Relativity Lite

Appendix | 93 APPENDIX 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, refer- ences 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 τ 2 2 2 1 2 = − Ar c t , (A1) (where A = GM/r 2 ) 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 ct r µ = { } ,  , (A2a) V dX d V V c dr d t r µ µ τ γ τ = = { } =       , ,   , (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: A dV d A A t r µ µ τ = = { } ,  . (A2c) * F. W. Sears and R. W. Brehme, Introduction to the Theory of Relativity (Addison-Wesley, Reading, MA, 1968), p. 102 ff.