**Stephen Paul King** (*stephenk1@home.com*)

*Thu, 25 Mar 1999 18:42:02 GMT*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 57] Re: [time 52] Orientation of time"**Previous message:**Stephen Paul King: "[time 55] Re: Euclidean vs Minkowskian"

On 23 Mar 1999 20:39:59 GMT, ranjitkuma@aol.com (RanjitKuma) wrote:

*>Response to comments on
*

*>Special Relativity as a Consequence of a-priori Axioms
*

*>Posted on Science.Physics.Research after/around December 24, 1998
*

*>
*

*>
*

*> (Please let me know if you wish to see the original paper (12 pages) again -
*

*> A. B. Ranjit Kumar, March 21, 1999 )
*

*>
*

*> The following response is about 9 pages long.
*

*>--------------------------------------------------------------------------
*

*>-------------------------------------
*

*>I want to thank the members and the moderators of this group for the
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*>opportunity to air my ideas on
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*>the axiomatic basis of Special Relativity. I am thankful to the few
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*>individuals that have reviewed them, and responded through the
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*>group and to my personal e-mail address. They were very thoughtful
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*>and considerate. My responses to their comments are given below. I
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*>have rearranged their comments into the following issues to minimize
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*>repetitions:
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*>
*

*>----------------------------------------------------------------------------
*

*>Comment (1) Toby Bartels, Kevin Brown:
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*>
*

*>>"The presented development allows both Special Relativity and
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*>>Galilean Relativity:
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*>
*

*>Response: I agree. In Lemma 4, I should have said "This can happen
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*>only if "a" is greater than or equal to zero". The first paragraph in the
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*>section on "Potential Implications" in the paper discusses this to
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*>some extent, and points out that an experiment is needed to
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*>determine if the constant c is finite or infinite and if any physical entity
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*>can travel at that speed. The purpose of this paper is to point out that
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*>the logical possibility of special relativity "could have been foreseen"
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*>by Newton if he started from my axioms (i.e., without any reference to
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*>an experiment). Of course I am saying it with the benefit of hindsight.
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*>
*

*>----------------------------------------------------------------------------
*

*>Comment (2) Toby Bartels:
*

*>
*

*>> "I see some flaws..."
*

*>
*

*>Response: I am sincerely thankful for the time that Toby already
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*>spent on my paper and hope that he can find additional time to point
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*>out the other "flaws" so that I can clean it up.
*

*>
*

*>------------------------------------------------------------------
*

*>Comment (3) Frank Wappler:
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*>
*

*>> "I read your (delayed) post in sci.physics.research with interest.
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*>> Unfortunately I seem to disagree with your entire approach.
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*>> Einstein's calibration procedure and the associated
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*>> distance definition give already some axiomatic basis to SR
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*>> (even without any specification for the notion "the middle
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*>> between" a pair of observers)".
*

*>
*

*>I agree that Einstein's original presentation had axiomatic basis, but
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*>he explicitly stated that he was using the independence of the
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*>velocity of light from its source as an axiom (he called it a postulate).
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*>And then he invoked measurements (perhaps axiomatically) using
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*>light signals in thought experiments. The essence of my paper is to
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*>separate logic (i.e. mathematics) and experiment (i.e., physics) and
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*>thus point out that humans had the potential to conceive special
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*>relativity without the benefit of experiment and then perform the
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*>experiment (i.e., physics) merely to determine the value of the
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*>constant. This is in contrast to the feeling of many (around the year
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*>1905), that special relativity was contrary to reason and humanity
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*>was forced to accept it by the "tyranny" of experimental results.
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*>
*

*>--------------------------------------------------------------------------
*

*>-------
*

*>Comment (4) Frank Wappler:
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*>
*

*>>"Beyond that, I prefer to derive measurement procedures
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*>>between observer_pairs_ from (only) the axiom that "an
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*>>observer can order own states". (see for instance
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*>>http://x4.dejanews.com/getdoc.xp?AN=421399809.1&CONT>
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*>>EXT=914976815.143327448&hitnum=0)."
*

