Faster-Than-Light Paradoxes
in Special Relativity
Brice N. Cassenti
Faster-Than-Light Paradoxes
in 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
•Assume transformation is linear
ct '  Act  Bx
x'  Cct  Dx
•Assume speed of light is the same in each frame
c dt  dx  c dt ' dx'  c d
2
2
2
2
2
2
2
2
Relative Motion
(x,ct)
v
(x’,ct’)
x'   ( x  ct )
ct '   (ct  x)
x   ( x' ct ' )
ct   (ct ' x' )
v
1


2
c
1 
(ct ) 2  x 2  (ct ' ) 2  x'2
Lorentz Transformation
Inverse transformation
(ct ) 2  x 2  (ct ' ) 2  x'2
1
v


c
1  2
  
 x'   
 
ct '  
    x 
 

  ct 
 x       x' 
 
 

ct     ct '
Use of hyperbolic functions
  tanh 
  cosh 
  sinh 
  
 x'   cosh 
 
ct '  sinh 
 x  cosh 
 
ct   sinh 
 sinh    x 
 

cosh   ct 
sinh    x' 
 

cosh   ct '
Coordinate Systems
ct‘
1.0
0.9
0.8
0.7
0.6
ct
x‘
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1.0
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:
v
1 
c
2
• The result holds in either frame.
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:
v
1 
c
2
• The result holds in either frame.
Paradoxes in Special Relativity
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•
•
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Pole Vaulter
Twin Paradox
Faster-than-Light Travel
Instant Messaging
Pole Vaulter & Barn
2
1
v
1   
2
c
15 ft
10 ft
Pole Vaulter & Barn
v
2
1
v
1   
2
c
7.5 ft
10 ft
Pole Vaulter & Barn
v
2
1
v
1   
2
c
15 ft
5 ft
Pole Vaulter Paradox
Barn View
• Front door opens
Pole Vaulter Paradox
Barn View
• Front door opens
• Forward end of pole enters barn
Pole Vaulter Paradox
Barn View
• Front door opens
• Forward end of pole enters barn
• Back end of pole enters barn
Pole Vaulter Paradox
Barn View
•
•
•
•
Front door opens
Forward end of pole enters barn
Back end of pole enters barn
Front door closes
Pole Vaulter Paradox
Barn 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 Paradox
Barn 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 Paradox
Barn View
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•
•
•
•
•
•
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 Paradox
Barn 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 Paradox
Barn 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 Paradox
Pole Vaulter View
• Front doors opens
Pole Vaulter Paradox
Pole Vaulter View
• Front doors opens
• Forward end of pole enters barn
Pole Vaulter Paradox
Pole Vaulter View
• Front doors opens
• Forward end of pole enters barn
• Back door opens
Pole Vaulter Paradox
Pole Vaulter View
•
•
•
•
Front doors opens
Forward end of pole enters barn
Back door opens
Front end of pole leaves barn
Pole Vaulter Paradox
Pole 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 Paradox
Pole 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 Paradox
Pole 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 Paradox
Pole 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 Paradox
Pole 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 Paradox
Resolution
• 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 Paradox
Resolution
• 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 of
Faster-than-Light Travel
•Recall
c 2 d 2  c 2 dt 2  dx 2
•Setting dx  vdt
2
2
2
2
2
•Then c d  c  v dt
•If
•Then
•Let
•And
•Or
vc
 c 2 d 2  c 2 d 2  v 2  c 2 dt 2  v 2 dt 2  c 2 dt 2
c 2 dt '2  v 2 dx 2
dx'2  c 2 dt 2
c 2 d 2  c 2 dt '2 dx'2
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 Energy
c 2 d 2  c 2 dt 2  dx 2
Relativistic Energy
c 2 d 2  c 2 dt 2  dx 2
2
 dt   dx 
1   

 d   cd 
2
Relativistic Energy
c 2 d 2  c 2 dt 2  dx 2
2
 dt   dx 
1   

 d   cd 
2
2
 dt 
2 2  dx 
m c  m c    m0 c  
 d 
 d 
2
0
4
2
0
4
2
Relativistic Energy
c 2 d 2  c 2 dt 2  dx 2
2
 dt   dx 
1   

 d   cd 
2
2
 dt 
2 2  dx 
m c  m c    m0 c  
 d 
 d 
E  m 0 c 2
2
0
4
2
0
pc  m0 c 2
4
2
Relativistic Energy
c 2 d 2  c 2 dt 2  dx 2
2
 dt   dx 
1   

 d   cd 
2
2
 dt 
2 2  dx 
m c  m c    m0 c  
 d 
 d 
E  m 0 c 2
2
0
4
2
0
2
4
pc  m0 c 2
1
 1 2
2
E  m 0 c 1     m 0 c  m 0 v 2
2
 2 
2
Relativistic Energy
• There is a relativistic rest mass energy
2
– m0 c & constant in all reference frames
• For
2
 dx   c
,
m
c
 
0
 dt 
is imaginary
– Particles are tachyons
– Tachyons have never been observed
Relativistic Accelerating Rocket
Relativistic Accelerating Rocket
p  m0 c sinh 
Relativistic Accelerating Rocket
p  m0 c sinh 
dp
d
 m0 c cosh 
d
d
Relativistic Accelerating Rocket
p  m0 c sinh 
dp
d
 m0 c cosh 
d
d
In rocket frame   0
Relativistic Accelerating Rocket
p  m0 c sinh 
dp
d
 m0 c cosh 
d
d
In rocket frame   0
dp
d
 m0 c
 m0 a
d
d
Relativistic Accelerating Rocket
p  m0 c sinh 
dp
d
 m0 c cosh 
d
d
In rocket frame   0
dp
d
 m0 c
 m0 a
d
d
d
ac
d
Relativistic Accelerating Rocket
p  m0 c sinh 
dp
d
 m0 c cosh 
d
d
In rocket frame   0
dp
d
 m0 c
 m0 a
d
d
d
ac
d
dt
 cosh 
d
dx
 c sinh 
d
Constant Acceleration Rocket
d
ac
d
dt
 cosh 
d
dx
 c sinh 
d
dx
 c tanh 
dt
  1
c
 a 
t  sinh  
a
 c 
c2 
 a  
x
cosh    1

a 
 c  
t 
a 2
x
2
dx
 a
d
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
m02 c 4  E 2  p 2 c 2  0
Photon Rocket
m02 c 4  E 2  p 2 c 2  0
E  pc
Photon Rocket
m02 c 4  E 2  p 2 c 2  0
E  pc
dp 1 dE
d

 mc
d c d
d
Photon Rocket
m02 c 4  E 2  p 2 c 2  0
E  pc
dp 1 dE
d

 mc
d c d
d
d E  mc 2 
d
dm
2 d
2 dm

c
 mc
c
0
d
d
d
d
dE
2
Photon Rocket
m02 c 4  E 2  p 2 c 2  0
E  pc
dp 1 dE
d

 mc
d c d
d
d E  mc 2 
d
m  mi e

dm
2 d
2 dm

c
 mc
c
0
d
d
d
d
dE
2
f
mi
MR 
e
mf
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.