What is a time-like killing vector?

[kurious]

What is a time-like killing vector?

 

[Atheist]

Unless given further explanation I´d say it´s exactly what the name sais:

Killing vector: A vector that fulfies the Killing-equation $$v_{i;j} + v_{j;i} = 0$$. The existence of a Killing-vector implies the existence of a coordinate system where the metric tensor is independent of one of the coordiantes.

time-like: A vector v is timelike if $$g_{ij} v^{i} v^{j} >0$$.
EDIT: As pmb_phy correctly claims I should mention that above inequality assumes the signature of the metric to be (+,-,-,-).

 

[pmb_phy]

What is a time-like killing vector?

A few preliminaries - A coordinate transformation which leaves the components of the metric tensor invariant is called an isometry. This means that when the coordinates are change from the primed coordinates, x', to the unprimed coordinates x, the metric tensor remains unchanged, i.e. is the same function of the coordinates. This means

$$g'_{\alpha\beta}(x') = g_{\alpha\beta}(x')$$

For the components of the metric tensor invariant under the isometry we must have

$$g_{\mu\nu} (x) = \frac{\partial x'^{\alpha}}{\partial x^{\beta}}\frac{\partial x'^{\mu}}{\partial x^{\nu}}g(x'(x))$$

Consider the infinitesimal coordinate transformation

$$x' = x^{\alpha} + \epsilon \xi^{\alpha}$$

where $$\xi^{\alpha}(x)$$ is a vector field and $$\epsilon$$ -> 0. For this coordinate transformation to yield an isometry the $$\xi^{\alpha}$$ must satisfy the following equation

$$\xi_{\mu;\nu} + \xi_{\nu;\mu} = 0$$

As Atheist mentioned, this equation is called Killing's equation and the solutions Killing vectors.

time-like: A vector v is timelike if $$g_{ij} v^{i} v^{j} >0$$.

That depends on the signature of the metric tensor.

Pete

 

[selfAdjoint]

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.

 

[pervect]

What is a time-like killing vector?

Killing vectors are generated by isometries. Isometries are transformations which leave lengths unchanged. For a more technical definition, see.

http://mathworld.wolfram.com/Isometry.html

A time-like 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 time-like Killing vector.