Note To The Reader: Due to
formatting some equations that were originally
written on one line will be split into two lines. If the reader will keep that
thought in mind then he (or she) should have no trouble in following
the very informative and ground breaking presentation that follows:
******************************************************
Under truth there often lies
mysteries that wise eyes would love to discover
but until the truth is well known
mystery and truth will be alone
in their knowledge of each other.
RHB
******************************************************
The year of 2005 was the 100th anniversary of the publication of Albert
Einstein's paper on Special Relativity (S.R.). At the core of that theory
are the Lorentz Transformations which provide a straight forward and elegant
mechanism by which reference frames, moving with constant velocity with respect
to
each other, may be connected.
The author is well aware of the existence of the General Theory and it's
applications to
Astro physics, black holes, etc., but these
applications have not yet been verified to the (very high) level of accuracy
that characterize the experimental results in Quantum Electrodynamics. QED, the
reader will recall is a union of Quantum Mechanics and
S.R. Accordingly the author, being singularly impressed with the elegance and
the "canonical form" of the Lorentz Transformations, has chosen to
seek a straight forward generalization, of the Lorentz Transformations,
that will apply to Uniformly Accelerated Frames of Reference. What follows is a
preliminary draft of his results.
Cubic Complex Numbers
The subject matter of Cubic Complex Numbers is a natural generalization of the
ring of complex numbers. Recall that the field of complex numbers is also a
ring since every field is a ring (but not vice  versa). The ring of cubic
complex numbers contains the ring of complex numbers as a sub  ring.
The ring of Cubic Complex Numbers is a commutative ring that is endowed with a
multiplicative identity. In addition to that property,
each element of the ring has a multiplicative inverse except those
elements for which a certain 3 x 3 determinant vanishes. The set of elements
associated with the vanishing 3 x 3 determinant will
prove to have special physical implications.
The Algebra of Cubic Complex Numbers consists of the set of ordered triples of
(ordinary) complex numbers together with the operations of addition and
multiplication.
If each complex number in each ordered triple has a zero imaginary part then
the resulting set (of cubic complex numbers) corresponds to the set of ordered
triples of
real numbers. The latter set is of course isomorphic to three
dimensional Vector
Space.
Recall that the imaginary unit i of complex numbers may be defined by
i = (1)^(1/2) = ( (1,0))^(1/2) where (1,0) is identified with 1,
the multiplicative
identity of the ring of complex numbers. Therefore
i^2 = 1
We now define the entity j as follows:
j = ((1,0,0))^(1/3)
where 1 is identified with the unit vector (1,0,0). Later on it will be shown
that, based upon the rules of multiplication of ordered triples to be set
forth, it is indeed appropriate to identify 1
with (1,0,0). It immediately follows that
j^3 = 1
j^4 =  j
j^5 =  j^2
j^6 = 1
What immediately follows is a brief informal presentation of these ideas. Let
us consider the set C3 of all ordered triples of complex numbers.
C3 = { (a,b,c) such that a, b, and c are complex
numbers}
(eq.1)
An element of C3 will be called a Cubic Complex Number. Two cubic complex
numbers (a,b,c) and (x,y,z) will be said to be equal if and
only if
a = x, b = y, c =
z
(eq.2)
Given two cubic complex numbers (a,b,c) and (x,y,z), their sum and
product will
be respectively defined by
(a,b,c) + (x,y,z) = (a+x, b+y,
c+z)
(eq.3)
(a,b,c)(x,y,z) = (axbzcy, ay+bxcz, az+cx+by)
(eq.4)
It will be noted that the relation of equality and the operation of addition
(of
members of the set C3) have been defined in accordance with the usual rules
of vector algebra. But, as can be easily demonstrated, multiplication in C3
is commutative.
It is not difficult to show that the identities of addition and multiplication,
respectively, are (0,0,0) and (1,0,0). For brevity we will
identify 0 with
(0,0,0) and 1 with (1,0,0). We write
0 <==>
(0,0,0)
(eq.5a)
1 <==>
(1,0,0)
(eq.5b)
The axioms associated with closure, associativity, commutativity of
addition
and multiplication and the distributive axiom may be shown to be satisfied in
a straight forward fashion.
