Well my main questions are colored in red....(adsbygoogle = window.adsbygoogle || []).push({});

Set of problem

Let's suppose we have a metric g_{ab}(x)

and the Killing vectors

I_{a}

( I_{a;k}=0, g^{μα}I_{μ}I_{α}= 0)

question

Show that those killing vectors are the gradient of a Scalar field, and that it satisfies the equation I_{α}R^{α}_{βγρ}=0

Show that the new metric bellow describes the same space.

G_{ab}= g_{ab}+I_{a}I_{b}

Attempt to solve

1. Since I_{a;k}=0, I_{k;a}=0

we get the equations

I_{a,k}- Γ^{ρ}_{ak}I_{ρ}= 0 (1)

I_{k,a}- Γ^{ρ}_{ka}I_{ρ}= 0 (2)

from (1)-(2) we get

I_{a,k}-I_{k,a}=0

can I say now that the I_{α}(x) is the gradient of a scalar?

I think I can.

2.

I_{α}R^{α}_{βγρ}=0

by using the commutation of covariant derivative:

I_{a;k;c}-I_{a;c;k}=I_{d}R^{d}_{akc}

but we know that I_{a;b}=0 so

0_{;c}- 0_{;k}=I_{d}R^{d}_{akc}

I_{d}R^{d}_{akc}=0

I think that this was an easy question.

3.

I have that

G_{ab}= g_{ab}+I_{a}I_{b}

how can I show that it describes the same space?

one way I thought of is to calculate the R' riemann tensor of metric G and show that it is equal to R tensor of the metric g. But things get lost in calculations...

what is your opinion?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Killing Vectors (question on Problem)

Loading...

Similar Threads - Killing Vectors question | Date |
---|---|

A Why is Killing vector field normal to Killing horizon? | Jan 26, 2018 |

Killing tensor/vector very basic context question about consevation | Jan 6, 2015 |

Killing Vectors Question | Jan 27, 2014 |

Killing Vector Question | Jan 30, 2011 |

Question about Killing vectors in the Kerr Metric | Jan 6, 2011 |

**Physics Forums - The Fusion of Science and Community**