Extra  notes  on  rela,vity

Wade  Naylor

Over 105 years since Einstein’s Special theory of relativity

A. Einstein, 1879-1955

The postulates of special relativity

1. The principle of relativity (Galileo) states that

1. The laws of physics are the same to all observers (in all inertial reference frames)

2. Universality of the speed of light c:

1. c = 186 000 miles per second or 300 000 000 meters per second!

• From these 2 postulates we found that for moving objects

• lengths contract

• clocks slow down

• mass and energy are related

© 2012 Pearson Education, Inc.

Newton’s law of gravitation states gravity is due to action at a distance (F=GmM/r2). But if light takes 8 minutes to go around the sun and nothing travels faster than light; how can the Earth know how to act?

A. gravity is not a force

B. because of free fall

C. gravity interacts faster than light

D. the first two above

E. none of the above

Q2.

© 2012 Pearson Education, Inc.

Newton’s law of gravitation states gravity is due to action at a distance (F=GmM/r2). But if light takes 8 minutes to go around the sun and nothing travels faster than light; how can the Earth know how to act?

A. gravity is not a force

B. because of free fall

C. gravity interacts faster than light

D. the first two above

E. none of the above

Q2.

General Relativity

© 2012 Pearson Education, Inc.

What is the reason for the free fall of an astronaut in space?

A. There is no gravity in space.

B. This is a consequence of Newton’s third law.

C. The astronaut has no normal (upward) force acting upon him.

D. Two of the above three statements are correct.

E. All of the first three statements are correct.

Q1.

© 2012 Pearson Education, Inc.

What is the reason for the free fall of an astronaut in space?

A. There is no gravity in space.

B. This is a consequence of Newton’s third law.

C. The astronaut has no normal (upward) force acting upon him.

D. Two of the above three statements are correct.

E. All of the first three statements are correct.

Q1.

Copyright © 2012 Pearson Education Inc.

Free fall means what it says

Astronauts always

feel sick at first when

in space for this very

reason!

Last  week’s  ques,on

1. Can  you  see  a  problem  with  “A  Journey  to  VEGA”  discussion?

2. How  is  it  that  some  Quasars/galaxies  have  been  observed  with  redshiHs  z  >  8.2  (high  redshiH  is  z>0.1)?

1 + z =�obs

�emit

=

rc+ v

c� v

Galilean Transformations

• Inertial Frame

• A frame of reference that is moving at constant velocity

• Any other frame moving uniformly (constant velocity) with respect to an inertial frame is also inertial

• Only non-accelerated frames are inertial!

t = t� y = y� z = z� x = x� + vt or x� = x� vty

xz

y’

z’x’

v

No Ether ) speed of light constant

• Consider the Earth going around the sun?

v = 30 km/sor

108 000 km/h

Ether

SunEarth

As the Earth goes around the sun we expect to see the speed of light change as it moves with or against the Ether!!!

Michelson-Morley Experiment

• The Michelson-Morley experiment consisted of an interferometer which measurers the time taken for light to travel along the two arms (see picture)

Taken from http://en.wikipedia.org/wiki/Michelson-Morley_experiment

The light beams as they travel in different directions would be expected to interfere if light travels in the ETHER, because they will take different times to reach the light detector

Galileo & Newton versus Einstein?

• The concept of “Now” is very different

Now Now

Future Future

Past

Past

Elsewhere Elsewhere

Future of A

Past of A

CommonPast

Past of B

Future of B

CommonFuture

AB

In Einstein’s special theory of relativity there is no concept of “now” or “simultaneity;”

Only “relative simultaneity”

All observer agree on now

• Different observers disagree on “NOW”

• Observer O uses two clocks to measure ’s single clock, and vice versa ) disagreement on “NOW”

• There is only “relative simultaneity”

• Spacetime diagrams help a great deal!

Subtleties?

Fig: http://en.wikipedia.org/wiki/Twin_paradox

Asymmetry “relative simultaneity”

� =�

1� v2 =�

1� (0.96)2 = 0.28

Terence Stella

V=24/25=0.96

Terence stays on Earth while Stella makes a 14 yr. round trip into space; 7 yr. outward journey.

Assuming that Stella is moving, then Terence sees Stella’s proper time Δτ as

�t =��⇥1� v2

=14

0.28= 50yrs

Twin paradox

• However, can’t Stella argue that the Earth was traveling with respect to her ship?

Terence

Stella

V=24/25=0.96

• Usual answer: SR does not say that all frames of references are equivalent, only inertial frames!

• Stella must accelerate to v=0.96 then change direction and then slow down to v=0 back at Earth.

?

• For Stella, as she changes her frame she sees time jump from A to C

• As v increases the jump becomes larger because lines of simultaneity get steeper!

However, SR allows for infinite accelerations and we can assume that Stella instantaneously changes direction

(No GR required)!

v = 0.5/c

Fig: http://en.wikipedia.org/wiki/Twin_paradox

Spacetime diagrams

For Stella, Terence’s time is PA = 2 yrs., AC=46 yrs., CD=2 yrs.

For Terence, Stella’s time is PB=7 yrs., BD=7 yrs.

• At faster speeds this jump gets larger!

• Note that Stella only covers a very small part of the spacetime of

Terence:

Terence = ΔPBD

Stella = ΔPBA + ΔBCD

xP

t

A

B

CD

Tere

nce

Stell

a lea

ves

Stella returns

25yrs

Terence’s line of simultaneity

• Why is Stella is surprised that Terence has aged? Bad spacetime coordinates!

• Consider an example in 2D Euclidean space

• For Stella to realize this fact she must keep smb on the outward journey for (see previous page):

• AD/0.28 = 48/0.28 =171yrs!

