[tt] PhysOrg: Relativity Derived Without Calculus--Possibly Centuries Ago

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Relativity Derived Without Calculus--Possibly Centuries Ago
http://www.physorg.com/news111075100.html
7.10.8

By Lisa Zyga

After Einstein developed his theories of special and general
relativity, in 1905 and 1916, respectively, the world of physics
changed dramatically. The theories, with their groundbreaking ideas
on space and time, helped lead 20^th century scientists to unlock
the secrets of the atom and unleash the power of nuclear energy.

Einsteinian relativity seemed to be a modern breakthrough: he had
derived his theories from ideas and mathematics that were new at
the time. The Lorentz transformations had just been discovered in
1895, and he derived a new velocity addition law using calculus
(both of these concepts describe how observers in different
reference frames perceive each other). Further, Einstein based his
theories on the assumption that the speed of light, c, is constant,
and used gedanken ("thought") experiments involving light rays to
reach his conclusions.

Now Joel Gannett, a Senior Scientist in the Applied Research Area
of Telcordia Technologies in Red Bank, New Jersey, has found that
Einstein didn't have to do the work the hard way. A researcher in
optical networking technologies, Gannett has shown that the Lorentz
transformations and velocity addition law can be derived without
assuming the constancy of the speed of light, without thought
experiments, and without calculus. In this case, Einsteinian
relativity could have been discovered several centuries before
Einstein.

"Einsteinian Relativity is difficult to wrap your mind around,"
Gannett told PhysOrg.com. "It does not help that Einstein's seminal
1905 paper, and many discussions of the topic since, start off with
the wildly counterintuitive assumption that the speed of light is
constant in all inertial frames.

"My work shows that the essential strangeness of Einsteinian
Relativity falls out of simple, intuitive assumptions using simple
math. A pre-calculus high school student could have derived
Einsteinian Relativity. Admittedly, some of the math in my paper
might seem beyond the high school level, but that was because I was
proving continuity from a boundedness assumption. One could bypass
this math by simply assuming continuity, a logical step that would
probably feel comfortable to most any high schooler or 17th century
scientist."

Gannett is not the first person to suggest that a simpler path to
modern relativity might exist. In 2003, Palash Pal of the Saha
Institute of Nuclear Physics in Calcutta, India showed that the
Lorentz transformations could be derived without assuming the
constancy of c and without thought experiments; in fact, scientists
had noted this possibility as far back as 1910.

To reach his derivation, Pal invoked the ideas that spacetime is
homogeneous and isotropic. Pal titled his paper "Nothing but
relativity"; after reading it, Gannett has called his paper
"Nothing but relativity, redux," which is published in a recent
issue of the European Journal of Physics. However, Gannett explains
that his derivation actually bypasses the principle of relativity
altogether--instead, he assumes the simpler idea of reciprocity.
"The current paper might have been titled `Nothing but relativity,
and not that either,' or perhaps `Nothing but reciprocity," he
writes, emphasizing the point.

"One of the issues I raise in my paper is, why make a heavyweight
assumption such as relativity when in fact all you need for the
derivation is reciprocity?" he explains. "I don't need the fact
that the laws of physics are the same for you on the speeding train
and me on the platform (i.e., relativity). All I need is
reciprocity."
Gannett uses the common analogy of the train to explain
reciprocity: "Suppose you are on a train and I am on the platform
waving goodbye. Suppose I measure your speed relative to me as 30
mph. Looking back at me, you would judge that I am moving away from
you at 30 mph as well. If we both had police radar guns to measure
our relative speeds with great accuracy, we would both come up with
the same number (say, 29.6 mph). That's reciprocity. In the
presence of isotropy, which is one of my other key assumptions,
relativity implies reciprocity."

Instead of using calculus to derive the spacetime transformations,
Gannett uses two basic concepts from mathematical topology: density
and continuity. Using these concepts, he demonstrates how the
spacetime coordinates of one reference frame (e.g. the train) can
be mapped to the spacetime coordinates of a second reference frame
(e.g. the platform), accounting for the distorted lengths and times
that occur at high speeds.

The density concept means that any irrational number (such as the
number p) can be approximated to arbitrary closeness by a rational
number (those with a finite number of digits or digits that
repeat). The second concept, continuity, means that a function maps
"close-by" points to "close-by" points. From these concepts he
derives a linear homogeneous function, or a matrix, to connect the
coordinates of the two reference frames.

Gannett explains that proving that spacetime is linear is a vital
point to make before reaching Pal's derivation. He also notes that
Einstein had glossed over this important point in his paper on
special relativity.

"Einstein merely stated that homogeneity (i.e., the uniformity of
space and time) clearly implied linearity," he says. "With the
level of mathematics I was applying in my paper, I could fairly
easily get to the point where one could assert linearity in a
mythical universe where coordinates exist only as rational numbers,
and we consider only rational scalings. But since the days of
Pythagoras this would not be considered adequate. Because rational
numbers can approximate irrational numbers to any desired degree of
accuracy, continuity provides the final logical link that lets you
assert linearity of the spacetime mapping."

Finally, invoking the ideas of the cosmological principle that the
universe is isotropic and reciprocal, Gannett demonstrates that
four basic properties of the mapping functions directly follow.