For a given ordered triple of complex numbers (a,b,c), ( a cubic complex
number),
the additive inverse is (a, b, c) and the multiplicative inverse is given by
(a, b, c)^(1) = (L/S, N/S,
M/S)
(eq.6)
where
L = a^2 +
bc
(eq.7a)
M = ac 
b^2
(eq.7b)
N = c^2 +
ab
(eq.7c)
and where S which is given by
S = a^3  b^3 + c^3 +
3abc
(eq.7d)
is different from zero. So then every cubic complex number (a, b, c) for which
S
does not vanish has a unique multiplicative inverse.
*******************************************************
Parallel lines run in the same direction
but do not meet in the ordinary plane.
But problem  solving minds must somehow
make a connection
or the problems of the world will always remain.
RHB
********************************************************
It may be recalled from the literature that a ring is defined as follows: a
non
empty set A will be said to form a ring with respect to the binary operations
of
addition ( + ) and multiplication (*), if the following conditions are
satisfied:
(please note that for brevity we will usually denote the product of the cubic
numbers u = (a,b,c) and v = (x,y,z) by uv instead of by the somewhat more
cumbersome u*v.
1) A is a commutative group with respect to addition.
2) The operation of multiplication is associative and has a unit element.
3) For all u, v, and w in A it is true that
(u +v)w= uw + vw
w(u + v) = wu +wv
The last item is called the Distributive Property.
From the previous discussion it is clear that the set C3 as defined by equa.(1)
together with the operations of addition and multiplication ( as defined by
eq.(3)
and eq.(4) respectively) constitute a ring.
It will be seen that, by following the multiplication rule given by
eq.(4), the product
of (0,1,0) and (0,1,0) is given by
(0,1,0)(0,1,0) = (0,1,0)^2 =
(0,0,1)
(eq.8)
also
(0,1,0)^3 = ( (0,1,0)^2 )( 0,1,0) = (0,0,1)(0,1,0) = (1,0,0)
that is
(0,1,0)^3 =
(1,0,0)
(eq.9)
Now let us identify the symbol j with the unit vector (0,1,0)
j <==>
(0,1,0)
(eq.10a)
Since (0,1,0)^2 = (0,0,1) we are motivated to make the following
identifications
j^2 <==>
(0,0,1)
(eq.10b)
And finally, making use of eqs.(9) and (10a) , we are led to the following:
j^3 <==> (1,0,0) <==>
1
(eq.10c)
The one to one correspondences in relation (10c) provides motivation for the
following definition
j = ( 
(1,0,0))^(1/3)
(eq.11a)
or more briefly
j =
(1)^(1/3)
(eq.11b)
From the above definition it immediately follows that
j^3 =
1
(eq.12a)
j^4 =
j
(eq.12b)
j^5 =
j^2
(eq.12c)
j^6 =
1
(eq.12d)
It is of course true that (1,0,0), (0,1,0) and (0,0,1) are unit vectors in a
Cartesian Coordinate system. Therefore if x, y and z are real numbers
then
(1,0,0)x + (0,1,0)y + (0,0,1)z = 1*x + jy + (
j^2)z = x + jy + (j^2)z
is an arbitrary point in 3D space.
**********************************************************
If something is true
then it can be trusted
but if it is only new
then it may become rusted.
RHB
***********************************************************
Minkowski's contribution to the theoretical refinement of the theory of Special
Relativity is well known. His starting point was essentially to take an
orthogonal coordinate transformation of a 2  dimensional coordinate system
(x,y) and convert it to the Lorentz Transformation by setting
y =
ict
(eq.13)
where i is the purely "imaginary" unit from complex variables,
t is the time coordinate and c is the speed of light. The basics of SR, which
describes the transformation laws between coordinate systems moving with
uniform relative velocity, can be fully
described in a coordinate system that consists of one spatial and one time
coordinate. The 2 dimensions of the complex plane along with eq.(13) proved to be a perfect starting point from which to
erect a canonical mathematical framework for the theory of Special Relativity.
Analogously, the basics of (nonrelativistic) uniformly accelerated motion can
be fully described in an (x,y) cartesian coordinate
system along with a time variable t. Although a uniformly accelerated object
can move in a straight line, more often its path is that of a parabola which is a plane figure. Following the lead of Minkowski, we
propose to give the time coordinate an equal
footing with the spatial variables x and y by writing
z =
(1/2)*j*L*t^2
(eq.14)
where j is as defined in eq. (11b), L is a parameter associated with optical
phenomena that has the physical dimensions of acceleration and t is
the time coordinate. It therefore follows that z will have the physical
dimensions of length.
