October 2006 SR/GR

Paradoxes in Special Relativity

Dr. Naylor

Paradoxes in Special Relativity

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Week 4SR/GR

102 years after Einstein’s

Special theory of relativity

A. Einstein, 1879-1955

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Paradoxes?

Twin Paradox

Time dilation

Barn-poleParadox

Lorentz Contraction

Relativity of simultaneity

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Subtleties?

• 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!

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Week 4SR/GR

Asymmetry ) “relative simultaneity”

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

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! =!

1! v2 =!

1! (0.96)2 = 0.28

Week 4SR/GR

Twin paradox

Terence Stella

V=24/25=0.96

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

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

!t =!!!1" v2

=14

0.28= 50yrs

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Week 4SR/GR

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

Terence

Stella

V=24/25=0.96

• Conventional 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.

?

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Week 4SR/GR

Spacetime diagrams

• 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 ¼ 0.5

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

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Week 4SR/GR

• For Stella, Terence’s time is PA = 2yrs, PC=48yrs, CD=2yrs• For Terence, Stella’s time is PB=7yrs=BD

• At faster speeds this jump gets larger!

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

Terence = ΔPBDStella = ΔPBA + ΔBCD

xP

t

A

B

CD

Tere

nce

Stella

leave

s

Stella returns

25yrs

Terence’s line of simultaneity

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• 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

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Digression: Relativistic Doppler effectNext week will will give a more rigorous derivation!!!

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

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!1 + v/c!1! v/c

Week 4SR/GR

Ste

lla to

Ter

ence

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

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Down to bad coordinates?• 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 page 8):

• AD/0.28 = 48/0.28 ¼ 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

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Week 4SR/GR

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The problem with Lorentz… contraction

Length contracted pole/ladder Length contracted garage/barn

This leads to…P.T.O.

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

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Week 4SR/GR

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”

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lS=20m

bT=15m

v=0.8c

Barn

Terence

Stella

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Barn-Pole: double door variation

• 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) frame Pole (Stella’s) frame

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

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Double door spacetime diagram• 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

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Week 4SR/GR

Barn-pole: single door variation

• 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 (v=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

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Week 4SR/GR

Single door spacetime diagramIn 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

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Week 4SR/GR

1. What length does Terence measure for the pole?[Hint, Lorentz length contraction formula:]

2. After Stella enters the barn how long does Terence measure before the pole hits the wall?

3. Is the interval in Terence’s frame “timelike or spacelike?”

4. According to Stella the barn length is 1. bS= ?

Questions on the barn-pole

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Week 4SR/GR

1. What length does Terence measure for the pole?Using the Lorentz contraction formula )

2. After Stella enters the barn how long does Terence measure before the pole hits the wall?

In c=1 units, (15-12)/0.6=3.75m But 1m = 1/(3£108) s ) 1.25 £ 10-8 s

3. The interval to Terence is “spacelike”:

ΔsB2 =-Δt2 + Δx2=-(3.75)2+(15)2=211m2 > 0

4. According to Stella the barn length is bS= 0.6£15 = 9m

Answ

ers !T =!

1! v2!S = 20!

1! (0.8)2 = 20" 0.6 = 12m

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Week 4SR/GR

References and final comment• 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 101 years later, SR still causes much

debate and sometimes controversy• However, this is only due to our Newtonian view of

the universe!

References and final com

ment

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