### Citations

224 |
Classical Electrodynamics (3rd edition
- Jackson
- 1998
(Show Context)
Citation Context ...ical laws of superenergy in General Relativity 8 equation (18): U is the electromagnetic energy density, P a is the Poynting vector and Tab is the stress tensor of the electromagnetic field (see e.g. =-=[26]-=- for detailed explanations about the role of each of these quantities). We must note at this point that equations (19)-(20) are usually presented under the assumption that the spacetime is flat and n ... |

122 |
Hoenselaers C and Herlt E 2003 Exact solutions of Einstein’s field equations 2nd ed
- Stephani, Kramer, et al.
(Show Context)
Citation Context ... 1 2 habW − 2PaPb W , Qabc = 3h (bcPa) − 16PaPbPc W 2 , tabcd = − 64PaPbPcPd W 3 + 12 W h (abPcPd), Pa ̸= 0, PaP a 2 W = 4 . The two independent repeated principal directions of Weyl tensor (see e.g. =-=[40]-=-) can be calculated explicitly yielding k a 1 = −P a + 1 2 naW, k a 2 = P a + 1 2 naW. (46) (ii) For Petrov type N we have tab = PaPb W , Qabc = PaPbPc W 2 , tabcd = PaPbPcPd W 3 , Pa ̸= 0, PaP a = W ... |

43 |
Spinors and space-time, Vol 1
- Penrose, Rindler
- 1984
(Show Context)
Citation Context ... are indeed null and that they fulfill the properties Tabcdk a 1k b 1k c 1k d 1 = 0, Tabcdk a 2k b 2k c 2k d 2 = 0 which implies that k a 1 and k a 2 are the Weyl tensor repeated null directions (see =-=[37]-=- p. 328). – Petrov type N: In this case, we obtain in the frame of (B.5) the other components vanishing. Hence Q123 = P1 = −t11 = −4(B 2 22 + E 2 22). Qabc = PaPbPc 16(B 2 22 + E2 22 ), Tcd = PcPd 4(B... |

23 |
der Burg M
- Bondi, van
(Show Context)
Citation Context ...es with dimensions of energy which tell us when a gravitational system is radiating. This has been performed for the important case of isolated systems where the quasilocal quantity is the Bondi mass =-=[9, 38, 35]-=-. If instead of energy, we use superenergy as a replacement, then the aforementioned problem disappears and one can use the same ideas as in electromagnetism to define radiating gravitational fields o... |

8 |
A G and Senovilla
- Bonilla
- 1997
(Show Context)
Citation Context ...rrent vanishes in which case it becomes a c Cbpdf − 1 ∇aT a bcd = 0. A full account of the properties reviewed here of Bel and Bel-Robinson tensors together with their proofs can be found in [39] and =-=[10]-=-. In the former reference a generalization of (14) and (37) valid for any tensor is put forward. Tensors resulting from this generalization are called superenergy tensors and they all fulfill the gene... |

7 |
González G A, Pachón L A and Rueda
- Herrera
(Show Context)
Citation Context ...that the flux of electromagnetic radiation produces a measurable effect on rotating gyroscopes. An effect similar to this was pointed out in a particular case in [12] and this was latter confirmed in =-=[22]-=-. In the former reference it was shown that gyroscopes placed in the spacetime generated by a nonrotating charged magnetic dipole would precess. As an explanation for this “surprising” result it was s... |

4 |
Cargése Lectures in Physics vol 6
- Ellis
- 1973
(Show Context)
Citation Context ... and formalisms in General Relativity and has been extensively studied in the literature but since it will be used in this work many times we review next its essentials (good accounts can be found in =-=[18, 32]-=-). Let na be any vector field with nana = −1 and define the spatial metric hab by hab ≡ gab + nanb, habh b c = hac, h a a = 3. (1) The tensor hab has the properties of an orthogonal projector. We shal... |

4 |
Naber The geometry of Minkowski spacetime
- L
- 1992
(Show Context)
Citation Context ...ectromagnetic field Fab is singular or null whichDynamical laws of superenergy in General Relativity 20 means that it can be written as the exterior product of a null and a spatial vector. (see e.g. =-=[34]-=-). If we try to follow the same procedure to define gravitational radiation in General Relativity we are immediately confronted to the fact that, due to the equivalence principle, we can always find a... |

