## Four-vectors and conservation laws in relativity

This lecture was delivered to the final year honors class of 3 year science degree students on 21 November 2017 as part of the Classical Dynamics paper.

In this lecture we will discuss some of the important tools of relativistic mechanics. We will discuss the idea of proper-time, 4-velocity, 4-acceleration, 4-momentum, 4-force and related conservation law of the 4-momentum.

A. Proper-time.
The proper time is the time interval in the rest-frame of any event. The proper time is related to time-interval in other inertial frame by: tau = (1/gamma)t where gamma  > 1 always.

Gamma is the Lorentz factor or Lorentz boost factor directly related to the speed of an object in speed-of-light units, i.e. beta.

gamma = 1/sqrt{1-v^2/c^2}

Hence proper-time is the smallest possible time interval for an object in motion in among all possible inertial frames of reference and it occurs in the rest frame.

d(tau) < dt

Proper-time is necessary to define other basic quantities in theory of relativity if we are to preserve their basic meaning in terms of the non-relativistic mechanics definitions.

B. Four velocity.
Four velocity of a particle is the rate of change of 4-displacement …

So, …  is the position vector — or space-time interval in the Minkowski  space — akin to the difference of two 3-dimensional vector in coordinate space, this time with 4 coordinates rather than 3.

The proper-time interval d(tau) is a Lorentz invariant i.e. when we move between arbitrary inertial frames of references given by the Lorentz factor beta or  gamma this interval retains its value — because it retains its form. Any variable which would retain its form under such transformation are said to be Lorentz invariant quantities.

## 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.

## Simple explanation of OPERA Anomaly of FLT

Simple explanation of OPERA Anomaly of FLT I just wrote two tweets, one of which, is a concise explanation of OPERA anomaly of Faster than Light neutrinos. (FLT neutrino). Einstein's Relativity Theory would be invalidated if neutrinos move faster than the photons, which is what OPERA experiment suspected it obtained, but Quantum Mechanics Uncertainty relations would save the grace of Relativity of Einstein, from falling off as an invalid theory. Its a bit tricky, but I explained it in 140 characters. Here are the tweets. Heisenberg would have tweeted in 1925: 1. when Q. Mech came physical variables got hats 2wear and were called as operators rather than variables, Heisenberg wanted2 tweet so in 1925. OPERA anomaly would also be explained by Heisenberg in 1925 via tweeter. Look guys. 2. Q. Mech uncertainty; E=f(t), p=f(x) >> E-t, p-x fuzzy, mixing of variables E-t, p-x, if E-x mix, eg E=f(x), E-v fuzzy, as v=f(x) > OPERA FLT

## An important lesson of Relativity.

An insight of why time and space are equivalents or unified in Relativity. Because its time which makes it possible for someone to crawl from x point to a y point, you can't simultaneously move bifurcated, into two orthogonal axes. Time mediates, hence its the time axis that mixes with all of the space axis, x, y, z. (Just came across this idea, originally I wrote in July 2012, and seems a bit non-trivial to share) An important lesson of Relativity: you can't go from x to y or y to z or z to x or z to y. They are perpendicular ... But you can go from x to t and then t to y. In effect you can go from anywhere to anywhere else, because there is anytime. Relativity has limitations though. Also Limitations are not to be confused with Laws of Nature or Physics, which are valid restrictions. So Anytime means anytime with restrictions and limitations. Imagine an x-axis point (x, 0) slipping back in through, to your joint, than your joint moving up there. Then this point is now, (x, t). While that was happening, the point (y, 0) also kept on moving towards your joint, to reach your damn thing, in time t, (y, t). (x, y, 0) evolved to (x, y, t) and you didn't do a thing, x, y, just met. At t = 0, x, y were on different axis, at t = t, they are at the same point. Thats how you moved from (0, 0, 0) to (x, y, t), Cool ?

## Equivalence from simple notions of Geometry ? Yes.

Now it may also be related that light bends in a denser media compared to a rarer media because an additional rotational force is working. In other words, the definition of straight line has to change in the medium, that is of different density, because path of light is changing. Light is the guy who suffers the least when something tries to buzz it, because its inertial property of mass is zero. The curvature of light or the bending or deflection known as Refraction is thus a measure of the sideways force or energy.

Thus speed of light in different media is a measure of this bending or curvature and is known as Snail’s law ( — Pun intended, its actually; Snell’s Law — ) . Automatically when distance and time have to readjust, to produce an angle or bending, known as refraction ( — possible because distance and time can produce an angle if they are equivalents — ) the speed must also change.

All these are inter related. Light refracts and its speed changes, in relative change in density of media because there are rotational or non-inertial effects.

( — or additional energy is available, or a force is acting to bring a curvature in light’s path, perhaps the electromagnetic effects of the molecules? Its not only distance or time that are equivalents, read one article of mine “All You Need To Know About Relativity” to understand; how energy and time and distance and in short all Physical Variables are equivalents of each other — )