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The metric for the product spacetime $AdS^2 \times S^2$ is given by $$ ds^2 = \dfrac{-dt^2 + dy^2}{y^2} + d\theta^2 + \sin^2 \theta \, d\phi^2.$$

Writing out the Killing equations yields a set of 10 PDE's, which is also the maximal number of Killing vectors one could have, in the case of a maximally symmetric space.

How could one determine beforehand how many Killing vectors this spacetime will have, and is there a more clever way to determine them than by bruteforcing the equations, or is there a particularly useful ansatz that leads to a quick solution for the Killing vectors?

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In this instance the answer is easily obtained from the fact that the spacetime is a direct product of two spaces, thus the number of independent Killing vector fields (KVFs) is $6=3+3$ where each of the $3$s is the dimension of isometry group (and thus also the number of KVFs) of corresponding factor.

Without knowing the structure of spacetime beforehand, we could have arrived at the answer via the following chain of reasoning:

  • Coordinates of the metric falls into two groups: $(t,y)$ and $(\theta,\phi)$, so that $(dt,dy)$ part does not depend on $(θ,ϕ)$ coordinates and vice versa. This means that the spacetime is a direct product, Killing equation for it separates and the number of KVFs is the sum of KVF numbers in each of the factors.

  • If we calculate the scalar curvature for each of the factors we obtain constants. So each of the factors has $3$ independent KVFs since for two-dimensional spaces constant scalar curvature means that the space is locally homogeneous.

For a more general spacetime the number of independent KVFs could be obtained from running Cartan–Karlhede algorithm for classifying manifold (see e.g. the book [1]).

Here is also an example of an algorithm specifically for counting the number of KVFs in a 2D space/spacetime (image taken from [2], the paper also provides an algorithm for 3D Lorentzian manifolds):

Algorithm for 2D from Nozawa & Tomoda

For a general 4D spacetime Cartan–Karlhede algorithm requires evaluation of a lot of curvature invariants, however depending on what is known about the spacetime significant simplifications are possible. Also, one should keep in mind that the Lorentzian manifolds present additional subtleties absent in case of Euclidean metric signature since it is possible to have nontrivial “null” curvature for which all scalar curvature invariants are zero.

References

  1. Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003). Exact Solutions of Einstein's Field Equations (2nd edn.). Cambridge: Cambridge University Press. ISBN 0-521-46136-7, sec. 9.2.

  2. Nozawa, M., & Tomoda, K. (2019). Counting the number of Killing vectors in a 3D spacetime. Classical and Quantum Gravity, 36(15), 155005, doi:10.1088/1361-6382/ab2da7, arXiv:1902.07899.

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