3 |
Feinstein A & Ibañez J
- Breton
- 1993
(Show Context)
Citation Context ...gy just mentioned and a lot of work has been devoted to study the behaviour of gravitational systems by studying the super-energy density and the super -Poynting vector in the system –see for example =-=[25, 13, 43, 23]-=-. The results obtained are very suggestive but we must note that (65)-(66) are not equivalent to (63)-(64) which in fact contain more information. Therefore, if we are to study gravitational radiation... |

2 |
Choquet-Bruhat Y and York
- Anderson
(Show Context)
Citation Context ...1 2 εafbj fb eZ ae +Y ae J ae+4A a Pa = 0.(62) Equations similar to this one have been used in different places of the literature principally with the aim of controlling the evolution of the scalar W =-=[1, 27]-=-. 7.1. Dynamical laws of superenergy in vacuum Theorem 4 adopts a far more simpler form in vacuum because the covariant divergence of Bel tensor takes the simpler form ∇aT a bcd = 0. The specific resu... |

2 |
Cahiers de Physique 16 59-80 (English translation
- Bel
- 1962
(Show Context)
Citation Context ...Some of these quantities have been obtained before and have found diverse applications. The scalar W (superenergy density) and the spatial vector P a , called super-Poynting vector were first used in =-=[5]-=- to define intrinsic radiation states in gravitation theory (see section 8 for more details about this) and the tensor tab was used in [11] to show the causal propagation of gravity in vacuum. We esta... |

2 |
Eriksson I and Senovilla
- Bergqvist
(Show Context)
Citation Context ...ssible superenergy tensor is the Chevreton tensor which was first introduced in [17] and recently stimulating results about its symmetries and the covariant divergence of its trace have been obtained =-=[8]-=-. Chevreton tensor, like Bel-Robinson tensor, is a four rank tensor and its covariant divergence couples the Riemann tensor with terms which contain covariant derivatives of the Faraday tensor. This s... |

2 |
Carot J and Di Prisco A 2007 Class
- Herrera, Barreto
(Show Context)
Citation Context ...gy just mentioned and a lot of work has been devoted to study the behaviour of gravitational systems by studying the super-energy density and the super -Poynting vector in the system –see for example =-=[25, 13, 43, 23]-=-. The results obtained are very suggestive but we must note that (65)-(66) are not equivalent to (63)-(64) which in fact contain more information. Therefore, if we are to study gravitational radiation... |

2 |
T and Schimdt
- Horowitz
(Show Context)
Citation Context ...logies) for W, typically spheres. In this last case it is possible to relate the limit of the quotient between many of these quasilocal energies and certain power of the sphere radius to the scalar W =-=[24, 7, 6, 15]-=-. To investigate further the meaning of E(W) we calculate this quantity in a particular but relevant case. Example 9.1 (Schwarzschild interior solution) The Schwarzschild interior solution is a static... |

2 |
Lichnerowicz A
- Mat
- 1960
(Show Context)
Citation Context ...R a a e p − 2RcapeRd b + a e d p , (25) ∇a∇ a Rdcbp = ∇bJdcp − ∇pJdcb − 2R ae Rbape − R a +2R becaR which can be shown to be a hyperbolic equation for the Riemann tensor. A result due to Lichnerowicz =-=[29]-=- proves that if the Cauchy data of (25) satisfy (24) then so does the solution of the hyperbolic equation. Hence, with the provision imposed by Lichnerowicz result, we can regard (23) and (24) as equi... |

2 |
Batelle Rencontres (New York
- Penrose
- 1968
(Show Context)
Citation Context ...es with dimensions of energy which tell us when a gravitational system is radiating. This has been performed for the important case of isolated systems where the quasilocal quantity is the Bondi mass =-=[9, 38, 35]-=-. If instead of energy, we use superenergy as a replacement, then the aforementioned problem disappears and one can use the same ideas as in electromagnetism to define radiating gravitational fields o... |

2 |
A 1977 Phys
- Wheeler
(Show Context)
Citation Context ...gy just mentioned and a lot of work has been devoted to study the behaviour of gravitational systems by studying the super-energy density and the super -Poynting vector in the system –see for example =-=[25, 13, 43, 23]-=-. The results obtained are very suggestive but we must note that (65)-(66) are not equivalent to (63)-(64) which in fact contain more information. Therefore, if we are to study gravitational radiation... |