So then starting with an (x,y,z) cartesian coordinate system, with z
defined as in (eq.14), we will derive a set of "Second Order Lorentz
Transformation" equations that are endowed with certain fundamental
invariance properties associated with uniform acceleration.
Let us now show how an orthogonal coordinate transformation in the plane
can be obtained from the fundamental (invariance) properties of complex
variables
and then we will present the corresponding case for cubic complex variables.
Let
X + iY = (A + iB)(x +
iy)
(eq.15)
after performing the indicated operations on the right side of the above
equation
and using the definition for the equality of complex numbers we have
X = Ax 
By
(eq.16a)
Y = Ay +
Bx
(eq.16b)
A basic identity associated with equations (16a & 16b) is
X^2 + Y^2 = (A^2 + B^2)(x^2 +
y^2)
(eq.17)
where the symbol ^ is used to denote exponentiation.
Now if we choose A and B so that
A = cos(a)
(eq.18a)
B =
sin(a)
(eq.18b)
where a is a continuous parameter
then
A^2 + B^2 =
1
(eq.19)
and (eq.17) reduces to
X^2 + Y^2 = x^2 +
z^2
(eq.20)
With A and B representing cos(a) and sin(a), respectively, the reader
will recognize eqs. (16a and 16b) as an orthogonal coordinate
transformation and (eq.20) as the fundamental
invariance property.
Now let's consider the corresponding set of transformations from
Cubic Complex Variables. Let
U + j*V + (j^2)*W = (p + j*q + (j^2)*r)(u + j*v +
(j^2)*w)
(eq.21)
Now after performing the indicated operations on the right side
of the above equation, making use of equas.(12) and the definition
of equality for Cubic Complex numbers we have the following:
U = pu  qw 
rv
(eq.22a)
V = pv + qu 
rw
(eq.22b)
W = pw + qv +
ru
(eq.22c)
It will also be convenient to write these equations as follows:
U = pu  rv 
qw
(eq.22d)
V = qu + pv 
rw
(eq.22e)
W = ru + qv +
pw
(eq.22f)
It may be shown by a straight forward but somewhat tedious
computation that
U^3  V^3 + W^3 + 3UVW = ( p^3  q^3 + r^3 + 3pqr)( u^3  v^3 + w^3 +
3uvw))
{the equation above is
(eq.23) }
It is very convenient (and corresponds in a natural way with the complex
variable analogue) to define the variables p, q and r in such a way that the
expression in the first set of parenthesis, on the right side of (eq.23),
satisfies the following
p^3  q^3 + r^3 + 3pqr =
1
(eq.24)
Therefore eq.(23) reduces to
U^3  V^3 + W^3 + 3UVW = u^3  v^3 + w^3 +
3uvw
(eq.25)
Equation (eq.25) is the cubic complex analogue to (eq.20) from (ordinary)
complex variable theory.
Now let
U =
X^2
V = XY
W = Y^2
and let
u = x^2
v = xy
w = y^2
then the reader can easily verify that substituting these quantities into
(eq. 25) will result in
(X^3 + Y^3)^2 = (x^3 +
y^3)^2
(eq.25a)
Also if you substitute the expressions, given above,
for U, V, W, u, v and w into (eqs. 22d, 22e and 22f),
the result will be
X^2 = px^2  rxy 
qy^2
(eq.25b)
XY = qx^2 + pxy 
ry^2
(eq.25c)
Y^2 = rx^2 + qxy +
py^2
(eq.25d)
which resemble "linear transformations of the quadratic forms
x^2, xy and y^2 into (expressions for) the quadratic
forms X^2, XY and
Y^2".
See Note Below << == >>
(eq.26a)
Note: For (eq.26a) and the other equations with like
numerical designations, such as those not beginning
with T or A, see the section on Second Order
Lorentz Transformations which follows the
sections on Tensor Product and on Proper
(Second Order) Time.
TENSOR PRODUCT
In this section we will show that the fact that The Algebra of Cubic Complex
Numbers, denoted by C3, contains certain elements
that don't possess unique multiplicative inverses and/or the fact that C3 has
zero divisors is actually appropriate in light of the
fact that C3 will be seen to be also associated with tensor and matrix forms as
well as
with vectors.
