# Integration trouble (integral over a 2-sphere)

• I
• etotheipi
In summary, we can use the Killing equation and antisymmetry properties to simplify the original integral into a more manageable form, which will hopefully lead to a solution.
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.

Last edited by a moderator:
JD_PM, Twigg and Dale
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?

etotheipi
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:
Twigg
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:

## 1. What is integration over a 2-sphere?

Integration over a 2-sphere involves calculating the volume or surface area of a three-dimensional object that is bounded by a two-dimensional sphere. It is a type of integration used in mathematics and physics to solve problems related to spherical objects.

## 2. How is integration over a 2-sphere different from regular integration?

The main difference is that integration over a 2-sphere involves calculating the volume or surface area of a three-dimensional object, while regular integration deals with finding the area under a curve in two dimensions. Additionally, integration over a 2-sphere requires the use of spherical coordinates instead of Cartesian coordinates.

## 3. What are some common applications of integration over a 2-sphere?

Integration over a 2-sphere is commonly used in physics to calculate the electric field and potential of a charged spherical object. It is also used in mathematics to solve problems related to spherical geometry, such as finding the area of a spherical triangle or the volume of a spherical cap.

## 4. What are some challenges or difficulties in performing integration over a 2-sphere?

One of the main challenges is converting the integral to spherical coordinates, which can be complex and time-consuming. Another difficulty is dealing with singularities, where the integrand becomes infinite, which requires special techniques to handle.

## 5. Are there any techniques or strategies for solving integration over a 2-sphere?

Yes, there are several techniques that can be used, such as substitution, integration by parts, and using symmetry properties of the integral. It is also helpful to have a good understanding of spherical coordinates and their properties. Practice and familiarity with these techniques can make integration over a 2-sphere easier and more efficient.

• Special and General Relativity
Replies
1
Views
390
• Special and General Relativity
Replies
7
Views
1K
• Special and General Relativity
Replies
11
Views
1K
• Special and General Relativity
Replies
2
Views
874
• Special and General Relativity
Replies
21
Views
2K
• Special and General Relativity
Replies
4
Views
738
• Special and General Relativity
Replies
16
Views
2K
• Special and General Relativity
Replies
2
Views
972
• Special and General Relativity
Replies
2
Views
825
• Special and General Relativity
Replies
1
Views
897