Physics 2D Lecture Slides Lecture 6 : Jan 11th 200 5 Vivek Sharma - - PDF document

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Physics 2D Lecture Slides Lecture 6 : Jan 11th 200 5 Vivek Sharma - - PDF document

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Physics 2D Lecture Slides Lecture 6 : Jan 11th 200 5 Vivek Sharma UCSD Physics First Quiz This Friday ! Bring a Blue Book, calculator; check battery Make sure you remember the code number for this couse given to you (record it some

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Physics 2D Lecture Slides Lecture 6 : Jan 11th 2005

Vivek Sharma UCSD Physics

First Quiz This Friday !

  • Bring a Blue Book, calculator;

check battery

– Make sure you remember the code number for this couse given to you (record it some place safe!)

  • No “cheat Sheet” please, I will give

you equations and constants that I think you need

  • When you come for the quiz, pl.
  • ccupy seats in the front first.
  • Pl. observe one seat distance in the

back rows (there is plenty of space)

  • Academic Honesty is for you to
  • bserve and for me to enforce:

– Be a good citizen, in this course and forever !

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Time Dilation Example: Relativistic Doppler Shift

  • Light : velocity c = f λ, f=1/T
  • A source of light S at rest
  • Observer S’approches S with

velocity v

  • S’ measures f’ or λ’, c = f’λ’
  • Expect f’ > f since more wave

crests are being crossed by Observer S’ due to its approach direction than if it were at rest w.r.t source S

Relativistic Doppler Shift

Examine two successive wavefronts emitted by S at location 1 and 2 In S’ frame, T’ = time between two wavefronts In time T’, the Source moves by cT’ w.r.t 1 Meanwhile Light Source moves a distance vT’ Distance between successive wavefront λ’ = cT’ – vT’

'=cT'-vT', now use f = c / f ' = c (c-v)T' ; but T ' = T 1- (v/c)2 = T substituting for T', use f = 1/T f ' = 1- (v/c)2 1- (v/c) f ' = 1+(v/c) 1-(v/c) f better remembered as: f obs= 1+(v/c) 1-(v/c) fsource f obs = Frequency measured by

  • bserver approching light source
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SLIDE 3

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  • bs

source

1+(v/c) f = f 1-(v/c)

Relativistic Doppler Shift Doppler Shift & Electromagnetic Spectrum

←RED BLUE→

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SLIDE 4

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Fingerprint of Elements: Emission & Absorption Spectra

Example : The Atomic Energy levels of Hydrogen

Doppler Shift in Spectral Lines and Motion of Stellar Objects

Laboratory Spectrum, lines at rest wavelengths Lines Redshifted, Object moving away from me Larger Redshift, object moving away even faster Lines blueshifted, Object moving towards me Larger blueshift, object approaching me faster

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Seeing Distant Galaxies Through Hubble Telescope

Through center of a massive galaxy clusters Abell 1689

Edwin Hubble, Mount Palomar & Expanding Universe

Hale 100 inch Telescope, Mount Palomar Edwin Hubble 1920

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SLIDE 6

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Galaxies at different locations in Universe moving away at different velocities

Spectral lines are shifted from Laboratory (at rest) Specimen

Hubble’s Measurement of Recessional Velocity of Galaxies Reccesional Velocity V ∝ distance; V = H d

Farther things are, faster they go

H = 75 km/s/Mpc (3.08x1016 m) Play the movie backwards! Our Universe is about 10 Billion Years old

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SLIDE 7

7 Cosmological Redshift & Discovery of the Expanding Universe: [ Space itself is Expanding ]

New Rules of Coordinate Transformation Needed

  • The Galilean/Newtonian rules of transformation could

not handles frames of refs or objects traveling fast

– V ≈ C (like v = 0.1 c or 0.8c or 1.0c)

  • Einstein’s postulates led to

– Destruction of concept of simultaneity ( Δt ≠ Δt’ ) – Moving clocks run slower – Moving rods shrink

  • Lets formalize this in terms of general rules of coordinate

transformation : Lorentz Transformation

– Recall the Galilean transformation rules

  • x’ = (x-vt)
  • t’ = t

– These rules that work ok for ferraris now must be modified for rocket ships with v ≈ c

