I Integration trouble (integral over a 2-sphere)

  • I
  • Thread starter Thread starter etotheipi
  • Start date Start date
  • Tags Tags
    Integration
etotheipi
There's an integral over a 2-sphere ##S## with unit normal ##N^a## within a hypersurface orthogonal to a Killing field ##\xi^a##$$F = \int_S N^b (\xi^a / V) \nabla_a \xi_b dA = \frac{1}{2} \int_S N^{ab} \nabla_a \xi_b dA, \quad N^{ab} := 2V^{-1} \xi^{[a} N^{b]}$$which follows because the Killing equation is ##\nabla_{a} \xi_b = \nabla_{[a} \xi_{b]}## and we can also write ##\xi^a N^b \nabla_{[a} \xi_{b]} = \xi^a N^b \delta^{[c}_{a} \delta^{d]}_b \nabla_c \xi_d = \xi^{[c} N^{d]} \nabla_c \xi_d##. The original integral is supposed to transform into$$F = \frac{-1}{2} \int_S \epsilon_{abcd} \nabla^c \xi^d$$but I don't see how yet. Can anyone provide a hint? Thanks. :smile:
 
Last edited by a moderator:
  • Like
Likes JD_PM, Twigg and Dale
Physics news on Phys.org
Did part of the last equation get lost to a typo? The final result is rank 2 (a and b are free) but the original integral is a scalar. Am I missing something?
 
As far as I can tell they're the same as in the book; the indices in this case are abstract, so I reckon the second should be understood as the integral of a 2-form over the submanifold.
 
Last edited by a moderator:
After some helpful discussions with @Twigg, here's a possible idea: first we will use that ##\nabla_a \xi_b = \nabla_{[a} \xi_{b]}##, and also use that the volume form ##\epsilon_{ab}## on the 2-sphere is totally antisymmetric, i.e. ##\epsilon_{ab} = \epsilon_{[ab]}##,\begin{align*}F = \frac{1}{2} \int_S N^{ab} \nabla_a \xi_b \mathrm{d}A &= \frac{1}{2} N^{ab} \nabla_{[a} \xi_{b]} \epsilon_{cd} \\

&= \frac{1}{2} \int_S N_{[ab]} \nabla^a \xi^b \epsilon_{[cd]} \\

&= \frac{1}{2} \int_S \nabla^a \xi^b \delta^{[e}_a \delta^{f]}_b \delta^{[g}_c \delta^{h]}_d N_{ef} \epsilon_{gh}

\end{align*}However, since ##\delta^{[e}_a \delta^{f]}_b \delta^{[g}_c \delta^{h]}_d = \frac{1}{4} \delta^{e}_a \delta^{f}_b \delta^{g}_c \delta^{h}_d = 6 \delta^{[e}_a \delta^{f}_b \delta^{g}_c \delta^{h]}_d##, this is simply\begin{align*}

F &= \frac{1}{2} \int_S \nabla^a \xi^b \cdot 6 N_{[ab} \epsilon_{cd]} \\

&= \frac{-1}{2} \int_S \nabla^a \xi^b \epsilon_{abcd} \\

\end{align*}where the last line follows because ##\epsilon_{abcd} = -6N_{[ab} \epsilon_{cd]}##
 
Last edited by a moderator:
Thread 'Can this experiment break Lorentz symmetry?'
1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
According to the General Theory of Relativity, time does not pass on a black hole, which means that processes they don't work either. As the object becomes heavier, the speed of matter falling on it for an observer on Earth will first increase, and then slow down, due to the effect of time dilation. And then it will stop altogether. As a result, we will not get a black hole, since the critical mass will not be reached. Although the object will continue to attract matter, it will not be a...
Back
Top