Faster-Than-Light Paradoxesin Special Relativity

Brice N. Cassenti

Faster-Than-Light Paradoxesin Special Relativity

• Principle of Special Relativity

• Paradoxes in Special Relativity

• Relativistic Rockets

• Conclusions

Principle of Special Relativity

• Postulates

• Lorentz Transformation

• Lorentz Contraction

• Time Dilation

Postulates of the Special Theory of Relativity

• The laws of physics are the same in all inertial coordinate systems

• The speed of light is the same in all inertial coordinate systems

Inertial Reference Systems

Lorentz Transformation Derivation

DxCctx

BxActct

'

'

22222222 '' dcdxdtcdxdtc

•Assume transformation is linear

•Assume speed of light is the same in each frame

Relative Motion

v

(x’,ct’)

(x,ct)

2222

2

')'()(

1

1

)''(

)''(

)('

)('

xctxct

c

v

xctct

ctxx

xctct

ctxx

Lorentz Transformation

'

'

'

'

1

1

')'()(

2

2222

ct

x

ct

x

ct

x

ct

x

c

v

xctxct

'

'

coshsinh

sinhcosh

coshsinh

sinhcosh

'

'

sinh

cosh

tanh

ct

x

ct

x

ct

x

ct

x

Inverse transformation Use of hyperbolic functions

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ct

x

Coordinate Systemsct‘

‘x

Lorentz Contraction

• Length is measured in a frame by noting end point locations at a given time.

• When observing a rod in a moving frame, with its length in the direction of motion the frame, the length is shorter by:

• The result holds in either frame.

2

1

c

v

Time Dilation

• Time is measured in a frame by watching a clock fixed in the moving frame.

• When observing a clock in a moving frame the clock runs slower by:

• The result holds in either frame.

2

1

c

v

Paradoxes in Special Relativity

• Pole Vaulter

• Twin Paradox

• Faster-than-Light Travel

• Instant Messaging

Pole Vaulter & Barn

10 ft

2

11

2

c

v

15 ft

Pole Vaulter & Barn

10 ft

2

11

2

c

v

7.5 ft

v

Pole Vaulter & Barn

5 ft

2

11

2

c

v

15 ft

v

Pole Vaulter ParadoxBarn View

• Front door opens

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn• Back end of pole enters barn

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn• Back end of pole enters barn• Front door closes

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn• Back end of pole enters barn• Front door closes• Pole entirely inside barn

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn• Back end of pole enters barn• Front door closes• Pole entirely inside barn• Back door opens

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn• Back end of pole enters barn• Front door closes• Pole entirely inside barn• Back door opens• Front end of pole leaves barn

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn• Back end of pole enters barn• Front door closes• Pole entirely inside barn• Back door opens• Front end of pole leaves barn• Back end of pole leaves barn

Pole Vaulter ParadoxBarn View

• Front door opens• Forward end of pole enters barn• Back end of pole enters barn• Front door closes• Pole entirely inside barn• Back door opens• Front end of pole leaves barn• Back end of pole leaves barn

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn• Back door opens

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn• Back door opens• Front end of pole leaves barn

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn• Back door opens• Front end of pole leaves barn• Both ends of pole never in barn simultaneously

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn• Back door opens• Front end of pole leaves barn• Both ends of pole never in barn simultaneously• Back end of pole enters barn

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn• Back door opens• Front end of pole leaves barn• Both ends of pole never in barn simultaneously• Back end of pole enters barn• Front door closes

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn• Back door opens• Front end of pole leaves barn• Both ends of pole never in barn simultaneously• Back end of pole enters barn• Front door closes• Back end of pole leaves barn

Pole Vaulter ParadoxPole Vaulter View

• Front doors opens• Forward end of pole enters barn• Back door opens• Front end of pole leaves barn• Both ends of pole never in barn simultaneously• Back end of pole enters barn• Front door closes• Back end of pole leaves barn

Pole Vaulter ParadoxResolution

• Use Lorentz transformation• In barn view, time interval from back door closing

to front door opening is less than time light needs to cover distance

• In pole vaulter view, time from front of pole to leave and back to enter is less than time light needs to cover distance along pole

• Lorentz transformation correctly predicts both views correctly from both the barn and pole vaulter frames

Pole Vaulter Paradox - Conclusion

Faster-than-light signals would allow contradictions in

observations

Twin Paradox

• One of two identical twins leaves at a speed such that each twin sees the other clock running at half the rate of theirs

• The traveling twin reverses speed and returns

• Does each think the other has aged half as much?

Twin ParadoxResolution

• In order to set out turn around and stop at the return the traveling twin accelerates.

• The traveling twin feels the acceleration.

• Hence, the traveling twin is not in an inertial frame.

• The stationary twin ages twice as much as the traveling twin.

