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Alternatively, can I define the metric only with the killing vector of the space time?

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Alternatively, can I define the metric only with the killing vector of the space time?

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Sure (strictly speaking, you should say "Killing vectorIs it possible to have a space-time with no killing vector?

If you mean, is knowledge of the Killing vector fields alone sufficient to determine the metric, in general, no. You will need additional information, because the Killing vector fields alone can tell you about symmetries of the spacetime, but there will be many possible spacetimes that share a given set of symmetries.Alternatively, can I define the metric only with the killing vector of the space time?

For example, one of the most common cases of a set of Killing vector fields that is useful in relativity is the set of 3 KVFs that defines spherical symmetry: a spherically symmetric spacetime is one that has 3 KVFs that are the same as those possessed by a 2-sphere. But there are many spacetimes which are spherically symmetric; for example, both Schwarzschild spacetime, which is used to model the vacuum region around an isolated gravitating mass, and the FRW spacetimes, which are used to model the entire universe in cosmology, are spherically symmetric. So you need more information to determine which one you are working with. (For example, if you knew the spacetime was vacuum, as well as spherically symmetric, then you *would* know it was Schwarzschild spacetime; this result is known as Birkhoff's theorem. But the knowledge that the spacetime is vacuum is additional knowledge, over and above knowledge of the KVFs.)

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Thank you very much!!

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You're welcome! (And welcome to PF!)

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