A Synopsis of Relative Expansion

By Jack Martinelli, August 9, 1998

In most theoretical treatments, Physicists typically do not consider a frame’s intrinsic unit length in constructing their theories. Instead, quantities relating to length are expressed as meters or refer directly or indirectly to the speed of light (and time) as a reference length. Neither one is a direct representation of a frame’s natural unit length. Simply defining c=1 is a step in the right direction but it is not clear how this relates to physical structure in a fundamental way.

It is plausible that a fundamental understanding of a frame’s intrinsic or natural unit length could be useful. Defining this intrinsic length as an abstraction is easy... relating it to a meter -- a unit of material length – is not.

The following summarizes the development of length introduced in http://www.martinelli.org/fundamental to represent new expressions (isomorphs?) for length, velocity and acceleration as:

Length: sr

Velocity: sc

Acceleration: sa

The s represents the scale or linear density of a frame, r, represents measured length, c represents velocity and a is acceleration. In [ref] it is shown that:

sr 0 0

= sr

i i (1)

sc 0 0

= sc

i i (2)

sa 0 0

= sa

i i (3)

where the subscript denotes some frame number. s represents the scale of a frame. r represents a unit length, c an expansion/contraction velocity with respect to frame i and a the acceleration of the expansion.

We can use (1) and (2) to calculate an acceleration relationship expressed in (3).

s Write (1) as: s

1

= 0 r

0 r 1

and substitute it

into (2) to get:

s 0 c

0

= src

0 0 r 1

1

(4)

Which is constant.

Then take the derivative with respect to time to get:

a

1

= c

1

2

r 1

(5)

In other-words, the representation for the acceleration of an expanding frame is exactly equivalent to the acceleration of uniform circular motion – i.e., the simplest possible oscillation that we know of. But it is not oscillating... there is only the abstract equivalence.

Now if we plug this back into (3) we have:

s 0 a

0

= s 1 c

1

2

r 1

And from (4) we have:

s a

= src

0 0 r 1

2 1

(6)

When we treat the expansion of a frame via special relativity, we account for its length contraction. (note that (6) is constant.)

l

0 0

2

'=

lc

0

(6.1) c 0

+

H l

defines the relationship of arbitrary lengths in the primed and unprimed frames under relative expansion.

Our expansion velocity for some expansion constant H then becomes:

c H

2 2 2

=

lc

0

(7) c 0

+

H l

Note that as l approaches infinity that c

1 approaches c

0

1

2 2 2

. Also note that for this relatively large l that the space of the frame is spherically curved with respect to the frame that it is contracting with respect to. Then, using this result we can write (6) as:

2 sa

=

srH 0 c 0 0

c

0

2

+

H r

(8)

By inspection, you can see that this expression is not constant for all r

1

2 2

2 1

(as was the case in (6)), so we drop the subscripts from the sa term. If you plot this function you get:

Plot of sa over r shows a hump. (set r=0 &you get a maximum)

Note that the most significant feature of this field is that it is solitonic in form. I.e., a one hump wave.

Note that for Hr

1

>;;;>;;; c

0

that this becomes:

sa

= src

0 0 r 1

2

2 0

(9)

Which is the familiar form of the inverse square law (note the constant numerator and non- constant denominator).

Then integrating (9) over r we get the expression for work done by a contracting frame as:

E = −

sc 0

2

Then, since the acceleration of a contracting frame is abstractly equivalent to uniform circular motion (5), we can express radius r 1 = c

0

ω ... as if angular rotation were present. Then, substituting this into our field equation and substituting some ħ (...just a label for some constant) for src

0 0 0

we can calculate the second form of energy as: E = − hf , where h = 2πħ . This gives something that looks like that famous equation that Einstein found: hf =

sc 0

2 (Except that in Einstein’s equation he used mass instead of scale.). That is, the “energy” of a contracting frame has two equivalent forms. One form is due to relativistic expansion, the other is due to a mathematical equivalence to angular rotation. Expanding h we can write: 2

πsrc 0 0 0 f =

sc 0 2 canceling c’s and from (1) we have: 2

πs 0 r 0

= sc f . This can also be written as: s 0 λ 0 = s

1 λ 1

. In other words, the usual idea of wavelength is equivalent to a simpler idea: a frames’s natural unit of length. See also the derivation of Snell’s law in [ref]

So here’s the essence of the problem I’m facing

now. (6.1) describes how length is compressed. (9) expresses how “points” in a frame accelerate with respect to some origin. The question is: “how do you arbitrary represent motion of one frame with respect to another?”