*>
*

*>Response: In the context of this paper, "measurement procedure" is
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*>a dirty word that must be avoided. See my response to Sebastien
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*>Kruk below. However, I tried to read the above referenced
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*>communication. This was my first time at dejanews and was
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*>overcome by the breadth and the depth of info available at the site. I
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*>shall be very interested in reading a "paper" (even in a draft form) on
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*>the approach you favor. Based on the little info I gleaned, I think I
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*>like the approach. Although, each observer can order his own states
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*>(I assume that here the word "states" includes all his measurements
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*>about his own neighborhood as well as other entities in the universe),
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*>there must be some relationship with the observations of others to
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*>facilitate meaningful communication between various observers.
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*>Axiom 7 of my paper requires a particular relationship between the
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*>orders observed by different observers. One can define and measure
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*>relative velocities between pairs of observers and develop "empirical"
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*>procedures (including thought experiments) to identify the formula for
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*>the composition of velocities. It is perfectly valid physics and is the
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*>approach used by Einstein's original presentation of Special
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*>Relativity. However, the purpose of my paper was to push
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*>Mathematics as far as it can go towards the formula for the
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*>composition and minimize the use of Physics. I believe that
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*>consideration of at least three observers is required for this purpose.
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*>In fact the argument used in Lemma 9 of the paper requires at least a
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*>fourth entity.
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*>
*

*>----------------------------------------------------------------------------
*

*>Comment (5) Frank Wappler:
*

*>
*

*>> However, (after having spent some time trying :) I also have a
*

*>> direct comment on your derivations. AFAIU, the seven axioms
*

*>> that you stipulate are for instance also satisfied by
*

*>
*

*>> 0 = f( x, y, z ) = p^x + p^y + p^z - p^x p^y - p^x p^z - p^y p^z
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*>> where p>0... and uniqueness:
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*>> z = log_p_( (p^x + p^y - p^x p^y)/( p^x + p^y - 1) )
*

*>
*

*>Response: I like to complement Frank on this intriguing counter
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*>example. I give a four part response to this.
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*>
*

*>Part 1: This expression satisfies all my axioms only for a finite set of
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*>real values of x and y for which a unique (real value of) z exists. As
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*>pointed out by Sebastien, consider the case when x=y. then
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*>
*

*>z = log_p_( (2(p^x) + - (p^x)^2 )/( 2(p^x) - 1) )
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*>
*

*>and has a real value only for a finite set of values of x. Sebastien
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*>concluded that when p^x <= 0.5 or p^x >= 2, z (i.e. log_p_(a negative
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*>real number) does not exist and thus the given expression violates
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*>Axiom 2. However, Frank countered that in such cases z exists
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*>although it will be imaginary. But then z would have the form
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*>e^((2n+1)iPI/2) where r is a real number and n is any integer.
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*>Therefore z would not be unique and thus the given expression
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*>violates Axiom 6.
*

*>
*

*>
*

*>Part 2: Some people may find the above argument tenuous, because
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*>the multiple imaginary solutions for z actually represent the same
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*>point in the complex plane. This response is directed to them. Here,
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*>we can show that Frank's expression violates the Axiom 7 (order).
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*>The proof follows the same steps as that of Lemma 9 of the paper.
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*>
*

*>z = log_p_( (p^x + p^y - p^x p^y)/( p^x + p^y - 1) )
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*>
*

*>For a given y (i.e. considering y as a constant), we evaluate the
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*>derivative of z with respect to x as
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*>
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*>p^x (p^y - p^(2y) -1) / ((p^x + p^y - 1)(p^x + p^y - p^x p^y))
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*>
*

*>Then to obey the axiom 7 (order) the above expression must be
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*>positive (or negative) semi-definite for any given real value of y. We
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*>notice that the numerator is always positive for real values of x.
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*>Therefore, the denominator must be positive (or negative) semi-
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*>definite. The following two cases exist:
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*>
*

*>Case (i): y>0- In this case, the first factor (p^x + p^y - 1) is positive,
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*>but the second factor changes sign, with
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*>p^x = p^y/(p^y-1) at the boundary resulting in the violation of Axiom
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*>7.
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*>
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*>Case (ii): y<0- In this case, the second factor (p^x + p^y - p^x p^y) is
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*>negative, but the first factor changes sign, with p^x = 1- p^y at the
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*>boundary resulting in the violation of Axiom 7.
*

