Second Order Lorentz Transformations with 2 Spatial Dimensions (SOLT-2SD)

In the case of a particle moving with constant velocity, a one spatial dimensional coordinate system
can always be chosen to coincide with the path of that particle. Likewise for the  motion of a particle
undergoing uniform acceleration,  a  2 -spatial dimensional coordinate system can always be chosen
to coincide with the plane of the motion of the particle.

Let the M frame of reference, with coordinates (x, y, t) and origin o, be considered to be fixed. Let [M]
be a frame of reference, whose origin [o] is moving along the x - axis of the M frame with constant
velocity v. Let the coordinates of [M] be ([x], [y], [t]) and let the motion be such that  the [x] and [y]
axis of the [M] frame are always parallel to the x and y axis, respectively, of the M frame.

Now let R denote a frame of reference with origin O and coordinates (X, Y, T). Let the origin O of the
R frame be moving with uniform acceleration a along the [y] axis of the [M] frame. Let the motion be
such that the X and Y axis of the R frame are always parallel to the  [x] and [y] axis, respectively, of
the [M] frame.  

So we are assuming that the R frame is uniformly accelerated with respect to the [M]
frame and the (constant) rate of acceleration is a.

Now let the origins of all three of the frames of reference coincide  when t = [t] = T = 0. In other words
let it be true that

(x, y, t) = ([x], [y], [t]) = (X, Y, T) = (0, 0, 0)

when t = 0.

Now we will show how the Second Order Lorentz Transformations with 2 Spatial Dimensions
(SOLT-2SD) are derived. In what follows c is the speed of light in a vacuum as measured by
an inertial observer with apparatus such that no part of it is accelerating with respect to any
other part. Let L be a parameter (possibly of an optical nature) with the physical dimensions
of acceleration. Let it be measured by an observer at rest in the R frame.

Actually in this theory of  "Mathematical Phiction" we are postulating that all observers who
have uniform relative acceleration with respect to each other, in a  sufficiently restricted region of
space - time, will measure the same value of the parameter L which has the physical dimensions of
acceleration. For example in a region near a star such as the sun, L could be the acceleration of
gravity at the  the surface. Just as the speed of light c is the maximal speed attainable by a particle
(massless or not), L (on a preliminary basis) is defined as the maximal acceleration attainable and
sustainable (over extended intervals of time) by a particle (massless or not).

One more fine point. The relative uniform acceleration referred to above could have the value
of zero or be negative. So the reference to "maximal acceleration" actually refers to the "maximal
absolute value of accelerations".

The ordinary Lorentz Transformation from the M frame to the [M] is given by

[x]   =   (x - vt)/G                                                                                                                   (eq.1a )

[y]  =    y                                                                                                                                (eq.1b )

[t]  =    (t - (v/c^2)x)/G                                                                                                           (eq.1c )

where G is given by

G = (1 - (v/c)^2)^(1/2).                                                                                                          (eq.1d )

The SOLT from the frame [M] to the frame R is given by

X  =  [x]                                                                                                                                 (eq.2a )

Y  = ([y]  - (1/2)a[t]^2)/(D^(2/3))                                                                                            (eq.2b )

T  =  (+/-){([t]^2 - 2(a^2/L^3)[y])^(1/2)}/D^(1/3)                                                                  (eq.2c)

where the symbol (+/-) means " + or - ".

where D is defined by

D = 1 - (a/L)^3                                                                                                                       (eq.2d )

Now if we substitute the expressions for [x], [y] and [t], from (eqs. 1a, 1b and 1c),
 into  (eqs. 2a, 2b and 2c)  we obtain  

X = (x -vt)/G                                                                                                                          

Y = [y - (1/2)a{(t - (v/c^2)x)^2/G^2}]/(D^(2/3))                                                                   

T =  (+ / -)[(t - (v/c^2)x)^2/G^2 - 2(a^2/L^3)y]^(1/2)/(D^(1/3))                                         

where the symbol (+/-) means " + or - ".

By convention we will choose the + sign so we have

T =  [(t - (v/c^2)x)^2/G^2 - 2(a^2/L^3)y]^(1/2)/(D^(1/3))                                                                                         

For easy reference lets write these equations for X, Y and T all

X = (x -vt)/G                                                                                                                           (eq.3a)

Y = [y - (1/2)a{(t - (v/c^2)x)^2/G^2}]/(D^(2/3))                                                                   (eq.3b)

T =  [(t - (v/c^2)x)^2/G^2 - 2(a^2/L^3)y]^(1/2)/(D^(1/3))                                                  (eq.3c)                                         

These are the transformation equations that express the coordinates of the uniformly
accelerated R frame in terms of those of the fixed M frame. We will refer to them as
the Second Order Lorentz Transformations with 2 Spatial Dimensions, (SOLT-2SD).

