How Do You Find Isometries in Relativity?

[Flaneuse]

This is really more of a general question about the process of solving a certain type of problem than it is an actual problem.

Concerning invariance of the interval, ds, in relativity, say you have an interval ds^2, ℝ^(1,3), and a certain equation for a metric space. I know that there are isometries that must exist such that the interval is rendered invariant, and that these are basically the Poincare group of transformations. But how do you find the isometries for a specific space (defined by whatever ds² is equal to, some combination of dx², dy², and dz²), say ds² = -dx²-dy²-dz²? How exactly do you express these isometries, and the transformations that would constitute invariance of the interval?

 

[Matterwave]

Given a metric tensor g, an isometry is associated with each Killing vector field, i.e. a vector field $\vec{v}$ such that:

$$\mathcal{L}_\vec{v}g=0$$
(The Lie derivative of the metric along this vector field is 0)
To find these isometries in general, one must in general solve Killing's equation:

$$\nabla_\mu v_\nu + \nabla_\nu v_\mu=0$$

You can often find some of these isometries by inspection. If your metric is independent of any coordinate, then translations along that coordinate is an isometry.

 

[Flaneuse]

Is there any way you can find the isometries with just the equation for the metric space though (instead of using a metric tensor)?