Relativistic vs. Euclidean geometry

[Danny Ross Lunsford]

This occured to me at lunch, and is sort of fun...

We know that the Lorentz transformation can be derived from a
group-theoretic analysis of space and time, based on simple
assumptions of isotropy, linearity etc. There exist both synthetic and
analytic versions of this derivation that have been posted here. The
end result is, there is a parameter with the dimensions of a velocity
that is either finite, or infinite, that characterizes the geometry.
Of course, it is C.

One might ask, is Euclidean geometry so characterized? The answer is
yes!

Metric geometry sits inside projective geometry by positing a
fundamental quadric. In relativity it is of course the light cone - in
1+1 dimensions

x^2 - (ct)^2 = 0

which can be factored

(x - ct)(x + ct) = 0

so x/t = +-c - this shows how the fundamental quadric is related to
the characteristic parameter.

What about Euclidean plane geometry? The fundamental quadric is

x^2 + y^2 = 0

which seems like an empty statement, but makes sense in the context of
projective geometry as the "circular points at infinity". We can now
factor this as

(x - iy)(x + iy) = 0

The characteristic parameter of Euclidean geometry is the imaginary
unit! So "i" plays the role of the "speed of imaginary light" in
Euclidean geometry :)

This has a beautiful interpretation. Intuitively, one knows that, on
the Euclidean plane, one can imagine a thing called "infinity" which
can never be got closer to, from which all regular points are
"infinitely" distant. No matter how far you go, you're always
"infinitely" far away from "infinity". This is the *exactly analogous*
result to the impossibility of attaining the speed C in relativity.

This may be the most basic way complex numbers enter into physics.

-drl

[Tom Capizzi]

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This information needs to be combined with eigenvector decomposition based
on the Lorentz matrix. In the first place, calling it the Lorentz matrix makers
it appear to be a physics thing. Wrong. It was a simple, hyperbolic rotation
long, long before Lorentz's name was attached to it.

"> We know that the Lorentz transformation can be derived from a
> group-theoretic analysis of space and time, based on simple
> assumptions of isotropy, linearity etc."

Nothing more sophisticated than Euclidean geometry is needed. Indeed,
the hyperbolic geometry that underpins relativity was first used by the
map-maker, Mercator, centuries before it was claimed by physics.
Centuries before group math and relativity.

The details of hyperbolic trigonometry were published in the mid-1700's.
The "Lorentz" Transformation is nothing more than a restatement of the
hyperbolic identities for the cosh and sinh of the sum of two hyperbolic
angles. Given a hyperbolic angle, w, that is the sum of two independent
variables, A and B, those identities are cosh(w) = cos(A+B) =
cosh(A)cosh(B)+sinh(A)sinh(B), and sinh(w) =
sinh(A)cosh(B)+cosh(A)sinh(B). A point, (x,y), on a unit hyperbola with
horizontal symmetry satisfies the equation xТВ-yТВ = 1. The coordinates
of this point are equal to (cosh(A),sin(A)). Since the addition of
hyperbolic angles is commutative and linear, it doesn't matter which of
the two angles refers to the initial state. The final state is just the
sum of an initial state and an increment. There is a simple proof, using
geometric algebra, that the sum of hyperbolic rotation angles is linear.
So, we can state that the point (cosh(w),sinh(w)) = (cosh(B+A),
sinh(B+A)). Using the above identities, cosh(B+A)=
cosh(B)cosh(A)+sinh(B)sinh(A) sinh(B+A) = sinh(B)cosh(A)+cosh(B)sinh(A)
The astute observer will notice that this is nothing more than an
expansion of the linear algebra equation: т_cosh(w)т_ т_cosh(B)
sinh(B)т_т_cosh(A)т_ т_ sinh(w)т_=т_sinh(B) cosh(B)т_т_ sinh(A)т_ In
Greek symbols and more specific coordinates: т_ct'т_ т_ ЮГ
-ЮВЮГт_т_ctт_ т_ r' т_=т_-ЮВЮГ ЮГт_т_ r т_ Using the same identities,
we can derive the hyperbolic identity of the tanh of the sum of two
hyperbolic angles: v3 = c tanh(w) = c tanh(A+B) =
c(sinh(A)cosh(B)+cosh(A)sinh(B))/(cosh(A)cosh(B)+sinh(A)sinh(B)) =
c(sinh(A)/cosh(A)+sinh(B)/cosh(B))/(1+sinh(A)/cosh(A)*sinh(B)/cosh(B)) =
c(tanh(A)+tanh(B))/(1+tanh(A)*tanh(B)) In its more familiar form, v1 = c
tanh(A) and v2 = c tanh(B): (v1+v2)/(1+v1/c*v2/c), the velocity addition
"rule" of special relativity. It's just a hyperbolic identity. Physics
did not invent it, even if they take credit for it.

