Wednesday, June 1, 2022

Special theory of Relativity notes

Albert Einstein's theories of relativity are some of the most talked about, and important, topics in physics. Indeed, their seemingly counter-intuitive results provoke much thought and discussion amongst the general public, students, teachers, and academics.

 There are two theories of relativity. The first one, published by Einstein in 1905 is called the Special Theory of Relativity, and it is this subject with which we will deal in the following Notes on Kinematics, Dynamics, and application of special relativity. The second theory, published in 1915, is called General Relativity and is basically a theory of matter and gravity. The General theory requires knowledge of an extremely complicated mathematical formalism in order to be stated and appreciated in its full form.

Although the effects of Special Relativity are not manifest in our daily lives, this by no means diminishes its applicability to the real world; indeed, if the speed of light were closer to "everyday" speeds the weird effects associated with special relativity would not appear so weird to us at all. Of course there are some situations in which the effects of Special Relativity are clear: particle physicists working with very high-speed particles need to constantly take the effects of time dilation and length contraction into account. Moreover, Special Relativity is crucial for understanding the interaction between electric and magnetic phenomena and the propagation of electromagnetic waves. This brings us to another point: Special Relativity (or General Relativity for that matter) cannot be derived from Newtonian or Classical physics in any form; a study of Special Relativity must begin with postulates that, until verified empirically, must be taken on faith, rather than proved from any pre-existing laws. The postulates, and the results derived from them, can only be tested in the laboratory--the "proof" of relativity is ultimately experimental. There exists one other check on Special Relativity, and that lies in the so-called correspondence principle. This essentially states that any correct theory must reduce to the classical, everyday laws of physics in the appropriate limits. For Special Relativity this means that when small speeds (much less than the speed of light) are involved, the equations should reduce (approximately) to a familiar Newtonian form.

Relativity has a reputation for being extremely difficult and counter-intuitive, and it is true that some concepts can be somewhat perplexing at first. However, if one is willing to be patient and think carefully though the problems, always keeping the basic concepts of relativity in mind, the subject is not as difficult as it might first appear. Moreover, relativity contains many cool "paradoxes" that are interesting and rewarding to untangle.

Frame of reference:- A system of coordinates whose axes can be suitably chosen is said to be a frame of reference.

Inertial frame:- A frame of reference either at rest or moving with a uniform velocity (zero acceleration) is known as inertial frame.

Non-inertial or accelerated frame:- It is a frame of reference which is either having a uniform linear acceleration or is being rotated with a uniform speed.

Galilean transformation:-

(a) Transformation of position:-

x' = x – vt

y' = y

z' = z

t' = t

Inverse transformation,

x = x'+ vt '

y = y'

z = z'

t = t'

(b) Transformation of distance:-

l ' = l

Here, l = x2-x1 is the length of rod as observed in frame S.

Anything which remains unchanged when observed from the two Galilean frames of reference is known as Galilean invariant.

(c) Transformation of velocity:-

u'= u– v

Inverse transformation:- u = u'+ v

(d) Transformation of acceleration:-

a' = a

Thus, the acceleration of body, as observed by two observers siting in two inertial frames, is same. Hence, acceleration is said to be Galilean invariant.

Law of conservation of momentum:-

It states that the total momentum of an isolated system (no external force) always remains constant.

In S frame:- m1u1 + m2u2 = m1v1 + m2v2

In S ' frame:- m1u'1 + m2 u'2  = m1 v'1  + m2 v'2

Thus, the law of conservation of momentum is valid in S ' also, indicating that the law is Galilean invariant.

Ether and velocity of light:- c' = c ±v

Postulates of special theory of relativity:-

(a) The laws of physical phenomenon are same when stated in terms of two systems of reference in uniform translator motion relative to each other.

(b) The velocity of light in vacuum is constant, independent not only of the direction of propagation but also of the relative velocity of the source and the observer.

Lorentz transformation equations:-

x' = [x - vt]/√[1- v2/c2]

y'= y

z'= z

t' = [t – [(v/c 2) x]]/√[1- v2/c2]

Lorentz inverse transformation equations:-

x= [x' + vt] /√[1- v2/c2]

y = y'

z= z'

t= [t' + [(v/c 2) x']]/√[1- v2/c2]

Length contraction:-

l = l0√[1- v2/c2]

So, l<l0

Here, l0 is the proper or original length in S frame and l is the relativistic length in S' frame.

Time Dilation:-

??t = ??t0/√ [1-v2/c2]

So, ?t > ??t0

Here, ??t0 is the proper or original time in S frame and ?t is the relativistic time in S' frame.

?Thus, the time interval as observed by the moving observer appears to be lengthened.

Frequency:- If f 0 is the natural frequency of a process in frame S, then the frequency f as observed from S ' given by,

f = f0√[1- v2/c2]

Relativistic velocity addition theorem:-

ux' = [ux – v]/[1- [(v/c2)ux]]

uy' = [uy√[1- v2/c2]]/[1- [(v/c2)ux]]

uz' = [uz√[1- v2/c2]]/[1- [(v/c2)ux]]

Relativistic variation of mass:-

m= m0/√[1-v2/c2]

Rest mass energy:-

E0 = m0c2

Total energy (Mass-Energy Equivalence):-

E= mc2 = [m0/√[1-v2/c2]] c2

Kinetic energy:-

EK= E - E0 = (m - m0) c2

Here c is the speed of light, m0 is the rest mass and m is the relativistic mass.

Relativistic momentum:-

p= mv

   = [m0/√[1-v2/c2]] v

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