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Discovering The Correct Transformation Rule

' guess ' ( ) ( ' ' ' ' guess ) x x v G x x vt x x vt t x vt G x =

  • =

= + +

  • =
  • Need to figure out the functional form of G !0
  • G must be dimensionless
  • G does not depend on x,y,z,t
  • But G depends on v/c
  • G must be symmetric in velocity v
  • As v/c→0 , G →1

Guessing The Lorentz Transformation

Rocket in S’ (x’,y’,z’,t’) frame moving with velocity v w.r.t observer on frame S (x,y,z,t) Flashbulb mounted on rocket emits pulse of light at the instant origins of S,S’ coincide That instant corresponds to t = t’ = 0 . Light travels as a spherical wave, origin is at O,O’ Do a Thought Experiment : Watch Rocket Moving along x axis Speed of light is c for both observers: Postulate of SR Examine a point P (at distance r from O and r’ from O’ ) on the Spherical Wavefront The distance to point P from O : r = ct The distance to point P from O : r’ = ct’ Clearly t and t’ must be different t ≠ t’

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Discovering Lorentz Transfromation for (x,y,z,t)

Motion is along x-x’ axis, so y, z unchanged y’=y, z’ = z Examine points x or x’ where spherical wave crosses the horizontal axes: x = r , x’ =r’

2 2 2 2 2 2 2

' ' ( - ) , ' ( - ) ( ) [ ] 1

  • r

= 1 ( / ) ( ' ') ( ' ') '

( )

x ct G x vt G t x vt c v ct G ct t c c x ct G x vt x ct G G c v ct vt G v vt v c t

x x vt

  • =

=

  • =
  • =
  • +
  • =
  • =

= + = =

  • +

=

  • =
  • 2

2 2 2 2 2 2 2 2

1 since 1 , ( ' ') ( ( ) ') ' 1 ' 1 , ' ( ) ' ' [1 x x vt x x vt vt x x vt x x vt vt x x t x x t t v v v v x t t v x v v v c v t v c t

  • =

+

  • =
  • +
  • =
  • +

=

  • +
  • =
  • =

+

  • =
  • +

=

  • =
  • +
  • 2

1 vx t c

  • =
  • Lorentz Transformation Between Ref Frames

2

' ' ' ' ( ) y y z z v t t x t c x v x

  • =
  • =
  • =
  • =

Lorentz Transformation

2

' ' ' ') ' ' ( v t x x vt t c y z x y z

  • =

+

  • =

+

  • =
  • =

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

Notice : SPACE and TIME Coordinates mixed up !!!

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Not just Space, Not just Time New Word, new concept !

SPACETIME

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 Δt’ did S’ measure ? two events occur at same place in S frame => Δx = 0

Δt’ = γ Δt (in this example Δt = proper time)

S

x

S’

X’

Length Contraction: Ruler measured in S between x1 & x2 : What Δx’ did S’ measure ? two ends measured at same time in S’ frame => Δt’ = 0

Δx = γ (Δx’ + 0 ) => Δx’ = Δx / γ ( in this example Δx = proper length)

x1 x2 ruler

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Lorentz Velocity Transformation Rule

' ' ' 2 1 x' ' ' ' 2 1 x' 2 x' 2 x' 2 '

In S' frame, u , u , u 1 For v << c, u (Gali divide by dt' ' lean Trans. Restor ( ) ( ed) )

x x x

x x dx t t dt dx vdt v dt dx c v dt dt dx dx v dx c u u u d v v t c v

  • =

=

  • =
  • =

= = =

  • S

S’ v

u

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

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 y y y

u u dy dy dy v dt dt dx dy c u u v dy dt dx c v c

  • =

= =

  • =

=

  • Velocity Transformation Perpendicular to S-S’ motion

' 2

Similarly Z component of Ant' s velocity transforms (1 ) as

z z x

u u v c u

  • =
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Inverse Lorentz Velocity Transformation

' x ' ' ' 2 ' 2 ' 2

Inverse Velocity Transform: (1 u ) 1 1 ( )

y y z x z x x x

u v vu u u v c u v c u c u u

  • =

+ = + + = +

As usual, replace

v ⇒ - v