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An Undergraduate Seminar for APHY 199, University of the Philippines-Los Banos Author: Karl Simon Revelar, BS Applied Physics

An Undergraduate Seminar for APHY 199, University of the Philippines-Los Banos Author: Karl Simon Revelar, BS Applied Physics

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- 1. Or How to Construct Your Own Time Machine
- 2. Special Relativity Universal speed limit, c No special frame Space and time merger, space-time Space-time diagram Lorentz contraction Time dilation (“Twin Paradox”) Relativity of Simultaneity
- 3. Future and Past Light cones The light cone represents all the possible world lines forwards or backwards in time in the universe since nothing can travel faster than light according to SR. Taken from Michio Kaku’s Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension, p. 239
- 4. Non-Euclidean Geometry and General Relativity Metric Tensor Connection coefficients Geodesics and the Geodesic Equation Null, Time-like, Space-like space-time curves Riemann Curvature Tensor Curvature α Energy density: Einstein Field Equations Schwarzschild Geometry
- 5. Time Machine any object or system that transports an observer or particle to the past or the future1 Seriously far from HG Wells’ (1885) time machine 1. Visser, M. 204
- 6. The “Physically Probable” Time Machine Makes use of concepts in General Relativity and Quantum Theory (or Quantum Gravity) Most of the speculative “machines” are certain geometries or solutions to the Einstein Field Equations where a closed time-like [(-) metric for η=(-1,1,1,1) ] curve or CTC exists e.g. : Kerr black hole Wormhole Godel Universe, etc. 1. Visser, M. 204
- 7. Solutions to the Einstein Field Equations that yield closed time-like curves 1. van Stockum Geometry Describes space-time around an infinitely long rotating cylinder of dust Time travel by traveling around the cylinder where you meet your old self at your starting point. The backward time-jump is given by where . CTC occurs when L is (-). Light cones tilt over so that world lines can point to the past The time-jump can be made as large as possible by going around the curve N times! and The CTCs in this geometry cover the whole space-time!!
- 8. Solutions to the Einstein Field Equations that yield closed time-like curves Problems of the van Stockum Geometry: Unphysical ( An infinitely long cylinder? CTCs are everywhere?) Mathematical gibberish (A solution to a differential equation need not mean physically meaningful.) The geometry is not asymptotically flat. (Space-time is curved everywhere.) Side note: You cannot travel into the future in van Stockum space- time. (my interpretation) CTCs can exist even in flat space-time
- 9. Solutions to the Einstein Field Equations that yield closed time-like curves 2. Gödel Universe van Stockum geometry where cosmological constant is non-zero Same method of travelling through time (going around the cylinder) Same problems as van Stockum (unphysical, just a mathematical exercise)
- 10. Solutions to the Einstein Field Equations that yield closed time-like curves 3a.Kerr Geometry (Case 1: radius < mass) Space-time due to a rotating black hole that becomes a ring by virtue of EFE CTCs are curves in the event horizon where r and θ are constant (r<zero but still meaningful) and all curves in the inner horizon Problems of this geometry: Chronology violations are hidden from us by the event horizon (the surface where even light cannot escape, therefore you cannot transfer information to outside) Inner horizon is unstable.
- 11. Solutions to the Einstein Field Equations that yield closed time-like curves 3b. Kerr Geometry (Case 2: radius > mass) The ring singularity has no event horizon.(It is naked.) Chronology violations can now be viewed anywhere outside CTCs are also curves in the event horizon where r and θ are constant (r<zero but still meaningful) and all curves in the inner horizon Problem with this geometry: Cosmic Censorship Conjecture due to Penrose Tidal gravity near horizon can kill you. (You would be stretched upwards and downwards, like water on Earth’s surface as pulled by the moon.)
- 12. The Kerr Blackhole From outside to center: Event horizon, inner horizon, (innermost) ring singularity. This is a 2D embedding diagram, and therefore when extended to 3D becomes a sphere. From Visser, M. Lorentzian Wormholes…p.76
- 13. Solutions to the Einstein Field Equations that yield closed time-like curves 4. Space-time due to Spinning Cosmic Strings A rotating infinite line mass Rotation curves space-time such that when one flies around the string one notices a deficit in subtended angle (frame gets dragged-my interpretation) One goes backward in time (proportional to rotation) CTCs are the integral curves of φ when r< a constant. Problem with this geometry: The usual. (Unphysical=infinitely long)
- 14. Solutions to the Einstein Field Equations that yield closed time-like curves 5. Gott Geometry Almost the same idea as 4. where now a system of two infinite line masses rotate around an axis to produce CTCs CTCs cannot be produced for very light strings, only for very massive and speedy strings. Time travel to infinite past and future is possible Problems in this geometry: Unphysical (infinite length) (-) Infinite time Calculated total mass of string is too large! (my calculation, weak argument)
- 15. Solutions to the Einstein Field Equations that yield closed time-like curves 5. Gott Geometry Cosmic Censorship Conjecture: According to Penrose when a star implodes into a singularity (hole in space-time) the implosion always leaves a horizon so that we cannot see what’s inside or in other words, there are no naked singularities. A bet was made between Kip Thorne, John Preskill and Stephen Hawking. Hawking, months later, discovered that it is probable that after a black hole evaporates, the singularity is left behind. He did not concede on the ground that evaporation is a quantum effect. But this is still insufficient proof against the conjecture.
- 16. Bet Between Hawking, and Thorne, Preskill Hawking after discovering that naked singularities probably exist did not concede on the ground that the bet was about naked singularities due to classical physics. From Kip Thorne’s Black Holes and Time Warps…, p. 482