Presently we shall informally discuss the concept of a Tensor Product. A
practical and
useful definition of a tensor product is given
on p.76 of GRAVITATION (Misner, Thorne and Wheeler). A vector is a first rank
tensor and the tensor product of two
vectors (of the same dimension) is a tensor of rank two.
Let R = (x, y) and S = (a, b) be two arbitrary two component vectors. If we
denote the
tensor product of R and S by R(x)S, then we have
_
_
 xa
xb 
R(x)S
=


(eq. T1)
_ ya yb _
Note that the symbol (x) denotes the Tensor Product and has
no reference to the variable x.
Note that on the right side of (eq. T1) xa denotes the product of x and a, and
xb, etc., have similar meanings.
Please note that the equation #'s in this section all begin with a T.
Now let us write the tensor product of the vector R = (x, y) with itself. We
write
_
_
 xx
xy 
R(x)R
=


(eq. T2)
_ yx yy _
Since the matrix representation of R(x)R is symmetric we may abbreviate
it's notation as follows:
_
_
 xx
xy 
R(x)R
=


(eq. T3)
_
yy _
We will now define two different types of Cubic Forms.
Type I Cubic Form:
Let X denote an n dimensional vector with components
x_k for k = 1 to n.
X = (x_1, x_2, . . . , x_n
)
(eq. T4)
In general the x_k may be complex.
We will define the First (type of) Cubic Form as a
function, to be denoted by F_1(X), that maps the set of
n  dimensional vectors into C (the set of ordinary
complex numbers). For (n > or = 2), we write
F_1(X) = (x_1)^3 + (x_2)^3 + . . . +
(x_n)^3
(eq.T5)
Type II Cubic Form
The procedure for symmetrizing a given matrix is well known.
Moreover, it is not difficult to show that the matrix represen
tation of the tensor product of an n dimensional vector R,
with itself, is symmetric. In general the 2nd cubic form will be
defined for an n x n symmetric matrix M, but due to the problem
of displaying the subscripts, of the matrix elements, using the avail
able typeset, we will presently limit the discussion to the special
cases when n =2 and n =3.
Let
_
_
 m_11
m_12 
M
=


(eq. T6)
_
m_22 _
(where the elements m_ij (in general) are complex)
be a 2 x 2 symmetric matrix.
Definition Of Second Cubic Form
We will now define the Second (type of) Cubic Form, denoted
by F_2, as a function that maps the set of 2 x 2 symmetric
matrices into the set of complex numbers. By definition,
F_2(M) = (m_11)^3  (m_12)^3 + (m_22)^3
+
3*m_11*m_12*m_22
(eq. T7)
For M = R (x) R where R =(x, y),
it can easily be shown that
F_2(R(x)R) = (F_1(R))^2 = (x^3 +
y^3)^2
(eq. T8)
The quantity in the third member of (eq.T8) is symmetrical,
is > or = to zero for all real x and y and is associated with
important invariance properties that will be discussed
in detail later.
Now let us consider the case when M is 3 x 3
symmetrical
matrix. Let
_
_
 m_11
m_12 m_13


M
=

m_22 m_23

(eq. T9)

_
m_33 _ 
Now we will define the Second Cubic Form for M in the
3 x 3
case
F_2(M) = (m_11)^3 + (m_22)^3 + (m_33)^3  (m_12)^3
 (m_13)^3  (m_23)^3 + 3*m_11*m_12*m_22
+ 3*m_11*m_13*m_33+
3*m_22*m_23*m_33
(eq.T10)
The reader can easily verify that for M = R(x)R where
R = (x, y, z), then
F_2(M) = (x^3 +y^3 + z^3)^2 =
(F_1(M))^2
(eq.T10a)
Note that in (eq.T10) the sign preceding (m_ij)^3 is +
if i = j (the element is on the diagonal) and (  ) if i is
different from j. The remaining items can be written as
3*(SUM (m_ii*m_ij*m_jj)  i < j))
Therefore if X is an n dimensional vector as defined in
(eq.T4) and if M is the tensor product of X by itself, M = X(x)X
then we define F_2(M) as follows
F_2(M) = SUM ( (m_ii)^3  i = 1 to n))  SUM( (m_ij)^3  i < j)
+ 3*(SUM (m_ii*m_ij*m_jj)  i <
j))
(Eq.T11)
The following conjecture will now be set forth:
If X is an n dimensional vector as defined in
(eq.T4) and M is the tensor product of X by itself, M = X(x)X
then
F_2(M) =
(F_1(X))^2
(Eq.T12)
We will now define a function P, that maps C3 into the
set of symmetrical 2 x 2 matrices, as follows: Let A be
