faster-than-light paradoxes in special relativity brice n. cassenti
TRANSCRIPT
![Page 1: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/1.jpg)
Faster-Than-Light Paradoxesin Special Relativity
Brice N. Cassenti
![Page 2: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/2.jpg)
Faster-Than-Light Paradoxesin Special Relativity
• Principle of Special Relativity
• Paradoxes in Special Relativity
• Relativistic Rockets
• Conclusions
![Page 3: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/3.jpg)
Principle of Special Relativity
• Postulates
• Lorentz Transformation
• Lorentz Contraction
• Time Dilation
![Page 4: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/4.jpg)
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
![Page 5: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/5.jpg)
Inertial Reference Systems
![Page 6: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/6.jpg)
Lorentz Transformation Derivation
DxCctx
BxActct
'
'
22222222 '' dcdxdtcdxdtc
•Assume transformation is linear
•Assume speed of light is the same in each frame
![Page 7: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/7.jpg)
Relative Motion
v
(x’,ct’)
(x,ct)
2222
2
')'()(
1
1
)''(
)''(
)('
)('
xctxct
c
v
xctct
ctxx
xctct
ctxx
![Page 8: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/8.jpg)
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
![Page 9: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/9.jpg)
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
![Page 10: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/10.jpg)
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
![Page 11: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/11.jpg)
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
![Page 12: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/12.jpg)
Paradoxes in Special Relativity
• Pole Vaulter
• Twin Paradox
• Faster-than-Light Travel
• Instant Messaging
![Page 13: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/13.jpg)
Pole Vaulter & Barn
10 ft
2
11
2
c
v
15 ft
![Page 14: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/14.jpg)
Pole Vaulter & Barn
10 ft
2
11
2
c
v
7.5 ft
v
![Page 15: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/15.jpg)
Pole Vaulter & Barn
5 ft
2
11
2
c
v
15 ft
v
![Page 16: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/16.jpg)
Pole Vaulter ParadoxBarn View
• Front door opens
![Page 17: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/17.jpg)
Pole Vaulter ParadoxBarn View
• Front door opens• Forward end of pole enters barn
![Page 18: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/18.jpg)
Pole Vaulter ParadoxBarn View
• Front door opens• Forward end of pole enters barn• Back end of pole enters barn
![Page 19: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/19.jpg)
Pole Vaulter ParadoxBarn View
• Front door opens• Forward end of pole enters barn• Back end of pole enters barn• Front door closes
![Page 20: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/20.jpg)
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
![Page 21: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/21.jpg)
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
![Page 22: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/22.jpg)
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
![Page 23: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/23.jpg)
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
![Page 24: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/24.jpg)
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
![Page 25: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/25.jpg)
Pole Vaulter ParadoxPole Vaulter View
• Front doors opens
![Page 26: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/26.jpg)
Pole Vaulter ParadoxPole Vaulter View
• Front doors opens• Forward end of pole enters barn
![Page 27: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/27.jpg)
Pole Vaulter ParadoxPole Vaulter View
• Front doors opens• Forward end of pole enters barn• Back door opens
![Page 28: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/28.jpg)
Pole Vaulter ParadoxPole Vaulter View
• Front doors opens• Forward end of pole enters barn• Back door opens• Front end of pole leaves barn
![Page 29: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/29.jpg)
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
![Page 30: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/30.jpg)
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
![Page 31: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/31.jpg)
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
![Page 32: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/32.jpg)
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
![Page 33: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/33.jpg)
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
![Page 34: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/34.jpg)
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
![Page 35: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/35.jpg)
Pole Vaulter Paradox - Conclusion
Faster-than-light signals would allow contradictions in
observations
![Page 36: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/36.jpg)
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?
![Page 37: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/37.jpg)
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.
