Relativity Lite

Gravity Lite | 59 is fascinating to note that a dark-complected scientist and parson named John Mitchell predicted this result using Newtonian arguments in 1783. * GRAVITATIONAL TIME DILATION In the section of chapter 2 entitled “The Low-Velocity Limit,” we read the Pythagorean theorem off of figure 1, c t c v t 2 2 2 2 2 2 = + τ , to give the exact expression for the time-dilation factor of special relativity: γ τ = ≡ − t v c 1 1 2 2 We wish to rewrite this in a pattern that will be useful for general relativity and for cosmology by squaring this and rearranging to t v c 2 2 2 2 1 1 = − τ For general relativity, the equivalent diagram, figure 1c, is much more difficult to solve for the length of t relative to τ . A great deal of mathematical work to find the length of this hyperbolic curve (see the appendix) gives t r GM c 2 2 2 1 1 1 2 = − × τ Inthe context of ablackhole, t is the timemeasuredby the astronautnear theSchwarzschild radius, and τ is that for an astronaut infinitely far away. The Earth-Sun distance is generally far enough away enough to give τ to good approximation. In chapter 2, we saw that coordinate time t became infinitely dilated when v = c or, alter- natively expressed, when v 2 = c 2 . If you compare the above two equations, looking for pat- terns as an artist might, you will notice that the position occupied by v 2 in the first equation is occupied by 1 2 r GM × in the second equation. Since there is order in mathematics, t must become infinitely dilated when 1 2 2 r GM c × = , or alternatively expressed, when 1 2 2 c GM r × = . But this value of the distance is precisely the Schwarzschild radius r GM c s = 2 2 we just derived. What this means is that the wristwatch on an astronaut approaching the Schwarzschild * “The Country Parson Who Conceived of Black Holes,” American Museum of Natural History, https://​www​.amnh​.org/​ learn​-teach/​curriculum​-collections/​cosmic​-horizons/​case​-study​-john​-michell​-and​-black​-holes (accessed April 29, 2020).