- #1

binbagsss

- 1,278

- 11

## Homework Statement

Killing Equation is: ##\nabla_u K_v + \nabla_v K_u =0 ##

In flat s-t this reduces to:

##\partial_u K_v + \partial_v K_u =0 ##

With a general solution of the form:

##K_u= a_u + b_{uc} K^c ##

where ##a_u## and ##b_{uv}## are a constant vector and a constant tensor

QUESTION

Show that ##b_{uv}## is antisymmetric: ##b_{uv}=-b{vu}##

## Homework Equations

see above

## The Attempt at a Solution

[/B]

subbing in the general solution form I have:

## \partial_u (a_v+ b_{va}x^a) + \partial_v (a_u+ b_{ua}x^a) ##

From Kiling's equation it is obvious that ##\partial_v K_u ## is antisymmetric in ##u,v ##

Here I need to look at ##v##and ##a## in the first term and ##u## and ##a## in the second term, ?where the ##a## is summed over in both terms, which is throwing me of a bit and I'm not sure what to do?

Or if I use the antisymmetry of ##u## and ##v## and substitute in the general form of ##K^u## I have:

##\partial_u(a_v+b_{va}x^a)=-\partial_v(a_u+b_{ua}x^a)##

and now I am not sure what to do

Many thanks