theory of relativity

Introduction to special theory of relativity.

Special Theory of Relativity:
Galilean Transformations,. Newtonian Relativity.

This was a lecture delivered to physics-elective class of a 3 year non-physics degree students on 10th April 2017. This is also a good exposition to honors students and anyone at an introductory level of the special theory of relativity, with requisite mathematical background. 

Let us consider an inertial frame of reference S. The space and time coordinates of any event occurring in frame S are given by x, y, z, t.

Now let us consider another frame of reference S’ which is inertial but moves wrt frame S at speed v, along +x direction.

The coordinates of the same event in the S’ frame are given as: x’, y’, z’, t’. The relationship among the coordinates of any event in two different frames of reference both of which are inertial frames, is known as Galilean Coordinate Transformation or Galilean Transformation.

If we assume that time passes by at the same rate in both S and S’ frames, the resulting laws satisfy Newtonian Relativity. We say time is an absolute quantity in an infinitude of equivalent inertial frames of references as the rate of time change is independent of the particular inertial frame of reference we have chosen. Consequently: t = t’.

The above equation is known as velocity addition rule in Newtonian Relativity. This is valid only for classical mechanics in the sense of speed of objects and speed of frame of reference, which are quite insignificant with respect to the speed-of-light value.

Velocity addition is nothing but a relation of velocities of objects in different frames among each other. So its exactly what we call “relative velocities” in elementary mechanics. Relative velocity, velocity addition and velocity transformation are the exact same thing. Read more about these here and here. The second link also expounds on what happens when speeds approach that of light.

Waves, particles and Einstein !

Waves are something that have no mass and move at the maximum speed, mass m = 0. speed c = 1. So whats their momentum? p = m.v = 0? Right?

No. For pure waves; momentum does not come from mass. It comes only from motion.

(pure wave; they do not have mass)

For matter waves, on the other hand, momentum comes in two ways, mass as well as motion.

(impure, now they have mass)

Albert Einstein recognized this fact and derived his relation; $latex E = \sqrt {(pc)^2+(mc^2)^2}$

This relation is called as Einstein’s relativistic equation, also Einstein’s mass-energy relation. But more appropriately mass-energy-momentum relation.

Let us consider E as the hypotenuse, p and m; as base or perpendicular as is your choice.

triangle_copyThen $latex E = \sqrt {(pc)^2+(mc^2)^2}$ is Pythagoras Theorem; when p is momentum and m is mass.

For pure waves such as photon … the quanta of light, m = 0.

Hence the Pythagorean Triangle is now one, where the mass side is arbitrary small. Thus E = p.

How to add speeds; Galileo and Einstein won’t agree.

How to calculate the speed of anything, when their speed becomes closer to the speed-of-light. 
This article was originally a comment in the linked article;  Why Nothing Moves Faster Than Light.

— In order to correct the comment I have made earlier  ” unless something is completely mass-less in its rest-frame ” I also add the following. This is a fact which I have realized lately — or rather trapped myself to commit an inconsistent remark, by following the same comment in making other remarks elsewhere. 

But it’s better late than never to realize; when something is mass-less, it will never have a rest-frame, because by Einstein’s transformation rules, known as Theory of Relativity, to be consistent, a mass-less particle will always move at the speed of light c, no matter which frame we are looking at it from. This then leads to the velocity addition formula of Einstein.

Now we will discuss in a slightly more detail the two kind of velocity addition formula, one prior to Einstein and one that came from Einstein’s work. 

Prior to Einstein. 
According to Newton and Galileo ( Galileo Project ), known by a name Galilean Relativity, the following follows; if C moves at speed

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven Weinberg .. review

A highly sofisticated and an elegant text book on general Relativity and its application. I would say a super book. I have read this book and solved for now a few daunting problems in Physics by applications of the content in this book. My research is available on my website but for context I recommend this book for serious practising physicists.

What is Theory of Relativity?

I have read so many complicated description of whats theory of relativity that I tend to think such complicacy serves two purposes;

Misleading: it takes us away from the real concepts.

Complicated: It does not make the concepts easy at all to grasp.

Ineffective: It ill-prepares even very good students — as their precious time is wasted and the mediocre rejoice.

Here is an appropriate description of relativity theory.

Relative as suggests hinges on in reference to what? So you may call it a reference-dependent theory. But if you understand what theory of relativity is you may as well call it an equivalence theory.

This theory establishes for the first time in greater depth the equivalence of many ideas, concepts, physical laws and methods. This is the work of Albert Einstein. But if you dig deeper you will see that most of the concepts were known before, and classical physics is capable of producing these ideas 1st hand.

It is for no strange reason relativity theory is included as classical physics, in greater conceptual frameworks. Its because its merely a more refined calculation of certain problems, some of which were known and some which were not, but never to the extent and power of the methods of Einstein.

Einstein eg solved the problem of perihelion of mercury very exactly which was not as exactly understood in the canonical formulations of the old classical theory, so theory of relativity is the new classical theory.

But there are two episodes of this story.

1. Special theory of relativity: this is a regime of the theory of relativity where it pertains exclusively to kinetic energy: