Transformations! - Brown Universitygaitskell.brown.edu/courses/PH0008_2002/SpecRel/...PH0008 Gaitskell Class Spring2002 Rick Gaitskell Video (3)- Lorentz Transformation • From “

Gaitskell

PH0008Quantum Mechanics and Special Relativity

Lecture 3 (Special Relativity)

Transformations! Lorentz-Einstein Transformation

Prof Rick Gaitskell

Department of PhysicsBrown University

Main source at Brown Course Publisher

background material may also be available at http://gaitskell.brown.edu

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Video

• “The Mechanical Universe” Video Serieso (See Gerry’s list on web site - I am using boiled down versions for time being)

o 41 - Michelson-Morley Experiment

o 42- The Lorentz Transformation

• Lanou Fall ‘94 Tapeo I won’t use this edit (see Lanou #2)

• Jim Valles ‘96 Tapeo Not useful

• From “The Mechanical Universe” Video Series (Lanou #2 Tape)o This is useful - see my edits on next page

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Video (2)

• Michelson-Morley Material

• From “The Mechanical Universe” Video Series (Lanou #2 Tape)o Nodeso 02:30 “Just as Newton was born…”[See Maxwell]o 03:15 [Queen Victoria * 1887]

• GOOD PLACE TO START

o 4:15 “Many physicists believe it is all over …”o 4:40 Michelson & Morleyo 8:26 Detailed technical discussion of M&M Experimento 15:46 “Light apparently doesn’t need an Aether”

• [STOP HERE]

o 15:58 “For next 50 years tried to …”o 16:50 “In one attempt after another …”

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Video (3)- Lorentz Transformation

• From “The Mechanical Universe” Video Series(Lanou #2 Tape)

o 00:16:50 “In one attempt after another “• START - Good place (bit of preamble, but allow them

to get used to video)

o 00:17:20 [FitzGerald] introduced, then onto[Lorentz]

• These theories were alternative to Einstein’s finalview/theory

o 00:19:35 [Big Train - discuss observers on..]

o 00:20:30 “Consider 2 observers in relative motion…” (SIMULTANEITY )

• SIMULTANEITY (2:10)

• 22:38 “This is not semantics of a petty debate…”

o 00:22:40 [Poincaret] Laws same in all frames• Lorentz developed formula that showed how much

dx and dt would be needed to keep c constant

o BREAK

o START AGAINo 00:24:12 [Light Clocks] - Henry & Albert ride with

clocks• USE THIS - To show idea of clocks

o SKIPo 00:26:40 [Examples of contractions on Earth]

• NOT USE - Examples of length contraction on Earth(too verbose to be useful)

o START AGAINo 00:28:14 [Lorentz Equations - explicit derivation]

• Lorentz’s Equations - but he uses them to sayelectrons shrink

• Can’t talk about a time, without mentioning aposition

o 00:30:30o BREAK FOR DISCUSSIONo 00:30:30 [Space - Time Diagrams]

• Detailed graphics of Space-Time Diagrams

o 00:37:50 [S-T End]

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Video (4)

• From “The Mechanical Universe” Video Series (Lanou #2 Tape)o Nodes hh:mm:ss (The times may differ slightly because I used different video decks)

o 00:37:50 [Twins] - Which tape does this material come from?o 00:43:14 [Space Billiards]

• Kinematics - Mass also depends on frame of reference

o 00:49:50 [E=mc^2]• Show how E=mc2 drops out

o 00:55:20 [ENDS]

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Exam I: Waves (See “Class Schedule” for diary)

• Exam I on classical wave phenomena will be on Friday 1 Marcho Note early start time 8.30 am - 9.50 am (Room 168 - lecture theatre)

o Will those who requested exam at rescheduled time please see me after this lecture

• To help you prepare …o Conferences on Tues & Wed this week

o Previous Exam Questions and Solutions are posted in Homework ‘Assignments’page on web

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Section: Special Relativity Week 2

• Homework 4 (hand in M 2/25)• (see”Assignments” on web pages)

• Pretty modest this time to allowrevision for Mid-Term I

• Reading (Prepare for 2/25)

o SpecRel (also by French)• Ch3 Einstein & Lorentz Transforms

• Ch4 Realtivity: Measurement of Length andTime Inetrvals

• Lecture 3 (M 2/25)o Lorentz Transformation

• Examples:

