Jan 16, 2008

Lorentz Velocity Transformation Rule
S
S and S’ are measuring ant’s speed u along x, y, z axes
S’ v u
In S' frame, u x'
= x
2
' t
2
'
!
x
1
'
!
t
1
'
= dx ' dt ' dx '
= " ( dx !
vdt ) , dt ' = " ( dt !
v c 2 dx )
u
u x' x'
=
= dx !
vdt dt !
u x v c 2
!
v
1 !
v u x c 2 dx
For v << c, u x'
, divide by dt
= u x
!
v
(Galilean Trans. Restored)

Velocity Transformation Perpendicular to S-S’ motion dy ' = dy , dt ' = !
( dt " u y
'
= dy ' dt '
=
!
( dt dy
" v c 2 divide by dt on RHS dx ) v c 2 dx ) u ' y
=
!
(1 " u y v u x
) c 2
There is a change in velocity in the direction # to S-S' motion !
u z
'
=
!
(1
u z v c 2 u x
)

Inverse Lorentz Velocity Transformation
x u y u z
=
= u x '
+ v
=
+ vu x ' c 2 u y
'
!
(1 + v c 2
!
(1 + u z
' v c 2 u x
' ) u x
' )
As usual, replace v
- v

y S
Does Lorentz Transform “work” For Topgun ?
A y’ S’
0.75c
O (Earth guy)
O’ x
-0.85c
B x’
Two rockets A &B travel in
opposite directions
An observer on earth (S) measures speeds = 0.75c
And 0.85c for A & B respectively
What does A measure as
B’s speed?
Place an imaginary S’ frame on Rocket A
⇒ v = 0.75c relative to Earth Observer S
Consistent with Special Theory of Relativity

Example of Inverse velocity Transform
Speed of ball relative to stationary observer u
X
?
Biker moves with speed = 0.8c
past stationary observer
Throws a ball forward with speed = 0.7c
What does stationary observer see as velocity of ball ?
Place S’ frame on biker
Biker sees ball speed u
X’
=0.7c

Hollywood Yarns Of Time Travel !

Terminator : Can you be seen to be born before your mother?
A frame of Ref where sequence of events is REVERSED ?!!
S
S’ u
I arrive in SF
(,)(,) xtxt 22''22
(,)(,) xtxt
!
t
= t '
2
" t
1
'
= # ,
*
+
!
t "
%&
$ v !
x ' c 2
()
/
-
.
0 !
t ' < 0

I Cant ‘be seen to arrive in SF before I take off from SD u
S S’
11''11
(,)(,) xtxt 22''22
(,)(,) xtxt
!
t ' = t '
2
" t
1
'
= #
*
,
+
!
t "
$
%& v !
x c 2
'
()
-
/
.
For what value of v can !
t ' < 0
!
0 t ' v c
< 0
>
0 !
t < v c
!
2 x
0 1 < v !
x c 2
!
t
= c u
0 v > c : Not allowed v u c 2

Relativistic Momentum and Revised Newton’s Laws pmu = rr and the Special theory of relativity: Example :
S P = mv –mv = 0 P = 0
Before
1 v v 2
V=0 1 2
After v
1
'
= v
1
1 !
!
v v
1 v c 2
= 0, v
2
'
= v
2
1 !
!
v v
1 v c 2
=
!
2 v
1 + v 2 c 2
, V ' =
V !
v
1
!
S’ p ' before
= mv
1
'
+ mv
2
'
p
=
!
2 mv
1 +
' before v c
2
2
, p ' after
" p ' after
= 2 mV ' = !
2 mv
Vv c 2
= !
v v
1
’=0
1
Before v
2
’
2 V’ 1 2
After

Definition (without proof) of Relativistic Momentum p = m u = γ m u
√
1 - (u/c)2
With the new definition relativistic momentum is conserved in all frames of references
: Do the exercise
New Concepts
Rest mass = mass of object measured
In a frame of ref. where object is at rest
2
11(/) !
u uc !
=

m
Nature of Relativistic Momentum u r
= m u r
1 !
( u / c ) 2
= " m u r
With the new definition of
Relativistic momentum
Momentum is conserved in all frames of references

