Stephen Paul King (stephenk1@home.com)
Thu, 25 Mar 1999 18:42:02 GMT
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
>opportunity to air my ideas on
>the axiomatic basis of Special Relativity. I am thankful to the few
>individuals that have reviewed them, and responded through the
>group and to my personal e-mail address. They were very thoughtful
>and considerate. My responses to their comments are given below. I
>have rearranged their comments into the following issues to minimize
>repetitions:
>
>----------------------------------------------------------------------------
>Comment (1) Toby Bartels, Kevin Brown:
>
>>"The presented development allows both Special Relativity and
>>Galilean Relativity:
>
>Response: I agree. In Lemma 4, I should have said "This can happen
>only if "a" is greater than or equal to zero". The first paragraph in the
>section on "Potential Implications" in the paper discusses this to
>some extent, and points out that an experiment is needed to
>determine if the constant c is finite or infinite and if any physical entity
>can travel at that speed. The purpose of this paper is to point out that
>the logical possibility of special relativity "could have been foreseen"
>by Newton if he started from my axioms (i.e., without any reference to
>an experiment). Of course I am saying it with the benefit of hindsight.
>
>----------------------------------------------------------------------------
>Comment (2) Toby Bartels:
>
>> "I see some flaws..."
>
>Response: I am sincerely thankful for the time that Toby already
>spent on my paper and hope that he can find additional time to point
>out the other "flaws" so that I can clean it up.
>
>------------------------------------------------------------------
>Comment (3) Frank Wappler:
>
>> "I read your (delayed) post in sci.physics.research with interest.
>> Unfortunately I seem to disagree with your entire approach.
>> Einstein's calibration procedure and the associated
>> distance definition give already some axiomatic basis to SR
>> (even without any specification for the notion "the middle
>> between" a pair of observers)".
>
>I agree that Einstein's original presentation had axiomatic basis, but
>he explicitly stated that he was using the independence of the
>velocity of light from its source as an axiom (he called it a postulate).
>And then he invoked measurements (perhaps axiomatically) using
>light signals in thought experiments. The essence of my paper is to
>separate logic (i.e. mathematics) and experiment (i.e., physics) and
>thus point out that humans had the potential to conceive special
>relativity without the benefit of experiment and then perform the
>experiment (i.e., physics) merely to determine the value of the
>constant. This is in contrast to the feeling of many (around the year
>1905), that special relativity was contrary to reason and humanity
>was forced to accept it by the "tyranny" of experimental results.
>
>--------------------------------------------------------------------------
>-------
>Comment (4) Frank Wappler:
>
>>"Beyond that, I prefer to derive measurement procedures
>>between observer_pairs_ from (only) the axiom that "an
>>observer can order own states". (see for instance
>>http://x4.dejanews.com/getdoc.xp?AN=421399809.1&CONT>
>>EXT=914976815.143327448&hitnum=0)."
>
>Response: In the context of this paper, "measurement procedure" is
>a dirty word that must be avoided. See my response to Sebastien
>Kruk below. However, I tried to read the above referenced
>communication. This was my first time at dejanews and was
>overcome by the breadth and the depth of info available at the site. I
>shall be very interested in reading a "paper" (even in a draft form) on
>the approach you favor. Based on the little info I gleaned, I think I
>like the approach. Although, each observer can order his own states
>(I assume that here the word "states" includes all his measurements
>about his own neighborhood as well as other entities in the universe),
>there must be some relationship with the observations of others to
>facilitate meaningful communication between various observers.
>Axiom 7 of my paper requires a particular relationship between the
>orders observed by different observers. One can define and measure
>relative velocities between pairs of observers and develop "empirical"
>procedures (including thought experiments) to identify the formula for
>the composition of velocities. It is perfectly valid physics and is the
>approach used by Einstein's original presentation of Special
>Relativity. However, the purpose of my paper was to push
>Mathematics as far as it can go towards the formula for the
>composition and minimize the use of Physics. I believe that
>consideration of at least three observers is required for this purpose.
>In fact the argument used in Lemma 9 of the paper requires at least a
>fourth entity.
>
>----------------------------------------------------------------------------
>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
>> where p>0... and uniqueness:
>> 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
>example. I give a four part response to this.
>
>Part 1: This expression satisfies all my axioms only for a finite set of
>real values of x and y for which a unique (real value of) z exists. As
>pointed out by Sebastien, consider the case when x=y. then
>
>z = log_p_( (2(p^x) + - (p^x)^2 )/( 2(p^x) - 1) )
>
>and has a real value only for a finite set of values of x. Sebastien
>concluded that when p^x <= 0.5 or p^x >= 2, z (i.e. log_p_(a negative
>real number) does not exist and thus the given expression violates
>Axiom 2. However, Frank countered that in such cases z exists
>although it will be imaginary. But then z would have the form
>e^((2n+1)iPI/2) where r is a real number and n is any integer.
>Therefore z would not be unique and thus the given expression
>violates Axiom 6.
>
>
>Part 2: Some people may find the above argument tenuous, because
>the multiple imaginary solutions for z actually represent the same
>point in the complex plane. This response is directed to them. Here,
>we can show that Frank's expression violates the Axiom 7 (order).
>The proof follows the same steps as that of Lemma 9 of the paper.
>
>z = log_p_( (p^x + p^y - p^x p^y)/( p^x + p^y - 1) )
>
>For a given y (i.e. considering y as a constant), we evaluate the
>derivative of z with respect to x as
>
>p^x (p^y - p^(2y) -1) / ((p^x + p^y - 1)(p^x + p^y - p^x p^y))
>
>Then to obey the axiom 7 (order) the above expression must be
>positive (or negative) semi-definite for any given real value of y. We
>notice that the numerator is always positive for real values of x.