— Time Dil.,

— Lorentz Cont.,

— Relativity of Simultaneity

• Lecture 4 (W 2/27)o Reciprocity of Time Dilation

• No Lecture (F 3/1)o (Exam Mid-Term I)

• Reading (Prepare for 3/4)

o SpecRel (also by French)• Ch5 RelativisticKinematics

• Ch6 Relativistic Dynamics: Collisions andConservation Laws

• Homework (None schedule M 3/4)o This is because of exam on previous

Friday

o However - I strongly suggest you startHomework #7(see web “Assignments”)

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Review SectionReview Section

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Question SpecRel L03-Q1

•How much longer does an airplane (v=300 m/s=10-6c)get to due to special relativistic effects viewed fromground?(Order of magnitude estimate fine)

o(1) ~1 Å

o(2) ~1 µm

o(3) ~1 mm

o(4) None of above (within factor 10)

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Question SpecRel L03-Q2

•How much shorter does an airplane (v=300 m/s=10-6c)get to due to special relativistic effects viewed fromground? (Order of magnitude estimate fine)

o(1) ~1 Å

o(2) ~1 µm

o(3) ~1 mm

o(4) None of above (within factor 10)

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Lorentz TransformationsLorentz Transformations•Also referred to as Einstein Lorentz Transformations

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

RecapRecap•Attempts to explain negative aether drag results

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Theoretical Explanation (1892)

•Does M-M mean that Aether doesn’t existoWhat if aether exerts some kind of influence on arms ofinterferometer?

oFitzGerald & Lorentz propose that arm || to motion through aether iscompressed by factor (1-v2/c2)-1/2

• Why would this help?

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Michelson-Morley Experiment (Calculation in Detail)

†

We won't make final approximations this timeTime for round trip in Arm 1 t1

t1 =l

c - v+

lc + v

=lc

11- v c

+1

1+ v cÊ

Ë Á

ˆ

¯ ˜

=2lc

11- v 2 c 2

Ê

Ë Á Á

ˆ

¯ ˜ ˜

Time for round trip in Arm 2 t2

Using path length in aether frame

¢ l = l2 + v 2t 2 = ct

fi l2 + v 2t 2 = c 2t 2

fi t 2 =l2

c 2 - v 2

fi t2 = 2t =2lc

11- v 2 c 2

Ê

Ë Á Á

ˆ

¯ ˜ ˜

Time for roundtrip t1

Time forroundtrip t2

vt

l

In frame of aether

Apparatus is moving through aether with velocity v

Source

Image

l’

020225.2.rjg

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Theoretical Explanation (1892)

• Does this mean that Aether doesn’t existo What if aether exerts some kind of influence on arms of interferometer?

o FitzGerald & Lorentz propose that arm || to motion through aether is compressedby factor (1-v2/c2)-1/2

• Why would this help? (1892)

• What mechanism?

—The newly discovered electron is influenced by the aether, and everything is made of electrons(Recent JJ Thompson discovery)

• Lorentz developed Transforms based on this argumento Designed to preserve observed speed of light in frames other than that of aether inthe face of M-M results

• However, consequences seem very strange …o Poincaré objects that Lorentz’s interpretation is not deep enough - what about therelativity of physics? - Maxwell’s Equations?

†

l = l01

1- v 2 c 2

Ê

Ë

Á Á

ˆ

¯

˜ ˜

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Video #2

• 00:16:50 -> 00:24:12

o 00:16:50 “In one attempt after another “• START - Good place (bit of preamble)

o 00:17:20 [FitzGerald] introduced, then onto [Lorentz]• These theories were alternative to Einstein’s final view/theory

o 00:19:35 [Big Train - discuss observers on..]

o 00:20:30 “Consider 2 observers in relative motion …” (SIMULTANEITY )• SIMULTANEITY (2:10)

• 22:38 “This is not semantics of a petty debate…”

o 00:22:40 [Poincaret] Laws same in all frames• Lorentz developed formula that showed how much dx and dt would be needed to keep c constant

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Frames of Reference - Loss of Simultaneity

• “This is not semantics, or a petty debate…”

• We have lost simultaneityo Events have become relative, and will be reordered in different frames