"> Metric geometry sits inside projective geometry by positing a
> fundamental quadric. In relativity it is of course the light cone - in
> 1+1 dimensions
> x^2 - (ct)^2 = 0
> which can be factored
> (x - ct)(x + ct) = 0"

Minkowski's initial foray into relativity identified time as an
imaginary component, so that the quadric becomes x^2 + (ict)^2 = x^2 -
(ct)^2 = 0. This equation generates the rectangular hyperbola. In
general, x^2 - (ct)^2 = ТБs^2, where the choice of sign depends on an
arbitrary convention. The two values, s and w, are coordinates in a
Cartesian grid. They represent hyperbolic coordinates in which they are
invariant in the absence of external forces. A shift of either
coordinate has no effect on the other coordinate, a standard property of
orthogonal coordinates. Since w can represent arbitrary combinations of
rotation angles, it is convenient to divide it into 2 components, one
being its initial value, w0, at some arbitrary time, t=0, and the other,
its value at some future time, w1. Then w = w1-w0. It makes no
difference if it is a reference to one frame which changes its
hyperbolic angle, or a given frame that is observed by a non co-moving
frame.

Since we choose non-hyperbolic geometry, these coordinates must be
transformed into our native coordinates to make sense to us. We have
already observed that ct = s cosh(w) and r = s sinh(w). From these
coordinates, we can restore the hyperbolic coordinates by using the
inverse transform. s^2 = (ct)^2-r^2, and tanh(w) = r/ct. So, in
Minkowski spacetime, the invariant is a coordinate transform to
hyperbolic coordinates. Note that the hyperbolic rotation angle is the
same for both frames. But is Minkowski's the only possible frame? No.

Minkowski is referred to as pseudo-Euclidean. Considering that Einstein
did not have the crutch of Minkowski trigonmetry when he first
published, it is worth investigating the Euclidean version. As similar
as it is to Minkowski, it is more productive to consider the Euclidean
eigenvector decomposition. A limited version of this can be found in
Hermann Bondi's k-calculus. The k in k-calculus, Bondi's "fundamental
ratio", stands for an eigenvalue of the Lorentz matrix, from which he
derives all the features of relativity. One of the tools that he uses
in his derivation is measurements by radar. This is another intrinsic
feature of eigenvector decomposition, because "> so x/t = +-c". These
are the eigenvectors of the Lorentz matrix, the worldlines of photons.
Bondi does not mention eigenvectors or eigenvalues, and misses some of
the bigger picture. The Euclidean nature of this approach allows us to
use conventional definition of the dot and cross-product, neither of
which applies in Minkowski trigonometry. In terms of an arbitrary
direction in space, r, for the frame velocity, the eigenvectors are ct=r
and ct=-r. (Note: I use the constant c to accommodate arbitrary units
for c. Because c in natural units may have a magnitude of 1, but it
still has units. Dropping the c because it is unit in magnitude also
drops the units. To maintain agreement of units on both sides of the
equation, some conversion factor, like c, is still needed to make the
units agree.) Coordinates on these two axes are (ct+r) and (ct-r). Their
Euclidean cross-product has magnitude (ct+r)(ct-r) = cТВtТВ-rТВ = sТВ,
the Einstein Interval. But this is the exact same s that is the
invariant hyperbolic magnitude. And if s^2 is invariant wrt hyperbolic
rotation, then I propose that the factored version does not belong to
Minkowski, It is Euclidean. And the Euclidean quadric, (ct)ТВ+(ir)ТВ
does not apply to real trigonometry, but Minkowski trig, (ct)^2-r^2. It
is, after all, pseudo-Euclidean.

