Skip to main content
Physics LibreTexts

Page Title

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    For the Instructor’s Use

    An appendix filled with nonstop mathematical gore may seem like a strange tag onto a book intended to teach people about relativity using primarily pictures, containing only as many equations as are absolutely required. I nevertheless include it for instructors and those few readers who do not mind the math. I do so for three reasons. First, an arc-length claim is made in chapter 4 of this book that is given without visual proof, and readers might wish to know that such a proof exists. Second, one may want to determine time-dilation factors exactly for the accelerated portions of our trip to Alpha Centauri in chapter 3. Third, references that lay out relativistic relations that include acceleration without tensor analysis are almost nonexistent. An out-of-print book by Francis Sears and Robert Brehme* is the only one I have found.

    To show that the arc length in figure 10 of chapter 4 is given by equation


    (where A = GM/r2) and, ultimately, to get the Schwarzschild radius, one must first know the relations between position, coordinate time, and proper time in an accelerated system. One first correctly defines acceleration in terms of proper-time derivatives of the four-velocity,

    \[X=\{c t, \vec{r}\}, tag{A2a}\]

    \[V=\frac{d X}{d \tau}=\left\{V_{t}, \vec{V}_{r}\right\}=c \gamma, \frac{d \vec{r}}{d \tau} , tag{A2b}\]

    where the superscript μ takes the values 0, 1, 2, and 3 referring to time and the three spatial coordinates. Notice that V is the derivative of position X with respect to the proper time and whose proper time derivative is the acceleration:






    • Was this article helpful?