In mathematics, a Killing vector field (often just Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.
For a complete description of Killing equation, Killing vector fields and how to find the Killing vector fields for a particular spacetime, please see the following book:
Zafar Ahsan: Tensors:Mathematics of Differential Geometry and Relativity. PHI Learning Pvt. Ltd., Delhi. Second Printing (December 2017).
This book can be obtained from the web site; www.phindia.com
Dear Ashok,
the condition, which a Killing vector field V of a manifold equipped with a metric g has to satisfy, is that the Lie derivative of g with respect to V vanishes. This condition results in a system of first order linear PDEs for the unknown components of V. A suitable coordinate system to write down these equations can be very useful to find their solution. For example if the metric tensor is independent of one coordinate variable, say x, then the partial derivative with respect to x is a Killing vector field. You can start off by practicing in a low-dimensional case, such as 3-manifolds (see for example the book of Schutz, Geometrical Methods of Mathematical Physics, p. 190), and then see what you can do in the space-time you are particularly interested.
Hope this was helpful.
The short answer is that in general there are no Killing vector fields.
Assuming that Killing vector fields exist, you maybe able to spot some of them (Nikos Kallinikos suggests one way of trying to spot some) and then you maybe able to generate others using their Lie bracket. (You need to check linear independence) But, you will not know when to stop looking in general; unless you reach the maximal number posissibe, of course and you in fact have a maximally symmetric Riemannian manifold.
Solving the Killing equation explicitly may be very difficult, and indeed generally impossible.
This is an interesting question.
A good overview of static space-times that are Lorentzian space-times possessing a timelike Killing vector field is given in
https://www.math.uni-tuebingen.de/arbeitsbereiche/geometrische-analysis-und-mathematische-relativitaetstheorie/personen/dr-carla-cederbaum-1/PosterCarla.pdf
An interesting application of killing vectors is given in
UNIQUENESS OF STATIC BLACK-HOLES WITHOUT ANALYTICITY:
http://inspirehep.net/record/850891/files/arXiv:1004.0513.pdf?version=1
See, also,
http://www.math.miami.edu/~galloway/papers/bhuniqueness.pdf
See, for example, Section 3 on non-rotating horizons and maximal hypersurfaces, starting on page 4.
The space of conformal Killing tensors is considered in
Higher symmetries of the Laplacian:
http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n3-p13.pdf
See, for example, Def. 3, on page 1647, for equations that define a conformal Killing vector, when s = 1 in formula (3) in Def. 3. A simple ambient interpretation of conformal Killing tensors is given on page 1651.
For a complete description of Killing equation, Killing vector fields and how to find the Killing vector fields for a particular spacetime, please see the following book:
Zafar Ahsan: Tensors:Mathematics of Differential Geometry and Relativity. PHI Learning Pvt. Ltd., Delhi. Second Printing (December 2017).
This book can be obtained from the web site; www.phindia.com
https://math.stackexchange.com/questions/1730051/identity-surrounding-killing-vector-field-on-a-spacetime-nabla-a-nabla-b-w-c
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