Relativistic Force & Acceleration r
= m r
1 !
( u / c ) 2
= " m
Relativistic
Force
And
Acceleration
Reason why you cant quite get up to the speed of light no matter how hard you try!
r
= d r dt
= d dt
"
$ m r
1 !
( u / c ) 2
%
' use d dt
= du dt d du
F =
+
+
)
+ m
1 !
( u / c ) 2 mu
+
( 1 !
( u / c ) 2 ) 3/ 2
( (
!
1
2
)(
!
2 u c 2
)
.
.
,
.
du dt
F =
+
+
)
+ mc 2 c 2
!
mu 2
+ mu 2
( 1 !
( u / c ) 2 ) 3/ 2 .
.
,
.
du dt
F =
+
+
)
+ m
( 1 !
( u / c ) 2 ) 3/ 2 .
.
,
.
du dt
: Relativistic Force
Since Acceleration r d dt
, [rate of change of velocity]
/ r
*
1 !
( u / c ) 2 ,
3/ 2 m
Note: As u / c 0 1, r
0 0 !!!!
Its harder to accelerate when you get closer to speed of light

q
A Linear Particle Accelerator
F
+
-E d
V
Parallel Plates
E= V/d
F= eE
Charged particle q moves in straight line in a uniform electric field r accelarates under force r r
= d dt r
= m
"
1 !
u c
2
2
%
3/ 2
= q r m
"
1 !
u c
2
2
%
3/ 2 larger the potential difference V across plates, larger the force on particle
Under force, work is done on the particle, it gains
Kinetic energy
New Unit of Energy
1 eV = 1.6x10
-19 Joules
1 MeV = 1.6x10
-13 Joules
1 GeV = 1.6x10-10 Joules

Your Television (the CRT type) is a Small Particle Accelerator !

Linear Particle Accelerator : 50 GigaVolts Accelating Potential r e r m #
" 1 !
( u / c ) 2
%
$
3/ 2

r
=
Relativistic Force & Acceleration m r
1 !
( u / c ) 2
= " m
Relativistic
Force
And
Acceleration r
= d r dt
= d dt
"
$ m r
1 !
( u / c ) 2
%
'
F =
(
*
* m
( 1 !
( u / c ) 2 ) 3/ 2
+
-
du dt
: Relativistic Force
Since Acceleration r d r
, [rate of change of velocity] dt
.
m )
1 !
( u / c ) 2 +
3/ 2
Reason why you cant quite get up to the speed of light no matter how hard you try!
Note: As u / c / 1, r
/ 0 !!!!
Its harder to accelerate when you get closer to speed of light

Relativistic Work Done & Change in Energy
X
2
, u=u
x
1
, u=0
W = x
1
!
x
2
.
d r
= x
1
!
x
2 d r
.
d r dt p = mu
1 " u 2 c 2
# d r dt
=
&
%
$
1 " m u c du dt
2
2
'
)
(
3/ 2
, substitute in W ,
# W = !
0 u m
&
%
$
1 " du dt udt u c
2
2
'
)
(
3/ 2
(change in var x * u)

Relativistic Work Done & Change in Energy
X
2
, u=u
x
1
, u=0
W = u
(
0 mudu
$
#
"
1 !
u 2 c 2
'
%
&
3/ 2
= ) mc 2
!
mc 2
= mc 2
$
#
"
1 !
u 2 c 2
'
%
&
1/ 2
!
mc 2
Work done is change in energy (KE in this case)
K = ) mc 2
!
mc 2 or Total Energy E= ) mc 2
= K + mc 2

But Professor… Why Can’t ANYTHING go faster than light ?
K
= mc 2
$
#
"
1 !
u 2 c 2
%
'
&
1/ 2
!
(
"
$
#
1 !
u 2 c 2
%
' =
& m 2 c 4 mc 2
K
(
+ mc
(
2
K
+
!
2 mc 2
2 )
=
)
+
+
+
+
$
#
"
1 !
mc 2 u c
2
2
%
'
&
1/ 2
,
.
.
.
.
2
( u = c 1 !
(
K mc 2
+ 1) !
2 (Parabolic in u Vs
K mc 2
)
As K / 0 , u / c
Non-relativistic case: K =
1
2 mu 2
( u =
2 K m

