# Killing vector tangents to geodesics

• Andre' Quanta
In summary, a null Killing vector implies that it is also a null geodesic, with the condition that ##\xi^{\nu}\nabla_{\nu}\xi^{\mu} = -\xi^{\nu}\nabla^{\mu}\xi_{\nu} = - \frac{1}{2}\nabla^{\mu}\xi^2 = 0##. This results in the integral curves of ##\xi^{\mu}## being null geodesics. Additionally, the null Killing vector has the property that ##\nabla_{[\mu}\omega_{\nu]} = 0 \Leftrightarrow \omega^{\mu} = 0##,
Andre' Quanta
Suppose to have a killing vector that its norm is null, so at the same time is also a null geodesic.
Does the metric have special propierty? What can i say about the Killing vector and its proprierties?

Andre' Quanta said:
Suppose to have a killing vector that its norm is null, so at the same time is also a null geodesic.

Why does the Killing vector being null imply that it's a null geodesic? Not all null curves are null geodesics.

PeterDonis said:
Why does the Killing vector being null imply that it's a null geodesic? Not all null curves are null geodesics.

Because ##\xi^{\nu}\nabla_{\nu}\xi^{\mu} = -\xi^{\nu}\nabla^{\mu}\xi_{\nu} = - \frac{1}{2}\nabla^{\mu}\xi^2 = 0## if ##\xi^{\mu}## is a null Killing field so the integral curves of ##\xi^{\mu}## are null geodesics.

To the OP, you need to be more specific. There is a myriad properties that such a space-time possesses, which you can find coherently interspersed throughout the text "Exact Solutions of Einstein's Field Equations" by Stephani et al.

As for the null Killing vector, the most important property is that ##\nabla_{[\mu}\omega_{\nu]} = 0 \Leftrightarrow \omega^{\mu} = 0## where ##\omega^{\mu}## is the twist of the null Killing field. I typed up a short calculation demonstrating the non-trivial direction; see the attached image.

#### Attachments

• null killing field twist.png
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Andre' Quanta

## 1. What are Killing vectors and geodesics?

Killing vectors are vector fields that preserve the metric and symmetries of a spacetime. Geodesics are curves that represent the shortest or longest path between two points in a given geometry.

## 2. What is the significance of Killing vectors tangents to geodesics?

When a Killing vector is tangent to a geodesic, it means that the vector field is parallel to the geodesic at every point along its path. This indicates that the spacetime has a symmetry that is preserved along the geodesic.

## 3. How are Killing vectors related to the concept of isometries?

Isometries are transformations that preserve the metric of a given geometry. Killing vectors are the generators of these isometries, meaning that they represent the infinitesimal transformations that preserve the geometry.

## 4. Can Killing vectors be used to find symmetries in spacetime?

Yes, Killing vectors can be used to identify the symmetries present in a given spacetime. If a spacetime has a Killing vector, it means that there exists a symmetry that is preserved throughout the spacetime.

## 5. How do Killing vectors and geodesics relate to the study of General Relativity?

In General Relativity, Killing vectors and geodesics are important tools for understanding the symmetries and curvature of spacetime. They help us analyze the behavior of particles and light in a given geometry, and provide insight into the underlying structure of the universe.

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