- #1
Omikron123
- 6
- 1
- Homework Statement
- I have an object orbiting in free fall with constant radius ##r## in the plane ##\theta = \frac{\pi}{2}##.
I am supposed to prove that the 4-velocity ##U = a\partial _t + b\partial _\phi## and find the values of ##a## & ##b## for a free falling object in the plane ##\theta = \pi/2##
- Relevant Equations
- The Schwarzchild line element is given by $$ ds^2 = (1- \frac{R_s}{r})dt^2 - (1- \frac{R_s}{r})^-1dr^2 - r^2d\Omega ^2$$
So the line element is given by $$ ds^2 = (1- \frac{R_s}{r})dt^2 - (1- \frac{R_s}{r})^{-1}dr^2 - r^2d\Omega ^2$$
The object is orbiting at constant radius ##r## in the plane ## \theta = \frac{\pi}{2}##. I am supposed to find the values of ##a## and ##b## in the 4-velocity given by: $$U = a\partial _t + b\partial _\phi$$.
Im pretty new the general relativity and Schwarzschild geodesics but here is my attempted solution:
For a massive particle the squared 4-velocity ##U^2 = -1##, space-like which i can expand with the Schwarzschild metric, which is diagonal:
$$ U^2 = U^tU_t + U^\phi U_\phi = (g^{ta}U_a)U_t + (g^{\phi a}U_\phi) = -1$$ Since the metric is diagonal only ##g^{tt}, g^{\phi \phi}## are non-zero with values $$g^{\phi \phi} = \frac{1}{g_{\phi \phi}} = -\frac{1}{r^2}, g^{tt} = ... = \frac{1}{1-\frac{R_s}{r}}$$ At this point I am not sure how to continue, because I am not sure if ##U_\phi = \partial _\phi## etc. One idea was to compare the following:
$$U^2 = g^{tt}U_tU_t + g^{\phi \phi}U_\phi U_\phi = -\frac{1}{r^2}\partial _t^2 + \frac{1}{1-\frac{R_s}{r}}\partial _\phi ^2$$ and $$U^2 = (a\partial _t + b\partial _\phi)^2 = a^2\partial _t ^2 + b^2\partial _\phi^2$$ give ## a = \sqrt{\frac{1}{1-\frac{R_s}{r}}}, b = \sqrt{-\frac{1}{r^2}}## (since ##2ab\partial _t\partial _\phi = 0 ## due to diagonal metric??) As I said I don't really know what I am doing here so there might be some major errors in my thinking..
The object is orbiting at constant radius ##r## in the plane ## \theta = \frac{\pi}{2}##. I am supposed to find the values of ##a## and ##b## in the 4-velocity given by: $$U = a\partial _t + b\partial _\phi$$.
Im pretty new the general relativity and Schwarzschild geodesics but here is my attempted solution:
For a massive particle the squared 4-velocity ##U^2 = -1##, space-like which i can expand with the Schwarzschild metric, which is diagonal:
$$ U^2 = U^tU_t + U^\phi U_\phi = (g^{ta}U_a)U_t + (g^{\phi a}U_\phi) = -1$$ Since the metric is diagonal only ##g^{tt}, g^{\phi \phi}## are non-zero with values $$g^{\phi \phi} = \frac{1}{g_{\phi \phi}} = -\frac{1}{r^2}, g^{tt} = ... = \frac{1}{1-\frac{R_s}{r}}$$ At this point I am not sure how to continue, because I am not sure if ##U_\phi = \partial _\phi## etc. One idea was to compare the following:
$$U^2 = g^{tt}U_tU_t + g^{\phi \phi}U_\phi U_\phi = -\frac{1}{r^2}\partial _t^2 + \frac{1}{1-\frac{R_s}{r}}\partial _\phi ^2$$ and $$U^2 = (a\partial _t + b\partial _\phi)^2 = a^2\partial _t ^2 + b^2\partial _\phi^2$$ give ## a = \sqrt{\frac{1}{1-\frac{R_s}{r}}}, b = \sqrt{-\frac{1}{r^2}}## (since ##2ab\partial _t\partial _\phi = 0 ## due to diagonal metric??) As I said I don't really know what I am doing here so there might be some major errors in my thinking..
Last edited: