The Physics Of Time Travel

Or How to Construct Your Own Time Machine

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

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

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

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

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

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!!

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


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)


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.


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 θare 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.)


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

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)


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)


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.


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

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)


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

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

A Summary of Presented Solutions that yield CTCs

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.

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

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

Wormholes before Thorne’s paper

Vacuum fluctuations and incoming radiation allow the wormhole to shrink instantly after creation

Cannot be produced naturally

Vacuum fluctuations near the horizon are negative average energy density material and can open the wormhole and de-focus incoming radiation

Quantum strategy and Semi-classical strategy

Traversable Wormholes according to Thorne

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

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

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 Minkowskiflat space-time and that the mouths are at rest with each other

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.

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!!

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.

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.

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.

Technology

The Physics Of Time Travel