###### Abstract

We show that the well-known most general static and spherically symmetric exact solution to the Einstein-massless scalar equations given by Wyman is the same as one found by Janis, Newman and Winicour several years ago. We obtain the energy associated with this spacetime and find that the total energy for the case of the purely scalar field is zero.

Janis-Newman-Winicour and Wyman solutions are the same^{1}^{1}1This paper is dedicated to the memory of Professor Nathan Rosen.

K. S. Virbhadra

Theoretical Astrophysics Group

Tata Institute of Fundamental Research

Homi Bhabha Road, Colaba, Mumbai 400005, India.

To appear in Int. J. Mod. Phys. A

Even before the general theory of relativity was proposed, scalar field has been conjectured to give rise to the long-range gravitational fields[1]. Several theories involving scalar fields have been suggested [2]. The subject of scalar fields minimally as well as conformally coupled to gravitation has fascinated many researchers’ minds ([3] -[10]). Since the last few years there has been a growing interest in studying the gravitational collapse of scalar fields and the nature of singularities in the Einstein-massless scalar (EMS) theory (see Ref. [4] and references therein). There has been considerable interest in obtaining solutions to the EMS as well as the Einstein-massless conformal scalar equations. Janis, Newman and Winicour (JNW) [5] obtained static and spherically symmetric exact solution to the EMS equations. Wyman[6] further obtained a static spherically symmetric exact solution to the EMS equations. There is no reference to the JNW solution in his paper. Agnese and La Camera[7] expressed the Wyman solution in a more convenient form. Roberts[8] showed that the most general static spherically symmetic solution to the EMS equations (with zero cosmological constant) is asymptotically flat and this is the Wyman solution. The Wyman solution is well-known in the literature[9].In the present note we show that the Wyman solution is the same as the JNW solution, which was obtained almost twelve years ago. We further obtain the total energy associated with this spacetime. We use geometrized units and follow the convention that Latin (Greek) indices take values ().

The EMS field equations are

(1) |

where , the energy-momentum tensor of the massless scalar field, is given by

(2) |

and

(3) |

stands for the massless scalar field. is the Ricci tensor and is the Ricci scalar. Equation with Eq. can be expressed as

(4) |

JNW[5] obtained static and spherically symmetric exact solution to the EMS equations, which is given by the line element

(5) |

with

(6) |

and the scalar field

(7) |

where

(8) |

and

(9) |

and are two constant parameters in the solution. gives the flat spacetime, whereas gives the line element

(10) |

which obviously represents the Schwarschild metric. The JNW solution can be put in a more convenient form, in coordinates , by the line element

(11) |

and the scalar field

(12) |

where

(13) |

and are constant parameters. The radial coordinates and are related through

(14) |

and the spacetime parameters and are related through

(15) |

The solution to the EMS equations, expressed by Eqs. , is exactly the well-known Wyman solution (see Ref. [7]). Thus, Wyman did not obtain a new solution, but he rediscovered the JNW solution independently. The JNW solution, in coordinates, has a curvature singularity at . Garfinkle, Horowitz and Strominger [10] obtained a nice form of charged dilaton black hole solution, characterized by mass, charge, and a coupling parameter (which controls the strength of the coupling of the dilaton to the Maxwell field). A particular solution of this (putting the electric charge parameter zero) yields the JNW (Wyman) solution. This fact is not noticed in their paper.

It is of interest to obtain the energy associated with the JNW spacetime. The energy-momentum localization has been a “recalcitrant” problem since the outset of the general theory of relativity. Though, several energy-momentum complexes have been shown to give the same and reasonable result (local values) for any Kerr-Schild class metric as well as for the Einstein-Rosen spacetime, the subject of energy-momentum localization is still debatable (see Ref. [11] and references therein). However, the total energy of an asymptotically flat spacetime is unanimously accepted. Using the Einstein energy-momentum complex we first obtain an energy expression for a general nonstatic spherically symmetric metric and then we calculate the total energy associated with the JNW spacetime. A general nonstatic spherically symmetric line element is

(16) |

where . The Einstein energy-momentum complex (see Refs. [11]) is

(17) |

where

(18) |

The Einstein energy-momentum complex satisfies the local conservation laws

(19) |

where

(20) |

is the symmetric energy-momentum tensor due to matter and all nongravitational fields and is the energy-momentum pseudotensor due to the gravitational field only. The energy and momentum components are given by

(21) |

where is the infinitesimal surface element and is the outward unit normal vector. stands for the energy and stand for the linear momentum components. As it is known that the energy-momentum complexes give meaningful result only if calculations are carried out in quasi-Minkowskian coordinates, we transform the line element in coordinates (according to

(22) |

where the prime denotes the partial derivative with respect to the radial coordinate . Using the above in we obtain the energy

(23) |

Substituting and we obtain the energy associated with the JNW spacetime.

