@Ibix, Last question for you: how can I get an expression for ##\frac{d\tau}{d\lambda}## that evolves concerning the trajectory? Because of how I have defined it, I am sure that ##\frac{d\tau}{d\lambda} = 1## or 0 for null geodesics. So, then, proper time evolves linearly with lambda. So I have...
@Ibix Is this because motion is purely in the r-direction for a radial trajectory with no angular displacement? so then ## \frac{d\phi}{d\lambda}=0 ## and we are effectively dividing by zero? How else might I write my system of equations, then? In the timelike case, use affine parameter ## \tau...
Hi all,
I am working through Sean Carroll's Textbook, particularly Chapter 5 regarding the Schwarzschild Solution. In this chapter, Energy and Angular Momentum are defined as follows:
$$
\begin{align}
E &= (1-\frac{2GM}{r})\frac{dt}{d\lambda} \Rightarrow \frac{dt}{d\lambda} = (1-...