Relativity Lite

60 | Relativity Lite radius will slow down and stop as she or he reaches the Schwarzschild radius. You may see the details in this footnote, * if you wish. On Earth we are not infinitely far away from a gravitating body, so we cannot actually measure proper time. But all we really need anyway is a relation that compares the time * The easiest way to get meaning out of an equation is to look at its limiting values. For speeds v approaching 0, the first equation goes to t 2 2 2 1 1 0 = − = τ τ That is, time is not stretched. The picture that corresponds to this equation is shown in the figure here. Since c t approxi- mately equals c τ in this picture and c is a constant, then t must approximately equal τ . v t → 0 c τ c t Likewise, if one goes an infinite distance from a black hole, 1/r goes to 0 in the second equation, and we have the same limiting equation above: time is not stretched. (The same limit applies if the mass M of the star is very small; there is not much spacetime warpage so we would expect the coordinate time to be about the same as proper time.) This is most easily visualized by returning to the equivalent picture of a rocket being accelerated to the left. When the acceleration is near zero, the average velocity of a rocket starting from rest is also near zero. This is shown in the second figure in this note. Far from the black hole, the gravitational bending of light is so slight that the hyperbolic path c t cannot be distinguished from the hypotenuse of the figure, which is nearly the same length as the vertical leg c τ : c τ c t as r → ∞ v t → 0 average If we go to the other extreme, for speeds v approaching the speed of light, the first equation goes to t 2 2 2 1 1 1 1 0 = − = = ∞ τ τ and coordinate time t becomes infinitely stretched compared to proper time τ as in the following picture: v t c τ c t Likewise, in the case of a black hole, the second equation says that as r approaches the value r GM c s = 2 2 , the Schwarzschild radius, the denominator again goes to 1 − 1 = 0 so that coordinate time at that radius, t s , goes to infinity. This is the place in figure 10 where the hyperbolic path length for accelerated light c t stretches infinitely far to the right, relative to the vertical path length given by proper-time c τ that is measured at r = ∞ at the top of figure 10. Note that in this limit also, the hyperbola is nearly superimposed over the hypotenuse of the triangle in the figure below. c τ v t average c t Infinite gravitational time dilation happens at a finite value of r rather than at r = 0 because the speed of light is finite in the second equation, just as infinite time dilation happens at a finite value of v rather than at v = ∞ because the speed of light is finite in the first equation.