### Abstract

We prove that given a solution of the Einstein equations g_{ab} for the matter field T_{ab}, an autoparallel null vector field l ^{a} and a solution (l_{a}l_{c}, T_{ac}) of the linearized Einstein equation on the given background, the Kerr-Schild metric g_{ac} + λl_{a}l_{c} (λ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor T_{ac} + λT_{ac} + λ^{2}l( ^{a}T^{c})_{b}l^{b} The mixed form of the Einstein equation for Kerr-Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed spacetime due to Gürses and Gürsey.

Original language | English |
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Pages (from-to) | 2515-2523 |

Number of pages | 9 |

Journal | Classical and Quantum Gravity |

Volume | 19 |

Issue number | 9 |

DOIs | |

Publication status | Published - May 7 2002 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**Linear Einstein equations and Kerr-Schild maps.** / Gergely, L.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 19, no. 9, pp. 2515-2523. https://doi.org/10.1088/0264-9381/19/9/313

}

TY - JOUR

T1 - Linear Einstein equations and Kerr-Schild maps

AU - Gergely, L.

PY - 2002/5/7

Y1 - 2002/5/7

N2 - We prove that given a solution of the Einstein equations gab for the matter field Tab, an autoparallel null vector field l a and a solution (lalc, Tac) of the linearized Einstein equation on the given background, the Kerr-Schild metric gac + λlalc (λ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor Tac + λTac + λ2l( aTc)blb The mixed form of the Einstein equation for Kerr-Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed spacetime due to Gürses and Gürsey.

AB - We prove that given a solution of the Einstein equations gab for the matter field Tab, an autoparallel null vector field l a and a solution (lalc, Tac) of the linearized Einstein equation on the given background, the Kerr-Schild metric gac + λlalc (λ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor Tac + λTac + λ2l( aTc)blb The mixed form of the Einstein equation for Kerr-Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed spacetime due to Gürses and Gürsey.

UR - http://www.scopus.com/inward/record.url?scp=0036274274&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036274274&partnerID=8YFLogxK

U2 - 10.1088/0264-9381/19/9/313

DO - 10.1088/0264-9381/19/9/313

M3 - Article

AN - SCOPUS:0036274274

VL - 19

SP - 2515

EP - 2523

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 9

ER -