Space time with no killing vector

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Discussion Overview

The discussion centers on the possibility of having a space-time without a Killing vector field and the implications of such a scenario. Participants explore the definitions and roles of Killing vector fields in the context of general relativity, particularly regarding the determination of metrics and symmetries in various spacetimes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that it is indeed possible to have a space-time with no Killing vector fields, noting that most solutions to the Einstein Field Equation lack them.
  • One participant clarifies that the term "Killing vector field" should be used, emphasizing that it refers to a mapping of vectors to points throughout the spacetime.
  • It is suggested that while Killing vector fields provide information about the symmetries of spacetime, they are insufficient alone to determine the metric, as multiple spacetimes can share the same symmetries.
  • An example is provided regarding spherically symmetric spacetimes, highlighting that both Schwarzschild and FRW spacetimes exhibit the same Killing vector fields but are distinct solutions requiring additional information for identification.
  • Reference is made to Birkhoff's theorem, which states that if a spacetime is both vacuum and spherically symmetric, it can be identified as Schwarzschild spacetime, indicating the need for supplementary knowledge beyond Killing vector fields.

Areas of Agreement / Disagreement

Participants generally agree that spacetimes can exist without Killing vector fields and that additional information is necessary to determine metrics. However, the discussion remains unresolved regarding the implications of these findings and the specific conditions under which they apply.

Contextual Notes

Participants note that numerical simulations may work with spacetimes lacking Killing vector fields, suggesting a distinction between analytical and numerical solutions. The discussion also highlights the complexity of defining metrics based solely on symmetries.

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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IEB said:
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.)

IEB said:
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.)
 
Thank you very much!
 

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