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Here is my first LaTeX attempt. I understand the warnings, but I don't understand the errors. I can't find anything on these questions in the guides:

  1. missing characters from the LaTeX script listed in LOG (whitespace, and 'a' with some mark over it--I didn't use anything like that. I did paste in the early part of the manuscript from a WORD document, without math)

  2. When I use $ or $$ for math expressions with Greek letters, it says I am not in the math environment, but when I use \[ and \], it says it doesn't understand these commands (undefined).

  3. I can't understand what's wrong with the second to last display equation (the one after the words 'hyperbolic functions that'.

  4. How do I do an inner product to look like ((x,it),(y,iu)). Is this right (with $ signs)?

  5. I read in one of the guides that Latex will compile even with errors, but if 'compile' means 'run', it does not run. I'm told to put latex rel3.tex or dvipdf on the command line to run. What do they mean by the 'command line'? Nothing happens when I type these into my LaTeX file. Where are the resulting files supposed to be?

  6. Where is an explanation of the color codes (pink, dark pink, green, yellow, red ,etc.?

I would really appreciate a detailed solution of these problems, not a rough indication or direction to another document). Best of all, please show me the corrected form of my document in addition to explaining how to fix the errors. I am at my wit's end. Research in relativity is a lot easier than figuring out LaTeX! --Thanks, Murray

PS: How can I attach the Latex file to this msg, so someone can edit it? I copied in the log below the latex.

\documentclass{article}

\title{Superluminal Velocities as Subluminal Motion Backward in Time}
\author{M.E.Denofsky}

\begin{document}

\begin{abstract}

A consideration of invariant complex angles in Minkowski space-time shows that space-like and time-like worldlines lie in orthogonally complementary planes. The imaginary Lorentz angle of rotation $i\phi$ and Majernik’s real angle $\theta$, where $\beta=v/c=\sin \theta$, combine into a consistent space-time representation of 4 complex dimensions, in a manner suggesting that rotation of the axes beyond $v=c$ corresponds to backward movement in time at a subluminal velocity. 
Many of the paradoxes associated with the Lorentz transformation can be resolved by such transformations of coordinate systems. 

\end{abstract}

\section{Introduction}
\label{sec-Introduction}
It appears to be generally assumed that the angle between the space and time axes of an inertial frame is a real right angle, as in Galilean coordinates, presumably from analogical and symmetry arguments, as there is no reason to select any other value. But this is far from obvious, in particular when we employ Minkowski coordinates, in which one of time or space is imaginary, and the other real. Nevertheless, we will show that, viewing a Lorentz transformation (LT) as a rotation, the angle is indeed a real right angle, despite the fact that the rotation angle between the two time axes is imaginary.

However, Majernik~\cite{maj} (1986), commented on in Wilkins and Williams (2000, appendix), shows that an alternative viewpoint has much to be said for it, viz.: that the angle between the two time axes is real, but that it already reaches $\pi/2$ when the moving time axis reaches that of a light ray ($v=c$). Under this approach, as the above authors point out, space-like intervals and superluminal velocities are not represented in the real $x-t$ plane, which is in keeping with physical reality. Space-like intervals can then be represented in the orthogonal complement plane, the $ix-it$ plane, where there is no temptation to interpret them as velocities. 

More important, this representation is consistent with the values of the complex angles that we shall show to exist between the space-like and time-like zones of Minkowski space-time, suggesting that rotation beyond $v=c$ corresponds to backward movement in time at a subluminal velocity. This is in keeping with what one might expect if acceleration is a process of rotation, rather than a limitless increase of velocity along a linear dimension. It is also consistent with the fact that a particle crossing a Schwarzschild radius, while continuing to accelerate due to the black hole’s gravity, is actually moving backward in time in our frame, and so has negative energy, as in the case of the infalling Hawking radiation particle.

Finally, if we

In this paper, we shall restrict ourselves to one space dimension, as that is sufficient to demonstrate our results, although I mention in the bibliography some papers that explore various unexpected difficulties in generalizing the standard LT equations to two and three spatial dimensions.

