Physics 2D Lecture SlidesJan 14

Vivek SharmaUCSD Physics

“Fixing” (upgrading) Newtonian mechanics • To confirm with fast velocities • Re-examine

– Spacetime : X,Y,Z,t– Velocity : Vx , Vy , Vz

– Momentum : Px , Py , Pz

– (Proper) Rest Mass – Acceleration: ax , ay , az

– Force– Work Done & Energy Change – Kinetic Energy– Mass IS energy

• Nuclear Fission• Nuclear Fusion • Making baby universes Learning about the first three minutes

since the beginning of the universe

Lorentz Transformation Between Ref Frames

2

''

'

' ( )y yz z

vt t

x t

c

x v

x

γ

γ

=

= −

= −

=

Lorentz Transformation

2

''

' )

''

(y yz

vt

x x v

tc

t

xz

γ

γ

= +

== +

=

Inverse Lorentz Transformation

As v→0 , Galilean Transformation is recovered, as per requirement

Notice : SPACE and TIME Coordinates mixed up !!!

Lorentz Transform for Pair of Events

Can understand Simultaneity, Length contraction & Time dilation formulae from this

Time dilation: Bulb in S frame turned on at t1 & off at t2 : What ∆t’ did S’ measure ?two events occur at same place in S frame => ∆x = 0

∆t’ = γ ∆t (∆t = proper time)

S

x

S’

X’

Length Contraction: Ruler measured in S between x1 & x2 : What ∆x’ did S’ measure ?two ends measured at same time in S’ frame => ∆t’ = 0

∆x = γ (∆x’ + 0 ) => ∆x’ = ∆x / γ (∆x = proper length)

x1 x2

ruler

Lorentz Velocity Transformation Rule ' ' '2 1

x' ' ' '2 1

x'

2

x'

2

x'

2'

In S' frame, u

,

u ,

u1

For v << c, u (Gali

divide by dt'

'

lean Trans. Restor

( )(

ed)

)

x

x

x

x x dxt t dt

dx vdtvdt dxc

vdt dtdx dx v dxc

uu

u

d

vv

t

cv

γγ

−= =

−

−=

−

−=

=

=

=− −

−

−

SS’

v

u

S and S’ are measuring ant’s speed u along x, y, z

axes

Does Lorentz Transform “work” ?Two rockets travel inopposite directions

An observer on earth (S) measures speeds = 0.75cAnd 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

y

x

S

x’

S’y’

0.7c-0.85c

AB

O

O’

2

'

2

'

2

divide by dt on

(1 )

There is a change in velocity in the direction to S-S' motion

' ,

'' (

H

)

'

S

!

R

( )

x

yy

y

uu

dy dy

dy

vdt dt dx

dyc

u

u vdy dt dxc

vc

γ

γ

γ=

= =

⊥

−

−

=

=

−

Velocity Transformation Perpendicular to S-S’ motion

'

2

Similarly Z component ofAnt' s velocity transforms

(1 )

as

zz

x

uu vcuγ

=−

Inverse Lorentz Velocity Transformation

'x

'

'

'

2

'

2

'

2

Inverse Velocity Transform:

(1

u

)

1

1

( )

yy

z

x

z

x

x

x

u vvu

uu v

cuv

c

u

c

u

u

γ

γ

=+

=+

+=

+

As usual, replace v ⇒ - v

Example of Inverse velocity Transform

Biker moves with speed = 0.8cpast stationary observer

Throws a ball forward with speed = 0.7c

What does stationary observer see as velocity of ball ?

Place S’ frame on bikerBiker sees ball speed

uX’ =0.7c

Speed of ball relative to stationary observer

uX ?

Can you be seen to be born before your mother?A frame of Ref where sequence of events is REVERSED ?!!

S S’

1 1' '1 1

( , )

( , )

x tx t

u

2 2' '2 2

( , )

( , )

x tx t

I tak

e off

from

SD

I arrive in SF

' '2 1 2

'

For what value of v can ' 0

v xt t t tct

γ ∆ ∆ = − = ∆ − ∆ <

I Cant ‘be seen to arrive in SF before I take off from SD

S S’

1 1' '

1 1

( , )

( , )

x tx t

u

2 2' '2 2

( , )

( , )

x tx t

'

2 22

'2 1 2

'

'

For what value of v

0

v

can

: Not al

lowe

u1 <

' 0

dc

v xt tc

c v cu

v x vc t c

v xt t t tct

γ ∆ ∆ = − = ∆ − ∆ <

∆∆

∆=

∆⇒<

⇒ >⇒

< ⇒ ∆

>

Relativistic Momentum and Revised Newton’s Laws Need to generalize the laws of Mechanics & Newton to confirm to Lorentz Transform

and the Special theory of relativity: Example : p mu=

1 2Before

v1’=0

v2’21

After V’

S’

S

1 2

Before v v 21

After V=0

P = mv –mv = 0 P = 0

' ' '1 2

' '1 21 2 2

1 122 2

2

2

2

'

' 'before after

20, , '

2

11 1

1

1

, 2 2

'

p p

afterbeforemvp mv m

v v v v v V vv v V vv v v v V

vvc

vvcc c c

p mV mv

− − − −= = = = =

−= +

= −−− − +

=

≠

+= = −

Wat

chin

g an

Inel

astic

Col

lisio

n be

twee

n tw

o pu

tty b

alls

Definition (without proof) of Relativistic Momentum

21 ( / )mup muu c

γ= =−

With the new definition relativistic momentum is conserved in all frames of references : Do the exercise

New ConceptsRest mass = mass of object measuredIn a frame of ref. where object is at rest

2

is velocity of the objectNOT of a referen

11 ( / )

!ce frame u

u cγ =

−