### Abstract

Many years ago Ehlers and Kundt showed that a spacetime M is an Einstein space if and only if the sectional curvatures of any pair of orthogonal non-null 2-spaces at any point of M are equal. This paper generalizes this result by first showing a very straightforward relation between the sectional curvatures of such orthogonal pairs of 2-spaces and the trace-free part of the Ricci tensor and then by establishing for each algebraic (Segre) type of the energy-momentum tensor precisely which orthogonal pairs of non-null 2-spaces have the same sectional curvature. The results are described in a manifold theoretic sense and are tabulated for each Segre type.

Original language | English |
---|---|

Pages (from-to) | 1493-1502 |

Number of pages | 9 |

Journal | Classical and Quantum Gravity |

Volume | 22 |

Issue number | May |

DOIs | |

Publication status | Published - 2005 |

### Keywords

- GENERAL-RELATIVITY
- CANONICAL FORMS

### Cite this

*Classical and Quantum Gravity*,

*22*(May), 1493-1502. https://doi.org/10.1088/0264-9381/22/9/001

**Sectional curvature and the energy-momentum tensor.** / Hall, Graham Stanley; McNay, L.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 22, no. May, pp. 1493-1502. https://doi.org/10.1088/0264-9381/22/9/001

}

TY - JOUR

T1 - Sectional curvature and the energy-momentum tensor

AU - Hall, Graham Stanley

AU - McNay, L.

PY - 2005

Y1 - 2005

N2 - Many years ago Ehlers and Kundt showed that a spacetime M is an Einstein space if and only if the sectional curvatures of any pair of orthogonal non-null 2-spaces at any point of M are equal. This paper generalizes this result by first showing a very straightforward relation between the sectional curvatures of such orthogonal pairs of 2-spaces and the trace-free part of the Ricci tensor and then by establishing for each algebraic (Segre) type of the energy-momentum tensor precisely which orthogonal pairs of non-null 2-spaces have the same sectional curvature. The results are described in a manifold theoretic sense and are tabulated for each Segre type.

AB - Many years ago Ehlers and Kundt showed that a spacetime M is an Einstein space if and only if the sectional curvatures of any pair of orthogonal non-null 2-spaces at any point of M are equal. This paper generalizes this result by first showing a very straightforward relation between the sectional curvatures of such orthogonal pairs of 2-spaces and the trace-free part of the Ricci tensor and then by establishing for each algebraic (Segre) type of the energy-momentum tensor precisely which orthogonal pairs of non-null 2-spaces have the same sectional curvature. The results are described in a manifold theoretic sense and are tabulated for each Segre type.

KW - GENERAL-RELATIVITY

KW - CANONICAL FORMS

U2 - 10.1088/0264-9381/22/9/001

DO - 10.1088/0264-9381/22/9/001

M3 - Article

VL - 22

SP - 1493

EP - 1502

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - May

ER -