- #1
alexriemann
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- Homework Statement
- I am asked to find the Killing vector fields on ##S^2## where the line element is given by ##ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi##.
- Solve the Killing equation for the ##(\theta,\theta)## components, that is ##(\mathcal L_\xi g)_{\theta\theta}=0## and show that it gives a Killing vector field ##\xi^{(1)}## that is only a function of ##\phi##, ##\xi^{(1)}=F(\phi)##. Show that this function has to be periodic and assume that it has the simple form ##\xi^{(1)}=A\sin(\phi-\phi_0)##
- Integrate the other Killing component equations and obtain two more Killing vector fields ##\xi^{(2)}## and ##\xi^{(3)}## that will depend on integration constant ##\phi_0, c_1,c_2##. By giving these constants simples values, recover the **Killing operators** ##\xi_1,\xi_2,\xi_3## given in the directions associated to the **Killing vector fields** ##\xi^{(1)},\xi^{(2)},\xi^{(3)}##.
- Relevant Equations
- $$\mathcal L_\xi g=0$$
$$\xi_1=(-\sin\phi,-\cot\theta\cos\phi)$$
$$\xi_2=(\cos\phi,-\cot\theta\sin\phi)$$
$$\xi_3=(0,1)$$
I know how to solve this problem by considering the Killing equation, namely ##\mathcal L_\xi g=0## that gives three differential equations involving the components of ##\xi=(\xi^\theta,\xi^\phi)## that can be integrated. The result I get, which I know to be true because this is a common result that can be found anywhere on the web, is:
$$\xi^\theta=\nu\cos\phi-\mu\sin\phi$$
$$\xi^\phi=\delta-\cot \theta(\mu\cos\phi+\nu\sin\phi)$$
Where ##\mu,\nu,\delta## are integration constants. By setting these constants to ##(\mu,\nu,\delta)=\{(1,0,0),(0,1,0),(0,0,1)\}##, I obtain three independent vector fields that constitute a basis for the Killing Lie algebra on ##S^2##. These are, in the chart:
$$\xi_1=(-\sin\phi,-\cot\theta\cos\phi)$$
$$\xi_2=(\cos\phi,-\cot\theta\sin\phi)$$
$$\xi_3=(0,1)$$
So, this is no big deal. However, the directions of my assignment insist that we strictly stick to the following procedure to find ##\xi_1,\xi_2,\xi_3## (these are given in the directions and match the vectors I wrote above) which is the source of my confusion:
- Solve the Killing equation for the ##(\theta,\theta)## components, that is ##(\mathcal L_\xi g)_{\theta\theta}=0## and show that it gives a Killing vector field ##\xi^{(1)}## that is only a function of ##\phi##, ##\xi^{(1)}=F(\phi)##. Show that this function has to be periodic and assume that it has the simple form ##\xi^{(1)}=A\sin(\phi-\phi_0)##
- Integrate the other Killing component equations and obtain two more Killing vector fields ##\xi^{(2)}## and ##\xi^{(3)}## that will depend on integration constant ##\phi_0, c_1,c_2##. By giving these constants simples values, recover the **Killing operators** ##\xi_1,\xi_2,\xi_3## given in the directions associated to the **Killing vector fields** ##\xi^{(1)},\xi^{(2)},\xi^{(3)}##.
I must admit that I am very confused with what my teacher is saying. He is basically saying that solving the three Killing component equations ##(\mathcal L_\xi g)_{\theta\theta}=(\mathcal L_\xi g)_{\theta\phi}=(\mathcal L_\xi g)_{\phi\phi}=0## give three Killing vector fields that depend on integration constants whereas, to my understanding, those three equation are solved for the two components of one generic vector field ##\xi=(\xi^\theta,\xi^\phi)## that give rise to three independent Killing vector fields when the integration constants are given some values.
For example, the first Killing component equation reads : ##\partial_\theta\xi^\theta=0##, which tells us that the ##\theta##-component of the Killing vector field is a function that only depends on ##\phi##. This is very different compared to saying that this equation gives a Killing vector field, isn't it?
Can someone make sense of what my teacher is trying to say or is that just wrong overall? I really do think that the directions are not only confusing but wrong. Any insight would be very much appreciated.
$$\xi^\theta=\nu\cos\phi-\mu\sin\phi$$
$$\xi^\phi=\delta-\cot \theta(\mu\cos\phi+\nu\sin\phi)$$
Where ##\mu,\nu,\delta## are integration constants. By setting these constants to ##(\mu,\nu,\delta)=\{(1,0,0),(0,1,0),(0,0,1)\}##, I obtain three independent vector fields that constitute a basis for the Killing Lie algebra on ##S^2##. These are, in the chart:
$$\xi_1=(-\sin\phi,-\cot\theta\cos\phi)$$
$$\xi_2=(\cos\phi,-\cot\theta\sin\phi)$$
$$\xi_3=(0,1)$$
So, this is no big deal. However, the directions of my assignment insist that we strictly stick to the following procedure to find ##\xi_1,\xi_2,\xi_3## (these are given in the directions and match the vectors I wrote above) which is the source of my confusion:
- Solve the Killing equation for the ##(\theta,\theta)## components, that is ##(\mathcal L_\xi g)_{\theta\theta}=0## and show that it gives a Killing vector field ##\xi^{(1)}## that is only a function of ##\phi##, ##\xi^{(1)}=F(\phi)##. Show that this function has to be periodic and assume that it has the simple form ##\xi^{(1)}=A\sin(\phi-\phi_0)##
- Integrate the other Killing component equations and obtain two more Killing vector fields ##\xi^{(2)}## and ##\xi^{(3)}## that will depend on integration constant ##\phi_0, c_1,c_2##. By giving these constants simples values, recover the **Killing operators** ##\xi_1,\xi_2,\xi_3## given in the directions associated to the **Killing vector fields** ##\xi^{(1)},\xi^{(2)},\xi^{(3)}##.
I must admit that I am very confused with what my teacher is saying. He is basically saying that solving the three Killing component equations ##(\mathcal L_\xi g)_{\theta\theta}=(\mathcal L_\xi g)_{\theta\phi}=(\mathcal L_\xi g)_{\phi\phi}=0## give three Killing vector fields that depend on integration constants whereas, to my understanding, those three equation are solved for the two components of one generic vector field ##\xi=(\xi^\theta,\xi^\phi)## that give rise to three independent Killing vector fields when the integration constants are given some values.
For example, the first Killing component equation reads : ##\partial_\theta\xi^\theta=0##, which tells us that the ##\theta##-component of the Killing vector field is a function that only depends on ##\phi##. This is very different compared to saying that this equation gives a Killing vector field, isn't it?
Can someone make sense of what my teacher is trying to say or is that just wrong overall? I really do think that the directions are not only confusing but wrong. Any insight would be very much appreciated.
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