>Therefore, the denominator must be positive (or negative) semi-
>definite. The following two cases exist:
>
>Case (i): y>0- In this case, the first factor (p^x + p^y - 1) is positive,
>but the second factor changes sign, with
>p^x = p^y/(p^y-1) at the boundary resulting in the violation of Axiom
>7.
>
>Case (ii): y<0- In this case, the second factor (p^x + p^y - p^x p^y) is
>negative, but the first factor changes sign, with p^x = 1- p^y at the
>boundary resulting in the violation of Axiom 7.
>
>Notice that the values of x that cause problems with Axiom 7 are
>exactly the same as those which cause problems with Axiom 6 in
>Part 1 of the response above. Basically, Frank's expression satisfies
>all my axioms (as originally presented) in the region bounded by the
>line X + Y = 1, and the hyperbola XY - X - Y = 0, and the quadrant
>X>0 and Y>0 where X=p^x and Y= p^y. It also allows for the
>possibility of unbounded relative velocity. See Part 4 below for
>modifications to the axioms.
>
>Part 3: Not withstanding the proof of part (2) above, it would still be
>interesting to consider such "counter examples". If such examples
>exist, on the negative side, they would indicate a mathematical flaw
>in my paper; but on the positive side, they indicate potential
>alternatives to Special Relativity which needs to be tested by
>experiment. I would be delighted (although I do not expect) to see the
>latter situation, because that would very much establish the value of
>my approach.
>
>Part 4: I propose to modify the axioms 2, 4 and 6 to make my implicit
>assumptions explicit as shown below. Then Frank's counter example
>would be excluded because of the following reason: Close to the
>boundary of the finite values of x and y for which a real z exists, z
>would approach infinity. If z is allowed infinite values, then to
>preserve symmetry, we should allow infinite values for x and y. This
>on the other hand precludes the possibility of finite bounds on the
>domain of x and y.
>
>Axiom 2: .., where x, y, and z are real numbers.
>
>Axiom 4: .., where the domains of x, y, and z are identical and
>convex.
>
>Note: If the domains are different, then we lose observer symmetry.
>Whether the domain is finite or infinite would not be addressed until
>the last Lemma (Lemma 9).
>
>Axiom 6: Given any arbitrary values (within their domain) for any two
>variables ....
>
>Note: Here the keyword is "arbitrary".
>
>
>----------------------------------------------------------------
>Comment (6) Sebastien Kruk:
>
>> "Axiom 3 (mirror symmetry) is not needed for obtaining your
>> result. ...
>> Let us consider a degenerate case for which x=y=0. In this case
>> all objects are moving at the same velocity and z=0. This comes
>> from the definition of relative velocity."
>
>Response: I used the above definition and the proof suggested by
>Sebastien in an earlier version of the paper. I was reluctant to use it
>and struggled for a while until Axiom 3 occurred to me. The reason
>for my reluctance was that I wanted to stay away from physics (and
>thus from measurement procedures) as far as possible. Notice that I
>did not define either relative velocity or any methods of measuring
>relative velocity. If I talk about a definition, then somehow it would
>imply methods of measurements (since physics deals with only
>measurable quantities, in a broad interpretation of the word
>measurable). I considered Axiom 3 as a much weaker proposition
>from the physical point of view. Even so, I was reluctant to use Axiom
>3, and took a cowardly escape out of my hang-up by using the
>phrase "a conceptual mirror and not a physical mirror". In my
>opinion, even Axiom 3 is much closer to "physical" than I would like it
>to be.
>
>An important aspect of my approach that did not come out loud and
>clear is that if you replace "relative velocity" with some meaningless
>phrase such as "gobbledygook" throughout, the paper would still be
>reasonably meaningful. This could not have happened if I used a
>"definition" of relative velocity. As a consequence, I could venture to
>suggest (see the paper, in the section on Potential Implications) the
>possibility of finite bounds on distance, acceleration and potential
>difference in a static electric field and any other "gobbledygook" that
>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
>had to invoke the "symmetry with respect to reversal of charge" in
>stead of the "mirror symmetry" used for velocity, distance and
>acceleration.
>
>However, I could possibly combine some of the statements used by
>Sebastian as an axiom, and use it to replace Axiom 3. I do not know
>if the new axiom would be simpler than Axiom 3 or not, and what
>implications it would have for applications to "gobbledygooks".
>Anyhow, even then the total number of axioms would still be seven.
>
>--------------------------------------------------------------------
>Comment (7) Kevin Brown:
>
>>Your presentation doesn't differ from the classical approach of
>>Whittaker et al, which stalls at this crucial point, i.e., just short
>>of identifying any underlying principle or symmetry that would
>>discriminate between what Minkowski called G_c and
>>G_infinity, the Lorentzian and Galilean transformation groups.
>
>Response: By the "approach of Whittaker", do you mean the
>approach in Reference 3 of the paper? If so, my approach is
>definitely different. Whittaker used "constancy of the speed of light"
>as an axiom, and I do not. The crucial idea advanced by my paper is
>the use of a third observer and the associated logical requirements. I
>was not aware of that reference until I finished a version of my paper
>and went to the library to select appropriate references. If you are
>referring to some other works of Whittaker, please provide a list of
>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
>>1905 to the present day) reluctantly concluding that something
>>like Einstein's empirically-justified second postulate is required
>>to distinguish between the two seemingly viable candidates for
>>the metric of space-time. (See Lucas and Hodgson's "Spacetime
>>and Electro- magnetism" for a particularly thorough review of
>>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
>provides such a fact. There can be other facts just as good. I am
>polishing a preliminary version of another paper titled "Special
>Relativity as a Consequence of Uncertainty" which presents an
>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
>>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
>>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.
>
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