Maxwell’s equations predicted that the speed of light in vacuum is a universal constant and is equal to C. If this is true in one coordinate system, it clearly will not be true in another coordinate system moving with a uniform velocity and defined by the Gallilean transformations. The pioneering attempts for the resolution of the problem of relativity in electrodynamics and mechanics were made by Lorentz and Poincare. The problem was finally resolved by Einstein in 1905 in the form of Postulates of Special Theory of Relativity namely,

(i) The Laws of physics are invariant in all inertial frames.

(ii) The velocity of light in free space is a universal constant independent of the frame of reference.

In Einstein’s world view space and time are still absolute entities but they are not independent. They form a four–dimensional Minkowski space.

2. Lorentz Transformations

Lorentz transformations have the fundamental property of keeping the velocity of light constant in all inertial frames which means that

The matrix tensor has the following property:

and we see that differentiation w.r.t. contravariant component of the coordinate vector transforms as a component of a covariant vector operator.

Centre of Mass (CM) Frame

It is more convenient to consider a scattering process in the CM frame, in which the CM is at rest and therefore the particles 1 and 2 approach each other with equal and opposite momenta and after scattering travel in opposite directions with equal momenta.

Now

Can also be expressed as

5. Summary:

• In special theory of Relativity, space and time are consider absolute and constitute coordinates in a 4 – D Minkowski space.
• The Lorentz transformations preserve the proper time element so that the velocity of light is independent of the inertial frame of reference.
• The scalar product of 4 – vectors is a Lorentz invariant whereas the 4 – vector itself transforms like the coordinates in an inertial coordinate system in Minkowski space.