the element of C3 given by
A = (U, V, W)
define P(A)
by
_
_

U
V 
P(A)
=


(eq. T13)
_
W _
Recalling (eq.6) and (eq.7d), from the section on Cubic
Complex Numbers, we may conclude that
A = (U, V, W)
has a unique inverse if and only if
U^3  V^3 + W^3 + 3UVW
does not vanish. But the above expression is just the
Second Cubic Form of P(A). We have
F_2(P(A)) = U^3  V^3 + W^3 +
3UVW
(eq. T14)
Recall that in the Tensor Product representation of R(x)R,
where R = (x, y), see (eq.T3), the symmetric matrix had
non  linear elements
of the second degree. Now lets assign to the matrix
of (eq. T13) linear elements. Moreover, we will
let
V be the simplest possible non  trivial linear combination
of the components U and W by writing
U = x, V = x + y and W =
y
(eqs.T15)
so that
V = U +
W
(eqs.T15a)
A simple calculation will show that
F_2(P(A)) = U^3  V^3 + W^3 + 3UVW =
0
For the U, V and W that are given in (eqs.T15).
Therefore the cubic complex number A = (U, V, W)
does not have a unique inverse. Well that is a "big deal"
considering that it corresponds to a symmetric matrix
whose elements are of the first degree and
have the simplest possible (non  trivial) linear dependence
on each other.
Linear transformations are usually associated with vectors
and/or matrices with linear components. We are looking
for a systematic (invariant) treatment of nonlinear,
(restricted to the second degree at present),
transformations. Such a set of non  linear invariant transformations
have been found for Uniformly Accelerated motion.
The Final Form of the Second Order Lorentz Transformation (SOLT)
do not involve elements of C3: only Real Numbers appear
in the equations.
Nevertheless, a more comprehensive
discussion of the zero divisors of C3 will be presented
later.
**********************************************************************
The plane is a two dimensional space
but a cube resides in 3D
so our search for truth will be in vain
if we restrict ourselves to the plane
and to transformations of the first degree.
RHB
**************************************************************
PROPER (Second Order) TIME
In this section we will show a direct approach to the
derivation of proper time in SR by following Minkowski's
use of complex variables. Then we will show an analogous
method, using cubic complex variables, of deriving a
(second order) proper time that is associated with
uniformly accelerated motion.
The Second Order Lorentz Transformations will be
presented as the fundamental relationships between
the coordinate systems of observers who are
uniformly
accelerated with respect to each other.
It is to be noted that the Final Form of the SOLT
equations will contain only Real Variables.
The Algebra of C3 (Cubic Complex Numbers) only served the
one (but important) purpose of "pointing us in the right
direction".
Let
d(x,y) = x^2 +
y^2
(eq.A1)
The reader will recall that the right side of (eq.A1)
is the square of the absolute value of the complex
variable x + iy and it is also the invariant associated
with orthogonal coordinate transformations in the
plane.
Now let
x =
vt
(eq.A2a)
y =
ict
(eq.A2b)
where i is the imaginary unit defined by
i^2 = 1
and where v and c are constants (with the physical
dimensions of velocity), t is the time and v < c.
Now substitute (eq.A2a) and (eq.A2b) into the
right side of (eq.A1) and obtain, after simplifying,
d(x, y) = (v^2 
c^2)*t^2
(eq.A2c)
Now let s be defined by
s =  d(x,
y)
(eq.A3a)
so that
s = (c^2 
v^2)*t^2
(eq.A3b)
If we now divide both sides of (eq.A3b) by
(c^2)*(t^2) we will arrive at the following:
s/((c^2)*(t^2)) = 1 
(v/c)^2
(eq.A4)
Now define the variable h
by
h = [ s/((c^2)*(t^2))]^0.5 = (1 
(v/c)^2)^0.5
(eq.A5a)
so that
h = (s)^0.5/ct = (1 
(v/c)^2)^0.5
(where ct denotes the absolute of ct) Or simply
h = (1 
(v/c)^2)^0.5
(eq.A5b)
The reader will recognize that
1/h = "the relativistic gamma" = 1/(1 
(v/c)^2)^0.5
(the above equation is...)
