  ## Relativistic Variations on a Theme Math Correction -- 23 September 2003

Inexplicable Addition -- 1 April 2005 (How appropriate)

In Relativity, the speed of light, c, has always been considered a limiting speed barrier for physical matter.  This derives from the Lorentz Contraction:

t  =  to / [ 1 - (v/c)2] 1/2

In the equation, to is the time as measured when a body is at rest, while t is the relativistic time, or the time when the body is traveling at a velocity v.  (Just as easily, we could use the same equation to represent the mass of a body at rest and it relativistic mass when in motion; or alternatively the length of the body at rest and the relativistic length.)

The challenge of Relativity as always been when the velocity of the body has the same value as the velocity of light, i.e. v = c.  Under these luminal circumstances, t (or m or L) each become infinite.   From the viewpoint of physicists, this constitutes a barrier, inasmuch as infinite time, mass or length are physically untenable (and very difficult to fathom mathematically).  It’s the great horror of physics: a singularity!

Using the results from (as provided in The Fifth Element, and from a mathematical viewpoint, Mathematical Theory), we can derive* what we will refer to as a Modified Lorentz Contraction (the “old MLC” you’ve heard so much about).  I.e.

1 / {1 - (v/c)2}1/2     ®   1 / {1 - (v/c)2 (1 + sin 2f) / (1 + D2  w2 )}1/2

Recall that D (or W) is the time delay constant associated with the mechanical (electrical) dimension of the object traveling at the speed, v, w is the frequency, and f is the phase angle given in the simplest form (assuming the viscosity and restoring forces to be zero) by:  [Note that the frequency is not squared as in the original version -- otherwise the units of time would not cancel and yield a pure number for the tangent.]

tan f  =  D

This Modified Lorentz Contraction can be considered to be of fundamental importance inasmuch as the additional terms in the equation [D2w2 and sin 2f)] have incorporated the effect of the non-simultaneity of imposed forces and their reaction!  (Because the product of D times w is a pure number, it is not subject to a Lorentz Contraction itself.)

Alternatively, if D w goes to zero (or is simply ignored and assumed to be zero), the Modified Lorentz Contraction term returns to its original form.

However, D is never zero!  In other words, simultaneity across space does not exist in the case of bodies with mass and dimensions.

In the case of an electron, D would be exceedingly small, but still not zero!  As the D w term becomes vanishingly small, the experimental evidence for the agreement of the traditional Lorentz Contraction theory with experiment will be quite good.  (Assuming of course, that the imposed frequency is not large.)

However, if the value of w is sufficiently large, a substantial “delay time” may not be required in order to keep the Lorentz Contraction term from approaching infinity.  In this case, the so-called light barrier might be more easily approached -- if not exceeded.  In particular, with a frequency large enough to make the D w term comparable in size to unity, the “infinity problem” or “very large mass, time, or length effects” of relativity as v approaches c, completely disappear!

For example, in the case where D w = 2, and v = c:

1 /  {1 - (v/c)2}1/2   ®   1 /  {1 - (1/1)2 (1 + 0.8) / (1 + 4 ) }1/2  =  1.250*

In this case, the Modified Lorentz Contraction does not increase the mass, time, or length to infinity!

At v = 2.15 c, the denominator becomes 0.0015, such that the light barrier has been pushed back, but not entirely dissipated!

By further increasing the frequency, such that D w = 6 and v = 5c, the ratio of relativistic time to rest time becomes 3.08.  If on the other hand, D w = 5, it will not be possible for the object to achieve a velocity of v = 5c, and in fact the “modified light barrier” will occur at a velocity of v = 4.333...c.

*Thanks to Kevin Trinder for pointing out the mathematical error.

The good news is that the conclusion is not modified by the error.  Whew!

The fundamental conclusion is that with respect to Connective Physics, the speed of light does not constitute an immovable barrier for a sufficiently large product of D w, and thereby superluminal communications become an ever-increasing reality.

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*Meanwhile, for those who really get into the mathematics, the Derivation!

Assume a driving force given by:

F  =  Fo cos (wt)

(Equation 1)

where the resulting displacement is of the form:

X  =  a cos wt  +  b sin wt  =  [(a2 + b2)]1/2 cos (wt - f)

where

tan f  =  b / a

(Equations 2)

with a and b being constants, and f being the phase angle.

In using Equations 1 and 2 to solve the traditional form of Newton’s Second Law (with the viscosity and restoring-forces assumed to be negligible), the particular solution is:

X  =  Fo cos (wt - f) / m w2

where

tan f  =  0

(Equations 3)

Using and the Mathematical Theory of The Fifth Element, which includes a Force term proportional to the rate of change of acceleration, i.e.

