Transformations! - Brown Universitygaitskell.brown.edu/courses/PH0008_2002/SpecRel/...PH0008...

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


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 …”


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]


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]


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


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”)


Review SectionReview Section


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)


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)


Lorentz TransformationsLorentz Transformations•Also referred to as Einstein Lorentz Transformations


RecapRecap•Attempts to explain negative aether drag results


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?


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


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

¯

˜ ˜


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


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


Light ClocksLight Clocks•Develop gedanken experiment to study transforms


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


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


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


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)


Video #2

• Video 00:24:12 -> 00:26:40

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


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


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



• Derivation of Transforms

†


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


g =1

1- v 2 c 2=

11- b 2




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

†


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


g =1

1- v 2 c 2=

11- b 2


†

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


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

Transformations! - Brown Universitygaitskell.brown.edu/courses/PH0008_2002/SpecRel/...PH0008...

Documents

Transcript of Transformations! - Brown Universitygaitskell.brown.edu/courses/PH0008_2002/SpecRel/...PH0008...