The document discusses various theoretical time machines that could be constructed based on solutions to Einstein's field equations, including rotating cylinders, wormholes, and ring lasers, but notes that most proposals are mathematically possible but physically improbable due to involving unphysical objects or violating causality.

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.