The classical mechanics of the special theory of [autosaved]

2. Introduction
Basic
Postulates of
Special Theory
of Relativity.
Lorentz
Transformation
Velocity
Addition and
Thomas
Precession

3.  At the end of nineteenth century, there are two mismatched
descriptive phenomena
 Newtonian Mechanics
 Maxwell’s Electromagnetic Theory
 Newtonian Mechanics assumed that
 Maxwell’s wave equation gave

 Any frame moving at constant velocity with respect to an
inertial frame is also an inertial frame.

4.  Consider two frames denoted by 𝑆 𝑎𝑛𝑑 𝑆′ with
𝑡, 𝑥, 𝑦, 𝑧 𝑎𝑛𝑑 (𝑡′
, 𝑥′
, 𝑦′
, 𝑧′
) the coordinates in 𝑆 𝑎𝑛𝑑 𝑆′ ,
respectively. The coordinate axes are aligned and,
𝑥 𝑎𝑙𝑜𝑛𝑔 𝑥′, and so on.
 Let 𝑆′ be moving relative 𝑆 in the +x-direction at a speed
𝑣, 𝑎𝑠 𝑠ℎ𝑜𝑤𝑛 𝑖𝑛 𝑓𝑖𝑔.

5.  The spacetime coordinates of 𝑆 and 𝑆′ are
related by simple expressions

6.  Transformation of this type is called Galilean transformations.
Under this assumption, it follows that Newton’s Second Law
 Relating to the applied force (𝑭) and the momentum (𝒑)
remains invariant, and
 The time in both 𝑆 and 𝑆′ frames is assumed to be 𝒕 = 𝒕′
.

7.  The Newtonian world view is that “The universe
consists of three spatial directions and one time
direction.” All the observers agree on the time
direction up to a possible choice of units.
 Under these assumptions, there are no universal
velocities. If u and u’ are the velocities of a particle
as measured in two frames moving with relative
velocity v, then

8.  On the other hand, Maxwell’s Electromagnetic
equations, there is a universal velocity 𝒄 , which is
interpreted as speed of light.
 Maxwell universal Velocity is inconsistent with the
Newtonian’s view having no universal velocity.
Then either Newtonian Mechanics or Maxwellian
mechanics would have to be modified.
 Then Albert Einstein came with Special theory of
relativity decided that Maxwell is right.

9.  Einstein’s Special Theory of Relativity Postulates are:
1. The laws of physics are the same to all inertial
observers.
2. The speed of light is the same to all inertial
observers.
 This means that time is now not an invariant quantity,
it is now covariant. So in order to relate time with the
space coordinates there must be some conversion
factor which is 𝑐𝑑𝑡 . In SI Systems of units, 𝑐𝑑𝑡 has
dimensions of meters. Hence, space and time in
special theory of relativity is a single entity and we
called is spacetime. This spacetime is the geometric
framework within which we perform physics.

10.  The square of distance in that space time,
△ 𝑠2
, between two points A and B is given by
If the separation is infinitesimal, the △ is
replaced by the differential symbol 𝑑. Such a
point in spacetime is known as event. The term
event is used because such a point has a
definite location and definite time in any frame.
……….. (1)

11.  The above equation can be written as
 Objects that travels on timelike path are called tardyons.
Hypothetical bodies that travels on spacelike path are called
tachyons, and the objects moving with speed of light are
called lightlike.
...........(1)a

12.  In the limits of small displacements in Cartesian coordinate
system,
 The four-dimensional space with an interval defined is often
called Minkowski space to distinguish it from a four
dimensional Euclidean space for which there would be no
minus sign.
………..(2)

13.  Let 𝑆 𝑎𝑛𝑑 𝑆′ are two different inertial frames, then
 Now, the time coordinate can no longer stand
independent of transformation. Now the time
measured in a Laboratory frame is different from
that measured by an observer at rest with respect to
the body under study.
z
y
x x'
y'
z' vS S’
………(3)

14.  Both times are not same so we distinguish them by calling “the time
interval measured by a clock at rest with respect to a body is the
proper time, while the other inertial observer uses a time that is
called Laboratory time.
 Consider the relation between the proper time 𝜏, measured by
observer at rest with coordinates (𝑡, 𝑥, 𝑦, 𝑧) and the Laboratory time 𝑡,
which is moving with velocity 𝒗 having coordinates (𝜏, 𝑥′, 𝑦′, 𝑧′). By Eq.
(2) and (3)
 This shows that 𝑑𝑡 > 𝑑𝜏. This effect is called “Time Dilation”. Moving
clock appears to run slower.

15.  The invariance of the interval expressed in Eq. (3) naturally
divides space-time into three regions relative to any event A
at time tA (A is located at 𝑥 = 𝑦 = 𝑡 = 0 in Fig.).