# Killing spinor equation [closed]

The Supersymmetry transformation is:

$$\delta \psi_\mu^i=(\partial_\mu +1/4 \gamma^{ab}\omega_{\mu ab})\epsilon^i -1/8\sqrt{2}\kappa \gamma^{ab}F_{ab}\epsilon^{ij} \gamma_\mu \epsilon_j$$ For the time part, we substitute $\mu$ by $t$ so equation (2) is:

$$\delta \psi_{tA} = \partial_t \epsilon _A + \frac{1}{4} \gamma^{ab} \omega_{tab} \epsilon_A - \frac{1}{8} \sqrt{2} \kappa F_{ab} \gamma^{ab} \gamma_t \epsilon_{AB} \epsilon^B =0$$

Automatically since the spin connections I found to be:

$\omega^{0i} = e^U \partial_iU e^0$ and $\omega ^{ij} = -dx^i\partial_jU+dx^j\partial_iU$

Implies that the ${a, b}$ indices in equation (2) should either be ${0,i}$ or ${i, j}$ $$\delta \psi_{tA} = \partial_t \epsilon _A + \frac{1}{4} \gamma^{0i} \omega_{t0i} \epsilon_A - \frac{1}{8} \sqrt{2} \kappa F_{0i} \gamma^{0i} \gamma_t \epsilon_{AB} \epsilon^B + \frac{1}{4} \gamma^{ij} \omega_{tij} \epsilon_A - \frac{1}{8} \sqrt{2} \kappa F_{ij} \gamma^{ij} \gamma_t \epsilon_{AB} \epsilon^B =0$$

From here, I am having difficulty in proceeding. Some papers like http://arxiv.org/abs/hep-th/0608139 see (4.3.27) separate the $\gamma^{0i}$ into $\gamma^{0}\gamma^{i}$.

This is the first time I encounter such problem, I have no idea how to proceed from here in order to find 2 conditions for ERN BH.

• This is a (admittedly very high level) homework question. Please check the homework policy. Specifically, why are you not sure how to proceed? What conceptual issues are you having? – Sean Dec 3 '14 at 14:23
• @Sean if I continued from where I stopped, this would be: $$\partial_t \epsilon_A + \frac{1}{4} \gamma^{0i} e^U \partial_iUe^0 \epsilon_A + \frac{1}{4} \gamma^{ij}(-dx^i \partial_jU+dx^j \partial_iU )\epsilon_A -\frac{1}{8} \sqrt{2} \kappa (-\partial_i A_t) \gamma^{0i} \gamma_t \epsilon_{AB} \epsilon^B = 0$$ But the final answer written in the book (actually this is not a homework but an exercise I was following) is: $$\delta \psi_{tA} = \partial_t \epsilon _A +1/2 e^{2U} \partial_i U\gamma^i \gamma^0 \epsilon_A -1/4 \sqrt{2}\kappa e^u \partial_i A_t \gamma^i \epsilon_{AB} \epsilon^b =0$$ – beyondtheory Dec 3 '14 at 14:31
• @Sean so how does my equation match the book's, I think there is something I am doing wrong.. That is why I am not sure how to proceed. – beyondtheory Dec 3 '14 at 14:33
• @beyondtheory: If it is not homework, but an exercise, it can still count as homework under our policies, since it is homework-like questions which don't meet certain criteria that are likely to be closed. – JamalS Dec 3 '14 at 14:38
• The issue is that this question is asking nothing more than "how do I proceed?" But I'm sure it could easily be edited into a good question. As Sean said, you should ask something about the concept that's confusing you. If you can find a reference that shows the steps to this sort of procedure, you could even ask how to justify going from one step to the next. Maybe if you showed that you very thoroughly searched for such a reference, and that it's just not out there, that might change things (no guarantees though). – David Z Dec 4 '14 at 12:25