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**Faster-Than-Light Paradoxes in Special Relativity**

Brice N. Cassenti

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**Faster-Than-Light Paradoxes in Special Relativity**

Principle of Special Relativity Paradoxes in Special Relativity Relativistic Rockets Conclusions

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**Principle of Special Relativity**

Postulates Lorentz Transformation Lorentz Contraction Time Dilation

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**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

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**Inertial Reference Systems**

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**Lorentz Transformation Derivation**

Assume transformation is linear Assume speed of light is the same in each frame

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Relative Motion (x,ct) v (x’,ct’)

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**Lorentz Transformation**

Inverse transformation Use of hyperbolic functions

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Coordinate Systems ct ‘ ct x ‘ x

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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.

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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.

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**Paradoxes in Special Relativity**

Pole Vaulter Twin Paradox Faster-than-Light Travel Instant Messaging

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Pole Vaulter & Barn 15 ft 10 ft

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Pole Vaulter & Barn v 7.5 ft 10 ft

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Pole Vaulter & Barn v 15 ft 5 ft

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**Pole Vaulter Paradox Barn View**

Front door opens

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**Pole Vaulter Paradox Barn View**

Front door opens Forward end of pole enters barn

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**Pole Vaulter Paradox Barn View**

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

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**Pole Vaulter Paradox Barn View**

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

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**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

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**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

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**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

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**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

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**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

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**Pole Vaulter Paradox Pole Vaulter View**

Front doors opens

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**Pole Vaulter Paradox Pole Vaulter View**

Front doors opens Forward end of pole enters barn

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**Pole Vaulter Paradox Pole Vaulter View**

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

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**Pole Vaulter Paradox Pole Vaulter View**

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

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**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

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**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

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**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

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**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

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**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

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**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

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**Pole Vaulter Paradox - Conclusion**

Faster-than-light signals would allow contradictions in observations

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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?

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**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.

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**Twin Paradox - Resolution Constant Acceleration**

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**Twin Paradox - Resolution Infinite Acceleration**

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**Twin Paradox - Conclusion**

Accelerating reference frames need to be treated with a more General Theory of Relativity

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**Faster-Than-Light Travel**

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**Faster-Than-Light Travel**

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**Faster-Than-Light Travel**

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**Faster-Than-Light Travel**

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**Faster-Than-Light Travel**

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**Faster-Than-Light Travel**

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**Faster-Than-Light Travel**

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**Faster-than-Light Travel – Conclusion**

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

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**The Mathematics of Faster-than-Light Travel**

Recall Setting Then If Then Let And Or

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Instant Messaging

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Instant Messaging

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Instant Messaging

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Instant Messaging

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Instant Messaging

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Instant Messaging

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Instant Messaging

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Instant Messaging

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**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

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**Relativistic Rockets Relativistic Energy Constant Acceleration Rocket**

Photon Rocket

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Relativistic Energy

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Relativistic Energy

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Relativistic Energy

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Relativistic Energy

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Relativistic Energy

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Relativistic Energy

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**& constant in all reference frames**

Relativistic Energy There is a relativistic rest mass energy For is imaginary Particles are tachyons Tachyons have never been observed & constant in all reference frames ,

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**Relativistic Accelerating Rocket**

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**Relativistic Accelerating Rocket**

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**Relativistic Accelerating Rocket**

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**Relativistic Accelerating Rocket**

In rocket frame

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**Relativistic Accelerating Rocket**

In rocket frame

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**Relativistic Accelerating Rocket**

In rocket frame

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**Relativistic Accelerating Rocket**

In rocket frame

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**Constant Acceleration Rocket**

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**Constant Acceleration Results**

In the reference frame of the rocket when sinhq>1, dx/dt>c. Accelerating at one gravity a crew could circumnavigate the universe within their working lifetime.

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Photon Rocket

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Photon Rocket

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Photon Rocket

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Photon Rocket

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Photon Rocket

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Photon Rocket

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Photon Rocket Results Velocity parameter replaces velocity in the ‘rocket equation’

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Photon Rocket Results Velocity parameter replaces velocity in the ‘rocket equation’ Circumnavigating the universe at one gravity requires an enormous mass ratio

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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

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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.

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