ÿþ<!-- saved from url=(0022)http://internet.e-mail --> <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD><TITLE>relativiti</TITLE> <META http-equiv=Content-Type content="text/html; charset=unicode"> <META content="MSHTML 6.00.2800.1561" name=GENERATOR></HEAD> <BODY><div align=left><BR><IMG height=63 src="http://wbabin.net///gs3.bmp" width=206></div><B> <P align=center>Special relativity and the theory of four-vectors</P></B> <P><BR></P><B><U> <P>Title Page</P></B></U> <P>Title: Special relativity and the theory of four-vectors</P> <P>By: d. e. matthieu-laudat </P> <P>E-mail Address: delmatlau@hotmail.com <P>2000 Math Subj Clas No.: 83A05 </P> <P>PACS No.: 03.03.+p</P> <BR><I> Special relativity and the theory of four-vectors</I><U> <P>Key words and phrases</U>: spacetime, Minkowski four-vector, Lorentz transformation, relativity<BR> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; principle, basis vectors, inertial frame, cosh - at the risk of being tedious<BR> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; we emphasize that cosh<FONT face=symbol>q</FONT> (=<FONT face=symbol>g</FONT>) is merely a name that does not affect any<BR> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; analysis in any regard for the physical Lorentz factor <FONT face=symbol>g</FONT>, as is sinh<FONT face=symbol>q</FONT>=<FONT face=symbol>g</FONT>v/c.</P> <P><BR><B><U> <P>Abstract</P></B></U> <P>In this straightforward analysis of the most basic fundamentals of the foundations of special relativity,<BR> we look at the effect of the hyperbolic functions <FONT face=symbol>g</FONT>, <FONT face=symbol>g</FONT>v/c on the structure and consistency of the special<BR> relativity (Minkowski) four-vectors. At the outset, relativity declared nonexistent both Newton's<BR> absolute space and Maxwell's aether, and even Lorentz's ether, the first two of which were accepted<BR> euclidean spaces. The newly discovered M<sub>4</sub> relativity space was decreed a priori euclidean just so<BR> without for a moment questioning whether M<sub>4</sub> spacetime satisfied Euclid's postulates, most particularly<BR> the parallel postulate - thus orthogonality/linear independence. After all, the brand new (to math and<BR> physics) M<sub>4</sub> (standard rectangular) frame, or metric, was never exactly the (staple cartesian) frame,<BR> or metric, of the discarded spaces. As there is no flaw of logic, error of analysis, faulty premise, or<BR> incompleteness of argument in the following basic analysis of the fundamentals of the foundations of<BR> special relativity, it means there is a general anomaly inherent in the structure and consistency of the<BR> four-vectors, and their relativity principle basis. </P> <BR><B><U>Manuscript</B></U><BR> <H4 align=center>Special relativity and the theory of four-vectors</H4> <BR><BR><BR><U>I</U>: <U>Introduction:</U><BR>&nbsp; &nbsp; We investigate a few of the standard basic properties of special relativity. The analysis is rigorous<BR> and complete and uncovers new unsuspected results and fatal flaws in special relativity theory.<BR>&nbsp; &nbsp; The question that drives the analysis is: <B><I>precisely what is the full dot product expression of the<BR> four-vectors of Minkowski space</i></b>? For the classical vectors of <I>euclidean</I> space the unique expression<BR> of the full dot product is <B>x.y</B>=[<FONT face=symbol size=5>S</FONT><SUB>i</sub>x<sub>i</sub>y<sub>i</sub>]=|<B>x</B>||<B>y</B>|<U>cos</U><font face=symbol>f</font>, the <B>sum of components</B> expression of the dot product<BR> being <B>provably</B> equal to the latter <B>magnitudes of the vectors</B> expression of the dot product. <BR>&nbsp; &nbsp; In section II and III we <I>derive</i> the magnitudes of the vectors expression of the L(<B>v</B>) four-vector dot<BR> product, the derivation duly demonstrating its inherent dependence on the hyperbolic cosine function,<BR> (or, that is, the <i>physical</i> Lorentz factor, <font face=symbol>g</font>). From this it is then shown in section IV and V that the<BR> relativity four-vector is not rigorously well-defined, and investigate the role of the relativity principle.<BR> In section VI and VII we investigate the generality of the results and look at specific cases in special<BR> relativistic mechanics. <BR>&nbsp; &nbsp; In the analysis we take the Lorentz frame and law of its orthogonal transformation to exist and<BR> show that the associated M<SUB>4</SUB> four-vector cannot exist. In the paper <I>Orthogonal transformations and<BR>Heisenberg analysis of Dirac hamiltonian </I>we again show that, among others, it is the Lorentz inertial<BR>frame that does not exist as defined, let alone the M<SUB>4</SUB> four-vector. The conceptual flaws of relativity<BR>are beyond repair and question: to cite one inconsistency - everyone knows <B>F.U</B>=0 in relativistic<BR>mechanics. Everyone knows from vector analysis <B>F.U</B>=0=|<B>F</B>||<B>U</B>|cos(?). Everyone knows the angle (?)<BR>in M<SUB>4</SUB>, or the Lorentz transformation equations of the Lorentz frame, is imaginary angle i<FONT face=symbol>q</FONT>. Therefore<BR>cos(?)=cosi<FONT face=symbol>q</FONT>=cosh<FONT face=symbol>q</FONT>=cosi<FONT face=symbol>q</FONT>=cos(?), (by definition). Therefore, unfortunately, |<B>F</B>|=0, and that as shown<BR> in section VIIA means every single component of <B>F</B> must vanish. It must surely be impossible to dismiss<BR> the train of logic and analysis that derives <B>F.U</B>=0=|<B>F</B>||<B>U</B>|<U>cosh</u><font face=symbol>q</font> but accept the standard textbook instance,<BR> <B>U.V</B>=c<sup>2</sup><font face=symbol>g</font>(v)=c<sup>2</sup>cosh<font face=symbol>q</font>=c<sup>2</sup>cosi<font face=symbol>q</font>=<U>|<B>U</b>||<B>V</B>|cosh<font face=symbol>q</font></u>=|<B>U</b>||<B>V</B>|cosi<font face=symbol>q</font>=<B>U.V</b>, up to +/-c<sup>2</sup> convention. For, that cosh<font face=symbol>q</font><BR> expression of the M<sub>4</sub> dot product of <B>U</b>, <B>V</b> is likewise an L(<B>v</b>) deduction. Nevertheless, from <u>either</u>, or<BR> both, the irrefutable and irremediable fact that it is cosh(<font face=symbol>q</font>) and not cos(<font face=symbol>q</font>) that is the space properties<BR> defining function of Minkowski space quite evidently implies that <I>there is absolutely no instance of<BR> orthogonality/linear independence in Minkowski space - and thus too parallelicity in Minkowski<BR> space</I>. <b><I>Thus, Minkowski space is simply not a euclidean/flat space of any kind</i></b>. <BR>&nbsp; &nbsp; Therefore we reiterate that the analysis that uncovers relativity's inherent contradictions is that of the<BR> standard Lorentz inertial frame and Lorentz transformation, the Minkowski space and the Minkowski<BR> four-vector/metric standard conception and formulation of relativity from its relativity principle Einstein<BR> invariance of the interval relation. <BR>&nbsp; &nbsp; For whether it is the interval or the length of the vector that is preserved, since Klein a vector is its set<BR> of transformation equations and its transformation equations must be the coordinate transformation of<BR> some specific coordinate reference frame in some specific