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I work in the Lorentzian manifolds, more generally in pseudo Riemannian manifolds and applications to general relativity. I know the definitions of conformal, Killing and homothetic vector fields in Riemannian hypersurfaces. Are these quantities defined in the same way in pseudo riemannian hypersurfaces? What is the physical significance of these quantities?

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  • $\begingroup$ Killing fields are definitely described the same way, and are physically deeply important, as much of the physics of a spacetime is governed by the symmetry of that spacetime, so a great many theorems will be dependent on the presence of a Killing field satisfying various properties. $\endgroup$ – Jerry Schirmer Jan 14 '14 at 15:05
  • $\begingroup$ thank youvery much I arrive to understand some of them, thank you again. $\endgroup$ – aboutammam Jan 20 '14 at 11:07
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The definitions work for arbitrary Riemannian manifolds (not just hypersurfaces) and generalize to the pseudo-Riemannian case - after all, they are pretty general statements about the Lie derivative of some tensor field.

In general relativity, there are many interesting vector fields (sometimes called collineations), considered infinitesimal symmetries of geometric structure or physical quantities like metric, curvature and energy-momentum tensors, geodesics and light cones.

They might be interesting for physical reasons, or just technical ones to make the equations more manageable.

Assuming I did not mess up (no promises), we get the following map where the sets of vector fields are related by inclusion:

         Killing
            |
      +-----+-----+
      |           |
      v           v
   matter     homothetic
                  |
            +-----+-----+
            |           |
            v           v
          affine    conformal
            |
      +-----+-----+
      |           |
      v           v
 projective   curvature
                  |
            +-----+-----+
            |           |
            v           v
         Ricci        Weyl

Killing vector fields preserve the metric and all derived structure, including the energy-momentum tensor assuming the Einstein field equations hold. A physical application would be the definition of Komar mass for spacetimes with a time-like Killing vector field.

A matter collineation is a vector field that preserves energy-momentum, but not necessarily any geometry.

Conformal symmetries preserves angles and in the pseudo-Riemannian case in particular the sign of scalar products, which is relevant for the light-cone structure and thus causality.

Affine vector fields preserve geodesics including their affine parameter (multiples of proper time), whereas projective vector fields don't necessarily preserve the latter.

Curvature collineations preserve the Riemann tensor, whereas Ricci and Weyl collineation only preserve Ricci and Weyl tensor, respectively.

I'll leave particular applications to people more knowledgeable than myself.

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  • $\begingroup$ thank you friend you gave me more that I hope, than you again I hope that we will keep us in contact $\endgroup$ – aboutammam Jan 20 '14 at 11:08

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