Minkowski reformulated his trigonometry so that none of the coordinates
were imaginary, but the metric remained the same as in the earlier
version. The transform used to generate Minkowski's light-cone axes is
the same one that generates the real eigenvectors of the Lorentz matrix.
Some contend that this disqualifies its use, because in Minkowski trig,
the word real means Minkowski-real, which the rest of us would
understand to be imaginary. And in Minkowski-speak, the dot product of
an eigenvector with itself is 0. Since orthogonal means Minkowski
orthogonal, the two light-cone "eigenvectors" (which are not real) are
actually parallel, and their product is 0. And since they are Minkowski
parallel, their cross-product does not exist. But the symmetry transform
applies to a wide range of physical processes and attributes. Physics
does not own it, nor do they get to decide where and when it can be
used.

The transform is almost auto-inverse, except for a scale factor. In the
for m that I have applied it, it is a combination of 3 simple
transforms. It is a scaling, a rotation and a mirror reflection.
Although it is a rotation, it is not a unit rotation matrix. More
important, its determinant is not positive. So, aside from the scaling,
it can never be a Lorentz Transform of any given frame. In this
application, the form was chosen so that the Euclidean/Minkowski
horizontal hyperbola maps to the diagonal eigenvector hyperbola with the
same invariant. But, geometrically, the invariant of the horizontal
hyperbola is a vector, the semi-major axis of the hyperbola, while the
invariant of the diagonal hyperbola is an invariant area. This can be
seen in the coordinates of the vertex. For the horizontal unit
hyperbola, the vertex is (1,0), and the invariant is the magnitude of
this vector. For the diagonal hyperbola, the vertex is (1,1), for an
area of the cross-product of 1. This is where the scale factor comes
from. The magnitude of the semi-major axis to the vertex is тЈ2, but in
this isomorphism, length is irrelevant. Although the shape of what
appears to be a bivector changes with hyperbolic rotation angle, its
area remains invariant, being the result of a squeeze mapping of
determinant, 1.

Euclidean eigenvectors of the Lorentz matrix are truly real, as are the
eigenvalues. Since real multiplication and real addition are both closed
operations, their is no real linear transform that will result in a
vector that is not in the same plane as its components. Minkowski
"eigenvectors" are not real, as they require 3 dimensions to embed the
light-cone hyperplane. As most critics make the assumption that if you
are writing about special relativity, you must be using Minkowski trig,
their irrelevant objections can simply be dismissed. We are talking
about Euclidean geometry, like Einstein had to when he first published.
And this is where it gets sticky for Einstein.

He postulated that the speed of light is invariant to the relative
velocity of either its source or the observer. He also asserted that the
properties of time dilation and length contraction are physical. In
terms of Euclidean geometry, these assertions are self-contradictory. In
his books, he always uses two frames, one a reference frame and the
other non co-moving. The essence of relativity is that it doesn't matter
which of the two frames is considered stationary. His description of
relativity only works for these two observers, because relative velocity
is just 1 equation in 2 variables. Even for 2 observers, there is the
infamous Twin Paradox. Based on only relative velocity, both observers
can legitimately claim that the other one is dilated or contracted.
While both observers can't actually be correct, one of them is correct,
but perhaps unable to confirm it.

If we introduce more observers at different relative velocities, we have
a problem. Assuming we start with two observers, then one of them is
correct. We can't be sure which one it is. But if we compare both of
them to any of the additional observers, the one thing we can be sure
of, is that none of the extra observers can be correct. According to
relativity, one of any pair of differently moving observers is correct.
Even if the original two observers can't agree about who is moving, they
can both agree on the magnitudes of the dilation and the contraction.
This applies to any pair formed from one of the first two and any of the
additional observers. But they will agree on measurements that are not
the same as the first pair. In an objective reality, this is a
contradiction. Rather than discard a theory based on a contradiction,
the mainstream physicists of the time discarded the objective reality.
By giving each frame its own subjective reality, they erased the
contradiction. It allows them to make fatuous statements like there is
no reason to expect any two observers to agree on any measurement. Great
science, isn't it?

Since it is frequently useful to examine a problem from a different
frame of reference, we shall look from the eigenvector frame. If we
apply the symmetry transform to the reference frame, there is no
relative velocity. Coordinates in this frame are determined by Bondi's
radar technique, an integral feature of real eigenvectors. That means
location in this plane is determined by two coordinates, both of which
are invariant wrt relative velocity of any kind. In this trigonometry,
the relative velocity is actually a third orthogonal coordinate. The
entire plane of the first two coordinates is translated completely
intact by the third coordinate. The interval between any two points on
this plane is thus unaffected by relative velocity. Physical objects do
not change size or shape due to relative velocity. This confirms that
Einstein's theory is self-contradictory in Euclidean geometry. More
important, it demonstrates that the measurement process itself is
defective.

