Lösningar/kommentarer till inlämningsuppgift 3, Mekanik B 1998

...Galilean Simplification

Frames in Relative Movement

by Ricardo L. Carezani

Let us examine two parallel frames of coordinates x, y, z, t and x', y', z', t'. Frame F is moving with relative velocity v with respect to frame F '. Using the Galilean transformation, the abscissa x of point P in F is given in the above figure by

x' = x + vt     y' = y    z' = z     t' = t   (1)
The velocity of P observed by frame F' is 


v' = v   (2) 


Introducing a third frame of reference F '' with respect to which frame
F is moving with relative velocity v1, the abscissa of P in F '' is



x'' = x' + vt

x'' = x + v t + v1 t = x + (v + v1) t   (3) 

We are using three coordinate reference frames, F, F ', and F '', and we
shall try to demonstrate that two of them are useless, inoperative, and
superfluous by doing the following analysis.



First: It is impossible to measure the abscissa of P in F from F ', 
independently
of the relative velocity v, supposing that P is moving with velocity -v
with respect to F, because P will be at rest with respect to F', and, in
this particular case, F' is useless because it cannot observe a phenomenon
at P.





Second: We can also suppose that P is moving within F with relative 
velocity
vx, or with a relative velocity u with respect to F', because no one in
F ' could distinguish the velocity vx from v, even though F' receives 
signals
from P and they are transmitted to F', the observer in F' will measure the
relative velocity u, because if the frame F first transmits v and then 
v+vx,
the observer in F ' will only think that the relative velocity v increases
until u. It would be easier for the observer in F ' to receive directly
the signals from P. So, the reference frame F is unnecessary.



It is formally possible to demonstrate that either the frame F '' or F is
useless and inoperative, as follows.



The velocity of the point P moving in F with velocity vx, measured from
F' is


v' = vx + v  (4)

x being the coordinate of P in F when t=0, its coordinate at a later time
t will be


x1 = x + vx t   (5)

Its coordinate in F' is



x' = x1 + v t

x' = x + vx t + v t = x + (vx + v) t   (6)

The equations (6) and (3) are totally equivalent. Is it not the same to
say that P moves with respect to F with relative velocity vx, giving 
equation
(6), as to say that P moves with respect to F' with relative velocity v,
giving equation (3)?



But equation (6) results only in applying "two systems of 
reference",
F and F'; F'' then is not operative, and is useless. Equation (3) is 
obtained
by applying "three systems of reference" but the point P is at
rest in F, v' = v, and consequently frame F is useless, and 
inoperative.




If point P is moving together with F, and observer in F will never observe
a phenomenon at P; that is to say, it is impossible to make the 
auto-observation
of point P.



Even so, we may insist upon the following: Can velocity vx and v be 
distinguished
from each other while we are observing the physical phenomenon. No. In each
phenomenon there exists the phenomenon itself and the observer. The 
relativity
of space and time is given by light speed, which is constant or, at least,
has a finite value. Can we divide space in such a manner as to have an 
arbitrary
quantity of reference frames?. A physical phenomenon naturally presupposes
two objects: the observer and the object under observation. If the process
is relative - and really it is - it should be to both, always due to their
mutual interaction.



Consequently we shall define as a "physical; system" one that
consists of a moving point P (the observed) and a "reference frame
of coordinates" (the observer). A brief, formal and complete 
exposition
is the following (1).



Let us consider three parallel frames F, F' F'', their origins coincident
at instant t=0, and point P of abscissa x on F. If F moves relative to F'
with velocity v and F' in respect to F'' with velocity v1, the coordinate
of P on F'' is


x'' = x' + v1 t    x'' = x + (v1 + v) t    (7)

During this exposition we have presumed a relation of P with F even though
P is at rest with respect to F. This, in our point of view, is not logical
due to the fact that self observation of a point is absolutely impossible.
In other words, with point P and a geometrical system F linked to it, we
have no physics whatsoever. We need another system, one not linked to P,
(and which we may individualize by its origin) in which we place an 
observer
describing the physical alternatives which P is undergoing. In this case,
frame F is superfluous.



Equation (7) maintains its value if we suppose that at time t=0, the point
passes the origin of F''. To make it easier, we shall run up the primed
values in the following manner: F' shall be F, and F'' will be F', v is
the velocity of P with respect to F, v1 the velocity of F with respect to
F' therefore (7) shall read"


x' = x + v1 t     x' = (v1 + v) t   (8)

which is equal to (7) when we only consider two systems. The preceding 
analysis
of the systems in relative movement leads to the following definition: a
physical system consists of two points in relative movement, P and F or
the origin of F. Up until now this has been known as a system F on one 
side,
and point P at rest on the other. This new concept or conclusion will be
used to analyze what happens in special relativity.












...FRAMES: 
Derivation
2






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by Ricardo Carezani


As with many powerful yet elegant theories, the derivation is often the
most powerful argument for the theory itself. Such is the case with 
Autodynamics.
Although it is impossible to discuss the background necessary to adequately
set up the AD derivation in this small article, the math is quite simple
and elegant.