We are now interested in the value of T when the[M] observer is in his rest
frame. That is when X = Y = 0. Setting the left side of (eq.3a) equal to zero

results in

x  = vt                                                                                                                                     (eq.3c1)                                         

Now set Y = 0 and make use of the result x = vt.  The result (after simplifying) will be

y = ((1/2)at^2)(1 - (v/c)^2)                                                                                                     (eq.3c2)                                         

Now substituting    

x = vt   and    y = ((1/2)at^2)(1-(v/c)^2)  

into (eq.3c) gives us, after simplifying,

T =  [(t){(1 - (v/c)^2)(1 - (a/L)^3)}^(1/2)]/D^(1/3)                                                                                                                    

This special value of T when X = Y = 0

is the "second order proper time." This is the time that an observer carries
with him as he is stationed at the origin of his  uniformly accelerating frame (the frame R
as described above). We will denote it by T_p   

T_p  =  [(t){(1 - (v/c)^2)(1 - (a/L)^3)}^(1/2)]/D^(1/3)                                                                  (eq.3d)                                              

and return to it later.

It will be immediately seen that when v = 0 and a = 0,  then G = 1 and D = 1,  
and so  equations (3a, b and c) reduce to simply the identity transformations.
When a = 0 (so that D = 1) and v is non - zero, then equations (3a, b and c)
reduce to the ordinary Lorentz Transformations. Finally when v = 0,  
(so that G =1) and a is non - zero, then equations (3a, b and c) are equivalent
(to the previously defined) SOLT  from the coordinates (y, t) to the coordinates
(Y, T) along with the equation  X = x.

                  In a region of 2 - spatial dimensions strings under varying tensions
                  vibrate in sync. But will a theory based upon the first order of
                  approximation allow theorists enough imagination to discover
                  the missing link: that will explain all known physical interactions
                  including gravitational attractions?

                  Just as the stock market can switch from it's highs to it's lows, a writer
                  can switch from verse to prose but not before characterizing high                  
                  frequencies as those... that can be associated with a vocal hum.
                  But frequencies that are low are sometimes produced by a
                  cellist's bow. But an even lower frequency is the beating of a drum.

                  So the two basic elements of music: 1) tone(s), which make up
                  melody and 2) rhythm, as in a quarter note gets one beat or an
                  actual drum beat, are unified by frequency. Relatively high frequencies
                  such as a soprano's 880 Hz high A (440 Hz for the tenor) gets
                  perceived as tone but the  1 Hz drum beat (in a performance by
                  a marching band or in rap music) gets interpreted as rhythm.
                  So frequency unifies the two basic elements of music.

                   To take this analogy a little further, when the Bolshoi Ballet
                   performs the Nutcracker Suite the artistic audio visual
                   experience is characterized by light, sound and rhythm.
                   The light is produced by high frequency electromagnetic waves.
                   The sound is associated with frequencies in the acoustic range.
                   And the rhythm of the dance are associated with frequencies
                   of the order of 1Hz. So light, music and dance are all unified
                   by frequency.
                  Frequency and wavelength are characteristics of strings. Topological
                  mathematical physicists may want to consider the possibility of
                  constructing an "topological mapping" of this argument into the realm
                  of the String Theory of Mathematical Physics": a theory in which an
                  attempt is made to unify all known physical interactions.
                   Cubic Complex Variables Versions of the Wave Equations of
                   Mathematical Physics will be presented soon.

Now as it is often desirable when giving a relativistic treatment of motion, we
want the physical dimensions of all of the coordinates to be the same.

So let s, [s] and S be defined by

s = ct                                                                                                                          (eq.4a)

[s] = c[t]                                                                                                                     (eq.4b)

S = (1/2)LT^2                                                                                                            (eq.4c)

So then it is convenient to redefine the three frames as follows: M is the frame with
origin o and coordinates (x, y, s); [M] is the frame with origin [o] and coordinates
([x], [y], [s]); and R is the frame with origin O and coordinates (X, Y, S). The motions
of the frames are still as described above.

We will shortly propose a definition for the "second order Lorentz invariant length".
But first we need some special notation. Some of the notation will also be used later
when a more detailed paper is presented.

Let x, y and t be the coordinate variables in the M frame. Let (x*) denote the first
derivative of the function x = x(t) with respect to time.
That is (x*) = dx/dt, let (x**) denote the second order time derivative of x  etc., and also
denote the time derivatives of  y = y(t) in similar fashions. Also  


will denote the value of (x*) when t = 0.

Also if f = f(t) is any (differentiable) function of t  we denote df/dt by (f*) and we
will assign the symbols

df(0)/dt   or  (f*(0))

to denote the value of df/dt when  t = 0.

Now consider [x], [y] and [t], the coordinates of the [M] frame. Let
[x*] and [y*]  denote, respectively, the derivatives of [x] and [y] with respect to u
where u is the proper time in the [M] frame. That is

[x*]  = d[x]/du,    [y*] = d[y]/du     where u is given by

u  =  tG                                                                                                            (eq.4d)

where G is defined by (eq.1d).

Also we denote second derivatives, of  [x] and [y],  with respect to u as follows:

[x**] = d/du([x*])     


[y**] = d/du([y*])

Now let the symbol I denote indefinite integration (with constants of integration
being set equal to zero unless stated otherwise) as follows:

I(f(t))dt = "the indefinite integral of f(t) with respect to t". And also

I{f(t)}dt could be used to denote the same thing. Also


will be used to denote the "double indefinite integral" or the iterated indefinite
integral of f(t) with respect to t.

