- #1
Morgoth
- 126
- 0
Well my main questions are colored in red...
Set of problem
Let's suppose we have a metric gab(x)
and the Killing vectors
Ia
( Ia;k=0, gμαIμIα= 0)
question
Show that those killing vectors are the gradient of a Scalar field, and that it satisfies the equation IαRαβγρ=0
Show that the new metric bellow describes the same space.
Gab= gab+IaIb
Attempt to solve
1. Since Ia;k=0, Ik;a=0
we get the equations
Ia,k - Γρak Iρ= 0 (1)
Ik,a - Γρka Iρ= 0 (2)
from (1)-(2) we get
Ia,k-Ik,a=0
can I say now that the Iα(x) is the gradient of a scalar?
I think I can.
2.
IαRαβγρ=0
by using the commutation of covariant derivative:
Ia;k;c-Ia;c;k=IdRdakc
but we know that Ia;b=0 so
0;c - 0;k =IdRdakc
IdRdakc=0
I think that this was an easy question.
3.
I have that
Gab= gab+IaIb
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?
Set of problem
Let's suppose we have a metric gab(x)
and the Killing vectors
Ia
( Ia;k=0, gμαIμIα= 0)
question
Show that those killing vectors are the gradient of a Scalar field, and that it satisfies the equation IαRαβγρ=0
Show that the new metric bellow describes the same space.
Gab= gab+IaIb
Attempt to solve
1. Since Ia;k=0, Ik;a=0
we get the equations
Ia,k - Γρak Iρ= 0 (1)
Ik,a - Γρka Iρ= 0 (2)
from (1)-(2) we get
Ia,k-Ik,a=0
can I say now that the Iα(x) is the gradient of a scalar?
I think I can.
2.
IαRαβγρ=0
by using the commutation of covariant derivative:
Ia;k;c-Ia;c;k=IdRdakc
but we know that Ia;b=0 so
0;c - 0;k =IdRdakc
IdRdakc=0
I think that this was an easy question.
3.
I have that
Gab= gab+IaIb
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?