In its attempt to legitimize the profession from the quacks of alchemy
and metaphysics, the scientific method placed a premium on the repeated
measurements of the same property. Agreement between multiple
measurements allowed a consensus reality to be crafted. The simple truth
is that this paradigm only applies to Newtonian physics. In plain
language, this is WYSIWYG. But just like Newtonian physics breaks down
at relativistic velocities, so does the standard of measurement. The
self-contradictory assertions of time dilation and length contraction
are mere artifacts of a false measurement. Euclidean geometry has
another standard for comparing the magnitude of an unknown with the
magnitude of a reference unit. It is the dot product. This does not
apply to Minkowski geometry, where the dot product of eigenvector units
with themselves is 0, and the cross-product of two different
eigenvectors is not defined, because the eigenvectors are parallel.

While the dot product is clearly defined for Euclidean geometry (and
extended into inner and outer products), it can be expressed simply as
the product of the magnitudes of the two factors with the cosine of the
angle between them. For Newtonian physics, this reduces to the product
of the unknown magnitude with the magnitude of the reference unit, 1,
and the cosine of 0, also 1. In other words, WYSIWYG. For relativistic
physics, this is not valid, because the angle between to moving frames
is no longer 0. While it could be anything, experimental evidence
confirms that the phase angle between two inertial frames that have been
synchronized is not 0, despite the fact that all the axes (observable,
that is) are parallel. It is defined by relative velocity as v/c =
sin(phase).

And this brings us to the core of special relativity. The phase angle
defined this way is the gudermannian of the rapidity, the hyperbolic
rotation angle. While it was studied extensively in the 18th century,
its first application was much older. It is the basis of the brilliant
achievement of the map-maker, Mercator. In hindsight, we can understand
where the transform applies. Mercator, perhaps protecting his
proprietary algorithm for map-making, did not reveal his secret. If we
call the longitude angle Я, and the latitude angle Я, then the
map-making rule is dЯ/dЯ = ЮГ, the Lorentz factor, defined centuries
before his birth. Я is defined geometrically as the gudermannian of Я.
We can't see it or measure it, because it is the angle between
dimensions of spacetime that are outside of the box that 4D spacetime
traps us in. As with much of physics, they appropriated it, as the
parametric angle used to manipulate velocity as a convenience.

The Lorentz factor was originally an empirically determined fudge factor
that made the predictions of Newton's law agree with experimental
evidence. Now, they claim that it was Newton's "Law" that was wrong, and
that the relativistic momentum is just what a moving object is supposed
to have. What a surprise. Newton's Law is short by exactly the same
factor as time dilation and length contraction, both shown to be
illusions. And by giving such a circular definition, there is no need
for the other relativistic canard, relativistic mass. But if we multiply
relativistic momentum by the same projection cosine as we used for time
and distance, we find the result to be measurable Newtonian momentum.
The dot product triumphs again.

So far, I haven't found any situation in which Euclidean geometry cannot
be used. As an isomorphism of Minkowski, it does not require that the
same operations get the same result. So, the Minkowski dot product
yields the same results as the Euclidean cross-product, another variant
of multiplication. In Euclidean geometry, the dot and cross-products
form an orthogonal basis, so pseudo-Euclidean Minkowski trig rotates the
basis 90 degrees and gets the same result. I think it is long past time
that we critically re-evaluate the core premises of special relativity.
Especially the artificial restriction to 4 real dimensions. I know that
this describes Euclidean geometry, but eigenvector geometry is different
from Euclidean in some important ways. In the same way that Minkowski
could reformulate his geometry with 4 real coordinates while retaining
the pseudo-Euclidean metric, I have found that there is an isomorphism
between ordered sets of n-tuples of real numbers and hypercomplex
numbers. The simplest is (a,b) т_ a+bi. This mapping extends at least to
bioctonions, which are ordered pairs of biquaternions, and are
equivalent to biquaternions of complex scalars instead of real scalars.
The bleatings of the relativity cult that no theory has been confirmed
more than relativity are totally irrelevant. Even Einstein agreed, "No
amount of experimentation can ever prove me right; a single experiment
can prove me wrong." How about a single gedanken experiment?