Relativistic Kinetic Energy Vs Velocity

A Digression: How to Handle Large/Small Numbers
• Example: consider very energetic particle with very large Energy E
!
=
E mc 2
= mc 2 mc
+ K
2
= 1 +
K mc 2
• Lets Say γ = 3x10 11 , Now calculate u from  u c
=
#
%
$
1 !
"
1
2
&
(
'
1/ 2 to limited precision.
• In fact u ≅ c but not exactly!, try to get this analytically
1
!
= =
1
1 "
Since # =
# u c
2
$ 1, 1 +
(1 " # )(1 + # )
# = 2
In Quizzes, you are
Expected to perform
Such simple approximations
1
!
%
2 1 " #
& 1 " # =
1
2 !
2
= 5 ' 10 " 24 , u = # c
& u = 0.999 999 999 999 999 999 999 995c !!
Such particles are routinely produced in violent cosmic collisions

When Electron Goes Fast it Gets “Fat”
E =
2
!
mc 2 Emc = !
K
= + mc 2 vAs 1, cApparent Mass approaches !
""##
New Concept
Rest Mass = particle mass when its at rest

Relativistic Kinetic Energy & Newtonian Physics
Relativistic KE K = !
mc 2
" mc 2
% $
%
%
# Remember Binomial Theorem for x << 1; (1+x) n = (1+ nx
1!
+ n(n-1)
2!
x 2 +smaller terms)
( '
(
(
& so K * mc 2 [1 + c 2 u 2
) When u << c , 1-
$
%
1
2 u 2
] " c 2 mc 2
(
&
'
"
1
2
=
* 1 +
1
2 mu 2
1
2 u c
2
2
+ ...smaller terms
(classical form recovered)
Total Energy of a Particle E = !
mc 2
= KE + mc 2
For a particle at rest, u = 0 " Total Energy E= mc 2

E=mc 2
⇒ Sunshine Won’t Be Forever !
Q:
Solar Energy reaches earth at rate of 1.4kW per square meter of surface perpendicular to the direction of the sun.
by how much does the mass of sun decrease per second owing to energy loss? The mean radius of the Earth’s orbit is 1.5 x 10 11 m.
r
• Surface area of a sphere of radius r is A = 4 π r 2
• Total Power radiated by Sun = power received by a sphere whose radius is equal to earth’s orbit radius

E= mc 2
⇒ Sunshine Won’t Be Forever !
r
Total Power radiated by Sun
= power received by a sphere with radius equal to earth-sun orbit radius( r in figure)
P sun lost
P sun lost
=
P Earth incident
A
A
= 4.0
# 10 earth !
sun
26 W
=
P Earth incident
A
4 " r 2 earth !
sun
= (1.4
# 10 3 W / m 2
So Sun loses E = 4.0
# 10 26 J of rest energy per second
)(4 " )(1.5
# 10 11 ) 2
Its mass decreases by m =
E c 2
=
4.0
# 10 26 J
(3.0
# 10 8 ) 2
= 4.4
# 10 9 kg per sec!!
If the Sun's Mass = 2.0 # 10 30 kg So how long with the Sun last ?
One day the sun will be gone and the solar system will not be a hospitable place for life

E = !
mc 2
" E 2
= !
2 m 2 c 4 Relationship between P and E p = !
mu " p 2 c 2
= !
2 m 2 u 2 c 2
" E 2
# p 2 c 2
= !
2 m 2 c 4
# !
2 m 2 u 2 c 2 m 2 c 2
=
1
# u 2 c 2
( c 2
# u 2 ) =
= !
2 m 2 c 2 ( c 2
# u 2 ) c 2 m 2 c 4
# u 2
( c 2
# u 2 ) = m 2 c 4
E 2
= p 2 c 2
+ ( mc 2 ) 2 ........important relation
For particles with zero rest mass like photon (EM waves)
E= pc or p =
E c
(light has momentum!)
Relativistic Invariance : E 2
# p 2 c 2
= m 2 c 4 : In all Ref Frames
Rest Mass is a "finger print" of the particle

Mass Can “Morph” into Energy & Vice Verca
• Unlike in Newtonian mechanics
• In relativistic physics : Mass and Energy are the same thing
• New word/concept : MassEnergy , just like SpaceTime
• It is the mass-energy that is always conserved in every reaction : Before & After a reaction has happened
• Like squeezing a balloon : Squeeze here, it grows elsewhere
– If you “squeeze” mass, it becomes (kinetic) energy & vice verca !
• CONVERSION FACTOR = C 2
• This exchange rate never changes !