This document discusses Einstein's theory of special relativity. It introduces Einstein's two postulates: 1) the laws of physics are the same in any inertial reference frame, and 2) the speed of light in vacuum is constant. It describes how the Galilean transformations do not account for electromagnetism, while the Lorentz transformations proposed by Hendrik Lorentz are consistent with Maxwell's equations. The Lorentz transformations relate space and time coordinates between inertial frames in motion. Relativistic addition of velocities is also covered.

1. The Theory of Special Relativity
 Einstein’s Postulates
 Galilean Transformations
 The Lorentz Transformation
 Time and Space in Special Relativity
 Relativistic velocity

2.  K is at rest and K’ is moving with velocity
 Axes are parallel
 K and K’ are said to be INERTIAL COORDINATE SYSTEMS
Inertial Frames K and K’

3. Einstein’s Postulates (1905)
 First Postulate
 The laws of physics are the same in any inertial
frame of reference (principle of relativity)
 Second Postulate
 The speed of light in vacuum is the same in all
inertial frames
 That is, the speed of light is c (~3x108
m/s) and is
independent of the motion of the source
 For example, If the light comes from the headlight of a
train moving at with velocity v
Newton: speed =c+v Einstein: speed still =c !!!
c
v

4. General Galilean
Transformations
'
'
'
tt
yy
vtxx
=
=
+=
11
'
'
'
'
'
'
=⇒=
=⇒=
+=⇒+=
dt
dt
dt
dt
vv
dt
dy
dt
dy
vvvv
dt
dx
dt
dx
samethearetandt
yy
xx
yy
yy
xx
xx
aa
dt
dv
dt
dv
aa
dt
dv
dt
dv
samethearetandt
'
'
'0
'
'
=⇒=
=⇒+=
inertial reference frame
∑∑ =⇒= FamFam '

11
'
'
'
'
'
'
=⇒=
=⇒=
+=⇒+=
dt
dt
dt
dt
tt
dt
dy
dt
dy
vuuv
dt
dx
dt
dx
samethearetandt
yy
xx
frame K frame K’
Newton’s Eqn of Motion the same at
face-value in both reference frames
PositionVelocityAcceleration

5. Limitations of the Galilean
Transformation
The Newton’s Second Law is invariant with respect to the
Galilean Transformation.
But the famous Maxwell Equations are NOT
INVARIANT with respect to the Galilean
Transformation!
The transformation, with respect to which
the Maxwell Equations were found to
invariant, was an “exotic” (interesting)
transformation formula discovered in 1905
by a Dutch physicist and mathematician,
Hendrik Lorentz.

6. Linear Transformation Equations
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
x a x a y a z a t
y a x a y a z a t
z a x a y a z a t
t a x a y a z a t
′ = + + +
′ = + + +
′ = + + +
′ = + + +
11 12 13 14
22 33
21 23 24 31 32 34
41 42 43 44
1
0
x a x a y a z a t
y y a a
z z a a a a a a
t a x a y a z a t
′ = + + + 
′ = = =
⇐ 
′ = = = = = = =
′ = + + + 
ˆ( )u u x i=

Principle of Relativity

7. Linear Transformation Equations
(cont.)
11 12 13 14
42 43
41 44
0 ( and )
x a x a y a z a t
y y
a a y y z z
z z
t a x a t
′ = + + + 
′ = 
⇐ = = ↔ − ↔ −
′ = 
′ = + 
Rotational symmetry
11
12 13
11 14
41 44
( )
0
0
0
x a x ut
t t
y y a a
x ut
z z a u a
x
t a x a t
′ = − 
′= =′ = = =  
⇐ ⇐ =   
′ = = −   ′ =′ = + 
Boundary conditions at origin

8. Linear Transformation Equations
(cont.)
Spherically symmetric wave front in S and S′
2 2 2 2 2 2
11 44
2 2 2 2 2
41 11
( ) 1/ 1 /
( ) /
x y z ct a a u c
x y z ct a ua c
+ + =  = = −
⇒ 
′ ′ ′ ′+ + = = − 
( )
( )
2 2
2
2
2 2
1 /
/
/
1 /
x ut
x x ut
u c
y y
z z
t ux c
t t ux c
u c
γ
γ
−
′ = = −
−
′ =
′ =
−
′ = = −
−
Lorentz Transform
( )
( )
2 2
2
2
2 2
1 /
/
/
1 /
x ut
x x ut
u c
y y
z z
t ux c
t t ux c
u c
γ
γ
′+
′ ′= = +
−
′=
′=
′ ′+
′ ′= = +
−
Inverse Lorentz Transform
2 2
1
1 /u c
γ ≡
−

9. Relativistic Velocity Transformation
)1(
'
2
c
uv
vuu
+
+=

10. V’pg
= velocity of police relative to ground
vbp
= velocity of bullet relative to police
V’og
= velocity of outlaws relative to ground
Vpg
= 1/2c Vog
= 3/4cVbp
= 1/3c
police outlawsbullet
Example: As the outlaws escape in their really fast car at (3/4)c, the
police follow in their pursuit car at a velocity of (1/2)c, firing a
bullet, whose speed relative to the gun is (1/3)c. Question: does the
bullet reach its target a) according to Galileo, b) according to
Einstein?

11. In order to find out whether justice is met, we need to compute
the bullet's velocity relative to the ground and compare that
with the outlaw's velocity relative to the ground.
In the Galilean transformation, the velocity of the bullet relative
to the ground is simply the sum of the bullet’s velocity and the
police car’s velocity. Since
Galileo’s addition of velocities
Vb’= 1/3 c+ ½ c = 5/6 c > ¾ c. Justice served!

12. Einstein’s addition of velocities
Due to the high speeds involved, we really must relativistically
add the police ship’s and bullet’s velocities:
5 3
7 4 !c c< → justice is not served
( ) ( )
1 1
3 2 5
721 1
3 2
v
1 /
bg
c c
c
c c c
+
⇒ = =
+