F  =  m d2X/dt2  +  D m d3X/dt3

(Equation 4)

The same logic or methodology can be used to obtain a modified version of the particular solution for an applied oscillatory force (again given by F = Fo cos (w t), and again assuming the viscosity and restoring forces to be negligible), i.e.:

X  =  Fo cos (wt - f) / [ m w2 ( 1 + D2 w2)1/2 ]

tan f  =  D w

(Equations 5)

D is assumed to be a constant, and can be interpreted as the delay time or response time of the system to an imposed force.  This delay time is related to the time required for a signal of an imposed force at one point on the mass of a body (i.e. a body with dimensions) to reach the mass of all parts of the body (and thereafter react in an equal and opposite manner in accordance with Newton’s Third law).  D w is a dimension-less number, which cannot be zero without requiring simultaneity of imposed forces and their reaction.

If we assume a solution of the form given in Equations 3, and calculate the velocity, we obtain:

v  = dX/dt  =  - (Fo/mw) sin (wt)

(Equation 6)

and similarly, for Equations 5,

v*  =  - (Fo/mw) sin (wt - f) / (1 + D2 w2 )1/2

where, again

tan f  =  D w

(Equations 7)

If we assume the velocity term in the Lorentz Contraction is more accurately described by Equation 7 (in lieu of equation 6), the Lorentz Contraction becomes a Modified Lorentz Contraction, i.e.:

1 / {1 - (v/c)2}1/2   ®    1 / {1 - (v/c)2 [sin (wt - f) / sin (wt)]2 / (1 + D2 w2)} 1/2

(Equation 8)

Equation 8 can be simplified by noting that the ratio of sine terms can be transformed by standard trigonometric identities to read:

sin (wt - f) / sin (wt)  =  cos f  -  sin f / tan (wt)

Utilizing Equations 3 and 6, it can be shown that:

1 / tan (wt)  =  - X w / v

where X and v are the displacement and velocity when D w is essentially zero.

One possibility is to consider the X term as representing a reaction distance or shock wave dimension, and v the velocity of that shock wave.  Food for thought...

By defining w‘ = v / X, we can write:

[sin (wt - f) / sin (wt)]2   =  [cos f  +  sin f ( w / w‘ )]2

(Equation 9)

Clearly, the first term in Equation 9 is always £ 1, and the second term is always £ 1 provided that the ratio of frequencies is £ 1.  But the imposed frequency is not a function of the reactivity of the mass upon which the frequency is being imposed; leading to the condition where this ratio is identically equal to 1.

Using other trigonometric identities, Equation 9 can be simplified by completing the square of the right hand side of the equation, which yields:

[sin (wt - f) / sin (wt)]2   =  [ 1 +  sin 2f ]

In the special case where D w becomes large, f will approach p/2, and the ratio of sine terms squared  will approach unity.

In the more general case, Equation 8 becomes:

1 / {1 - (v/c)2 }1/2    ®   1 /  {1 - (v/c)2 (1 + sin 2f) / (1 + D2  w2) }1/2

(Equation 10)

This result constitutes the modification of the Lorentz Contraction to that of the MLC.

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Now for those who really get into the strange and inexplicable, i.e. for the April Fools everywhere:

Consider the case where our intrepid explorers are traveling at the speed of light with the "Omega Propulsion System" (w). Suddenly the system shuts off, such that both w and f go to zero mathematically, the Equation 10 goes the other direction. In this case, the traditional Lorentz transformation shifts into gear with v = c, and lo and behold the mass of the explorers' spaceship approaches infinity!

In addition to being a serious weight gain problem, the spaceship and its contents would now attract all of the mass in the universe into what would have to be considered as the mother of all black holes. This implies total universal collapse -- just because some mechanic forgot to tighten a bolt and the vaunted Omega Propulsion System shut down at the worse possible moment. The real bummer, of course, is that while the mechanic would initially be drawn to the spaceship -- along with the rest of the universe -- it is unlikely he or she would be able to rectify the problem.

The next step would be that the universe's mass collapsing would then encounter the nuclear counterforces of entirely too much mass in one place, and would then explode in a counter effect. And whoever arrived late for the party would see a Big Bang.

Which makes one wonder if perhaps the universe as we know it is simply the failure of a propulsion unit of a bygone era.

In an equally weird alternative, we might assume that the Omega Propulsion System was simply allowing the spacecraft to exceed the speed of light when it provides a reason for the ultimate manufacturer's recall. In this case, the mass does not approach infinity, but instead merely becomes imaginary (in the mathematical sense). In this case, the "you're outta here" is one of an imaginary or illusory nature. But then the universe is probably all one gigantic illusion anyway, so it's more like homeweek.

Aren't mathematical results just wonderful?

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