A

B

C

x

y

θ

D Imagine measuring the line AD in x-y frame, but at point B you

rotate the axes by an angle θ to frame

Clearly then you would begin at point C and measure CD

Total will be AB+CD≠AD

Analogy taken from

Schutz’s book

Down to bad coordinates?

�1 + v/c�1� v/c

Ste

lla t

o Te

renc

e

Imagine Stella and Terence send laser light pulses to each other every second ) fe=1

Replace v by –v for blue-shifts

red-shift

Terence to Stella

Fig: http://en.wikipedia.org/wiki/Twin_paradox

• Stella sees more blue-shifted light

• Terence see more red-shifted light

• Thus, Terence ages more!

• Still confused?

�1� v/c�1 + v/c

Last  week’s  ques,on

1. Can  you  see  a  problem  with  “A  Journey  to  VEGA”  discussion?

2. How  is  it  that  some  Quasars/galaxies  have  been  observed  with  redshiHs  z  >  8.2  (high  relaOvisOc  redshiH  is  z>0.1)?

1 + z =�obs

�emit

=

rc+ v

c� v

Twin Paradox

Time dilation

Barn-poleParadox

Lorentz Contraction

Relativity of simultaneity

Homework: Consider Paradoxes?

Appendix: Spacetime diagrams

• Draw t-axis against x -axis

• A 45 degree line is the light cone (speed of light = c [=1])

• An constant velocity inertial object (no acceleration)

• An accelerating non-inertial object is a curved line

• Note that two inertial frames O with coordinates (t,x) and O’ with coordinates (t’,x’) are related as in the following cartoon:

tan� = v

�

-a

a

�

Appendix: The interval (metric)

Important definitions�s2 < 0� timelike�s2 > 0� spacelike�s2 = 0� null or lightlike

The spacetime interval is defined as

�s2 = ��t2 + �x2 + �y2 + �z2 = ��t2 + �r2

and it can be shown that the interval is invariant:

�s2 = �s2

for any two di⇥erent frames O and O

• However, due to time dilation Stella will measure the time between waves as

• Thus, Stella observes frequency

�te =�e

v � c=

1(1� v/c)fe

⇥ �e =c

fe

fo =⇤

1� v/c⇤1 + v/c

fe ⇥�1� v

c

⇥fe ⌅ v

c⇤ 0

Stella

Light source

(f e)

vc

Next wave meets at time delay

λe

Non-relativistic limit

�to = �te ⇥�

1� v2/c2 =�

1� v2/c2

(1� v/c)fe=

1fo

Appendix: Relativistic Doppler effect

28

Length contracted pole/ladder Length contracted garage/barn

These lead to…P.T.O.

Ref: http://en.wikipedia.org/wiki/Ladder_paradox

Lorentz length contraction

Lorentz contraction paradoxes?

• Various kinds have been devised

• We shall look at barn-pole (or ladder-garage) type paradoxes

• Key point is that length and time are linked so length contraction leads to time dilation and hence “relative simultaneity”

lS=20m

bT=15m

v=0.8c

Barn

Terence

Stella

• Problem is only with concept of “NOW,” there is only “relative simultaneity”

• As we can see Stella and Terence disagree on the times when both doors are actually open and shut!

Barn (Terence’s) framePole (Stella’s) frame

Ref: http://en.wikipedia.org/wiki/Ladder_paradox

Barn-Pole: double door variation

• Blue and red bands show the barn & pole spacetime, respectively.

• Front of the pole hits back of barn at event A.

• D is the point where the end of the pole enters the barn

• AB is simultaneous in barn frame so this will be what the barn sees as the pole length at the time of event A and thus, the pole fits in the barn

• However, from the point of view of the pole, AC is the pole length and thus, the back of the pole is outside the barn.

The above diagram is in the rest frame of the barn, with x and t being the barn frame. The pole frame is for a person

sitting on the front of the pole (axes x’ and t’).

Ref: http://en.wikipedia.org/wiki/Ladder_paradox

Double door spacetime diagram

• Consider a 20m pole which an Olympic athlete (Stella) runs with at speed v=0.8c into a barn of length 15m?

lS=20m

bT=15m

v=0.8c

Barn

Terence

Stella

Finite transmission speed (=c) of the shock wave prevents the pole from behaving rigidly and thus, Stella and Terence disagree on the time the door shuts; however, both

agree that the door does shut!Pole fitting into length contracted barn.

Ref: http://en.wikipedia.org/wiki/Ladder_paradox

Barn-pole: single door variation

• In barn frame rod stops simultaneously all along its length. Barn frame sees the ladder as AB, but the pole frame sees the pole as AC.

• When the back of the pole enters the garage at point D, it has not yet felt the effects of the impact.

• According to someone at rest with respect to the back of the pole, the front of the ladder will be at point E and will see the ladder as DE.

• The length in the pole frame is not the same as CA which is the rest length of the pole before impact. (See previous slide.)

Spacetime diagram when one of the doors remains shut: Ref: http://

en.wikipedia.org/wiki/Ladder_paradox

Single door spacetime diagram

• John Baez’s web page for many useful discussions on physics http://math.ucr.edu/home/baez/physics/

• go to SR and twin paradox

• Wikipedia has many nice diagrams

http://en.wikipedia.org/wiki/Twin_paradox

• Both of these web cites discuss a myriad of paradoxes in SR including the Barn-pole paradox, e.g., http://en.wikipedia.org/wiki/Ladder_paradox

• For criticism of Rindler’s “Man in grate” paradox see

http://www.iop.org/EJ/abstract/0143-0807/26/1/003

• Even over 105 years later, SR still causes much debate and sometimes controversy!

• However, this is only our Newtonian view of the universe! J

References and final com

ment

References and final comment