CAN EINSTEIN BE FINESSED?
(C)1994 Alan M. Schwartz
Probably. Precedents exist evading the same mathematical
constraint, that of impending division by zero.
The foundation of Einstein's Special Theory of Relativity is
simply stated: Information cannot be transferred at speeds
exceeding that of light in a vacuum. That was simple, wasn't it?
The devil lies in the details.
The Weltanschauung derived from this boundary condition defines
space-time geometry. Academics with their research grants then
logically derived the self-consistent mathematical consequences.
Euclid rigorously defined plane geometry, the geometry of zero-
curvature space, from five such postulates. He was uneasy about
the fifth - that only one line exists parallel to a given line.
Bolyai and Lobachevsky assumed no lines parallel to a given line
exist to prove Euclid's Fifth Postulate and got elliptic geometry
(of positive-curvature space). Riemann assumed an infinity of
lines parallel to a given line exist to prove Euclid's Fifth
Postulate and got hyperbolic geometry (of negative-curvature
space), completing the set.
Each geometry is unassailable, an intricately crafted and fitted
perfection of detail constructed of uncompromising logic and
irrevocable cause and effect. Whether any of them correspond to
reality's observables is irrelevant. That mathematics does
accurately model reality is beyond our explanation. Perhaps the
link is through Euler's Equation - e to the (i)(pi) power equals
one - uniting algebra and geometry. Perhaps it is the substance
of God.
Special Relativity deduces the factor sqrt[1-(v^2/c^2)] to scale
length, mass and time observed twixt reference frames possessing
constant relative velocity. Length multiplies by sqrt[1-(v^2/c^2)]
to approach zero at lightspeed. Mass and time are divided by
sqrt[1-(v^2/c^2)] to approach infinite mass and infinitely slow
time passage at lightspeed. Einstein's geometric derivation is
unassailable. Experimental evidence is prodigious: If it works,
then it all works. Division by zero is not contained within
mathematics. The region infinitesimally close by it is not
technologically accessible. Lightspeed forever quarantines us
Terrans within an inconsequential volume of space-time.
Must we accept the boundary conditions of his geometry? Can
Einstein be evaded, opening the universe to human conquest?
Factors approaching zero appear in denominators erecting
impenetrable barriers in another fundamental physical doctrine:
the Third Law of Thermodynamics (absolute zero may be arbitrarily
closely approached but never reached). It has been finessed.
Positive absolute temperatures down to 280 picokelvins
(trillionths of an absolute degree) and negative absolute
temperatures up to -750 picokelvins have been achieved in rhodium
foil [Science 265(5180) 1821 (1994)], consistent with the
mathematics of how temperature is defined. The Third Law, which
thwarts such niceties as perpetual motion machines, emerges
remarkably unscathed. At the moment when The Third Law
denominator should touch zero... the statistical distribution of
energy states which constitutes temperature is undefined. The
system emerges with negative absolute temperature having gone the
long way around - "through T= plus infinity = minus infinity,
without crossing absolute zero." Is it hand waving?
Experimental observations of the negative temperature samples
were mathematically consistent with theory.
Circle-squarers and angle-trisectors have squawked their dogmas
since Euclid. People have worshipped everything from quartz
crystals to human sacrifice to Scientific Socialism. Einstein's
Relativity was denied by classical physicists as Einstein himself
denied the indeterminacies of quantum mechanics. Like a diamond
cleaved with a razor blade and a tiny mallet, the unyielding
substance of reality is prone to unsuspected manipulation. If
you travel from Sol to Centauri and back in a pair of eye blinks,
returning with samples, the evanescence of the trip is not mere
hand waving. Avoiding reality is not the same as evading it.
Certainly there remain no logical errors to be corrected in
Relativity. Is it a subset of a more inclusive geometry of
space-time, allowing a deft evasion in the manner that a left-
handed glove becomes right-handed when everted through 3-space?
Could we avoid a damning division by zero by "taking the long way
around?" I hope so; my species' phylogeny depends upon it!
Let us neither damn Einstein nor acquiesce to his limitations
forced upon us. Let us subvert him. In the Third Law case, a
nicely crafted sudden inversion of spin states sends the sample's
absolute temperature for a negative temperature Burton. Consider
how the asymptotes of a hyperbola can be tamed by graphing it on
(1/x,1/y) reciprocal axes. What would be the corresponding
manipulation pertinent to the Relativistic structure of reality,
and when do we leave?