Basics of special theory of relativity

ABHISHEK SAVARNYA
IIT KANPUR
BASICS OF
SPECIAL THEORY OF RELATIVITY

Frame of reference [निर्देश तंत्र]
x
Y
zAxes [अक्ष] system
• Length scale (position)
• Time scale (rest/motion)

Inertial[जड़त्वीय] frame Non-Inertial frame
Newton’s first law of motion
If, F =0 then a =0 holds
Newton’s first law of motion not valid.
Hence, we introduce pseudo-force [छद्म
बल] to use Newton’s law of motion.

Non-inertial frame
1.Centrifugal [अभिके न्द्रीय] force
2.Coriolis force

Frame notation
If S’ moves with uniform
velocity with respect to S, then
S’ is also inertial.
1
2
v
s S’
• X and X’ axis are on same line.
• Y and Y’ axis are parallel.
• Z and Z’ axis are parallel.
• Event of origin O crossing O’ is at t=0 and t’=0.
X
Y
Z
X’
Y’
Z’
o O’

Galilean transformation
• x = x’ + vt
• y = y’
• z = z’
• t = t’
s
X
Y
Z
o
v
S’
X’
Y’
Z’
O’
(x,y,z,t)
(x’,y’,z’,t’)
 x’ = x - vt
 y’ = y
 z’ = z
 t’ = t

• x = x’ + vt
• y = y’
• z = z’
• t = t’
s
X
Y
Z
o
v
S’
X’
Y’
Z’
O’
(x,y,z,t)
(x’,y’,z’,t’)
 x’ = x - vt
 y’ = y
 z’ = z
 t’ = t
 vx
’ = vx- v
 vy
’ = vy
 vz
’= vz
 ax
’ = ax
 ay
’ = ay
 az
’= az
सिी जड़त्वीय तंत्रों में ककसी कण का त्वरण समाि ोतता ो |
{ Galilean transformation के तोत}

Newton’s law of motion [गनतनियम]
बल = रव्यमाि * त्वरण
S F=ma
S’ F’=m’a’
Newton का गनतनियम सिी
जड़त्वीय तंत्रों में एक समाि
(invariant) रोता ो |
प्रकृ नत के नियम सिी जड़त्वीय तंत्रों में एक ोी रूप के ोतते ोैं
Principle of relativity
आपेक्षक्षकता का भसद््ांत

Principle of relativity
Newton का गनतनियम सिी जड़त्वीय तंत्रों में एक समाि (invariant) रोता ो |
Laws of nature in all inertial frames have the same form.
But, Maxwell equations are non-invariant under Galilean
transformation.
Maxwell equations [1862]

Michelson-Morley experiment [1887]
Objective: To find the speed of earth in the frame of ‘Aether’ (the absolute inertial frame where
Maxwell equations hold) Aether
S
S’
v
vx
’ = vx- v
ta1=L/(c- v)
L
Ta=
2𝐿
𝑐
[1/ (1 −
v2
𝑐2)]
vx
’ = vx- v
ta2=L/(c+v)
M1
M2
For M1

S
S’
vvx
’ = vx- v
vx
2+ vy
2 = v2 +(vy
’)2
vy
’2
= 𝑐2 − v2
L
Tb =
2𝐿
𝑐
[
1
1 −
v2
𝑐2
]
vx = v
vy = vy
’
L
M1
M2
For M2

S
S’
v
L
2 Tb =
2𝐿
𝑐
[
1
1 −
v2
𝑐2
]
L
Ta=
2𝐿
𝑐
[1/ (1 −
v2
𝑐2)]
∆𝑡 = Ta − Tb
M1
M2
1

S
S’
v∆𝑡 = Ta − Tb
L
L
M1
M2

Maxwell equations hold)
∆𝑡 = Ta − Tb
L
L
M1
M2
∆𝑡′ = Tb − Ta
∆𝑡 − ∆𝑡′
No shift in fringe pattern was observed
V=0
Null result of Michelson-Morley experiment

Faraday’s coil magnet experiment [1831]
Laws of electromagnetic induction
1. Whenever there is change in
magnetic flux linked with coil an
electromotive force (emf) is
induced.
2. The magnitude of induced emf is
directly proportional to the rate
of change of magnetic flux linked
with the coil.
Motivated by these observation in 1905
Einstein gave the postulates of ‘special
theory of relativity’
Time dilation, Length contraction, Twin paradox,
Relativity of momentum, kinetic energy

Special theory of relativity
Postulates of special theory of relativity [ववभशष्ट आपेक्षक्षकता का भसद््ांत के अभिगृहोत]
1 Laws of nature in all inertial frames have the same form.
2 Speed of light (in vacuum/निवाात) in all inertial frames is same.
s
X
Y
Z
o
v
S’ X’
Y’
Z’
O’

Time dilation [समय ववस्तार]
M1
M2
S frameEvent 1: Light beam travels from mirror M1
Event 2: Light beam is reflected to mirror M1
L
∆𝑡 =
2𝐿
𝑐

S’ frameEvent 1: Light beam travels from mirror M1
M1
M2
L
M1
M2
L

S’ frameEvent 1: Light beam travels from mirror M1
M1
M2
L
M1
M2
L
M1
M2
vt
∆𝑡 ′ =
2𝐿
𝑐
[
1
1 −
v2
𝑐2
]
∆𝑡 ′ =
∆𝑡
1 −
v2
𝑐2
The time interval between two events depends upon the frame of reference.

∆𝑡 ′ =
∆𝑡
1 −
v2
𝑐2
s
X
Y
Z
o Moving clock runs slow
Improper
time interval Proper time
interval

Derivation of time dilation from Lorentz transformation

Relativity of simultaneity [समकालीिता की आपेक्षक्षकता]
1 2
s
X
Y
Z
o’
Events
(x1,y,z,t) (x2,y,z,t)
Order of events depends on frame of reference.
If two events occur at the same location in a frame then their order remains unchanged in all other frames.
The order of events related by cause and effect remain unchanged (Principle of causality).
∆𝑡 ′ =
∆𝑡
1 −
v2
𝑐2

Synchronization of clocks
v
S’
X’
Y’
Z’
O’
C C’
s
X
Y
Z
o
Front clock in the direction of motion shows
less time.

Length contraction
v
s
X
Y
Z
o
S’
E1 : x’1 , t1
E2 : x’2 , t2
E2
E1
S
E1 : x1 ,t
E2 : x2 ,t
x’2 - x’1 =
x2−x1
1−
𝑣2
𝑐2
Moving length = Rest/proper length * 1 −
𝑣2
𝑐2

Transformation of velocity
• Relativistic linear momentum
(Relativistic mass)
𝑝 =
𝑚0 𝑣
1 −
𝑣2
𝑐2
• Relativistic kinetic energy
𝐾 = 𝑚𝑐2
− 𝑚0 𝑐2
• Mass-energy conversion
𝑒+ + 𝑒− = 2𝛾
• Energy-momentum relation
𝐸2
− 𝑝𝑐 2
= (𝑚0 𝑐2
)2

Thank you
Reference:
Video lectures of Prof. H.C.Verma

Basics of special theory of relativity

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