vector space. Thus, when a Minkowski space<BR> four-vector of any Einstein interval/length whatsoever is shown to be inconsistent then that's necessarily<BR> because its set of (Lorentz) transformation equations is necessarily inconsistent, in turn because the<BR> coordinate transformation equations of the (Lorentz inertial) frame it must transform like, and indeed the<BR> reference frame itself, must necessarily be inconsistent, which in turn means even the Minkowski space<BR> itself, spannable by the reference frame, must necessarily be inconsistent - and vice versa. <BR>&nbsp; &nbsp; The L(<B>v</B>) dot product is merely the simplest device with which to demonstrate these inconsistencies<BR> of relativity. <P><BR><BR><U>II</U>: <U>The Four-Vector Transformations:</U><BR>&nbsp; &nbsp; To show that cosh<font face=symbol>q</font>=<font face=symbol>g</font> is a fundamental parameter in special relativity analysis we utilize relativity's<BR> own precisely defined Lorentz transformation which defines the relativity (Minkowski) four-vectors,<BR> (c=1), <DIV align=center><BR><FONT face=symbol>D</FONT>x'=<FONT face=symbol>D</FONT>xcosh<FONT face=symbol>q</FONT>-<FONT face=symbol>D</FONT>tsinh<FONT face=symbol>q</FONT>, <FONT face=symbol>D</FONT>t'=-<FONT face=symbol>D</FONT>xsinh<FONT face=symbol>q</FONT>+<FONT face=symbol>D</FONT>tcosh<FONT face=symbol>q</FONT> &nbsp; &nbsp; (1) </DIV><BR>and<BR> <DIV align=center><FONT face=symbol>D</FONT>x=<FONT face=symbol>D</FONT>x'cosh<FONT face=symbol>q</FONT>+<FONT face=symbol>D</FONT>t'sinh<FONT face=symbol>q</FONT>, <FONT face=symbol>D</FONT>t=<FONT face=symbol>D</FONT>x'sinh<FONT face=symbol>q</FONT> +<FONT face=symbol>D</FONT>t'cosh<FONT face=symbol>q</FONT> &nbsp; &nbsp; (2) </DIV><BR>for relative motion along x, x'.<BR>&nbsp; &nbsp; Taking the coordinate increments as the components of four-vectors <FONT face=symbol>D</FONT><B>R</B>' and <FONT face=symbol>D</FONT><B>R</B>, whose <I>relative<BR>orientation in the same frame</I> (accepted of orthogonal transformations decades prior to and since)<BR>[1] pp 182, 232, is the velocity/rapidity parameter <FONT face=symbol>q</FONT>, it is straightforward to show from (1), (2) <DIV align=center><BR><FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B>=<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t=[(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)]<SUP>1/2</SUP>cosh<FONT face=symbol>q</FONT> &nbsp; &nbsp; (3).<BR></DIV><BR>&nbsp; &nbsp; Thus, for arbitrary <FONT face=symbol>q</FONT> orientation (or, from cosh<FONT face=symbol>q</FONT>=cosi<FONT face=symbol>q</FONT>, rotation i<FONT face=symbol>q</FONT>) in Minkowski space, M<SUB>4</SUB>, the<BR>dot/inner product <FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B> definition analogous to the transform relations (1) and (2) is, generally, <DIV align=center><BR><FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B>=<FONT face=symbol>D</FONT>R'<FONT face=symbol>D</FONT>Rcosh<FONT face=symbol>q</FONT>=[(<FONT face=symbol>D</FONT>r'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)(<FONT face=symbol>D</FONT>r<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)]<SUP>1/2</SUP>cosh<FONT face=symbol>q</FONT> &nbsp; &nbsp; (4), </DIV><BR> since <FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B>=<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>y'<FONT face=symbol>D</FONT>y+<FONT face=symbol>D</FONT>z'<FONT face=symbol>D</FONT>z-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t is universally accepted to validly exist for four-vectors. <I>Thus<BR> the absolute significance and importance of that</i> cosh<font face=symbol>q</font>=<font face=symbol>g</font> <I>factor of</i> (3,4) <I>to the four-vectors of<BR> special relativity simply cannot be overstressed</i> - <I>nor its reality as a fundamental fact of the<BR> foundation of four-vector relativity ever, ever be questioned. It is a <u>derived</u> fact of four-vector<BR> relativity for <U>all</u> intervals</i> <FONT face=symbol>D</FONT><B>R</B><sup>2</sup>&lt0, <FONT face=symbol>D</FONT><B>R</B><sup>2</sup>=0, <I>or</i> <FONT face=symbol>D</FONT><B>R</B><sup>2</sup>&gt0 <I>in</i> (1,2).<BR>&nbsp;&nbsp; Incidentally, since, analogously to (3), the orthogonal transformation equations (1), (2) hold for <FONT face=symbol>D</FONT><B>R</B>,<BR> <FONT face=symbol>D</FONT><B>R</B>' in the same reference frame by (the Klein) definition of a vector, we may also calculate (inferring<BR> we can in the manner of the euclidean space classical - real - vector) a Minkowski two-dimensional<BR> 'area' |<FONT face=symbol>D</FONT><B>R</B>x<FONT face=symbol>D</FONT><B>R</B>'| from (1) and (2). From (1) we find (<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>t-<FONT face=symbol>D</FONT>x<FONT face=symbol>D</FONT>t')=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)sinh<FONT face=symbol>q</FONT>=|<FONT face=symbol>D</FONT><B>R</B>|<SUP>2</SUP>sinh<FONT face=symbol>q</FONT>, and<BR> analogously from (2), (<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>t-<FONT face=symbol>D</FONT>x<FONT face=symbol>D</FONT>t')=(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)sinh<FONT face=symbol>q</FONT>=|<FONT face=symbol>D</FONT><B>R</B>'|<SUP>2</SUP>sinh<FONT face=symbol>q</FONT>. <BR>&nbsp;&nbsp; Thus (<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>t-<FONT face=symbol>D</FONT>x<FONT face=symbol>D</FONT>t')=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)<SUP><SUP>1/2</SUP></SUP>(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)<SUP><SUP>1/2</SUP></SUP>sinh<FONT face=symbol>q</FONT>=|<FONT face=symbol>D</FONT><B>R</B>||<FONT face=symbol>D</FONT><B>R</B>'|sinh<FONT face=symbol>q</FONT>. And thus, along with (3),<BR> [(<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>t-<FONT face=symbol>D</FONT>x<FONT face=symbol>D</FONT>t')/(<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t)]=[|<FONT face=symbol>D</FONT><B>R</B>||<FONT face=symbol>D</FONT><B>R</B>'|sinh<FONT face=symbol>q</FONT>/|<FONT face=symbol>D</FONT><B>R</B>||<FONT face=symbol>D</FONT><B>R</B>'|cosh<FONT face=symbol>q</FONT>]=[(<FONT face=symbol>b</FONT>'-<FONT face=symbol>b</FONT>)/(<FONT face=symbol>b</FONT>'<FONT face=symbol>b</FONT>-1)]=tanh<FONT face=symbol>q</FONT>=<FONT face=symbol>b</FONT><SUB>r</SUB>, one<BR> form of the usual relative velocity factor between primed and unprimed frames/vectors. Thus (1) and<BR> (2) do behave like orthogonal transforms of vectors <FONT face=symbol>D</FONT><B>R</B>, <FONT face=symbol>D</FONT><B>R</B>'. A definition of the cross product on<BR> four-dimensional vectors is possible only with tensors: but see <A href="http://www.byandbygones.com/paper0.htm"><FONT color=blue>paper0</FONT></A> ; it's neither essential nor<BR> necessary to deriving the analytic properties of <FONT face=symbol>D</FONT><B>R</B>, <FONT face=symbol>D</FONT><B>R</B>' dealt with in any further analysis. <P><BR><BR><U>III</U>: <U>Derivation of "<FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B>":</U> <P><U>A</U>: <U>L(<B>v</B>) Derivation of Dot Product:</U> <BR>&nbsp; &nbsp; The general Lorentz transform, L(<B>v</B>), between inertial frames in arbitrary configuration is