Soap Films and Time Machines
Kenneth Brakke
Mathematics Department
Susquehanna University
TGTS
Sept. 6, 2013
Outline
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Basics of soap films and soap bubbles
Examples
Wormhole model of soap films
Converting wormhole to time machine
Surface structure of a soap solution
Soap film structure
Properties of Soap Films
• Energy is proportional to area.
• Films try to shrink, to minimize energy.
• Surface tension is constant, for any extent, as
long as there are enough soap molecules.
• Soap films are thin, about 1/100 the thickness
of a sheet of paper for ordinary films, and
1/100 of that for the thinnest “black” films.
Complicated Soap Films
Soap films can get much more complicated than simple smooth surfaces,
which makes them interesting and more mathematically difficult. What
kinds of things can happen where soap films join?
Junctions of Soap Films
The equal-angle Y is the only possible way for soap
films to meet along a line or curve.
Junctions of Triple Lines
Multiple triple lines can only meet four at a time, at
equal angles, at a “tetrahedral junction”.
Films on a Cube Frame
A film on a cubical frame with 12 flat surfaces meeting at the center would
satisfy the triple-line criterion, but the center point is not tetrahedral, so it
would pop into the central rounded square on the right, with four
tetrahedral points.
Films on a Trefoil Knot
Several films can form
on a trefoil knot wire,
including one that
does not touch the
entire boundary.
A Conjecture
My Ph.D. adviser, Fred Almgren, conjectured in 1976 in Scientific
American magazine that a film could exist on an unclosed wire only if
the wire had thickness.
Soap Film Questions
• What is a good mathematical model for soap films?
• How can one prove the existence of a soap film on a
given boundary?
• How can one prove a given soap film has absolute
minimum area among all competitors?
Classic model of soap films
A soap film is an oriented manifold spanning an oriented closed curve.
But that can’t handle triple junctions.
Duality – the minimum area of a soap film spanning a wire
loop is the same as the maximum flux of a incompressible
fluid with maximum velocity 1. The flow vectorfield is said
to “calibrate” the surface.
Proof that a surface is minimal by calibration
Paired calibration – The minimum film area is equal to the maximum total flux of one
incompressible fluid flow for each region, with the velocity difference of each pair of
flows at most magnitude 1 at each point.
Proof that a general soap film has minimum area
Hypercube cones are minimal.
I was able to show using paired calibration that in space dimension 4 and higher,
the cone on a hypercube frame is the minimal area soap film.
Covering Space model of soap films
• The ambient domain is a covering space of the
complement of the boundary, with the boundary
wires being branch curves.
• One sheet is designated the “home” sheet.
• A soap film candidate is an oriented manifold
separating the home sheet from the others, with
the projected sum of 0. (double-layer film)
• The minimum area of a soap film is the maximum
flux from the home sheet into the others of an
incompressible flow whose difference between
sheets has magnitude at most 1.
Polycut Interlude
Wormhole
If the two cuts are in the same universe, we have a wormhole
or stargate or portal.
Wormhole
Using the other sides of the wormhole, the traveler can move
continuously straight forward, but still be in a loop.
Wormhole to Time Machine
Einstein’s Theory of Relativity says time slows for a moving object, so moving one
end of the wormhole slows its clock relative to the other end. But there is no
relative motion through the wormhole, so the two clocks agree for a traveler
through the wormhole.
Through the Time Machine
As the traveler moves back and forth through the wormhole, his personal clock
always agrees with the wormhole clock, so to an outside observer he
sometimes appears to be in two places at once, and sometimes nowhere.
Paradox
A traveler can exit one side and enter the other at exactly the same gate-time, so
the traveler is in a closed time-like loop, and his existence has no past cause.
Stargate Construction
• Needs negative energy cosmic string.
• Density must be about an Earth mass per inch.
• No way to hold it, so it must be left to oscillate
like a big rubber band in space.
• It can be made to oscillate with a twist so that
it avoids itself at its tightest point, so it can
oscillate indefinitely.
• No gravitational forces to rip you apart.
References
• K. Brakke, Minimal cones on hypercubes, J. Geom.
Anal. 1 (1991) 329-338.
http://www.susqu.edu/brakke/aux/downloads/papers/hyper.pdf
• K. Brakke, Soap films and covering spaces, J. Geom.
Anal. 5 (1995) 445-514.
http://www.susqu.edu/brakke/aux/downloads/papers/covering.pdf
• K. Brakke, Numerical solution of soap film dual
problems, Exp. Math. 4 (1995) 269-287.
http://www.susqu.edu/brakke/aux/downloads/papers/soapdual.pdf
• Polycut program http://www.susqu.edu/brakke/polycut/polycut.htm