- 17. Solutions to the Einstein Field Equations that yield closed time-like curves 6. Mallett’s Earth-Based Time Machine As seen on the documentary on Discovery Science, “The World’s First Time Machine” Based on a paper submitted by Ronald Mallett to Physics Letters A that a rotating ring of laser induces inertial frame-dragging on a massive spinning particle on the center and produces CTCs outside the cylinder Criticisms of this machine (all due to Olum & Everett): Energy of laser is not enough to twist space-time Hawking’s Chronology Protection Conjecture Mallett’s space-time has a singularity (incorrect analysis)
- 18. Mallett's Time Machine (Stationary) Mallett’s machine is a system of rotating half-silvered mirrors that guide the laser around. From Mallett’s Physical Letters A article, Weak Gravitational Field of the Electromagnetic Radiation in a Ring Laser, p.215
- 19. A Summary of Presented Solutions that yield CTCs Most of the Presented Solutions to EFEs: Involve cylindrical symmetry e. g. infinitely long cylinders, very massive and rapidly rotating strings, rotating lasers Involve unphysical objects e.g. infinitely long cylinders and strings Do not mirror the space-time in our universe i.e. CTCs are everywhere, not asymptotically flat, negative infinite time (time before Big Bang? Not for now.) Are impossible for human time travel (for now or near future) i.e. intense tidal gravity, very far away from Earth
- 20. The Wormhole An example of a wormhole that is 1 kilometer long and connects Earth and Vega, which is 26 light years away in normal space travel. Diagram assumes universe is 2D. From Black Holes and Time Warps…, p.485.
- 21. The Wormhole: Can be inter-universe or intra-universe Two singularities that meet in hyperspace Also a solution to EFE (discovered by Einstein himself in 1916) known as the Einstein-Rosen bridge Parts: Mouths and Throat Mostly are “diseased” i.e. unstable or have unphysical quirks and die out as soon as they are made (due to radiation) Quite impossible to be created by virtue of Cosmic Censorship and that they would find each other in hyperspace or be produced naturally
- 22. Traversable Wormhole A solution presented by Kip Thorne to Carl Sagan to smoothen out the science in Sagan’s novel, Contact, where the heroine travelled to Vega in just one hour using a black hole (instead of a worm hole) Incoming accelerating radiation and vacuum fluctuations in the black hole can destroy the rocket ship
- 23. Wormholes before Traversable Wormholes Thorne’s paper according to Thorne Vacuum fluctuations Vacuum fluctuations and incoming radiation near the horizon are allow the wormhole to negative average energy shrink instantly after density material and can open the wormhole and creation de-focus incoming radiation Cannot be produced Quantum strategy and naturally Semi-classical strategy
- 24. Traversable Wormhole and Vacuum fluctuations The “real” vacuum is not empty. If we rid it of EM fields, some parts outside that have less grab fields from the other parts with excess, and then grab it back, these fields oscillate randomly In flat space-time, the average energy density is zero In curved space-time, it is negative as seen by a light beam traveling through a wormhole. Negative energy density defocuses the light beam so that they do not cause damage to the wormhole
- 25. Traversable Wormhole Creation Strategies The quantum strategy is to go down in vacuum at Planck length making use of gravitational vacuum fluctuations (space is erratic and can produce tiny wormholes) and enlarge the wormhole to classical size (quantum gravity is far, far ahead) Classical strategy—tear down space-time by intense energy. But classical strategy creates a singularity (QG). Solution: Singularity-free construction—twisting space-time during construction and become a time machine
- 26. Step 1. Acquire a traversable Wormhole Assume that we are an infinitely advanced civilization (by virtue of last slides’ construction strategies) that maintain a traversable wormhole Assume further that the hole is embedded in Minkowski flat space-time and that the mouths are at rest with each other
- 27. Step 2. Induce a time shift Leaving one mouth to your assistant, take one mouth, bring it inside a space ship, travel at near light speed, come back to earth after some time and bring the mouth back. The assistant will see you arrive on earth through the other mouth, but in their time, you are still travelling [Twin Paradox] Then after a very long time, he sees you arrive and just age maybe for a day.
- 28. Step 3. Bring the mouths together Push the two mouths towards one another. (Slowly.) A time machine forms when the distance is smaller than the time shift Presto! You now have a time machine! Simply let your (now old) assistant peek through one mouth and see his younger self awaiting your return. Finally, let the assistant go inside the mouth and give his younger self the fright of his life!!
- 29. Time travel to the past cannot occur before the construction of the time machine. Time travel paradoxes!!! Or the Death of Causality. Chronology Protection Conjecture: “Whenever one tries to make a time machine, just before it becomes a time machine, a beam of vacuum fluctuations will circulate through the device and destroy it.” “Keeping the world safe for historians.”—S. Hawking.
- 30. Books Visser, M. (1996). AIP Series in Computational and Applied Mathematical Physics. Lorentzian Wormholes: From Einstein to Hawking. New York: Springer-Verlag Inc. Thorne, K. S. (1994). Black Holes and Time Warps, Einstein's Outrageous Legacy. New York: W. W. Norton & Co. Kaku, M. (1995). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps and the 10th Dimension. New York: Anchor Books. Journal Articles Mallett, R. L. (2000). Weak Gravitational Field of Electromagnetic Radiation in a Ring Laser. Physical Letters A, 214-217.
- 31. Internet Articles Chronology Protection Conjecture. Wikipedia: The Free Encyclopedia Ronald Mallett. Wikipedia. The Free Encyclopedia. Time travel and time machine. The Stanford Online Encyclopedia of Philosophy.

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