L(<B>v</B>)=e<SUP><FONT face=symbol>xq</FONT></SUP><BR>in which the 4x4 matrix <FONT face=symbol>x</FONT>(<FONT face=symbol>l</FONT>,<FONT face=symbol>h</FONT>,<FONT face=symbol>e</FONT>) is such that elements <FONT face=symbol>x</FONT><SUB>mn</SUB>=0, except <FONT face=symbol>x</FONT><SUB>14</SUB>=i<FONT face=symbol>l</FONT>=<FONT size=3>-</FONT><FONT face=symbol>x</FONT><SUB>41</SUB>, <FONT face=symbol>x</FONT><SUB>24</SUB>=i<FONT face=symbol>h</FONT>=<FONT size=3>-</FONT><FONT face=symbol>x</FONT><SUB>42</SUB>,<BR><FONT face=symbol>x</FONT><SUB>34</SUB>=i<FONT face=symbol>e</FONT>=<FONT size=3>-</FONT><FONT face=symbol>x</FONT><SUB>43</SUB>. From L(<B>v</B>) we derive the generalization of (1), (2) <DIV align=center><BR><FONT face=symbol>lD</FONT>x<SUB>1</SUB>'+<FONT face=symbol>hD</FONT>x<SUB>2</SUB>'+<FONT face=symbol>eD</FONT>x<SUB>3</SUB>' = (<FONT face=symbol>lD</FONT>x<SUB>1</SUB>+<FONT face=symbol>hD</FONT>x<SUB>2</SUB>+<FONT face=symbol>eD</FONT>x<SUB>3</SUB>)cosh<FONT face=symbol>q</FONT>+i<FONT face=symbol>D</FONT>x<SUB>4</SUB>sinh<FONT face=symbol>q</FONT> &nbsp; &nbsp; (5) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <BR><BR>i<FONT face=symbol>D</FONT>x<SUB>4</SUB>'=(<FONT face=symbol>lD</FONT>x<SUB>1</SUB>+<FONT face=symbol>hD</FONT>x<SUB>2</SUB>+<FONT face=symbol>eD</FONT>x<SUB>3</SUB>)sinh<FONT face=symbol>q</FONT> + i<FONT face=symbol>D</FONT>x<SUB>4</SUB>cosh<FONT face=symbol>q</FONT> &nbsp; &nbsp; (6)</DIV>and the inverse relations.<BR>&nbsp; &nbsp; Taking <FONT face=symbol>lD</FONT>x<SUB>1</SUB>'+<FONT face=symbol>hD</FONT>x<SUB>2</SUB>'+<FONT face=symbol>eD</FONT>x<SUB>3</SUB>'=<FONT face=symbol>D</FONT>x', <FONT face=symbol>lD</FONT>x<SUB>1</SUB>+<FONT face=symbol>hD</FONT>x<SUB>2</SUB>+<FONT face=symbol>eD</FONT>x<SUB>3</SUB>=<FONT face=symbol>D</FONT>x, <FONT face=symbol>D</FONT>x<SUB>4</SUB>'=ic<FONT face=symbol>D</FONT>t', <FONT face=symbol>D</FONT>x<SUB>4</SUB>=ic<FONT face=symbol>D</FONT>t, and rewriting (5),<BR>(6) as <DIV align=center><FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t'=e<SUP>-<FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t), &nbsp; &nbsp; &nbsp; <FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t'=e<SUP><FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t) &nbsp; &nbsp; (7)</DIV>we obtain<BR><BR> <DIV align=center>(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)e<SUP><FONT face=symbol>q</FONT></SUP>=(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')e<SUP>-<FONT face=symbol>q</FONT></SUP></DIV>which yields<BR><BR> <DIV align=center>e<SUP>2<FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')<SUP>2</SUP>(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)<SUP>2</SUP>=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>) &nbsp; &nbsp; (8)<BR><BR>e<SUP>-2<FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)<SUP>2</SUP>(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')<SUP>2</SUP>=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>) &nbsp; &nbsp; (9)</DIV><BR>From (9) - (or correspondingly from (8)) - we also derive <DIV align=center><BR>(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')e<SUP><FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>) &nbsp; &nbsp; (10)<BR><BR>(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')e<SUP>-<FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)=(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>) &nbsp; &nbsp; (11)</DIV><BR>and<BR> <DIV align=center>(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')e<SUP>-<FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>) &nbsp; &nbsp; (12)<BR><BR>(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')e<SUP><FONT face=symbol>q</FONT></SUP>(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)=(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>) &nbsp; &nbsp; (13)</DIV><BR>Adding (10) and (11) gives <DIV align=center>2(<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t)=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)e<SUP>-<FONT face=symbol>q</FONT></SUP>+(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)e<SUP><FONT face=symbol>q</FONT></SUP></DIV>which gives <DIV align=center><BR>4(<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t)<SUP>2</SUP>=(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)<SUP>2</SUP>(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)<SUP>2</SUP>e<SUP>-2<FONT face=symbol>q</FONT></SUP>+(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')<SUP>2</SUP>(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')<SUP>2</SUP>e<SUP>2<FONT face=symbol>q</FONT></SUP>+2(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)<BR><BR>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; =[{(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)/(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')}<SUP>2</SUP>+{(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')/(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)}<SUP>2</SUP>+2](<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)<BR><BR>=[e<SUP>-2<FONT face=symbol>q</FONT></SUP>+e<SUP>2<FONT face=symbol>q</FONT></SUP>+2](<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>) &nbsp; &nbsp; (14) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </DIV><BR>Also, (to show symmetry in (5) and (6) isn't analyzed out), adding (12) and (13) gives <DIV align=center><BR>2(<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t)=(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)e<SUP><FONT face=symbol>q</FONT></SUP>+(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)e<SUP>-<FONT face=symbol>q</FONT></SUP></DIV>which gives <DIV align=center><BR>4(<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t)<SUP>2</SUP>=(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)<SUP>2</SUP>(<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t)<SUP>2</SUP>e<SUP>2<FONT face=symbol>q</FONT></SUP>+(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')<SUP>2</SUP>(<FONT face=symbol>D</FONT>x'-<FONT face=symbol>D</FONT>t')<SUP>2</SUP>e<SUP>-2<FONT face=symbol>q</FONT></SUP>+2(<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)<BR><BR>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; =[{(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)/(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')}<SUP>2</SUP>+{(<FONT face=symbol>D</FONT>x'+<FONT face=symbol>D</FONT>t')/(<FONT face=symbol>D</FONT>x+<FONT face=symbol>D</FONT>t)}<SUP>2</SUP>+2](<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>)<BR><BR>=[e<SUP>2<FONT face=symbol>q</FONT></SUP>+e<SUP>-2<FONT face=symbol>q</FONT></SUP>+2](<FONT face=symbol>D</FONT>x<SUP>2</SUP>-<FONT face=symbol>D</FONT>t<SUP>2</SUP>)(<FONT face=symbol>D</FONT>x'<SUP>2</SUP>-<FONT face=symbol>D</FONT>t'<SUP>2</SUP>) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </DIV><BR>which, like (14), is the square of (4).