(eq.A5c)
Assuming the usual initial conditions let u denote
the proper time of an observer moving with
constant velocity v along the x axis of a fixed
frame of reference, then we have
u = th from which we have
u = t*(1 
(v/c)^2)^0.5
(eq.A6)
Now we will begin a discussion in which D and S
will play roles in the cubic complex variable
derivation of "second order proper time" that
correspond to the role that (the small case)
d and s played in the previously discussion.
But first recall that the expression
(x^3 + y^3)^2, see ( eq.25a)
plays a role in C3 theory that corresponds to the role that
x^2 + y^2
plays in ordinary complex variable theory. Let us define
D(x, y) as follows:
D(x, y) = (x^3 +
y^3)^2
(eq.A7)
Now let
x =
1/2*a*t^2
(eq.A8a)
y =
1/2*j*L*t^2
(eq.A8b)
where the entity j is the cubic complex
unit and is defined by (eqs.11a and b), and
where a denotes the constant (uniform)
acceleration of an observer with respect
to a fixed frame of reference. And L will
be used to denote the absolute value of
the conjectured acceleration of photons.
It is postulated that all observers, who
are uniformly accelerated relatively to
each other, will measure the same value
of L.
Exactly how is the quantity L to be measured?
Later in the discussion we will propose an indirect
method of measuring it. We do indeed
have very good reasons, based
upon symmetry and other theoretical
considerations , to believe that it does
exist and is measurable. The reader, we are sure,
is well aware of the important place that
symmetry plays in Theoretical Physics.
For the present time let us briefly
call attention to
the possibility that a photon, after being
emitted from the electron of a hydrogen
atom, may have a very large ( but finite)
acceleration [for a (very) short period of
time] before it (the photon) reaches the
full speed of light c. This may be a con
stant of nature and we will present a
method of measuring it [ or for mea
suring the magnitude of some
other quantity, (that may be a
constant of nature), that is associated
with optical phenomena and has the
physical dimensions of acceleration].
Now if we substitute (eq.A8a) and (eq.A8b) into
the right side of (eq.A7) the result will be,
after making use of the
fact that j^3 =  1 and simplifying,
D(x, y) = 1/64*t^12*(a^6  2*a^3*L*3 +
L^6)
(eq.A9)
Multiplying both sides of (eq.A9) by 64 and
dividing by L^6*t^12 will give us
64*D(x, y)/(t^12*L^6) = ((a/L)^6  2*(a/L)^3 + 1)
Now if we make use of the fact that the quantity on the right
side of the above equation is a perfect square, we may
write
64*D(x, y)/(t^12*L^6) = ( (a/L)^3 
1)^2
or
64*D(x, y)/(t^12*L^6) = (1  (a/L)^3
)^2
(eq.A10a)
Now define S by
S = 64*D(x,
y)
(eq.A10b)
Therefore (eq.A10a) may now be written as
S/(t^12*L^6) = (1  (a/L)^3
)^2
(eq.A10c)
Now since t^12 appears in the denominator of
the left side of (eq.A10c) and we are looking for
a quantity that is related to (the first power of)
the time t, we take the 12th root of both sides
of (eq.A10c) and obtain the following:
(S)^(1/12)/t*L^0.5 = (1  (a/L)^3
)^(1/6)
(eq.A11)
We will now define H as being equal to the left
side of (eq.A11) so that H is also given by
H = (1  (a/L)^3
)^(1/6)
(eq.A12)
Later on it will be shown that s "the 2nd order proper
time" of an observer, moving with uniform acceleration a
with respect to a fixed frame of reference, will be
s = tH
s = t*(1  (a/L)^3
)^(1/6)
(eq.A13)
Remark
Note that the small case s of (eq.A13) is not the same as
the s of (eq.A3a) that was used in the illustration of how
"first order" proper time could also be derived.
SECOND ORDER LORENTZ TRANSFORMATIONS
Below we will state the Second Order Lorentz Transformations (SOLT)
for one spatial dimension and time. The more general two spatial
dimensional case (which includes the Lorentz Transformation as a
special case) will be presented later.