\section{Invariant Complex Angles}
The standard approach to the LT emphasizes  the invariant measure $\tau$ of a space-time interval, but neglects that, where there are invariant measures for lengths, there must also be invariant measures for angles. Interval invariance implies a metric for both intervals and angles, the latter obtainable from the former via the cosine law~/footnote{As in the customary representation of a LT as a rotation through an imaginary angle, we take the length of a complex segment $z$ to be its actual complex value, rather than its absolute value, $\sqrt{z\ast z}$, the practice when using complex amplitudes in quantum mechanics to define probabilities, which have to be real positive. Thus we are using analytic expressions, and complex analysis tells us that the laws of real analysis still hold, including common geometric and trigonometric results, such as Euclid’s theorem on the sum of the angles of a triangle, the Pythagorean theorem, and the cosine law.} Thus, taking the letters denoting the vertices to also denote the (possibly) complex angles at these vertices, a complex triangle $UVW$ with sides \mbox{$u$, $v$, $w$} obeys
  \[ w^{2} = u^{2} + v^{2} - 2uv \cos W \].

Now consider the special case $u^{2} + v^{2} = w^{2}$, as for a proper time $\tau$ corresponding to a time $it$ and a distance $x$.~\footnote{Although $t$ and $x$ depend on the frame, we may take them as invariant here, since they are the proper time and proper distance for our original frame, which we are not changing. They are invariant as long as we use them only to measure their own intervals, not the time and distance of the moving body.  The angle between the frame axes, measured in this same frame, may be defined as the \emph{proper frame angle}, also invariant, as are the other angles in our right triangle, as they are determined by the invariant measures of the sides.} Letting $W = a+ib$, and expanding $\cos W$ by the cosine addition formula, noting that $\cos ib  = \cosh b$ and $\sin ib  =i \sinh b$, it is easy to show from the functional equations of the circular and hyperbolic functions that                          
  $$\cos W = 0  \mbox{ iff  /( a=\pm \pi /2 \) and \(b=0\)}$$

This proves our claim at the beginning of Section, that Einstein’s invariant interval formula implies that the angle (proper frame angle) between the time and space axes is real, and is $\pi/2$. Thus, if the angle between $it$ and $i\tau$ is $i\phi$\footnote{A good exposition of the LT in terms of $i\phi$ is given in \cite{cpxangle}.}, then the angle between $i\tau$ and the space axis $x$ must be $\pi /2-i\phi$. (The angle sum theorem may  be verified for the right triangle by adding the three angles we now know; the imaginary components add to zero, as they must for any complex triangle.) 

Since our frame was an arbitrary one, and since the same reasoning can be applied to the angles between the frame axes and the space axis of the moving body, it follows that in Minkowski space the angle between two worldlines of the same kind (space-like\footnote{but in the same space direction; it will be complex if 3-space is used} or time-like) is purely imaginary if they both go in the same time (or space) direction from their common vertex, while the angle between two lines of opposite kind is an imaginary angle plus $\pi ⁄2$. It follows also that the angle between two time-like (space-like) vectors going in opposite time (space) directions is an imaginary angle plus $\pi$, since the angle is the sum of two complex angles like that between the $\tau$-vector and the $x$-axis. These two angles add, as they are in the same plane. These results can be verified by taking the inner product of two general space-time vectors:
$$((x,it),(y,iu))$$


or the fact that a distant particle approaching us from beyond our cosmic event horizon at the speed of light, before crossing which it must have been accelerating to accelerate due to tour expansion of the universe)
\end{document}

Log file:

This is pdfTeX, Version 3.1415926-2.4-1.40.13 (MiKTeX 2.9 64-bit) (preloaded format=pdflatex 2014.2.9)  13 FEB 2014 13:34
entering extended mode
**rel3.tex
(C:\Users\User\Downloads\rel3.tex
LaTeX2e <2011/06/27>
Babel <v3.8m> and hyphenation patterns for english, afrikaans, ancientgreek, ar
abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic, 
croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga
lician, german, german-x-2012-05-30, greek, gujarati, hindi, hungarian, iceland
ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian,
 lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n
german-x-2012-05-30, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman
ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm
an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl
ishmax, welsh, loaded.

! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              

l.1 d
     vipdfm
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no d in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no m in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 1--2
[] 
 []

("C:\Program Files\MiKTeX 2.9\tex\latex\base\article.cls"
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
("C:\Program Files\MiKTeX 2.9\tex\latex\base\size10.clo"
File: size10.clo 2007/10/19 v1.4h Standard LaTeX file (size option)
)
\c@part=\count79
\c@section=\count80
\c@subsection=\count81
\c@subsubsection=\count82
\c@paragraph=\count83
\c@subparagraph=\count84
\c@figure=\count85
\c@table=\count86
\abovecaptionskip=\skip41
\belowcaptionskip=\skip42
\bibindent=\dimen102
 [1{C:/ProgramData/MiKTeX/2.9/pdftex/config/pdftex.map}]) (C:\Users\User\Downlo
ads\rel3.aux)
LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 8.
LaTeX Font Info:    ... okay on input line 8.
LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 8.
LaTeX Font Info:    ... okay on input line 8.
LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 8.
LaTeX Font Info:    ... okay on input line 8.
LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 8.
LaTeX Font Info:    ... okay on input line 8.
LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 8.
LaTeX Font Info:    ... okay on input line 8.
LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 8.
LaTeX Font Info:    ... okay on input line 8.
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <9> on input line 12.
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <6> on input line 12.
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <5> on input line 12.
Missing character: There is no â in font cmr9!
Missing character: There is no  in font cmr9!
Missing character: There is no  in font cmr9!


LaTeX Warning: Citation `maj' on page 2 undefined on input line 21.

LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <7> on input line 21.
Missing character: There is no â in font cmr10!
Missing character: There is no  in font cmr10!
Missing character: There is no  in font cmr10!
Missing character: There is no â in font cmr10!
Missing character: There is no  in font cmr10!
Missing character: There is no  in font cmr10!
[2

]
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <8> on input line 33.

! Missing $ inserted.
<inserted text> 
                $
l.34 ...mbox{ iff  /( a=\pm \pi /2 \) and \(b=0\)}
                                                  $$
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

Missing character: There is no â in font cmr10!
Missing character: There is no  in font cmr10!
Missing character: There is no  in font cmr10!

LaTeX Warning: Citation `cpxangle' on page 3 undefined on input line 36.

Missing character: There is no â in font cmr10!
Missing character: There is no  in font cmr10!
Missing character: There is no  in font cmr10!
[3] [4] (C:\Users\User\Downloads\rel3.aux)

LaTeX Warning: There were undefined references.

 ) 
Here is how much of TeX's memory you used:
 242 strings out of 493921
 2630 string characters out of 3147286
 56031 words of memory out of 3000000
 3606 multiletter control sequences out of 15000+200000
 8257 words of font info for 29 fonts, out of 3000000 for 9000
 841 hyphenation exceptions out of 8191
 23i,8n,24p,1148b,158s stack positions out of 5000i,500n,10000p,200000b,50000s
<C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmbx10.pfb><C:/Pr
ogram Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmbx12.pfb><C:/Program Fi
les/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmbx8.pfb><C:/Program Files/MiKTe
X 2.9/fonts/type1/public/amsfonts/cm/cmbx9.pfb><C:/Program Files/MiKTeX 2.9/fon
ts/type1/public/amsfonts/cm/cmmi10.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1
/public/amsfonts/cm/cmmi8.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/a
msfonts/cm/cmmi9.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/c
m/cmr10.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmr6.pf
b><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmr7.pfb><C:/Prog
ram Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmr8.pfb><C:/Program Files/
MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmr9.pfb><C:/Program Files/MiKTeX 2.9
/fonts/type1/public/amsfonts/cm/cmsy10.pfb><C:/Program Files/MiKTeX 2.9/fonts/t
ype1/public/amsfonts/cm/cmti8.pfb>
Output written on rel3.pdf (4 pages, 163177 bytes).
PDF statistics:
 71 PDF objects out of 1000 (max. 8388607)
 0 named destinations out of 1000 (max. 500000)
 1 words of extra memory for PDF output out of 10000 (max. 10000000)
share|improve this question

closed as unclear what you're asking by Joseph Wright Sep 7 at 17:28

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

    
Delete the .aux file and re-compile. Why? The .aux file is read as part of \document and the errors seem to include stuff that is not part of your actual code. –  Werner Feb 13 at 18:54
    
one obvious thing missing is the command \maketitle which should appear before the abstract. other than that, in the display on l.32, there is a /( instead of \( which will get the math out of synch, and, since it's in an \mbox, will result in the error ! Missing $ inserted. –  barbara beeton Feb 13 at 19:07
    
Welcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. Incase you are interested: Debugging can be made by creating MWE as described by Nicola Talbot book –  texenthusiast Feb 13 at 19:55
    
No one really understands all the error messages. What is important is the line the error occurs on. –  John Kormylo Feb 14 at 3:13
    
As it stands this isn't a good question. It seems you need to read some introductions to LaTeX then asked focussed questions on specific topics. –  Joseph Wright Sep 7 at 17:29

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