<BR>&nbsp; &nbsp; Like (5,6) the dot product (14)/(4) characterizes/defines Minkowski space. There is one and only one<BR> Einstein invariance of the interval relation and it is governed in every possibility and particular by the<BR> Lorentz transformation. The invariance relation, and thus L(<B>v</b>), are independent of whether <B>R</b><sup>2</sup>&lt0, <B>R</b><sup>2</sup>=0,<BR> or <B>R</b><sup>2</sup>&gt0; neither is the reason <B>R</b><sup>2</sup>&lt0, <B>R</b><sup>2</sup>=0, or <B>R</b><sup>2</sup>&gt0; and thus L(<B>v</b>) at (5,6) transforms a vector <B>R</b>, a la<BR> Klein, regardless of its interval <B>R</b><sup>2</sup>&lt0, <B>R</b><sup>2</sup>=0, or <B>R</b><sup>2</sup>&gt0. Thus the dot product (14) is unique and is the<BR> derived consequence of L(<B>v</b>) at (5,6) regardless of whether <B>R</b><sup>2</sup>&lt0, <B>R</b><sup>2</sup>=0, or <B>R</b><sup>2</sup>&gt0. Thus when L(<B>v</b>) is<BR> inconsistent for one of <B>R</b><sup>2</sup>&lt0, <B>R</b><sup>2</sup>=0, <B>R</b><sup>2</sup>&gt0 it is therefore inconsistent for all. <P><BR><U>B</U>: <U>To Show <FONT face=symbol>D</FONT>x=<FONT face=symbol>D</FONT>r:</U><BR>&nbsp; &nbsp; [Here, we drop the <FONT face=symbol>D</FONT> to avoid the above clutter.]<BR>&nbsp; &nbsp; It is clear from the invariance of the interval that (x'<SUP>2</SUP>-r'<SUP>2</SUP>)=(x<SUP>2</SUP>-r<SUP>2</SUP>). Thus, unless the presence of the<BR>direction cosines <FONT face=symbol>l</FONT>,<FONT face=symbol>h</FONT>, <FONT face=symbol>e</FONT> in (x'<SUP>2</SUP>-t'<SUP>2</SUP>)=(x<SUP>2</SUP>-t<SUP>2</SUP>) somehow changes the physical situation (r'<SUP>2</SUP>-t'<SUP>2</SUP>)=(r<SUP>2</SUP>-t<SUP>2</SUP>),<BR>then in the primed frame x'<SUP>2</SUP>=r'<SUP>2</SUP> and in the unprimed frame x<SUP>2</SUP>=r<SUP>2</SUP>, since the time t', t is the same in the<BR>invariant interval of each transformation.<BR>&nbsp; &nbsp; Algebraically the same result is derived from substitution of the actual values of the direction cosines<BR><FONT face=symbol>l</FONT>, <FONT face=symbol>h</FONT>, <FONT face=symbol>e</FONT> in terms of only x<SUB>i</SUB>', x<SUB>i</SUB>, i=1,2,3, into (x'-x)(x'+x) to obtain (x'<SUP>2</SUP>-x<SUP>2</SUP>)=(r'<SUP>2</SUP>-r<SUP>2</SUP>), a result that,<BR>algebraically, does suggest but not necessarily demand x'<SUP>2</SUP>=r'<SUP>2</SUP> and x<SUP>2</SUP>=r<SUP>2</SUP>. However, when it does not,<BR>it does require x'<SUP>2</SUP>-r'<SUP>2</SUP> and x<SUP>2</SUP>-r<SUP>2</SUP> to be the same invariant distance (I<SUB>x'</SUB><SUP>2</SUP>-I<SUB>r'</SUB><SUP>2</SUP>)=(I<SUB>x</SUB><SUP>2</SUP>-I<SUB>r</SUB><SUP>2</SUP>).<BR>&nbsp; &nbsp; Clearly such a distance will have to be the proper length of the point particle, or nothing. Since it<BR>cannot be such, it either is trivial, or is to be treated as an ordinary invariant interval with t', t canceling<BR>out. Since t' is the same for (or can now be put in as such with) x', r', and so too t for x, r, it is evident<BR>that x', r' are now the space components of two four-vectors both in the same primed inertial frame<BR>now in standard rather than general configuration with their Lorentz transform components x, r both<BR>in the unprimed frame - or all in the one frame or the other [1].<BR>&nbsp; &nbsp; That is, the relation is one of four four-vectors (x'<SUP>2</SUP>-r'<SUP>2</SUP>-t'<SUP>2</SUP>+t'<SUP>2</SUP>)=(I<SUB>x'</SUB><SUP>2</SUP>-I<SUB>r'</SUB><SUP>2</SUP>)=(I<SUB>x</SUB><SUP>2</SUP>-I<SUB>r</SUB><SUP>2</SUP>)=(x<SUP>2</SUP>-r<SUP>2</SUP>-t<SUP>2</SUP>+t<SUP>2</SUP>)<BR>which relation we recast as (given of course (5), (6), orthogonality of the relevant basis vectors) a<BR>scalar dot/inner product of two four-vectors from the difference and sum of the said vectors:<BR>[(x'+r')(x'-r')-(t'+t')(t'-t')]=(I<SUB>x</SUB><SUP>2</SUP>-I<SUB>r</SUB><SUP>2</SUP>)=[(x+r)(x-r)-(t+t)(t-t)], which are by the analysis of (14) only,<BR>{[(x+r)<SUP>2</SUP>-(t+t)<SUP>2</SUP>][(x-r)<SUP>2</SUP>-(t-t)<SUP>2</SUP>]}<SUP>1/2</SUP>cosh<FONT face=symbol>f</FONT>=(I<SUB>x</SUB><SUP>2</SUP>-I<SUB>r</SUB><SUP>2</SUP>), and its primed counterpart, in which the invariant<BR>interval (I<SUB>x</SUB><SUP>2</SUP>-I<SUB>r</SUB><SUP>2</SUP>) is the difference of the invariant x't'xt-interval I<SUB>x'</SUB><SUP>2</SUP>=(x'<SUP>2</SUP>-t'<SUP>2</SUP>)=(x<SUP>2</SUP>-t<SUP>2</SUP>)=I<SUB>x</SUB><SUP>2</SUP> and the<BR>r't'rt-interval I<SUB>r'</SUB><SUP>2</SUP>=(r'<SUP>2</SUP>-t'<SUP>2</SUP>)=(r<SUP>2</SUP>-t<SUP>2</SUP>)=I<SUB>r</SUB><SUP>2</SUP>, each of the (I<SUB>x'</SUB><SUP>2</SUP>-I<SUB>r'</SUB><SUP>2</SUP>), (I<SUB>x</SUB><SUP>2</SUP>-I<SUB>r</SUB><SUP>2</SUP>) devised from the match or mix or<BR>the mix and match of primed and unprimed intervals as necessary.<BR>&nbsp; &nbsp; In those equations it is clear we are always free to set (x'-r')=(x-r)=0. For the other factor, if <FONT face=symbol>f</FONT>, the<BR>velocity parameter between the sum and difference of the xt-vector and the rt-vector, is such that<BR>cosh<FONT face=symbol>f</FONT>=1, then for positive roots evidently t=t'=0, or x=r, x'=r' and (4) is valid without further analysis.<BR>&nbsp; &nbsp; For |x-r|&gt;0, |x+r|&gt;0 and cosh<FONT face=symbol>f</FONT>&gt;1 in the equations associated with (I<SUB>x</SUB><SUP>2</SUP>-I<SUB>r</SUB><SUP>2</SUP>) we solve the unprimed<BR>equation for t<SUP>2</SUP>=f(x,r,<FONT face=symbol>f</FONT>) and the primed equation for t'<SUP>2</SUP>=f(x',r',<FONT face=symbol>f</FONT>) to derive the recurrent constraints<BR>|(x'+r')/(t'+t')|=|(x+r)/(t+t)|&gt;c, an expression typically of the cosh<FONT face=symbol>f</FONT> or (1/4)[(x'+r')<SUP>2</SUP>-(x+r)<SUP>2</SUP>]tanh<SUP>2</SUP><FONT face=symbol>f</FONT> sort<BR>for (t'<SUP>2</SUP>-t<SUP>2</SUP>), and find (t'/t)<SUP>2</SUP>=[(x'+r')/(x+r)]<SUP>2</SUP> or (t'/t)<SUP>2</SUP>=[(x'+r')(x-r)/(x'-r')(x+r)].<BR>&nbsp; &nbsp; Next, working with t't and (t'<SUP>2</SUP>+t<SUP>2</SUP>) in the square of each term c(t'-t)=(x'+x)tanh(<FONT face=symbol>q</FONT>/2) and<BR>c(t'+t)=(x'-x)ctnh(<FONT face=symbol>q</FONT>/2), obtained from the t', t transforms, we substitute in the (t'/t)<SUP>2</SUP> equation to find<BR>expressions for t'<SUP>4</SUP>, t<SUP>4</SUP>. We use t'<SUP>4</SUP> and t<SUP>4</SUP> in the difference of square roots form (t'<SUP>2</SUP>-t<SUP>2</SUP>) to find an<BR>expression for tanh<SUP>2</SUP><FONT face=symbol>f</FONT>, and in the form (t'<SUP>2</SUP>+t<SUP>2</SUP>)(t'<SUP>2</SUP>-t<SUP>2</SUP>) to get a second expression for tanh<SUP>2</SUP><FONT face=symbol>f</FONT>. We equate<BR>both around tanh<SUP>2</SUP><FONT face=symbol>f</FONT> to obtain (x'-x+r'-r)(x'-x)=(x'+x+r'+r)(x'+x)tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2), showing that <FONT face=symbol>q</FONT> is not<BR>arbitrary.<BR>&nbsp; &nbsp; Adjusting (t'<SUP>2</SUP>-t<SUP>2</SUP>) to yield each, the same procedure with the one or the other of equations<BR>(t/t')<SUP>2</SUP>=[(x'-r')/(x-r)]<SUP>2</SUP> and (t/t')<SUP>2</SUP>=[(x'-r')(x+r)/(x'+r')(x-r)] results in the analogue expression, or<BR>transforms the previous tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2) expression to, (x'-x-r'+r)(x'-x)=(x'+x-r'-r)(x'+x)tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2).