*>
*

*>Notice that the values of x that cause problems with Axiom 7 are
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*>exactly the same as those which cause problems with Axiom 6 in
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*>Part 1 of the response above. Basically, Frank's expression satisfies
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*>all my axioms (as originally presented) in the region bounded by the
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*>line X + Y = 1, and the hyperbola XY - X - Y = 0, and the quadrant
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*>X>0 and Y>0 where X=p^x and Y= p^y. It also allows for the
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*>possibility of unbounded relative velocity. See Part 4 below for
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*>modifications to the axioms.
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*>
*

*>Part 3: Not withstanding the proof of part (2) above, it would still be
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*>interesting to consider such "counter examples". If such examples
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*>exist, on the negative side, they would indicate a mathematical flaw
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*>in my paper; but on the positive side, they indicate potential
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*>alternatives to Special Relativity which needs to be tested by
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*>experiment. I would be delighted (although I do not expect) to see the
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*>latter situation, because that would very much establish the value of
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*>my approach.
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*>
*

*>Part 4: I propose to modify the axioms 2, 4 and 6 to make my implicit
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*>assumptions explicit as shown below. Then Frank's counter example
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*>would be excluded because of the following reason: Close to the
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*>boundary of the finite values of x and y for which a real z exists, z
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*>would approach infinity. If z is allowed infinite values, then to
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*>preserve symmetry, we should allow infinite values for x and y. This
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*>on the other hand precludes the possibility of finite bounds on the
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*>domain of x and y.
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*>
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*>Axiom 2: .., where x, y, and z are real numbers.
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*>
*

*>Axiom 4: .., where the domains of x, y, and z are identical and
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*>convex.
*

*>
*

*>Note: If the domains are different, then we lose observer symmetry.
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*>Whether the domain is finite or infinite would not be addressed until
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*>the last Lemma (Lemma 9).
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*>
*

*>Axiom 6: Given any arbitrary values (within their domain) for any two
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*>variables ....
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*>
*

*>Note: Here the keyword is "arbitrary".
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*>
*

*>
*

*>----------------------------------------------------------------
*

*>Comment (6) Sebastien Kruk:
*

*>
*

*>> "Axiom 3 (mirror symmetry) is not needed for obtaining your
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*>> result. ...
*

*>> Let us consider a degenerate case for which x=y=0. In this case
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*>> all objects are moving at the same velocity and z=0. This comes
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*>> from the definition of relative velocity."
*

*>
*

*>Response: I used the above definition and the proof suggested by
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*>Sebastien in an earlier version of the paper. I was reluctant to use it
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*>and struggled for a while until Axiom 3 occurred to me. The reason
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*>for my reluctance was that I wanted to stay away from physics (and
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*>thus from measurement procedures) as far as possible. Notice that I
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*>did not define either relative velocity or any methods of measuring
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*>relative velocity. If I talk about a definition, then somehow it would
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*>imply methods of measurements (since physics deals with only
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*>measurable quantities, in a broad interpretation of the word
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*>measurable). I considered Axiom 3 as a much weaker proposition
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*>from the physical point of view. Even so, I was reluctant to use Axiom
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*>3, and took a cowardly escape out of my hang-up by using the
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*>phrase "a conceptual mirror and not a physical mirror". In my
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*>opinion, even Axiom 3 is much closer to "physical" than I would like it
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*>to be.
*

*>
*

*>An important aspect of my approach that did not come out loud and
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*>clear is that if you replace "relative velocity" with some meaningless
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*>phrase such as "gobbledygook" throughout, the paper would still be
*

*>reasonably meaningful. This could not have happened if I used a
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*>"definition" of relative velocity. As a consequence, I could venture to
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*>suggest (see the paper, in the section on Potential Implications) the
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*>possibility of finite bounds on distance, acceleration and potential
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*>difference in a static electric field and any other "gobbledygook" that
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*>satisfies the seven axioms.
*

*>
*

*>The "physical" nature of Axiom 3 is apparent from the following fact:
*

*>To suggest the possibility of a finite bound on potential difference, I
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*>had to invoke the "symmetry with respect to reversal of charge" in
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*>stead of the "mirror symmetry" used for velocity, distance and
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*>acceleration.
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*>
*

*>However, I could possibly combine some of the statements used by
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*>Sebastian as an axiom, and use it to replace Axiom 3. I do not know
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*>if the new axiom would be simpler than Axiom 3 or not, and what
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*>implications it would have for applications to "gobbledygooks".
*

