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What is a timelike killing vector? 
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#1
Aug2504, 04:27 PM

P: 653

What is a timelike killing vector?



#2
Aug2504, 04:54 PM

P: 38

Unless given further explanation I´d say it´s exactly what the name sais:
Killing vector: A vector that fulfies the Killingequation [tex] v_{i;j} + v_{j;i} = 0 [/tex]. The existence of a Killingvector implies the existance of a coordinate system where the metric tensor is independent of one of the coordiantes. timelike: A vector v is timelike if [tex] g_{ij} v^{i} v^{j} >0 [/tex]. EDIT: As pmb_phy correctly claims I should mention that above inequality assumes the signature of the metric to be (+,,,). 


#3
Aug2504, 05:35 PM

P: 2,954

[tex]g'_{\alpha\beta}(x') = g_{\alpha\beta}(x') [/tex] For the components of the metric tensor invariant under the isometry we must have [tex]g_{\mu\nu} (x) = \frac{\partial x'^{\alpha}}{\partial x^{\beta}}\frac{\partial x'^{\mu}}{\partial x^{\nu}}g(x'(x))[/tex] Consider the infinitesimal coordinate transformation [tex]x' = x^{\alpha} + \epsilon \xi^{\alpha}[/tex] where [tex]\xi^{\alpha}(x)[/tex] is a vector field and [tex]\epsilon[/tex] > 0. For this coordinate transformation to yield an isometry the [tex]\xi^{\alpha}[/tex] must satisfy the following equation [tex]\xi_{\mu;\nu} + \xi_{\nu;\mu} = 0 [/tex] As Atheist mentioned, this equation is called Killing's equation and the solutions Killing vectors. Pete 


#4
Aug2504, 05:55 PM

Emeritus
PF Gold
P: 8,147

What is a timelike killing vector?
Perfectly true, Pete, but the definition is still good with the appropriate sign in. For newbies, the semicolon in Atheist's definition denotes covariant derivative, so the equation he gives, called Killing's equation, is a differential equation.



#5
Aug2504, 06:29 PM

Emeritus
Sci Advisor
P: 7,659

http://mathworld.wolfram.com/Isometry.html A timelike Killing vector means, roughly speaking, that the distances in the system are unchanged as time increases (i.e by a time translation). Since the distances are defined by the mteric tensor, g_ab, this means that the components of the metric tensor are unchanged by time. A stationary black hole is an example of a system with a timelike Killing vector. 


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