<BR>Adding this equation to, subtracting from, and multiplying with the previous expression yields<BR>(x'-x)<SUP>2</SUP>=(x'+x)<SUP>2</SUP>tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2), and (r'-r)(x'-x)=(r'+r)(x'+x)tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2), which are solved in section V,<BR>and the old familiar results x'=r', x=r, or t'<SUP>2</SUP>=t<SUP>2</SUP> and x'=r'=x=r. Thus, for |x-r|&gt;0, |x+r|&gt;0, and cosh<FONT face=symbol>f</FONT>&gt;1,<BR>either, once more, |(x'+r')/(t'+t')|=|(x+r)/(t+t)|&gt;c, or x<SUP>2</SUP>=r<SUP>2</SUP>, x'<SUP>2</SUP>=r'<SUP>2</SUP>, I<SUB>x</SUB><SUP>2</SUP>=I<SUB>r</SUB><SUP>2</SUP> since <FONT face=symbol>f</FONT> is invariant, implying<BR>we can always set x'<SUP>2</SUP>=r'<SUP>2</SUP> and x<SUP>2</SUP>=r<SUP>2</SUP> for all real <FONT face=symbol>l</FONT>, <FONT face=symbol>h</FONT>, <FONT face=symbol>e</FONT>, and arbitrary <FONT face=symbol>q</FONT> and <FONT face=symbol>f</FONT>; and that holds generally,<BR>even despite some truly convoluted expressions now resulting for <FONT face=symbol>l</FONT>, <FONT face=symbol>h</FONT>, <FONT face=symbol>e</FONT> when, for instance, <FONT face=symbol>l</FONT>, <FONT face=symbol>h</FONT>, <FONT face=symbol>e</FONT><BR>are derived from x=(r<SUP>2</SUP>)<SUP>1/2</SUP>, x'=(r'<SUP>2</SUP>)<SUP>1/2</SUP>, and the delineating definition <FONT face=symbol>l</FONT><SUP>2</SUP>+<FONT face=symbol>h</FONT><SUP>2</SUP>+<FONT face=symbol>e</FONT><SUP>2</SUP>=1. <BR>&nbsp; &nbsp; To show that, although not identities for all <FONT face=symbol>q</FONT>, <FONT face=symbol>f</FONT> and/or all admissible <FONT face=symbol>l</FONT>, <FONT face=symbol>h</FONT>, <FONT face=symbol>e</FONT>, x'<SUP>2</SUP>=r'<SUP>2</SUP><BR>and x<SUP>2</SUP>=r<SUP>2</SUP> imply <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>x<SUB><FONT face=symbol>u</FONT></SUB>'x<SUB><FONT face=symbol>u</FONT></SUB> of (4), we sort through the algebra slush coupled to the<BR>Lorentz transformation matrices L(<B>v</B>)=e<SUP><FONT face=symbol>xq</FONT></SUP> and L(<B>v</B>)<SUP>-</SUP>=e<SUP>-<FONT face=symbol>xq</FONT></SUP> to obtain the simple results<BR><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>x<SUB><FONT face=symbol>u</FONT></SUB>'x<SUB><FONT face=symbol>u</FONT></SUB>=[r<SUP>2</SUP>-x<SUP>2</SUP>+(x<SUP>2</SUP>+x<SUB>4</SUB><SUP>2</SUP>)cosh<FONT face=symbol>q</FONT>]=[r'<SUP>2</SUP>-x'<SUP>2</SUP>+(x'<SUP>2</SUP>+x<SUB>4</SUB>'<SUP>2</SUP>)cosh<FONT face=symbol>q</FONT>], showing that equation (14)/(4)<BR>necessarily results, precisely on setting - (i.e. putting x<SUB>1</SUB><SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>=<FONT face=symbol>l</FONT>r<SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>, x<SUB>2</SUB><SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>=<FONT face=symbol>h</FONT>r<SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>, x<SUB>3</SUB><SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>=<FONT face=symbol>e</FONT>r<SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>) - x'<SUP>2</SUP>=r'<SUP>2</SUP><BR>and x<SUP>2</SUP>=r<SUP>2</SUP>, which, by (14) and (4), must now necessarily imply (x'x+x<SUB>4</SUB>x<SUB>4</SUB>')=(r'r+x<SUB>4</SUB>x<SUB>4</SUB>')=<FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>x<SUB><FONT face=symbol>u</FONT></SUB>'x<SUB><FONT face=symbol>u</FONT></SUB>. <BR>&nbsp; &nbsp; We have taken the time to prove (4)/(14) analytically, but it is surely clearly recognizable as the<BR><I>intrinsic</I> fact that the dot product in any orthogonal transformation is the definition, to stay with<BR>M<SUB>4</SUB>, <B>R</B>'<B>.R</B>=<FONT face=symbol size=5>S</FONT><SUB>j</SUB>(x<SUB>j</SUB>'x<SUB>j</SUB>+x<SUB>4</SUB>'x<SUB>4</SUB>) on the <I>components</I> of a vector definition, equating identically the product<BR><B>R</B>'<B>.R</B>=|<B>R</B>'||<B>R</B>|<B>s</B>'<B>.s</B> on the <I>magnitude</I> (and projection) of a vector definition, <B>s</B>', <B>s</B> being (<B>R</B>'/|<B>R</B>'|)=<B>s</B>',<BR>(<B>R</B>/|<B>R</B>|)=<B>s</B>, respectively. Since Lorentz transformation is intrinsically an 'orthogonal' transformation<BR>on a pure imaginary angle (of rotation), the angle of relative orientation [1] of <B>R</B>', <B>R</B> is imaginary,<BR>and therefore |<B>R</B>'||<B>R</B>|<B>s</B>'<B>.s</B>=|<B>R</B>'||<B>R</B>|cos(i<FONT face=symbol>q</FONT>), which is (4)/(14). Therefore the cos(i<FONT face=symbol>q</FONT>)=cosh<FONT face=symbol>q</FONT>=<FONT face=symbol>g</FONT>, in<BR>fact, can never be rid of, by anything in mathematics, from the dot product intrinsic in the Lorentz<BR>transformation. In fact, let us remind ourselves that cosh<FONT face=symbol>q</FONT> is merely an analytically convenient substitute<BR>for <I>physical</I> <FONT face=symbol>g</FONT> - and, in fact, everyone writes the particle velocity product <B>U</B>.<B>V</B>=c<SUP>2</SUP><FONT face=symbol>g</FONT>(v), (which, in fact,<BR>=|<B>U</B>||<B>V</B>|cosh<FONT face=symbol>q</FONT>=c<SUP>2</SUP>cosh<FONT face=symbol>q</FONT>), and thinks no more of it since it's not conspicuously obvious how to contrive<BR><B>U</B>.<B>V</B>=0 without <B>U</B>=<B>0</B>, or <B>V</B>=<B>0</B>, or <B>U</B>=<B>V</B>=<B>0</B>. Without any claims of the 4-vector being identical to the<BR>3-vector, the M<SUB>4</SUB> dot/inner product proves itself to be, and is, (cf.[3],[1]), the analogue of the real angle<BR>vector dot product. To demonstrate that cos(i<FONT face=symbol>q</FONT>)=cosh<FONT face=symbol>q</FONT>=<FONT face=symbol>g</FONT> is actually in reality cos<FONT face=symbol>q</FONT>, as is the actual<BR>practice in analysis when cosh<FONT face=symbol>q</FONT>=<FONT face=symbol>g</FONT>, though logically necessary, is evidently physically incorrect and<BR>absurd, or is of no consequence, or can be transformed away in the theory of 4-vectors, is at the same<BR>time to demonstrate that real cos<FONT face=symbol>q</FONT> is of no consequence and can be transformed away in the theory of<BR>real 3-vectors. Its validity is not selective, or to be taken as convenient. The identity is intrinsic, and<BR>either half must necessarily carry the sum total of the properties of the identity. Relativity has always<BR>ignored the cos(i<FONT face=symbol>q</FONT>)=cosh<FONT face=symbol>q</FONT>=<FONT face=symbol>g</FONT> half precisely because ordinarily that fact makes it superfluous and<BR>redundant when its components half is analytically or algebraically preferred. That is to say, <I>it is<BR>mathematically/analytically impossible to logically define the components half dot product</I><BR><FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B>=<FONT face=symbol>D</FONT>x'<FONT face=symbol>D</FONT>x-<FONT face=symbol>D</FONT>t'<FONT face=symbol>D</FONT>t <I>while conveniently abolishing the magnitudes half</I> <FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B>=|<FONT face=symbol>D</FONT><B>R</B>'||<FONT face=symbol>D</FONT><B>R</B>|cosh<FONT face=symbol>q</FONT><BR><I>of the identity since merely <B>substituting</B> the absolutely indispensable transforms</I> (1), (2) (<I><B>back</B></I>)<BR><I>into the components half of</I> (3) <I>inevitably yields its magnitudes half</I> <FONT face=symbol>D</FONT><B>R</B>'.<FONT face=symbol>D</FONT><B>R</B>=|<FONT face=symbol>D</FONT><B>R</B>'||<FONT face=symbol>D</FONT><B>R</B>|cosh<FONT face=symbol>q</FONT>.