(24) |

The total energy of the JNW spacetime is given by the parameter . We have repeated these calculations using other energy-momentum complexes and have found the same result as given by Eq. (). The total energy of a purely scalar field (i.e. for ) is zero. It is of interest to investigate whether or not this is true in general, i.e. for any purely massless scalar field.

Acknowledgements

Thanks are due to Professor P. C. Vaidya for a careful reading of the manuscript.

## References

- [1] M. Abraham, Jahrb. Radioakt. Electronik 11, 470 (1914); O. Bergmann, Am. J. Phys. 24, 38 (1956); S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity (John Wiely & Sons, NY, 1972), p.157.
- [2] C. H. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961); C. G. Callan, S. Coleman and R. Jackiw, Ann. Phys. (N.Y.) 59, 42 (1970); J. H. Horne and G. T. Horowitz, Phys. Rev. D46, 1340 (1992); H. Yilmaz, Il Nuovo Cimento 107B, 941 (1992); I. Peterson, Science News 146, 376 (1994).
- [3] R. Penny, Phys. Rev. 174, 1578 (1968); J. D. Bekenstein, Ann. Phys. 91, 75 (1975); S. S. Bayin, F. I. Cooperstock and V. Faraoni, Astrophy. J. 428, 439 (1994); K. S. Virbhadra and J. C. Parikh, Phys. Lett. B331, 302 (1994); Erratum : Phys. Lett. B340, 265 (1994); K. S. Virbhadra, gr-qc/9408035, Pramana-J.Phys. 44, 317 (1995); A. E. Mayo and J. D. Bekenstein, Phys. Rev. D54, 5059 (1996); C. Martinez and J. Zanelli, Phys. Rev. D54, 3830 (1996); J. D. Bekenstein, Black hole hair: twenty-five years, gr-qc/9605059 (1996).
- [4] E. Malec, Selfgravitating nonlinear scalar fields, Commun. Math. Phys., (1996) (submitted).
- [5] A. I. Janis, E. T. Newman and J. Winicour, Phys. Rev. Lett. 20, 878 (1968).
- [6] M. Wyman, Phys. Rev. D24, 839 (1981).
- [7] A. G. Agnese and M. La Camera, Phys. Rev. D31, 1280 (1985).
- [8] M. D. Roberts, Astrophys. & Space Sc. 200, 331 (1993).
- [9] T. Papacostas, J. Math. Phys. 32, 2468 (1991); D. V. Galtsov and B. C. Xanathopoulos, J. Math. Phys. 33, 273 (1992); K. Schmoltzi and T. Schucker, Phys. Lett. A 161, 212 (1991); B. C. Xanthopoulos and T. E. Dialynas, J. Math. Phys. 33, 1463 (1992); P. Jetzer and D. Scialom, Phys. Lett. A 169, 12 (1992); J. Z. Li, J. Math. Phys. 33, 3506 (1992); A. Hardell and H. Dehnen, Gen. Relativ. Gravit. 25, 1165 (1993); V. Husain, E. A. Martinez and D. Nunez, Phys.Rev. D50, 3783 (1994); H. P. Deoliveira, J. Math. Phys. 36, 2988 (1995); Y. J. Kiem and D. Park, Phys. Rev. D53, 747 (1996); J. M. Salim and S. L. Sautu, Class. Quantum Grav. 13, 353 (1996).
- [10] D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D43, 3140 (1991); Erratum : Phys. Rev. D45, 3888 (1992).
- [11] K. S. Virbhadra, Phys. Rev. D41, 1086 (1990); Phys. Rev. D42, 1066 (1990); Phys. Rev. D42, 2919 (1990); N. Rosen and K. S. Virbhadra, Gen. Relativ. Grav. 25, 429 (1993); K. S. Virbhadra and J. C. Parikh, Phys. Lett. B317, 312 (1993); K. S. Virbhadra, Pramana- J. Phys. 45, 215 (1995); J. M. Aguirregabiria, A. Chamorro and K. S. Virbhadra, Gen. Relativ. & Gravit. 28, 1393 (1996).