# Homothetic vector field

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Vector field

In [physics](/source/Physics), a **homothetic vector field** (sometimes **homothetic collineation** or **homothety**) is a [projective vector field](/source/Projective_vector_field) which satisfies the condition:

- L X g a b = 2 c g a b {\displaystyle {\mathcal {L}}_{X}g_{ab}=2cg_{ab}}

where c is a real constant. Homothetic vector fields find application in the study of [singularities](/source/Gravitational_singularity) in [general relativity](/source/General_relativity). They can also be used to generate new solutions for Einstein equations by similarity reduction.[1]

## See also

- [Affine vector field](/source/Affine_vector_field)

- [Conformal Killing vector field](/source/Conformal_Killing_vector_field)

- [Curvature collineation](/source/Curvature_collineation)

- [Killing vector field](/source/Killing_vector_field)

- [Matter collineation](/source/Matter_collineation)

- [Spacetime symmetries](/source/Spacetime_symmetries)

## References

1. **[^](#cite_ref-1)** [*Exact Solutions of Einstein's Field Equations*](https://archive.org/details/exactsolutionsei00step). [Cambridge University Press](/source/Cambridge_University_Press). 2003. pp. [163](https://archive.org/details/exactsolutionsei00step/page/n194). [ISBN](/source/ISBN_(identifier)) [978-0-521-46136-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-46136-8).

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