Note that the equation numbering sequence, used in this section, is
a continuation of the sequence that was used in the first section
(on Cubic Complex Numbers).
X = p(x 
(1/2)a*t^2)
(eq.26a)
T = (p(t^2 
2(a^2/L^3)x))^0.5
(eq.26b)
where p is given by
p = 1/(1 
(a/L)^3)^(2/3)
(eq.27)
It may be shown that eq.(26a and 26b) satisfy the following
invariance relationship
(X^3  ((L/2)^3)*T^6)^2 = (x^3 
((L/2)^3)*t^6)^2
(eq.28)
Now let us give an informal discussion of the physical interpretation
of the SOLT. The frames of reference r (with coordinates (x,t)
) and
the frame of reference R (with coordinates (X,T)) are both to be
considered as consisting of one spatial dimension and a time
dimension. At the point in space  time
(x, t) = (X, T) = (0,
0)
(eq.29)
the R frame is moving with initial velocity = zero and constant
acceleration = a in a direction that is parallel to the x  axis of the
r frame. The parameter L has the physical dimensions of acceleration
and is associated with optical phenomena.
Now let us determine the Second Order Proper Time. We follow a
procedure that is similar to the corresponding case in S.R. If an
observer is fixed at the origin of the R frame then X = 0
and then from (eq.26a) it follows that
x =
(1/2)at^2
(eq.30)
If we now substitute the above result (along with the expression for p
as defined by (eq. 27)) into eq.(26b) we obtain,
after simplifying,
T = [t]*(1 
(a/L)^3)^(1/6)
(eq.31a)
where the symbol [ ]
indicates the absolute value of the quantity enclosed. We denote
this special value of T by s (the second order proper time) and we write
s = [t]*(1 
(a/L)^3)^(1/6)
(eq.31b)
******************************************************
If there is one
and if there is another
then surely one of them
must be the other.
RHB
******************************************************
Returning, for the moment back to Special Relativity, let u denote the
proper time of an observer (the invariant time carried by an observer
in his rest frame). We have
u = t(1 
(v/c)^2)^(0.5)
(eq.32)
where v is the absolute value of the velocity of the observer with
respect to a fixed frame with (conventional ) time coordinate t. Now
if we solve (eq.32) for c we obtain
c = [v]/( 1 
(u/t)^2)^(0.5)
(eq.33)
The following is the author's interpretation of the physical im
plications of (eq.33): (your comments will be appreciated)
At the instant that an observer, who is moving with constant
(absolute) velocity v along the positive x axis of a fixed frame
of reference, passes the origin of the fixed frame, let his clock show
the time of zero and also let the time as measured in the fixed frame
be zero. Then when the observer passes some arbitrary point whose
coordinate is x, in the fixed frame, his time will be indicated by u and
the time as measured in the fixed frame will be indicated by t. Now
if these values of v, u and t are plugged into the right side of (eq.33)
the result will be c ( the absolute value of the speed of light).
Now if we solve (eq.31b) for L we will obtain
L = a/(1 
(s/[t])^6)^(1/3)
(eq.34)
The following is the author's interpretation of the physical im
plications of (eq.34): (your comments will be appreciated)
At the instant that an observer, who is moving with constant
uniform acceleration a along the positive x axis of a fixed frame
of reference, passes the origin of the fixed frame, let his clock show
the time of zero and also let the time as measured in the fixed frame
be zero. Then when the observer passes some arbitrary point whose
coordinate is x, in the fixed frame, his time will be indicated by s and
the time as measured in the fixed frame will be indicated by t. Now
if these values of a, s and t are plugged into the right side of (eq.34)
the result will be L. As mentioned before, L has the physical
dimensions of acceleration and is presumed to be associated with
some form of optical phenomena.
Surely a series of experimental tests of the implications of (eq. 34),
using the high technology of today, will not be difficult or expensive to
conduct. If the same value of L was
determined, via (eq.34), as a
consequence of a series of experiments using a variety of values of the
parameter a, then one would not be remiss in attributing a certain measure
of scientific value to the ideas that have been set forth herein .
Just what is the optical phenomena associated with the quantity L?
Well first let us recall that it was once thought that the speed of light
was infinite but now we know it has a finite value which we denote by c.