<BR>Similarly for (4). And as far as actual logical application goes/ought to go the instance of the calculation<BR>of the velocity formula [(<FONT face=symbol>b</FONT>'-<FONT face=symbol>b</FONT>)/(<FONT face=symbol>b</FONT>'<FONT face=symbol>b</FONT>-1)]=<FONT face=symbol>b</FONT><SUB>r</SUB> from the 'area' |<FONT face=symbol>D</FONT><B>R</B>x<FONT face=symbol>D</FONT><B>R</B>'| and <FONT face=symbol>D</FONT><B>R</B>.<FONT face=symbol>D</FONT><B>R</B>' following (4) though<BR>not completely rigorous nevertheless makes it clear that cosh<font face=symbol>q</font>=<font face=symbol>g</font> and sinh<font face=symbol>q</font>=<font face=symbol>g</font><font face=symbol>b</font> simply cannot be<BR> replaced by cos<font face=symbol>q</font><FONT face=symbol>¹</FONT><font face=symbol>g</font> and sin<font face=symbol>q</font><FONT face=symbol>¹</FONT><font face=symbol>g</font><font face=symbol>b</font> in the two indispensable vector basics <FONT face=symbol>D</FONT><B>R</B>.<FONT face=symbol>D</FONT><B>R</B>' and <FONT face=symbol>D</FONT><B>R</B>x<FONT face=symbol>D</FONT><B>R</B>'. <P><BR><BR><U>IV</U>: <U>Four-Vector "Geometry":</U><BR>&nbsp; &nbsp; Having shown that <FONT face=symbol>D</FONT><B>R</B>.<FONT face=symbol>D</FONT><B>R</B>' is rigorously also <FONT face=symbol>D</FONT><B>R</B>.<FONT face=symbol>D</FONT><B>R</B>'=|<FONT face=symbol>D</FONT><B>R</B>||<FONT face=symbol>D</FONT><B>R</B>'|cosh<font face=symbol>q</font> it is shown from here on that<BR> due to this cosh<font face=symbol>q</font>=<font face=symbol>g</font> parameter special relativity consequently has basic logical and structural difficulties<BR> deep in its foundation - all the way from defining the Einstein interval ds<SUP>2</SUP>=c<SUP>2</SUP>d<FONT face=symbol>t</FONT><SUP>2</SUP> to linearity or/and the<BR> Lorentz transform factor.<BR>&nbsp; &nbsp; To this end we emphatically reiterate that the dot product at (3), (4)/(14) is a <u>derived</u> <u>consequence</u> of<BR> L(<B>v</b>) at (1,2) and (5,6) for <u>all</u> Minkowski space vectors - it is not speculation, conjecture, hypothesis,<BR> premise, or even analogy with that of classical vector analysis; there is one and only one dot product<BR> for Minkowski space. Thus regardless of the formulation the relativity principle, the L(<B>v</B>), and the four-<BR>vector all uncover the self-contradictory inconsistencies of relativity. <BR>&nbsp; &nbsp; Now, since (4) determines/is the general definition of <B>R</B>'.<B>R</B>, and is, by (14), a consequence of (7),<BR>which is a consequence of the relativity principle, we obtain from (7) <DIV align=center><BR>0=<FONT face=symbol size=5>S</FONT><SUB>k</SUB>(x<SUB>k</SUB>'<SUP>2</SUP>-x<SUB>k</SUB><SUP>2</SUP>)=<FONT face=symbol size=5>S</FONT><SUB>k</SUB>(x<SUB>k</SUB>'-x<SUB>k</SUB>)(x<SUB>k</SUB>'+x<SUB>k</SUB>) &nbsp; &nbsp; (15)<BR><BR>&nbsp; &nbsp; &nbsp; =(<B>R</B>'-<B>R</B>).(<B>R</B>'+<B>R</B>)</DIV><BR>for k=1,2, the subscripts retrieving not standard components but rather only the four values x<SUB>1</SUB><SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>=x<SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP><BR>and x<SUB>2</SUB><SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP>=ict<SUP><FONT size=1>(</FONT></SUP>'<SUP><FONT size=1>)</FONT></SUP> as in (7). Nevertheless, as defined following (6), x' is taken as the "space component"<BR>of <B>R</B>' and x is taken as that of <B>R</B>.<BR>&nbsp;&nbsp; With |<B>R</B>'|=|<B>R</B>| by invariance, (15), by (4), is <DIV align=center><BR>0=(<B>R</B>'-<B>R</B>).(<B>R</B>'+<B>R</B>)=[<FONT face=symbol size=5>S</FONT><SUB>k</SUB>(x<SUB>k</SUB>'-x<SUB>k</SUB>)<SUP>2</SUP>]<SUP>1/2</SUP>[<FONT face=symbol size=5>S</FONT><SUB>k</SUB>(x<SUB>k</SUB>'+x<SUB>k</SUB>)<SUP>2</SUP>]<SUP>1/2</SUP>cosh<FONT face=symbol>f</FONT> &nbsp; &nbsp; (16).</DIV> <P></P> <P>Geometrically, |<B>R</B>'|=|<B>R</B>| implies that (<B>R</B>'-<B>R</B>) and (<B>R</B>'+<B>R</B>) are the diagonals of an M<SUB>4</SUB> "rhombus" - i.e.,<BR> <B>R</B>'-<B>R</B> and <B>R</B>'+<B>R</B> ought to be linearly independent/orthogonal. It is this property of it being required<BR> of all vectors <B>R</B>'-<B>R</B> and <B>R</B>'+<B>R</B>, |<B>R</B>'|=|<B>R</B>|, to be necessarily mutually linearly independent/orthogonal<BR> (or else, <B>R</B>'//<B>R</B>; or, <B>R</B>'<font face=symbol>&#186;</font><B>R</B>) that is by (15,16) necessarily fundamental to <u>all</u> vectors defined/definable <BR>by a Klein orthogonal transformation - a property that necessarily evidently is the corollary of Euclid's<BR> parallel postulate, equally determining the metric nature and geometry of the vector space. In (16)<BR> euclidean orthogonality/linear independence of <B>R</B>'-<B>R</B> and <B>R</B>'+<B>R</B> must be in cosh<FONT face=symbol>f</FONT>=cosi<FONT face=symbol>f</FONT><FONT face=symbol>¹</FONT>cos<font face=symbol>f</font>, and<BR> |<B>R</B>'-<B>R</B>|=0 or |<B>R</B>'+<B>R</B>|=0, or perhaps |<B>R</B>'-<B>R</B>|=|<B>R</B>'+<B>R</B>|=0, must be euclidean parallelicity of <B>R</B>', <B>R</B>. Thus,<BR> for ordinary euclidean vectors, i.e. with cosh<FONT face=symbol>f</FONT><FONT face=symbol>®</FONT>cos<FONT face=symbol>f</FONT>, (16) is the elementary finding that the diagonals<BR>of a rhombus are mutually perpendicular, but is impossible for Minkowski M<SUB>4</SUB> four-vectors unless, in<BR>consequence of (15), and making use of the fact in (15), (16) that k=1, 2 only, both <FONT face=symbol size=5>S</FONT><SUB>k</SUB>(x<SUB>k</SUB>'-x <SUB>k</SUB>)<SUP>2</SUP>=0,<BR>and <FONT face=symbol size=5>S</FONT><SUB>k</SUB>(x<SUB>k</SUB>'+x<SUB>k</SUB>)<SUP>2</SUP>=0 - which constraints also must mean |(x'-x)/(t'-t)|=|(x'+x)/(t'+t)|=c, and<BR>|(x'/t')|=|(x/t)|=c. In this case that each sum expression can be trivial or meaningless implies that for<BR>|<B>R</B>'|=|<B>R</B>|, <B>R</B>'<SUP>2</SUP>=<B>R</B><SUP>2</SUP> or <B>R</B>'=±<B>R</B> real, or M<SUB>4</SUB>, <B>R</B>'<SUP>2</SUP>=<B>R</B><SUP>2</SUP>=<B>0</B><SUP>2</SUP> and <B>R</B>'.<B>R</B>=0, which for the euclidean vector<BR>implies the rhombus is a square, but for the Minkowski world vector, |<B>R</B>'|=|<B>R</B>|=|<B>0</B>|=<B>R</B>'=<B>R</B> identically,<BR>which restricts the relativity principle and the spacetime concept to velocities c only. The possibilities,<BR><B>R</B>'=±<B>R</B> are not ascribed to M<SUB>4</SUB> since <B>R</B>'=<B>R</B> means the relative velocity <B>v</B>=|<B>v</B>|=0 eternally, and <B>R</B>'=-<B>R</B><BR>means <B>v</B>, |<B>v</B>| whatever each now is, is such that <B>R</B> is boosted (booted?) through the lightcone to<BR>timetravel into the past in violation of causality. Thus, with the null vectors, although not yet trivial<BR>vectors, result <B>R</B>'=<B>R</B>=<B>0</B> it is to be noted that the velocity four-vectors are d<SUB><FONT face=symbol>t</FONT></SUB><B>R</B>'=<B>U</B>'=<B>0</B> and d<SUB><FONT face=symbol>t</FONT></SUB><B>R</B>=<B>U</B>=<B>0</B>,<BR>implying |<B>U</B>'|=0, |<B>U</B>|=0 for all massive particles and photons alike. The magnitudes are intrinsic<BR>properties of M<SUB>4</SUB>. The point is, unlike the real rotation group, L(<B>v</B>) does <U>not</U> point to a <I>unitary</I> group,<BR>and therefore short of discarding L(<B>v</B>), a theory of unitary groups of any sort cannot affect (16), and<BR>a theory of nonunitary groups can only preserve (16). And actually any orthogonality that