*>Anyhow, even then the total number of axioms would still be seven.
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*>
*

*>--------------------------------------------------------------------
*

*>Comment (7) Kevin Brown:
*

*>
*

*>>Your presentation doesn't differ from the classical approach of
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*>>Whittaker et al, which stalls at this crucial point, i.e., just short
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*>>of identifying any underlying principle or symmetry that would
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*>>discriminate between what Minkowski called G_c and
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*>>G_infinity, the Lorentzian and Galilean transformation groups.
*

*>
*

*>Response: By the "approach of Whittaker", do you mean the
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*>approach in Reference 3 of the paper? If so, my approach is
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*>definitely different. Whittaker used "constancy of the speed of light"
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*>as an axiom, and I do not. The crucial idea advanced by my paper is
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*>the use of a third observer and the associated logical requirements. I
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*>was not aware of that reference until I finished a version of my paper
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*>and went to the library to select appropriate references. If you are
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*>referring to some other works of Whittaker, please provide a list of
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*>them. I shall be eager to read them.
*

*>
*

*>
*

*>--------------------------------------------------------------------
*

*>Comment (8) Kevin Brown:
*

*>
*

*>>Since the non-negativeness of 1/c doesn't imply that it is
*

*>>positive, this leaves us (along with all the others who have
*

*>>tried their hand at rationally deducing special relativity, from
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*>>1905 to the present day) reluctantly concluding that something
*

*>>like Einstein's empirically-justified second postulate is required
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*>>to distinguish between the two seemingly viable candidates for
*

*>>the metric of space-time. (See Lucas and Hodgson's "Spacetime
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*>>and Electro- magnetism" for a particularly thorough review of
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*>>the extensive literature on this topic.)
*

*>
*

*>Response: I agree with you here. Some physical fact is required to
*

*>break the tie between the two choices. Michelson-Morley experiment
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*>provides such a fact. There can be other facts just as good. I am
*

*>polishing a preliminary version of another paper titled "Special
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*>Relativity as a Consequence of Uncertainty" which presents an
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*>alternative to that experiment.
*

*>
*

*>--------------------------------------------------------------------
*

*>Comment (9) Kevin Brown:
*

*>
*

*>>To emphasize the 2+1 symmetry of the composition rule, it's
*

*>>helpful to express it in the form w = f(u,v), where u,v are the
*

*>>commensurate primary speeds , and w is the secondary speed to
*

*>>be inferred. By the usual arguments (uniqueness, invertibility,
*

*>>etc) we can infer that f must be a linear fractional form, and
*

*>>then we impose the symmetries
*

*>>
*

*>>f(u,-u) = 0 f(0,u) = f(u,0) = -u f(u,v) = f(1/u,1/v)
*

*>>
*

*>>The first two symmetries are probably familiar, but it's the
*

*>>third that rules out Galilean relativity and uniquely selects
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*>>the composition rule f(u,v) = (u+v)/(1+uv) for some fixed units,
*

*>>which yields the complete Minkowski structure of spacetime
*

*>>(although one further ambiguity is encountered a few steps
*

*>>later, leading to some fascinating alternatives that actually
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*>>have physical applicability, but I digress...)
*

*>>
*

*>> ( Some very interesting discussion here - snipped by Ranjit)
*

*>>
*

*>>
*

*>>Unfortunately, to really convey any sense of this "deduction" of
*

*>>Lorentz covariance, and for the symmetry f(u,v) = f(1/u,1/v) not
*

*>>to be seen as ad hoc, it's necessary to take a longer running
*

*>>start, because it relies on a severely abstract view of physical
*

*>>phenomena, perception, and measurement. In addition, to
*

*>>convey the heuristic value of this interpretation (not only to
*

*>>special and general relativity but also to quantum mechanics)
*

*>>would take a whole book.
*

*>
*

*>Response: I am very much intrigued by the ideas discussed. Any
*

*>further information or references would be greatly appreciated.
*

*>
*

*>
*

*>Sincerely,
*

*>Ranjit, March 21, 1999.
*

*>
*

**Next message:**Stephen P. King: "[time 57] Re: [time 52] Orientation of time"**Previous message:**Stephen Paul King: "[time 55] Re: Euclidean vs Minkowskian"

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