# Relativistic astrophysics, mathematical problems in

Problems that arise in the study of astrophysical phenomena in which relativistic effects, i.e. effects of the special or general theory of relativity (cf. Relativity theory), are significant.

Mathematical problems in relativistic astrophysics are commonly divided into problems relating to cosmology — the science of the structure and the evolution of the Universe, and problems of relativistic astrophysics of individual celestial bodies. The solution of A.A. Friedman (cf. Cosmological models) is an example of a cosmological solution that describes the expansion (or contraction) of a homogeneous and isotropic Universe. Homogeneous anisotropic cosmological solutions have been classified (9 Bianchi types have been identified) and are well studied. Anisotropic and non-homogeneous solutions, being slight deviations from Friedman's solution (a linear approximation) have been studied in detail, and several simple non-linear solutions have been constructed.

An especially interesting problem is that of the presence of a singular point in the general cosmological solution at which infinite density of matter and an infinite space-time curvature is reached. Singularities have been shown to be unavoidable in the past under conditions that took place in the real Universe, and a general solution of the equations of the general theory of relativity with a singularity has been constructed. Active research is being conducted on the possibility of constructing cosmological solutions without a singularity, representing a departure from the framework of the traditional general theory of relativity.

A large class of problems involves the study of the interaction of relic radiation (which occupies space) with matter during the expansion of the Universe, and of the physical processes capable of generating such radiation.

The mathematical problems in relativistic astrophysics for individual celestial bodies concern the equilibrium and stability of stars and constellations. Equilibrium masses have been found in white dwarfs and neutron stars, and the relativistic collapse of more massive stars (which turn into so-called "black holes" — objects that are only observable through their gravitational field) is also being studied. In connection with the search for and the study of relativistic objects (neutron stars, "black holes" , etc.), the problem of the accretion in them of matter with a magnetic field is studied.

The mathematical problems in relativistic astrophysics also include research on gravitational radiation. In a weak gravitational field in empty space the perturbations, e.g. the invariants of curvature, satisfy the wave equation, and the field of gravity extends in space like electromagnetic waves.

#### References

[1] | Ya.B. Zel'dovich, I.D. Novikov, "Relativistic astrophysics" , 1 - Stars and relativity; 2 - Structure and evolution of the Universe , Chicago (1971–1983) (Translated from Russian) |

[2] | Ya.B. Zel'dovich, I.D. Novikov, "Relativistic astrophysics" , I. Stars and relativity , Chicago (1971) (Translated from Russian) |

[3] | Ya.B. Zel'dovich, I.D. Novikov, "Relativistic astrophysics" , II. Structure and evolution of the Universe , Chicago (1983) (Translated from Russian) |

[4] | P.J.E. Peebles, "Physical cosmology" , Princeton Univ. Press (1971) MR1216520 |

[5] | C.W. Misner, K.S. Thorne, J.A. Wheeler, "Gravitation" , Freeman (1973) pp. Chapt. 30 MR0418833 Zbl 1177.83013 Zbl 0078.19106 |

[6] | E.M. Lifshits, "On the gravitational stability of the expanding universe" Zh. Eksper. i Teor. Fiz. , 16 (1946) pp. 587–602 (In Russian) (English abstract) |

[7] | V.A. Belinskii, E.M. Lifshits, I.M. Khalatnikov, Uspekhi Fiz. Nauk , 102 (1970) pp. 463–500 |

[8] | V.A. Braginskii, Uspekhi Fiz. Nauk , 86 (1965) pp. 433–446 |

#### Comments

Cf. also Astrophysics, mathematical problems of.

#### References

[a1] | Ya.B. Zeldovich, A.A. Ruzmaikin, D.D. [D.D. Sokolov] Sokoloff, "Magnetic fields in astrophysics" , Gordon & Breach (1983) (Translated from Russian) |

[a2] | M.J. Rees, Phys. Rev. Letters , 28 (1972) pp. 1669–1671 |

[a3] | R. Penrose, "Singularities and time-asymmetry" S.W. Hawking (ed.) W. Israel (ed.) , General Relativity, an Einstein Centenary Survey , Cambridge Univ. Press (1979) pp. 581–638 |

[a4] | P.T. Landsberg, D.A. Evans, "Mathematical cosmology" , Clarendon Press (1977) MR0626340 |

[a5] | S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973) MR0424186 Zbl 0265.53054 |

[a6] | S. Weinberg, "Gravitation and cosmology" , Wiley (1972) pp. Chapt. 3 |

[a7] | S. Chandrasekhar, "The mathematical theory of black holes" , Oxford Univ. Press (1983) MR0700826 Zbl 0511.53076 |

[a8] | J. Ehlers (ed.) , Relativity theory and astrophysics , 1–3 , Amer. Math. Soc. (1967) Zbl 0203.28304 |

[a9] | I.D. Novikov, V.P. Frolov, "Physics of black holes" , Kluwer (1989) (Translated from Russian) MR1032763 Zbl 0688.53034 |

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