orthogonal<BR>matrices and groups supposedly prove is already posited in their structure by the analytic properties<BR>of the linear independence of <B>i</B>, <B>j</B>, <B>k</B>. <P><BR><BR><U>V</U>: <U>The Relativity Principle:</U><BR>&nbsp; &nbsp; The relativity principle is postulate in (3)/(4) since, given euclidicity, isotropy and invariance of<BR>causality the relativity principle necessarily implies (1), (2) - or more precisely their generalization<BR>(5), (6) for arbitrary relative velocity v not transformed to the usual standard configuration of (1), (2). <BR>&nbsp; &nbsp; Now, the fact is, the same procedure followed above can be applied to<BR>0=<FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'<SUP>2</SUP>-x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>)=<FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>), the more usual form of the relativity principle, to yield the<BR>very same results. By either of equation (4)/(16), we must have <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0, or <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0.<BR>Working with <FONT face=symbol>l</FONT><SUP>2</SUP>+<FONT face=symbol>h</FONT><SUP>2</SUP>+<FONT face=symbol>e</FONT><SUP>2</SUP>=1 we derive the general relation: <DIV align=center><BR><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=-(ct'-ct)<SUP>2</SUP>+(ct'+ct)<SUP>2</SUP>tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2) &nbsp; &nbsp; (17). </DIV><BR>For only the product <B>[</B><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP><B>][</B><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP><B>]</B>=0 we use <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP> to show that the<BR><FONT face=symbol>lhe</FONT>-generated equation (17) becomes [(t'-t)/(t'+t)]<SUP>2</SUP>=tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2), or the notable and familiar equation<BR>t'<SUP>2</SUP>-2tt'cosh<FONT face=symbol>q</FONT>+t<SUP>2</SUP>=0, which shows that it is <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0. Ordinarily that's an unwitting or ostentatious<BR>exercise, for, of course, naturally, at least one factor of the product must be trivial for ordinary<BR>variables/numbers, so multiplying (17) through by <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP> may be as good as postulating it<BR>as nontrivial; but it just may be that ordinary variables/numbers may not be all there is to (17). So,<BR>without posing whether the one term or the other, or neither, or both, may be 0 in the product<BR><B>[</B><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP><B>][</B><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP><B>]</B>=0 we multiply (17) through by <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>. The left hand side of (17) is<BR>now 0; its right hand side can be converted to <B>[</B>-(ct'-ct)<SUP>2</SUP>+(ct'+ct)<SUP>2</SUP>tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2)<B>][</B><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>4x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>-(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP><B>]</B>=0,<BR>which gets around the logical hurdles of multiplying by a possible <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0. Thus, either<BR><B>[</B><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>4x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>-{-(ct'-ct)<SUP>2</SUP>+(ct'+ct)<SUP>2</SUP>tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2)}<B>]</B>=0, or [-(ct'-ct)<SUP>2</SUP>+(ct'+ct)<SUP>2</SUP>tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2)]=0 with <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0<BR>of (17) a consequence rather than an implicit or explicit premise. From the first possibility and the fact<BR><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=<FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>2x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>(1-cosh<FONT face=symbol>z</FONT>) we find <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>2x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>(1+cosh<FONT face=symbol>z</FONT>)=0 which means |<B>R</B>'|=|<B>R</B>|=|<B>0</B>| and <B>R</B>'=<B>R</B>, or<BR>(1+cosh<FONT face=symbol>z</FONT>)=0. The latter is impossible, and |<B>R</B>|=|<B>0</B>| and <B>R</B>'=<B>R</B> contradict the premise/hypothesis of the<BR>relativity principle making it a far too narrow or circumscribed one vector or all nullvector postulate.<BR>Thus, [-(ct'-ct)<SUP>2</SUP>+(ct'+ct)<SUP>2</SUP>tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2)]=0 for ordinary or nonordinary variables/numbers, and therefore<BR><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0 even if <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0 or <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP><FONT face=symbol>¹</FONT>0. Definitely then only M<SUB>4</SUB> invalid cos<FONT face=symbol>f</FONT> in<BR>(16), with all that that implies in (1), (2) and (5), (6) can generalize the relativity principle to arbitrary<BR>vectors. <BR>&nbsp; &nbsp; When only <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0 the <FONT face=symbol>lhe</FONT>-generated equation becomes the tt'-equations; the general<BR>solutions are t'=e<SUP><FONT face=symbol>q</FONT></SUP>t or t'=e<SUP>-<FONT face=symbol>q</FONT></SUP>t, and x'=e<SUP><FONT face=symbol>q</FONT></SUP>x or x'=e<SUP>-<FONT face=symbol>q</FONT></SUP>x, implying x<SUB>i</SUB>'=e<SUP><FONT face=symbol>q</FONT></SUP>x<SUB>i</SUB> or x<SUB>i</SUB>'=e<SUP>-<FONT face=symbol>q</FONT></SUP>x<SUB>i</SUB>, |(x'/t')|=|(x/t)|=c.<BR>These <I>purely algebraic </I>results now necessarily imply the relations <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=(e<SUP><FONT face=symbol>q</FONT></SUP>-1)<SUP>2</SUP><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>=0<BR>and <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=(e<SUP><FONT face=symbol>q</FONT></SUP>+1)<SUP>2</SUP><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>, or the relations <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=(e<SUP>-<FONT face=symbol>q</FONT></SUP>-1)<SUP>2</SUP><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>=0,<BR><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=(e<SUP>-<FONT face=symbol>q</FONT></SUP>+1)<SUP>2</SUP><FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>. Thus, in this case too unless <FONT face=symbol>q</FONT>=0 both sums of each pair must evidently<BR>be zero, with |(x<SUB>i</SUB>'/t')|=|(x<SUB>i</SUB>/t)|=<FONT face=symbol>l</FONT>c, =<FONT face=symbol>h</FONT>c, =<FONT face=symbol>e</FONT>c, respectively, or x<SUB><FONT face=symbol>u</FONT></SUB>'=x<SUB><FONT face=symbol>u</FONT></SUB>, implying, like <FONT face=symbol>q</FONT>=0, that no such<BR>transformation exists, or that the relativity principle governs only photons. <BR>&nbsp; &nbsp; For the case when only <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0 the <FONT face=symbol>lhe</FONT>-generated standard equation, (17), is now<BR>reduced to the expression 2(x<SUB><FONT face=symbol>u</FONT></SUB>'<SUP>2</SUP>+x<SUB><FONT face=symbol>u</FONT></SUB><SUP>2</SUP>)=(x<SUB>4</SUB>'-x<SUB>4</SUB>)<SUP>2</SUP>-(x<SUB>4</SUB>'+x<SUB>4</SUB>)<SUP>2</SUP>tanh<SUP>2</SUP>(<FONT face=symbol>q</FONT>/2), from which we find<BR>(x<SUB>4</SUB>'+x<SUB>4</SUB>)<SUP>2</SUP>=-(r'<SUP>2</SUP>+r<SUP>2</SUP>)[(1+cosh<FONT face=symbol>q</FONT>)/cosh<FONT face=symbol>q</FONT>]. Along with this we use (r'<SUP>2</SUP>-r<SUP>2</SUP>)=(x<SUB>4</SUB><SUP>2</SUP>-x<SUB>4</SUB>'<SUP>2</SUP>) to find x<SUB>4</SUB>' and x<SUB>4</SUB>.