No-scalar-hair theorem for spherically symmetric reflecting stars
aa r X i v : . [ g r- q c ] D ec No-scalar-hair theorem for spherically symmetric reflecting stars
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: November 8, 2018)
Abstract
It is proved that spherically symmetric compact reflecting objects cannot support static bound-state configurations made of scalar fields whose self-interaction potential V ( ψ ) is a monotonicallyincreasing function of its argument. Our theorem rules out, in particular, the existence of massivescalar hair outside the surface of a spherically symmetric compact reflecting star. . INTRODUCTION The no-hair theorem of Bekenstein [1] (see also [2]) has revealed the intriguing fact that,in accord with Wheeler’s celebrated conjecture [3, 4], asymptotically flat black holes cannotsupport regular self-gravitating static matter configurations made of massive scalar fieldsin their external spacetime regions [5–9]. This important physical characteristic of blackholes is often attributed in the physics literature to the fact that the boundary of a classicalblack hole (its horizon) acts as a one-way membrane that irreversibly absorbs matter andradiation fields.One naturally wonders whether this no-scalar-hair behavior is a unique property of blackholes? In the present paper we shall explore the possibility of extending the no-scalar-hairtheorem to the regime of regular (that is, horizonless ) curved spacetimes. In particular,we here raise the following physically intriguing question: Can regular compact reflectingobjects (that is, reflecting stars [10] which possess no event horizons) support self-gravitatingmassive scalar field configurations in their exterior spacetime regions?In order to address this interesting question, in the present study we shall replace thestandard ingoing ( absorbing ) boundary condition which characterizes the behavior of classi-cal fields at the horizon of a black hole [1, 6] by a reflecting ( repulsive ) boundary condition atthe surface of the horizonless compact star. Our theorem, to be proved below, reveals the in-triguing fact that horizonless compact reflecting stars share the characteristic no-scalar-hairproperty with asymptotically flat black holes.
II. THE NO-SCALAR-HAIR THEOREM FOR SPHERICALLY SYMMETRICCOMPACT REFLECTING STARS
We consider a static spherically symmetric compact reflecting object (a reflecting star[10]) of radius R . Using the Schwarzschild coordinates ( t, r, θ, φ ), the line element of thecorresponding curved spacetime can be expressed in the form [11, 12] ds = − e ν dt + e λ dr + r ( dθ + sin θdφ ) , (1)where ν = ν ( r ) and λ = λ ( r ). An asymptotically flat spacetime is characterized by theasymptotic behaviors ν ∼ O ( r − ) and λ ∼ ( r − ) for r → ∞ .2he compact star is non-linearly coupled to a real scalar field ψ with a general self-interaction potential V = V ( ψ ) whose action is given by [11, 13] S = S EH − Z h ∂ α ψ∂ α ψ + V ( ψ ) i √− gd x . (2)We shall assume a positive semidefinite self-interaction potential for the scalar field whichis a monotonically increasing function of its argument. That is, V (0) = 0 with ˙ V ≡ d [ V ( ψ )] d ( ψ ) ≥ . (3)Note that the physically interesting case of a free massive scalar field with ˙ V = µ ≥ ∂ α ∂ α ψ − ˙ V ψ = 0 (4)for the self-interacting scalar field. Substituting the line-element (1) of the spherically sym-metric curved spacetime into (4), one obtains the characteristic radial differential equation[14] ψ ′′ + 12 ( 4 r + ν ′ − λ ′ ) ψ ′ − e λ ˙ V ψ = 0 (5)for the self-interacting static scalar field.The energy density of the self-interacting scalar field (2) is given by [11] ρ = − T tt = 12 [ e − λ ( ψ ′ ) + V ( ψ )] . (6)An asymptotically flat (finite mass) spacetime is characterized by an energy density ρ whichapproaches zero asymptotically faster than 1 /r [15]: r ρ ( r ) → r → ∞ . (7)Taking cognizance of Eqs. (3), (6), and (7), one deduces that the scalar field eigenfunctionis characterized by the asymptotic boundary condition ψ ( r → ∞ ) → r = R of the compact reflecting star [16, 17]: ψ ( r = R ) = 0 . (9)3aking cognizance of the boundary conditions (8) and (9), one concludes that the charac-teristic eigenfunction ψ of the scalar field must have (at least) one extremum point, r = r peak ,between the surface r = R of the reflecting star and spatial infinity [that is, in the interval r peak ∈ ( R, ∞ )]. At this extremum point the eigenfunction ψ of the external scalar field ischaracterized by the relations { ψ ′ = 0 and ψ · ψ ′′ < } for r = r peak . (10)Substituting (3) and (10) into the l.h.s of (5), one finds the inequality ψ ′′ + 12 ( 4 r + ν ′ − λ ′ ) ψ ′ − e λ ˙ V ψ < r = r peak , (11)in contradiction with the characteristic relation (5) of the self-interacting scalar field. III. SUMMARY
