- #1

Pencilvester

- 184

- 42

- TL;DR Summary
- I was working through problem #1 under section 2.6 (pg. 54) in “A Relativist’s Toolkit” by Poisson, and I basically just want to make sure I didn’t make a mistake in any of the hundreds of places that I could have. I really need to start making myself comfortable with the computer programs that can do tensor analysis for me, but until then, I’ll have to settle for someone who has the answers to tell me if I’m right or wrong.

Here’s the metric: $$ds^2 = -dt^2+dl^2+r^2(l)d\Omega^2$$where ##r(l)## is minimum at ##l=0## with ##r(0)=r_0## and ##r## approaching ##|l|## asymptotically as ##l## approaches ##\pm \infty##

Part a of the problem seemed pretty straightforward and intuitive, but part b asks which energy conditions are violated at ##l=0##, which required me to fill out a few pages of my notebook, but I finally ended up with the answer that WEC, NEC, and SEC are all violated since $$\rho+p_2 = \rho+p_3 = -\frac{1+r_0 r’’_0}{8\pi r^2_0} <0$$where indices 2 and 3 indicate the angular directions and ##r’’_0 \equiv \frac{d^2 r}{dl^2}## at ##l=0##. Finally, I have that the dominant energy condition is not necessarily violated as long as ##r_0 r’’_0 \geq 1## so that ##\rho \geq |p_i|##.

If someone has the answers and can give me a “look’s good” or a “doesn’t look good”, that’d be much appreciated. Or, you know, if someone wants to work through the problem themselves, I’d appreciate that even more!

Part a of the problem seemed pretty straightforward and intuitive, but part b asks which energy conditions are violated at ##l=0##, which required me to fill out a few pages of my notebook, but I finally ended up with the answer that WEC, NEC, and SEC are all violated since $$\rho+p_2 = \rho+p_3 = -\frac{1+r_0 r’’_0}{8\pi r^2_0} <0$$where indices 2 and 3 indicate the angular directions and ##r’’_0 \equiv \frac{d^2 r}{dl^2}## at ##l=0##. Finally, I have that the dominant energy condition is not necessarily violated as long as ##r_0 r’’_0 \geq 1## so that ##\rho \geq |p_i|##.

If someone has the answers and can give me a “look’s good” or a “doesn’t look good”, that’d be much appreciated. Or, you know, if someone wants to work through the problem themselves, I’d appreciate that even more!