这是我第一次尝试使用 LaTeX。我理解警告，但我不明白错误。我在指南中找不到有关这些问题的任何信息：
LOG 中列出的 LaTeX 脚本中缺少字符（空格和带有某些标记的“a”——我没有使用任何类似的东西。我确实从 WORD 文档中粘贴了手稿的早期部分，没有数学）
当我使用
$或$$表示带有希腊字母的数学表达式时，它说我不在数学环境中，但是当我使用\[和 时\]，它说它不理解这些命令（未定义）。我不明白倒数第二个显示方程（“双曲函数”后面的方程）有什么问题。
我如何使内积看起来像 ((x,it),(y,iu))。这是正确的吗（带有 $ 符号）？
我在一本指南中看到，即使出现错误，Latex 也会编译，但如果“编译”表示“运行”，它就不会运行。我被告知要在命令行上输入
latex rel3.tex或dvipdf才能运行。他们所说的“命令行”是什么意思？当我将这些输入到我的 LaTeX 文件中时，什么都没有发生。结果文件应该在哪里？哪里有关于颜色代码（粉红色、深粉红色、绿色、黄色、红色等）的解释？
我真的希望得到这些问题的详细解决方案，而不是粗略的指示或指向另一份文档的方向）。最好，除了解释如何修复错误之外，请向我展示我文档的更正形式。我已经束手无策了。研究相对论比弄清楚 LaTeX 容易得多！——谢谢，Murray
附言：我怎样才能将 Latex 文件附加到此消息，以便有人可以编辑它？我已将日志复制到 Latex 下方。
\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}
日志档案：
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