![Page 38: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/38.jpg)
Twin Paradox - Resolution Constant Acceleration
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html
![Page 39: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/39.jpg)
Twin Paradox - Resolution Infinite Acceleration
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html
![Page 40: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/40.jpg)
Twin Paradox - Conclusion
Accelerating reference frames need to be treated with a more
General Theory of Relativity
![Page 41: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/41.jpg)
Faster-Than-Light Travel
![Page 42: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/42.jpg)
Faster-Than-Light Travel
![Page 43: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/43.jpg)
Faster-Than-Light Travel
![Page 44: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/44.jpg)
Faster-Than-Light Travel
![Page 45: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/45.jpg)
Faster-Than-Light Travel
![Page 46: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/46.jpg)
Faster-Than-Light Travel
![Page 47: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/47.jpg)
Faster-Than-Light Travel
![Page 48: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/48.jpg)
Faster-than-Light Travel – Conclusion
Faster-than-Light motion reverses travel through time – a time machine
![Page 49: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/49.jpg)
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
![Page 50: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/50.jpg)
Instant Messaging
![Page 51: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/51.jpg)
Instant Messaging
![Page 52: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/52.jpg)
Instant Messaging
![Page 53: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/53.jpg)
Instant Messaging
![Page 54: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/54.jpg)
Instant Messaging
![Page 55: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/55.jpg)
Instant Messaging
![Page 56: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/56.jpg)
Instant Messaging
![Page 57: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/57.jpg)
Instant Messaging
![Page 58: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/58.jpg)
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
![Page 59: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/59.jpg)
Relativistic Rockets
• Relativistic Energy
• Constant Acceleration Rocket
• Photon Rocket
![Page 60: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/60.jpg)
Relativistic Energy
![Page 61: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/61.jpg)
Relativistic Energy22222 dxdtcdc
![Page 62: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/62.jpg)
Relativistic Energy22222 dxdtcdc
22
1
cd
dx
d
dt
![Page 63: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/63.jpg)
Relativistic Energy
222
0
242
042
0
d
dxcm
d
dtcmcm
22222 dxdtcdc 22
1
cd
dx
d
dt
![Page 64: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/64.jpg)
Relativistic Energy
222
0
242
042
0
d
dxcm
d
dtcmcm
20cmE
20cmpc
22222 dxdtcdc 22
1
cd
dx
d
dt
![Page 65: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/65.jpg)
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
![Page 66: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/66.jpg)
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,
![Page 67: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/67.jpg)
Relativistic Accelerating Rocket
![Page 68: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/68.jpg)
Relativistic Accelerating Rocket
sinh0cmp
![Page 69: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/69.jpg)
Relativistic Accelerating Rocket
sinh0cmp
d
dcm
d
dpcosh0
![Page 70: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/70.jpg)
Relativistic Accelerating Rocket
sinh0cmp
d
dcm
d
dpcosh0
0In rocket frame
![Page 71: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/71.jpg)
Relativistic Accelerating Rocket
sinh0cmp
d
dcm
d
dpcosh0
amd
dcm
d
dp00
0In rocket frame
![Page 72: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/72.jpg)
Relativistic Accelerating Rocket
sinh0cmp
d
dcm
d
dpcosh0
amd
dcm
d
dp00
d
dca
0In rocket frame
![Page 73: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/73.jpg)
Relativistic Accelerating Rocket
sinh0cmp
d
dcm
d
dpcosh0
amd
dcm
d
dp00
d
dca
sinh
cosh
cd
dxd
dt
0In rocket frame
![Page 74: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/74.jpg)
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
![Page 75: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/75.jpg)
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.
![Page 76: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/76.jpg)
Photon Rocket
![Page 77: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/77.jpg)
Photon Rocket
0222420 cpEcm
![Page 78: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/78.jpg)
Photon Rocket
0222420 cpEcm
pcE
![Page 79: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/79.jpg)
Photon Rocket
0222420 cpEcm
pcE
d
dmc
d
dE
cd
dp
1
![Page 80: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/80.jpg)
Photon Rocket
0222420 cpEcm
pcE
d
dmc
d
dE
cd
dp
1
0222
2
d
dmc
d
dmc
d
dmc
d
dE
d
mcEd
![Page 81: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/81.jpg)
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
![Page 82: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/82.jpg)
Photon Rocket Results
• Velocity parameter replaces velocity in the ‘rocket equation’
![Page 83: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/83.jpg)
Photon Rocket Results
• Velocity parameter replaces velocity in the ‘rocket equation’
• Circumnavigating the universe at one gravity requires an enormous mass ratio
![Page 84: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/84.jpg)
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
![Page 85: Faster-Than-Light Paradoxes in Special Relativity Brice N. Cassenti](https://reader035.vdocuments.us/reader035/viewer/2022062619/5516b857550346f0208b5554/html5/thumbnails/85.jpg)
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.