In this compact analysis, we have proved that if a spherically symmetric compact re-flecting star [10] can support self-gravitating massive scalar field configurations, then thecorresponding scalar field eigenfunction ψ must have an extremum point outside the reflect-ing surface of the star. At this extremum point, the scalar field eigenfunction is characterizedby the inequality (11). However, one realizes that this inequality is in contradiction withthe characteristic identity (5) for the self-interacting scalar field. Thus, there is no solutionfor the external scalar eigenfunction except the trivial one, ψ ≡ V ( ψ )is a monotonically increasing function of its argument. In particular, our theorem rules outthe existence of asymptotically flat massive scalar hair (regular self-gravitating massive scalarfield configurations) outside the surface of a spherically symmetric (horizonless) compactreflecting star.Our compact theorem therefore reveals the interesting fact that horizonless compactreflecting stars share the no-scalar-hair property with the more familiar asymptotically flatabsorbing [20] black holes. 4 CKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I would like to thank YaelOren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for helpful discussions. [1] J. D. Bekenstein, Phys. Rev. Lett. , 452 (1972).[2] J. E. Chase, Commun. Math. Phys. , 276 (1970); C. Teitelboim, Lett. Nuovo Cimento ,326 (1972); J. D. Bekenstein, Physics Today , 24 (1980).[3] R. Ruffini and J. A. Wheeler, Phys. Today , 30 (1971).[4] B. Carter, in Black Holes , Proceedings of 1972 Session of Ecole d’ete de Physique Theorique,edited by C. De Witt and B. S. De Witt (Gordon and Breach, New York, 1973).[5] As noted in [6], this interesting no-hair property of black holes can be extended to the regimeof self-gravitating scalar fields whose self-interaction potential V ( ψ ) is a monotonically in-creasing function of its argument [see Eqs. (2) and (3) below].[6] J. D. Bekenstein, arXiv:gr-qc/9605059 .[7] Interestingly, it has recently been proved [8, 9] that stationary configurations made of massivescalar fields can be supported in the external spacetime regions of spinning black holes.[8] S. Hod, Phys. Rev. D , 104026 (2012) [arXiv:1211.3202]; S. Hod, The Euro. Phys. Journal C , 2378 (2013) [arXiv:1311.5298]; S. Hod, Phys. Rev. D , 024051 (2014) [arXiv:1406.1179];S. Hod, Phys. Lett. B , 196 (2014) [arXiv:1411.2609]; S. Hod, Class. and Quant. Grav. , 134002 (2015) [arXiv:1607.00003]; S. Hod, Class. and Quant. Grav. , 114001 (2016); S.Hod, Phys. Lett. B , 181 (2016) [arXiv:1606.02306]; S. Hod and O. Hod, Phys. Rev. D , 061502 Rapid communication (2010) [arXiv:0910.0734]; S. Hod, Phys. Lett. B , 320(2012) [arXiv:1205.1872].[9] C. A. R. Herdeiro and E. Radu, Phys. Rev. Lett. , 221101 (2014); C. A. R. Herdeiro andE. Radu, Phys. Rev. D , 124018 (2014); C. A. R. Herdeiro and E. Radu, Int. J. Mod. Phys.D , 1442014 (2014); C. L. Benone, L. C. B. Crispino, C. Herdeiro, and E. Radu, Phys.Rev. D , 104024 (2014); C. Herdeiro, E. Radu, and H. R´unarsson, Phys. Lett. B , 302(2014); C. Herdeiro and E. Radu, Class. Quantum Grav. , 1542014 (2015); C. A. R. Herdeiro and E. Radu, Int. . Mod. Phys. D , 1544022 (2015); J. C. Degollado and C. A. R. Herdeiro, Gen. Rel. Grav. , 2483 (2013); P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. R´unarsson, Phys.Rev. Lett. , 211102 (2015); C. A. R. Herdeiro, E. Radu, and H. F. R´unarsson, Phys. Rev.D , 084059 (2015); Y. Brihaye, C. Herdeiro, and E. Radu, Phys. Lett. B , 279 (2016).[10] We use here the term ‘reflecting star’ to describe a physical compact object for which theexternal scalar field vanishes on its surface.[11] A. E. Mayo and J. D. Bekenstein, Phys. Rev. D , 5059 (1996).[12] We shall use natural units in which G = c = 1.[13] Here S EH is the Einstein-Hilbert action.[14] Here a prime ′ denotes a derivative with respect to the radial coordinate r .[15] S. Hod, Phys. Lett. B , 383 (2014) [arXiv:1412.3808].[16] It is worth noting that in the vast physics literature that deals with the famous ‘black-holebomb’ mechanism of Press and Teukolsky [17], one usually places a reflecting surface around ablack hole in order to prevent the scalar field from escaping to infinity. On the other hand, inthe present study the role of the reflecting surface is to prevent the scalar field from enteringthe central horizonless compact star.[17] W. H. Press and S. A. Teukolsky, Nature , 211 (1972); W. H. Press and S. A. Teukolsky,Astrophys. J. , 649 (1973).[18] As nicely emphasized by the anonymous referee, the result of the present paper can be framedin the familiar context of standard quantum mechanics. In particular, a stationary state of astandard one dimensional quantum mechanical problem is also characterized by the relation(10), which implies that the energy of the stationary quantum state is bounded from belowby the potential energy at the corresponding extremum point. Note that in the terminologyof quantum mechanics, our static scalar field corresponds to a zero-energy state, whereas theeffective radial potential is positive [see (3)].[19] As pointed out by the anonymous referee, it would be interesting to check whether a reflectingstar can support a stationary complex scalar field configuration around it.[20] It is worth emphasizing again that black holes, as opposed to the compact reflecting starsdiscussed in the present analysis, are characterized by the presence of absorbing one-waymembranes (event horizons).one-waymembranes (event horizons).