What is a time-like killing vector?

In summary: Since the metric tensor is invariant under a stationary time translation, any particles which enter a stationary black hole will be trapped forever. In summary, a time-like Killing vector is a vector which satisfies the Killing-equation v_{i;j} + v_{j;i} = 0 .
  • #1
kurious
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What is a time-like killing vector?
 
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  • #2
Unless given further explanation I´d say it´s exactly what the name sais:

Killing vector: A vector that fulfies the Killing-equation [tex] v_{i;j} + v_{j;i} = 0 [/tex]. 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 [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 (+,-,-,-).
 
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  • #3
kurious said:
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

[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.

Atheist said:
time-like: A vector v is timelike if [tex] g_{ij} v^{i} v^{j} >0 [/tex].
That depends on the signature of the metric tensor.

Pete
 
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  • #4
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
kurious said:
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.
 

1. What is a time-like killing vector?

A time-like killing vector is a type of mathematical object used in the study of spacetime geometry. It is a vector field that satisfies certain mathematical equations and has the property of preserving the spacetime metric along its flow. This means that if you move along the vector, you will not change the measurement of time or distance. In other words, it represents a symmetry in the spacetime that preserves the notion of time.

2. How is a time-like killing vector different from other types of vectors?

A time-like killing vector is specifically defined as a vector that is tangent to the world line of a particle moving through spacetime. It is also a special case of a Killing vector, which is any vector field that preserves the metric of a manifold. However, a time-like killing vector has the additional property of preserving the concept of time, making it uniquely suited for studying spacetime symmetries.

3. What is the significance of time-like killing vectors in physics?

Time-like killing vectors are important in physics because they represent symmetries in the underlying structure of spacetime. These symmetries can be used to solve equations and make predictions about the behavior of particles in spacetime. They are also essential for understanding the concept of energy conservation in general relativity.

4. How are time-like killing vectors used in the study of black holes?

Time-like killing vectors are particularly useful in the study of black holes because they represent symmetries that are preserved even in the extremely curved spacetime around a black hole. These vectors can be used to define a set of conserved quantities, such as energy and angular momentum, that are preserved as particles fall into a black hole.

5. Are time-like killing vectors related to time travel?

No, time-like killing vectors do not have any direct connection to time travel. While they do represent symmetries that preserve the concept of time, they do not allow for time travel or violate the laws of causality in any way. Time travel is a concept that goes beyond the realm of Einstein's theory of general relativity and is not related to the use of time-like killing vectors in physics.

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