Twin Paradox - Resolution Constant Acceleration

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html

Twin Paradox - Resolution Infinite Acceleration

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html

Twin Paradox - Conclusion

Accelerating reference frames need to be treated with a more

General Theory of Relativity

Faster-Than-Light Travel

Faster-Than-Light Travel

Faster-Than-Light Travel

Faster-Than-Light Travel

Faster-Than-Light Travel

Faster-Than-Light Travel

Faster-Than-Light Travel

Faster-than-Light Travel – Conclusion

Faster-than-Light motion reverses travel through time – a time machine

The Mathematics ofFaster-than-Light Travel

22222 dxdtcdc •Recall

•Then

•Setting vdtdx 22222 dtvcdc

•If cv •Then 22222222222 dtcdtvdtcvdcdc

•Or

•Let

•And

2222 ' dxvdtc 222' dtcdx

22222 '' dxdtcdc

Instant Messaging

Instant Messaging

Instant Messaging

Instant Messaging

Instant Messaging

Instant Messaging

Instant Messaging

Instant Messaging

Instant Messaging –Conclusion

Instant communications is a nonlinear process (exponential) and does not satisfy the postulates of Quantum Mechanics.

The number of messages must collapse to a single message.

Faster-than-light travel may result in the collapse of the wave function

Relativistic Rockets

• Relativistic Energy

• Constant Acceleration Rocket

• Photon Rocket

Relativistic Energy

Relativistic Energy22222 dxdtcdc

Relativistic Energy22222 dxdtcdc

22

1

cd

dx

d

dt

Relativistic Energy

222

0

242

042

0

d

dxcm

d

dtcmcm

22222 dxdtcdc 22

1

cd

dx

d

dt

Relativistic Energy

222

0

242

042

0

d

dxcm

d

dtcmcm

20cmE

20cmpc

22222 dxdtcdc 22

1

cd

dx

d

dt

Relativistic Energy

222

0

242

042

0

d

dxcm

d

dtcmcm

20cmE

20cmpc

20

20

220 2

1

2

11 vmcmcmE

22222 dxdtcdc 22

1

cd

dx

d

dt

Relativistic Energy

• There is a relativistic rest mass energy

• For is imaginary – Particles are tachyons– Tachyons have never been observed

20cm– & constant in all reference frames

cdtdx

2

0cm,

Relativistic Accelerating Rocket

Relativistic Accelerating Rocket

sinh0cmp

Relativistic Accelerating Rocket

sinh0cmp

d

dcm

d

dpcosh0

Relativistic Accelerating Rocket

sinh0cmp

d

dcm

d

dpcosh0

0In rocket frame

Relativistic Accelerating Rocket

sinh0cmp

d

dcm

d

dpcosh0

amd

dcm

d

dp00

0In rocket frame

Relativistic Accelerating Rocket

sinh0cmp

d

dcm

d

dpcosh0

amd

dcm

d

dp00

d

dca

0In rocket frame

Relativistic Accelerating Rocket

sinh0cmp

d

dcm

d

dpcosh0

amd

dcm

d

dp00

d

dca

sinh

cosh

cd

dxd

dt

0In rocket frame

Constant Acceleration Rocket

1cosh

sinh

2

c

a

a

cx

c

a

a

ct

tanhcdt

dx

ad

dx

ax

t

2

2

sinh

cosh

cd

dxd

dt

d

dca 1

Constant Acceleration Results

• In the reference frame of the rocket when sinh>1, dx/d>c.

• Accelerating at one gravity a crew could circumnavigate the universe within their working lifetime.

Photon Rocket

Photon Rocket

0222420 cpEcm

Photon Rocket

0222420 cpEcm

pcE

Photon Rocket

0222420 cpEcm

pcE

d

dmc

d

dE

cd

dp

1

Photon Rocket

0222420 cpEcm

pcE

d

dmc

d

dE

cd

dp

1

0222

2

d

dmc

d

dmc

d

dmc

d

dE

d

mcEd

Photon Rocket

0222420 cpEcm

pcE

d

dmc

d

dE

cd

dp

1

0222

2

d

dmc

d

dmc

d

dmc

d

dE

d

mcEd

emm if

f

i em

mMR

Photon Rocket Results

• Velocity parameter replaces velocity in the ‘rocket equation’

Photon Rocket Results

• Velocity parameter replaces velocity in the ‘rocket equation’

• Circumnavigating the universe at one gravity requires an enormous mass ratio

Photon Rocket Results

• Velocity parameter replaces velocity in the ‘rocket equation’

• Circumnavigating the universe at one gravity requires an enormous mass ratio

• But the mass required is not larger than the mass of the universe

Conclusions

Faster-than-light paradoxes in the Special Theory of Relativity are:

• More than curiosities •They can provide a better understanding of space and time

They can add insight into other paradoxes•Collapse of the wave function

They can lead to new physical theories•And may allow unlimited access to the universe.