• We will work the example of a light sphere in following lecture

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Light ClocksLight Clocks•Develop gedanken experiment to study transforms

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Time Dilation

• The Light Clock

†

1 tick in the rest frame ¢ S of the clock

D ¢ t =2Dc

D Frame S’

• Imagine the Clock is moving past us at speed v

†

1 tick in the lab frame ¢ S of the clock

Dt =2Hc

H 2 = D2 + 12 vDt( )2

DH

vD t

D t

Frame S

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Time Dilation (2)

• The Light Clock

D

DH

vD t

D t†

So in ¢ S we have D ¢ t =2Dc

and

in Lab S we have Dt =2Hc

Using H 2 = D2 + 12 vDt( )2

fi 12 cDt( )2

= 12 cD ¢ t ( )2

+ 12 vDt( )2

fi Dt 2 = D ¢ t 2 +vc

Dt 2

fi Dt =D ¢ t

1- v 2 c 2

Frame S’

Frame S

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Time Dilation (3)

The apparent clock tick Dt seen in the lab frame is longer than Dt’

MOVING CLOCKS APPEAR TO RUN SLOWER

†

Dt =D ¢ t

1- v 2 c 2

Dt > D ¢ t

†

b =vc

b ≥ 0

g =1

1- b 2≥1

So Dt = gD ¢ t

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Nota Bene

•Henry Lorentz (1904 - Transforms) and Albert Einstein(Special Relativity - 1905) are coming at this problemfrom very different directions (and were, in probability,working without knowledge of the other)

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Video #2

• Video 00:24:12 -> 00:26:40

• 00:24:12 [Light Clocks] - Henry & Albert ride with clocks

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Lorentz-Einstein Transforms

• Derivation of Transforms

†

¢ x = g x - bct( ) x = g ¢ x + bc ¢ t ( )¢ y = y¢ z = z

c ¢ t = g ct - b x( ) ct = g c ¢ t + b ¢ x ( )

b = v c ,v is velocity of frame ¢ S measured in S)

g =1

1- v 2 c 2=

11- b 2

Note the use of (ct) rather than t which accentuates the symmetry of the transforms

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Video #2

• Video 00:28:14 -> 00:30:30

o 00:28:14 [Lorentz Equations - explicit derivation]• Lorentz’s Equations - but he uses them to say electrons shrink• Can’t talk about a time, without mentioning a position

o 00:30:30

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Lorentz-Einstein Transforms

• Derivation of Transforms

†

¢ x = g x - bct( ) x = g ¢ x + bc ¢ t ( )¢ y = y¢ z = z

c ¢ t = g ct - b x( ) ct = g c ¢ t + b ¢ x ( )

b = v c ,v is velocity of frame ¢ S measured in S)

g =1

1- v 2 c 2=

11- b 2

Note the use of (ct) rather than t which accentuates the symmetry of the transforms

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Lorentz-Einstein Transforms

• In the limit of low speed “non-relativistic” -> Galilean Transforms

†

¢ x = g x - bct( ) x = g ¢ x + bc ¢ t ( )¢ y = y¢ z = z

c ¢ t = g ct - b x( ) ct = g c ¢ t + b ¢ x ( )

b = v c ,v is velocity of frame ¢ S measured in S)

g =1

1- v 2 c 2=

11- b 2

Note the use of (ct) rather than t which accentuates the symmetry of the transforms

†

b <<1 fi g ª1¢ x = g x - bct( ) Æ ¢ x @ x - bct = x - vt¢ y = y¢ z = z

c ¢ t = g ct - b x( ) Æ ¢ t @ t - b x c @ tNote that relation for time equivalencerequires x << ct

PH0008 Gaitskell Class Spring2002 Rick Gaitskell

Opposing Interpretations of Transforms

Implications of these Transforms are Hard to Believe

• Lorentz continued to argue that they reflected property of electrons (andthat the aether existed) - Michelson also believed this.

• Einstein embraces constancy of speed of light as a fundamental propertycombined with a restatement of principle of Galilean-Newtonian-PoincaréRelativity

o Physics (including Maxwell) is same in all frames

o We can’t determine our velocity (relative to absolute frame) by experiment (c constant)

o The transforms are now a direct consequence of these assumptions