Contents
1 Theroy of relativity
1.1 Inertial frame of reference: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Non-inertial frame of reference: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Galilean Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Galilean Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Differences between Newtonian Principle of Relativity & Special Theory of Relativity . .
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CONTENTS
IIT JAM
captain sharath chandra
September 2025
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CONTENTS
Chapter 1
Theroy of relativity
Unaccelerated reference frames in uniform motion of translation relative to one another are called
”Galilean frames” or ”inertial frames”.Accelerated frames are called ”non-inertial frames”.
1.1
Inertial frame of reference:
An inertial frame of reference is one in which Newton’s first law of motion holds. In such a frame, an
object at rest remains at rest and an object in motion continues to move at a constant velocity if no
force acts on it.
• Any frame of reference that moves at constant velocity relative to an inertial frame is itself an
inertial frame.
• Special theory of relativity deals with the problems that involve inertial frames of reference.
1.2
Non-inertial frame of reference:
A non-inertial frame of reference is the one in which the Newton’s laws of motion are not valid, i.e., a
body is accelerated when no external force acts on it.
1.3
Galilean Transformations
When physical phenomena is observed in two inertial frames moving with uniform velocity relative to one
place and at one time, registered in both the frames is the same, then the doctrine of results in one frame
of reference can be transformed to those in the second frame. This process is called as Galilean transformations.
The equations which connect the position vectors of a particle (or event) of
two inertial frames are known as Galilean transformations.
Let’s assume S be at rest and S’ move with uniform velocity ⃗v along
+x-direction. Let v¡¡C, and assume O, O’ coincide at t = 0. There is no
relative motion between S and S’ along the axes of Y and Z. The distance
moved by S’ in the +x-direction in time t is vt. So, the x-coordinates of the
two frames differ by vt.
⃗x′ = ⃗x − ⃗v t
(1.1)
′
⃗y = ⃗y
⃗z′ = ⃗z
(1.2)
(1.3)
t′ = t
(1.4)
′
x
d⃗
x
• Transformation of velocities: d⃗
v =⇒ ⃗u′ = ⃗u − ⃗v
dt = dt − ⃗
′
u
d⃗
u
• Transformation in acceleration: d⃗
a′ = ⃗a
dt = dt − 0 =⇒ ⃗
The acceleration is invariant under Galilean transformations.
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Figure 1.1: galilean transformation
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CHAPTER 1. THEROY OF RELATIVITY
1.4
Galilean Invariance
The basic laws of mechanics do not change under Galilean transformations. All the three fundamental
quantities, length, mass and time are invariant of the relative motion of the observer.
• Consider a rod at rest in the inertial frame S having its two ends at the points x2 and x1 , then for
the observer of S, length of rod = x2 − x1 .
• The observer of S’ will consider the rod to be moving with a velocity −v from o’ to o and will
observe the two ends to be at the points x′2 and x′1 at the same time.
=⇒ for S’,length of rod= x′2 − x′1 .
x′2 − x′1 = (x2 − vt2 ) − (x1 − vt1 )
(1.5)
= (x2 − x1 ) − v(t2 − t1 )
(1.6)
= x2 − x1
(1.7)
It shows that the space interval or the distance between two points, i.e., length is invariant under Galilean
transformations.
• Mass is absolute: It is same for all observers and it is independent of motion of observer.
• Time is absolute: The motion has no effect on time. If the clocks in two inertial frames which
are in uniform motion agree at one instance, they will agree at all later times.
• Length is absolute: The length of does not change due to relative motion of an inertial frame
with respect to another.
1.5
Differences between Newtonian Principle of Relativity &
Special Theory of Relativity
Feature
Newtonian Principle of Relativity
Applicability Applies to the laws of classical mechanics only.
TransforUses Galilean transformations to
mations
convert coordinates and velocities between inertial frames.
Speed
of The speed of light is not a fundamental
Light (c)
constant and changes for observers in
different reference frames.
Time
Space
Velocity
Addition
Validity
Absolute time: Time is the same for
all observers. Clocks tick at the same
rate in all inertial frames.
Absolute space: The length of an
object and the distance between two
points are the same for all observers.
Follows simple vector addition (u′ =
u−v). For example, if a car moves at 60
mph and a person walks at 2 mph inside
it, their speed relative to the ground is
62 mph.
Valid for objects moving at speeds
much less than the speed of light.
Special Theory of Relativity
Applies to all laws of physics (mechanics, electromagnetism, etc.).
Uses Lorentz
transformations,
which include time dilation and length
contraction effects.
The speed of light in a vacuum is a
universal constant, the same for all observers, regardless of their motion. This
is a central postulate.
Relative time: Time is not absolute.
Clocks in motion relative to an observer
tick slower (time dilation).
Relative space: Lengths of objects
appear to contract in the direction of
motion (length contraction).
Follows a more complex, relativistic formula that ensures no speed can exceed the speed of light. The formula
u−v
is u′ = 1−
uv .
c2
Valid for all speeds, including those approaching the speed of light.