<BR>From i(x<SUB>4</SUB>'-x<SUB>4</SUB>cosh<FONT face=symbol>q</FONT>)=rsinh<FONT face=symbol>q</FONT> and i(x<SUB>4</SUB>'cosh<FONT face=symbol>q</FONT>-x<SUB>4</SUB>)=r'sinh<FONT face=symbol>q</FONT> we find the equation (r'-r)=f(r',r,<FONT face=symbol>q</FONT>), and<BR>squaring that relation find r'<SUP>2</SUP>-2rr'cosh<FONT face=symbol>q</FONT>+r<SUP>2</SUP>=0 yielding the same solutions and implications of the<BR>identical equation in t', t, noted earlier. Therefore, for all |<FONT face=symbol>q</FONT>|&gt;0, <I>purely algebraically </I>and/or<BR>"geometrically", both <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u</FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'-x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0 and <FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>u </FONT></SUB>(x<SUB><FONT face=symbol>u</FONT></SUB>'+x<SUB><FONT face=symbol>u</FONT></SUB>)<SUP>2</SUP>=0, and x<SUB><FONT face=symbol>u</FONT></SUB>'=x<SUB><FONT face=symbol>u</FONT></SUB>=0 for all nonphotons; for<BR>v=<FONT face=symbol>q</FONT>=0 no Lorentz transformation exists, and x<SUB><FONT face=symbol>u</FONT></SUB>'=x<SUB><FONT face=symbol>u</FONT></SUB>=0, verifying once more that the relativity<BR>principle governs only particles moving at the invariant limit velocity.<BR>&nbsp; &nbsp; Analytically, it seems clear that although the result in each case is for the metric tensor<BR>g<SUB><FONT face=symbol>mu</FONT></SUB>=g<SUB><FONT face=symbol>uu</FONT></SUB>=(1,-1,-1,-1) it must needs hold for all spacetime vectors/tensors and Lorentz invariant<BR>metrics 0=<FONT face=symbol size=5>S</FONT><SUB><FONT face=symbol>m</FONT>,<FONT face=symbol>u</FONT></SUB>(g'<SUB><FONT face=symbol>mu</FONT></SUB>dx'<SUP><FONT face=symbol>m</FONT></SUP>dx'<SUP><FONT face=symbol>u</FONT></SUP>-g<SUB><FONT face=symbol>mu</FONT></SUB>dx<SUP> <FONT face=symbol>m</FONT></SUP>dx<SUP><FONT face=symbol>u</FONT></SUP>), <FONT face=symbol>m</FONT>=<FONT face=symbol>u</FONT>, whether or not the g'<SUB><FONT face=symbol>mu</FONT></SUB>=g<SUB><FONT face=symbol>mu</FONT></SUB>=g<SUB><FONT face=symbol>uu</FONT></SUB> are constants,<BR>or g<SUB><FONT face=symbol>mu</FONT></SUB> , <FONT face=symbol>m</FONT>=<FONT face=symbol>u</FONT>, is the more general tensor of general relativity. <P><BR><BR><U>VI</U>: <U>Arbitrary <FONT face=symbol>D</FONT><B>R</B>', <FONT face=symbol>D</FONT><B>R</B></U>: <P><U>A</U>: <U>Derivation of <FONT face=symbol>D</FONT><B>R.</B><FONT face=symbol>D</FONT><B>R</B>'</U>: <BR>&nbsp; &nbsp; In obtaining the result (14) the equality |<FONT face=symbol>D</FONT><B>R</B>'|=|<FONT face=symbol>D</FONT><B>R</B>| is deliberately not explicitly utilized to demonstrate<BR>the dot product <FONT face=symbol>D</FONT><B>R.</B><FONT face=symbol>D</FONT><B>R</B>'=|<FONT face=symbol>D</FONT><B>R</B>||<FONT face=symbol>D</FONT><B>R</B>'|cosh<FONT face=symbol>q</FONT>, but it is evidently unavoidably postulate in it. Nevertheless<BR>the result also holds for arbitrary vector magnitudes since <FONT face=symbol>q</FONT> relating <FONT face=symbol>D</FONT><B>R</B>', <FONT face=symbol>D</FONT><B>R</B> is independent of the<BR>relative magnitudes or length intervals and thus of <FONT face=symbol>D</FONT><B>R</B>'<SUP>2</SUP>=<FONT face=symbol>D</FONT><B>R</B><SUP>2</SUP>. Therefore, relations involving (methods<BR>of finding) <FONT face=symbol>q</FONT> must be independent of the relative lengths/magnitudes of the <FONT face=symbol>D</FONT><B>R</B>' and <FONT face=symbol>D</FONT><B>R</B> that <FONT face=symbol>q</FONT> relates. <BR>&nbsp; &nbsp; However, for vectors <B>R</B>, <B>R</B>' of arbitrary relative magnitudes we can always find [3], [1], by (the<BR>"parallelogram" or "triangle" rule of) vector addition, a "rhombus" <B>R</B>'=<B>R</B><SUB>1</SUB>+<B>R</B><SUB>2</SUB>, with |<B>R</B><SUB>1</SUB>|=|<B>R</B>|=|<B>R</B><SUB>2</SUB>|; or a<BR>"rhombus" <B>R</B>=<B>R</B><SUB>1</SUB>+<B>R</B><SUB>2</SUB>, now with |<B>R</B><SUB>1</SUB>|=|<B>R</B>'|=|<B>R</B><SUB>2</SUB>|, depending on the relative values of |<B>R</B>'| and |<B>R</B>|. If the<BR>parameter in <B>R</B>'<B>.R</B> is <FONT face=symbol>q</FONT> and the parameter in <B>R</B><SUB>1</SUB><B>.R</B><SUB>2</SUB> is 2<FONT face=symbol>f</FONT> then the parameter in <B>R</B><SUB>1</SUB><B>.R</B> or <B>R</B><SUB>1</SUB><B>.R</B>' is the<BR>one or the other of <FONT face=symbol>f</FONT>-<FONT face=symbol>q</FONT> or <FONT face=symbol>q</FONT>-<FONT face=symbol>f</FONT>, depending on whether <FONT face=symbol>f</FONT>&gt;<FONT face=symbol>q</FONT> or <FONT face=symbol>q</FONT>&gt;<FONT face=symbol>f</FONT> - that is, on whether <B>R</B>', <B>R</B> lies<BR>outside rhombus <B>R</B><SUB>1</SUB><B>R</B><SUB>2</SUB>, or bisects the half "vertex" <B>R</B><SUB>h</SUB><B>R</B>, or <B>R</B><SUB>h</SUB><B>R</B>', formed of <B>R</B><SUB>h</SUB>=<B>R</B><SUB>1,2</SUB> and resultant<BR>"diagonal" <B>R</B><SUB>1</SUB>+<B>R</B><SUB>2</SUB>=<B>R</B>, or <B>R</B><SUB>1</SUB>+<B>R</B><SUB>2</SUB>=<B>R</B>'.<BR>&nbsp; &nbsp; Thus, choosing with no loss of generality, <B>R</B>'=<B>R</B><SUB>1</SUB>+<B>R</B><SUB>2</SUB> with |<B>R</B><SUB>1</SUB>|=|<B>R</B>|=|<B>R</B><SUB>2</SUB>|, we obtain <DIV align=center><BR><B>R</B>'<B>.R</B>=|<B>R</B><SUB>1</SUB>||<B>R</B>|cosh(<FONT face=symbol>q</FONT>-<FONT face=symbol>f</FONT>)+|<B>R</B><SUB>2</SUB>||<B>R</B>|cosh(<FONT face=symbol>q</FONT>+<FONT face=symbol>f</FONT>)<BR><BR>=2|<B>R</B>||<B>R</B>|cosh<FONT face=symbol>q</FONT>cosh<FONT face=symbol>f</FONT> &nbsp; &nbsp; (18) &nbsp; &nbsp; &nbsp; </DIV><BR>From <B>R</B>'=<B>R</B><SUB>1</SUB>+<B>R</B><SUB>2</SUB> we derive <DIV align=center><BR><B>R</B>'<SUP>2</SUP>=<B>R</B><SUB>1</SUB><SUP>2</SUP>+<B>R</B><SUB>2</SUB><SUP>2</SUP>+2<B>R</B><SUB>1</SUB><B>.R</B><SUB>2</SUB>=2|<B>R</B>|<SUP>2</SUP>(1+cosh2<FONT face=symbol>f</FONT>)=4|<B>R</B>|<SUP>2</SUP>cosh<SUP>2</SUP><FONT face=symbol>f</FONT> &nbsp; &nbsp; (19)</DIV><BR>from which |<B>R</B>'|=2|<B>R</B>|cosh<FONT face=symbol>f</FONT> in (18) yields <DIV align=center><BR><B>R</B>'<B>.R</B>=|<B>R</B>'||<B>R</B>|cosh<FONT face=symbol>q</FONT> &nbsp; &nbsp; (20)</DIV><BR>as, by (4)/(14), is to be expected of arbitrary four-vectors since <B>R</B>'<B>.R</B>, regardless of the relative<BR>magnitudes of <B>R</B>', <B>R</B> is always |<B>R</B>'||<B>R</B>|<B>s</B>'<B>.s</B>=|<B>R</B>'||<B>R</B>|cos(i<FONT face=symbol>q</FONT>); and, evidently, also can be demonstrated<BR><I>purely algebraically</I> from components.<BR> &nbsp;&nbsp; Evidently (20) suffices for all four-vector dot products, but we'll take the time to show it. From (20)<BR> in the form <FONT face=symbol>D</FONT><B>R</B>.<FONT face=symbol>D</FONT><B>R</B>'=|<FONT face=symbol>D</FONT><B>R</B>||<FONT face=symbol>D</FONT><B>R</B>'|cosh<font face=symbol>q</font> we get (<FONT face=symbol