Discussing in detail the Galilean coordinate transformation principle and
comparing it with the Lorentz-Einstein transformation of frames in relative
motion, the author of AD demonstrated that it is possible to simplify the
Lorentz equations by recognizing that one of the two coordinate systems
used by Lorentz & Einstein were superfluous. Not only was one coordinate
system superfluous, but the relationship between the Lorentz coordinate
systems (one for the object and one for the observer) was arbitrarily fixed
by Einstein by setting their relative velocities equal in the acceleration
derivation and introducing a problem in the velocity sum equation.






SR: 2 coordinate systems



Below is the universal system of frames for relative movement.



(place figure here)



Figure 1 System of coordinates



We will first find the velocity v' of an object with respect to the 
observer
using the Lorentz equations. But it is important to note that the object
is at rest with respect to its coordinate system F, and in motion with 
respect
to the observer in F ' (this becomes very important further on).



The Lorentz equations are: 

 (1)

(where , v = particle velocity, and c = light velocity)




SR Velocity





The Lorentz equations in differential form, are 

(2)


In order to find the velocity v' we need to divide by : 

(3)

This equation was obtained working with two frames in relative motion and
with a point P at rest in frame F (see above figure). Yet equation (3) has
3 velocities even though it must only have 2.




What is remarkable, is that the velocity vx appears spontaneously through
the mathematical operation of the derivative from the coordinate x in 
equation
(1) to the derivative dx/dt in equation (2). 
This innocent operation has
no physical meaning. There is no energy to move point P (originally at rest
in frame F) with respect to the F coordinate system. This is the reason
why the SR sum velocities equation fails to maintain momentum and energy
conservation when the equation is applied to a dynamic physical phenomenon
observed by observers in two relative frames in motion.







SR Force



Now that we have the velocity v' for the object with respect to the 
observer,
we now can derive the SR equation for force.



In frame F, force is 

(4)

In frame F ', force is 

(5)


We need to find the acceleration . Using the chain 
derivative,
we now need to take the derivative of v' to get acceleration using equation
(3). 

(6)

and given = vx. we obtain the acceleration: 

(7)


But in order to obtain the well known SR formula 

(8)

it is necessary to set vx = v.




Once again, this simplification has no physical sense. There is no reason
for setting these two velocities equal. 

To confound the matter, the 
exponent
3/2 in equation (8) does not end up matching the 1/2 in target Lorentz's
equation. In other words, even with the strange assumption that vx must
equal v, it still does not match the desired Lorentz exponent.




An even more strange phenomena in Einstein's derivation is the question
of why he didn't set the velocities equal to each other back in equation
(3). The problem is if he did, the result would be very strange indeed:
either 0, or . This emphasizes again the utter 
irrelevance
of setting the two velocities equal. In reality, there are an infinite 
number
of choices for v and vx (either taking on any value between 0 and c).







AD: 1 coordinate system



These problems never arise in Autodynamics because there is only one frame
of reference.




There are two ways to derive the AD equations for a system in relative 
motion.
One involves using Galilean transformations without using calculus thus
avoiding any spontaneous generation of velocity requiring physical energy.
The result leads to a simplification of the Lorentz time dilation equation
and the observers coordinate position. This however, is too long and 
involved
for this type of article.



The second way to derive the AD equations is shorter and more 
classical.



Returning to the figure for frames of reference, we see below that AD 
"collapses"
the two SR coordinate systems into one.





In SR, position and time in one coordinate system are a function of 
position
and time with respect to another coordinate system: 

(x',y',z',t') = f(x,y,z,t)(9)

Without the extra coordinate system, AD describes position and time for
one frame of reference as only a function of time: 

(x',y',z',t') = f(t)(10)


To remain consistent with the simultaneity problem of the relationship of
two observers viewing the same object, let us describe the coordinates of
both observers for AD: 




(11)

(12)


Because Einstein's equations had four variables x, y, z, & t, he had
to solve four equations simultaneously. AD on the other hand has only one
variable t, and thus has to solve only one equation.



Both AD and SR assume y & z and y' & z' are parallel respectively for both
observers and therefore equal: 

y' = yz' = z (13)

But unlike SR, the AD function is dependent on only one variable t, and
thus x' and t' must be related by one coefficient a: 

x' = avtt' = at(14)


Substituting (13) and (14) into (12) and solving for a: 


   (15)


(16)
(17)
(18)


Substituting a in equation (14), we get the AD equations: 


(19)


In AD, these are called the "Simplified Lorentz 
Equations".








AD Velocity



Continuing, we find the velocity of the observer in frame F' by taking the
derivative of equation (19), 


(20)


and calculating v' by dividing the above two equations: 

(21)

The result is that v' = v: the observer velocity IS the object velocity.
This is not surprising given that in AD, frame F is collapsed onto F' and
therefore the velocity of the object is equal to the velocity of frame F
with respect to F' before collapsing.







AD Force



Taking the derivative of equation (21) yields the acceleration: 

(22)

Given the classical definition of force 


(23)


we substitute the result of (22) into (23) and get 


(24)


Thus, the mass in motion must be 


(25)


This makes complete physical sense: when energy is expended, mass is
expended! This is one of the elegant mathematical descriptions of the AD
theory.








AD Kinetic Energy



First we start out with rest mass energy: 


(26)


Next, we describe energy of a particle that begins to move from a decay
process: 


(27)


Substituting equation (25) for mass and (26) for energy, we get 


(28)

(29)


The equations for kinetic energy (29) and mass (25) are coherent: when
the mass decreases, the kinetic energy increases and there is energy 
conservation.