Now consider, for example,
the electron in orbit around the nucleus of a hydrogen atom. Now
if it is at all possible, let us just for a moment disregard such things as
Heisenberg's Uncertainty Principle, the curvature of space  time,
the path of geodesics, etc., and ask ourselves the simple question:
when the electron in the hydrogen atoms emits a photon, does the photon
INSTANTANEOUSLY start out with the velocity c? Can we allow
ourselves to dare to speak of the acceleration of a certain optical
entity without bringing the full weight of the General Theory into
play? We must remember that a basic principle of that theory is
that space  time is flat if a sufficiently small region is considered.
Note that we may also solve (eq. 34) for a and (eq. 33) for v and
obtain
a = L*(1 
(s/t)^6)^(1/3)
(eq.35)
v = c*( 1 
(u/t)^2)^(0.5)
(eq.36)
An examination of (equas. 26a & 26b) and (eq. 27) will show that the
quantities X and T will be come unbounded if
1  (a/L)^3 = 0
Therefore we must have
a not equal to
L
(ineq.37)
where " ineq. " denotes "inequality". However
(eq.26b) involves
the square root of the quantity t^2  2(a^2/L^3)x
. So then we
must have
t^2  2(a^2/L^3)x >
0
(ineq.38)
But for an observer at rest at the origin of the R frame
we must have X = 0, which on account of (eq. 26a) implies
that
x = (1/2)a*t^2
which when substituted into (ineq.38) gives us
(t^2)*(1  (a/L)^3) > 0
which implies that
L >
a
(ineq.39)
It is postulated that for a sufficiently small region of
space (time), all observers who are moving with
a constant uniform acceleration with respect to each other,
will measure the same value of the parameter L. But over
larger regions of space (time) L may possibly vary. Also the
uniform acceleration referred to above may have the
(constant) value of zero.
The author has rigorously verified that (eq. 23) holds true as a consequence of
the definitions of the variables involved. This was done early in the 1990's.
The author has recently done "empirical verifications" of (eq. 28)
using arbitrary values of the independent variables (and parameters) using GW
Basic. Due to the fact that
rounding errors do occur and some of the
computations involved the 12th powers
of certain quantities, it was not surprising that relatively small differences
in the absolute values of the sides of (eq. 28) sometimes occurred when
relatively large
values of the parameters were selected.
The author does not recalls explicitly if he
did or did not do an algebraic verification of (eq. 28), based upon the
definitions of the variables involved. Nonetheless, (eq. 28) was obtained from
the previous definitions and results using the standard ALGEBRAIC METHODS that
are associated with commutative rings. The reader will recall that all
the algebraic structures associated with mathematical physics are not algebraic
fields.
The author enthusiastically invites anyone with the appropriate software
(MATHEMATICA (?)) to verify (or refute) any of
the results stated in this draft presentation.
The reader will get just a hint of the rich symmetry that is associated with
cubic complex variables by referring to (eq. 23). It is seen that variations
of the cubic form U^3  V^3 + W^3 + 3UVW also appear
as factors on
the right side of (eq.23). This form U^3  V^3 + W^3 + 3UVW
may be considered
to be the cubic analogue of the form quadratic form x^2 + y^2 from (ordinary)
complex variables theory.
Now for an arbitrary point (x,y) in the plane, let
U = x^2, V = xy, W =
y^2
(eqs.40)
Then it can easily be shown that
U^3  V^3 + W^3 + 3UVW = (x^3 +
y^3)^2
(eq.41)
Similar results hold for points (x,y,z) in 3D and for the nth dimensional case.
*******************************************************
The First Verse was deep and profound
and when it was uttered it was the very first sound!
And though the last verse is yet to be spoken
the power of that First Verse is still unbroken.
RHB
*******************************************************
Addendum
The Algebra of Cubic Complex Variables was
formulated in the early 1980's. The author distributed several (informal)
papers featuring his concepts when he was an undergraduate student at
The author has extended the concepts of Cubic Complex Variables to systematic
generalizations of such entities as differentiation, integration, Cauchy 
Riemann equations, Cauchy Integral formulas, (cubic) complex exponentials and
many others.
In addition, he has generalized Cubic Complex Variables to the Quartic and the
nth
degree cases. Many of these concepts were reviewed by members of the faculty
and
the graduate student body at the
The author has also laid the foundations of a "cubic complex
inspired" generalization of Riemannian geometry and the calculus of
tensors.
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