Space time with no killing vector

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IEB
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Is it possible to have a space-time with no killing vector?
Alternatively, can I define the metric only with the killing vector of the space time?
 

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PeterDonis
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Is it possible to have a space-time with no killing vector?
Sure (strictly speaking, you should say "Killing vector field", since we're talking about a mapping of vectors to points everywhere in the spacetime, not just a vector at one point). Most solutions of the Einstein Field Equation have no Killing vector fields; the ones that do are unusual, mathematically speaking, even though they're the ones we use most often, at least when we're working with analytical solutions. (Numerical solutions are another matter: as I understand it, numerical simulations routinely work with spacetimes that have no Killing vector fields, although they may be "close", in some sense, to spacetimes that do.)

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

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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IEB
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Thank you very much!!
 
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PeterDonis
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You're welcome! (And welcome to PF!)
 

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