From: Wade Cherrington (wade_c@uniserve.com)
Date: Thu Nov 05 1998 - 00:32:03 MST
>Everybody worries about time travel with wormholes, but I am concerned
>about more mundane matters.
>
>How does conservation of energy and of momentum work with a wormhole?
>What if I shoot a ball into hole A and it comes out hole B moving in a
>different direction? Momentum is not conserved.
>
>Worse, what if I position the exit of hole B above the entrance to
>hole A, and drop in a ball. The ball falls through A, comes out B
>where it then drops into A again, and continues, going faster and
>faster as it falls forever. Where does the energy come from?
>
>(Note that in relativity, momentum and energy are closely related,
>being separate components of the four-vector which represents a
>particle's motion. So probably these two paradoxes are two aspects
>of a single phenomenon.)
>
>I can think of two possible resolutions.
>
>First, it could be that there are some compensating forces which serve to
>balance the books. When the ball comes out of B in a different direction
>than it went in A, maybe the "substance" of the wormholes (whatever
>that is) is given a kick which balances the momentum. And maybe when
>you try to go through a wormhole and come out higher than you were,
>maybe there is some kind of force opposing your motion and you have to
>force your way through, just enough to balance the potential energy gain.
>
>Second, it could be that wormholes simply break the rules. Hypothetical
>stable wormholes require negative energy, and it could be that with
>negative energy you already break conservation of momentum and energy.
>A negative-energy mass is attracted to a positive one, but the positive
>one is repelled by the negative one (or is it the other way around?).
>So the pair goes rocketing off across the universe, accelerating steadily
>without any input of energy. I think Robert Forward describes this
>in his novel, Timemaster.
>
>(Actually I suppose that example technically may not break the rules,
>since the negative mass acquires greater negative energy and negative
>momentum as it accelerates, balancing the positive energy and momentum
>of the positive mass. So the system as a whole has constant energy
>and momentum.)
>
>Hal
>
>
The point about conservation of momentum is interesting.
I think it's helpful to consider that the conservation of momentum means
anything only in the local Lorentz frame of an observer.
A good analogy is light being bent around the sun - locally
no (relativistic) momentum conservation is violated but globally
there is curvature. Pursuing the global/local distinction further,
if the universe is in fact closed (with positive curvature), then
if you go far enough you will end up where you came from, but going the
opposite direction (if the universe hasn't ended by then). This too might
superficially appear as a violation of momentum conservation, but
in fact isn't when only local frames are considered.
This is related to the fact that strict, cartesian x-y-z momentum
conservation in classical mechanics is derived in part from the translation
symmetry of space, something you lose in curved space-times over large
distances.
Presumably, by falling into a wormhole, like other forms of free-fall
in General Relativity all the particle is doing is following a geodesic
in spacetime; like a particle in a gravitational field, no force is involved,
but rather an intrinsic property of the geometry of space-time.
In regard to your infinite energy scenario, perhaps there is an energy required
to maintain the wormhole proportional to the amount of momentum passing
through it. That way, as an object picked up speed the wormhole would
require more energy to maintain stability. How this might work I have no idea.
Something I have a hard time imagining is how the exit velocity changes
as a function of the entrance trajectory. It can't be said that the
particle "bounces" off the "interior walls" of the wormhole; perhaps
a more realistic path would be a smooth, spiralling corkscrew or
straight through for a head-on entry. Given the 4-D geometry, you
could probably work out the motion of a particle in the reference
frame of the wormhole, given the initial velocity and position conditions.
Then there is always the problem of what determines where you come
out of a wormhole. Maybe it takes participation at both ends. If it does,
then it would be quite a problem keeping one end stationary with
respect to another one light years away. Once it's set up, of course,
nearly instantaneous communications might allow for real-time adjustment of
the appropriate boundary conditions by varying the amount and geometry of
your negative matter.
There would seem to be a lot of computation required, since on top
of adjusting for drift you would have to adjust for the distortions
caused by the objects entering and exiting.
The business about negative matter and its interactions
with positive matter sound interesting, but considerably
speculative.
I'm not sure how, but I once read or heard somewhere that
a very small negative energy density can be induced using
the Casimir effect, which is related to quantum fluctuations
of the vacuum.
As I understand it, the 4-dimensional geometry needed for a wormhole
implies a solution to Einstein's field equation that contains
entries of negative energy in the stress-energy tensor.
Mathematically it's a very small leap, but is there any reason to
expect the existence of negative matter any more than it would be
meaningful for a biologist to talk about negative
population?
BTW, Does anyone on the list know of any good papers of an
introductory/intermediate technical level on wormhole physics?
- Wade
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