Now let A and B be defined as follows:

A = [y**]/(w**)                                                                                                           (eq.5a)

where w = w(t) is a twice differentiable function of t. And let

B = (x*(0))/c                                                                                                                (eq.5b)

where c is the speed of light in vacuum.

Now define the quantity W  by

W = I(I([{(1 - B^2)(1 - A^3)}^(1/2)]/[(1 - A^3)^(1/3)])dt)dt                                    (eq.6 )

Now, in the expression for A (see (eq.5a)), let the 2nd order derivative of [y]  
with respect to u (the proper time in the [M] frame) be equal to a

[y**] = a                                                                                                                 (eq.7a)

where a is the uniform acceleration referred to above, [see footnote (*)].

Let w, in that same expression for A, be given by

w = (1/2)Lt^2                                                                                                         (eq.7b)

where t is the time coordinate in the M frame and L is the constant
with the physical dimensions of acceleration referred to above.


w** = L                                                                                                                  (eq.7c)

Also let the value of dx/dt, evaluated when t = 0, be  v

(x*(0)) = v                                                                                                               (eq.7d)

where v is the constant velocity with which the [M] frame is moving relative
to the fixed M frame as described above.

Therefore, with these substitutions, both A and B, as defined by (eqs. 5a and 5b),
 are both constant and all expressions, that depend solely upon them,
may be removed from under the double integral signs in (eq. 6). So we have

W = [{(1 - B^2)(1 - A^3)}^(1/2)]/[(1 - A^3)^(1/3)]I(Idt)dt                                              

evaluating the inner integral  (with constants of
integration set equal to zero)

W = [{(1 - B^2)(1 - A^3)}^(1/2)]/[(1 - A^3)^(1/3)]I(t)dt

Multiply both sides by L  (a constant)

WL  = [{(1 - B^2)(1 - A^3)}^(1/2)]/[(1 - A^3)^(1/3)]I(Lt)dt

WL  = [{(1 - B^2)(1 - A^3)}^(1/2)]/[(1 - A^3)^(1/3)]((1/2)Lt^2)

but by (eq.7(b))

w = (1/2)Lt^2                                                                                                         


WL  = [{(1 - B^2)(1 - A^3)}^(1/2)]/[(1 - A^3)^(1/3)]w

multiplying both sides by t and dividing by w

(W/w)Lt = [(t){(1 - B^2)(1 - A^3)}^(1/2)]/[(1 - A^3)^(1/3)]                                 (eq.8)

Now if we actually substitute the expressions for [y**], (w**)
and (x*(0)), from (eqs. 7a, 7c and 7d), appropriately into
the right sides of (eqs. 5a and b), we will obtain

A = a/L                                                                                                               (eq.9a)

B = v/c                                                                                                                (eq.9b)

Now substitute these values of A and B into the right of
(eq.8) and arrive at

 (W/w)Lt =  [(t){(1 - (v/c)^2)(1 - (a/L)^3)}^(1/2)]/[(1 - (a/L)^3)^(1/3)]                                      

Recalling (eq.2d) we have

D = 1 - (a/L)^3

substituting into the denominator on the right side of the above

 (W/w)Lt =  [(t){(1 - (v/c)^2)(1 - (a/L)^3)}^(1/2)]/[D^(1/3)]                                           (eq.10)                                      

Now comparing the right sides of (eq. 3d) and
(eq.10) we are led to

T_p  =  (W/w)Lt                                                                                                         (eq.11)                                                                                                                                                                             

Now referring again to (eq.3d), if a = 0 so that
D = 1, (eq.3d)
reduces to

T_p = t[1 -(v/c)^2]^(1/2)                                                                                       (eq.12)

which is the proper time for an observer in Special Relativity.

Equa.(6) is a special case of a more general result which will be
published later: as an introduction to Cubic Riemannian ( Differential
Geometry and Tensor Calculus).

 footnote (*)

Recall from (eq.1b) that

[y] = y

but in the rest frame of [M]   via (eq.3c2) we have   

y = ((1/2)at^2)(1 - (v/c)^2)     

By definition  

[y**] = d/du[y*] = d/du(d/du[y]) = d/du(d/du(y)) = d/du(d/du( ((1/2)at^2)(1 - (v/c)^2) )) = a

where we have made use of the fact that the proper time u in the [M] frame is given by

u = t(1 -(v/c)^2)^(1/2)

du = dt(1 -(v/c)^2)^(1/2)

which defines the differential operator d/du in terms of d/dt as follows:

d/du  = (1 - (v/c)^2)^(-1/2)(d/dt)

One should not be too surprised
if when all the unifying principles of all known interactions
are discovered
that new previously unknown interactions and forces of attractions
are also uncovered.

There must always be mystery
so that God can work in his mysterious way.
And even when man finds a unified theory
it will only represent an approximation of the real forces at play.

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