Killing vector in Stephani

In summary, in section 33.3 of Stephani's "Relativity", equation (33.9) presents the Killing equations for cartesian coordinates. Upon differentiation, three additional equations are obtained. The first equation is equivalent to \partial_b \xi_a + \partial_a \xi_b = 0, but the remaining two equations do not follow this pattern. However, in a flat space without torsion, the partial derivatives commute when applied to any covector, making it possible to permute the indices without changing the equations. This allows for the permutation of indices in all six possible cases without altering the equations.
  • #1
Pengwuino
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In Stephani's "Relativity", section 33.3, equation (33.9), he has the Killing equations for cartesian coordinates as

[tex]\xi_{a,b}+\xi_{b,a}=0[/tex]

From there he says upon differentiation, you can get the following three equations

[tex]\xi_{a,bc}+\xi_{b,ac}=0[/tex]
[tex]\xi_{b,ca}+\xi_{c,ba}=0[/tex]
[tex]\xi_{c,ab}+\xi_{a,cb}=0[/tex]

Now, I'm not use to the ,; notation, but doesn't the first equation mean

[tex]\partial_b \xi_a + \partial_a \xi_b=0[/tex]?

If so, I don't understand the other 3 equations then. If for example, the first one is suppose to be subsequent differentiation by [tex]\partial_c[/tex], then wouldn't it be[tex]\xi_{a,b,c}+\xi_{b,a,c}=0[/tex]?
 
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  • #2
I think that it is supposed to be a second derivative, and the second comma is omitted. So:

[tex] \xi_{b,ca} = \partial_a(\partial_c\xi_b)[/tex]

EDIT: If you assume that, then does it work?
 
  • #3
As far as i can tell, no. He seems to be permuting the indices but I don't know what about the killing vector allows one to do that.
 
  • #4
If you're working in a flat space without a torsion, then the partial derivatives commute when applied to any covector, be it Killing or not.

So from the Killing equation [itex] \xi_{(a,b)} = 0 [/itex], differentiating it by [itex] x^c [/itex], one obtains succesively

[tex] \xi_{(a,b)c} = \xi_{(a,bc)} = 0 {}[/tex] ,

thing which allows you, Stephani and everybody else to permute the indices in every of the 6 possible cases, without changing anything.
 
  • #5


I can clarify the concept of Killing vectors in Stephani's "Relativity". A Killing vector is a vector field that satisfies the Killing equations, which are a set of equations that describe the symmetries of a spacetime. In other words, a Killing vector represents a symmetry in a spacetime that leaves the metric tensor unchanged.

In Stephani's notation, the first equation, \xi_{a,b}+\xi_{b,a}=0, represents the Killing equations for cartesian coordinates. This equation means that the vector field \xi is invariant under a change of coordinates. The other three equations, \xi_{a,bc}+\xi_{b,ac}=0, \xi_{b,ca}+\xi_{c,ba}=0, and \xi_{c,ab}+\xi_{a,cb}=0, represent the subsequent differentiation of the Killing equations by the coordinates \partial_c. This is necessary because the Killing equations hold for all coordinates, not just cartesian coordinates.

To put it simply, the first equation represents the original Killing equation while the other three equations represent subsequent differentiation of the original equation. This allows us to fully understand the symmetries of a spacetime and how they are represented by Killing vectors.
 

What is a Killing vector in Stephani?

A Killing vector in Stephani is a vector field in a four-dimensional spacetime that preserves the metric of the spacetime. In other words, it is a vector that describes a symmetry of the spacetime.

What is the significance of Killing vectors in Stephani?

Killing vectors in Stephani play a key role in understanding the symmetries of a spacetime and determining the solutions to Einstein's field equations. They also have important applications in general relativity and cosmology.

How are Killing vectors in Stephani related to geodesics?

Killing vectors in Stephani are closely related to geodesics, which are the paths that particles follow in spacetime. In fact, for each Killing vector, there is a corresponding conserved quantity along geodesics, making it easier to solve for geodesics in a spacetime with symmetries.

Can Killing vectors in Stephani be used to solve the Einstein field equations?

Yes, Killing vectors in Stephani can be used to solve the Einstein field equations. They provide a powerful tool for finding exact solutions to these equations, especially in the case of highly symmetric spacetimes.

Are Killing vectors in Stephani unique to four-dimensional spacetimes?

No, Killing vectors can exist in any dimension of spacetime. However, in four-dimensional spacetime, there are a maximum of ten independent Killing vectors, known as the Killing-Yano tensors, which have special properties and are essential in constructing the most general solutions to Einstein's equations.

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