Abstract
The recently published no-hair theorems of Hod, Bhattacharjee, and Sarkar have revealed the intriguing fact that horizonless compact reflecting stars cannot support spatially regular configurations made of scalar, vector and tensor fields. In the present paper we explicitly prove that the interesting no-hair behavior observed in these studies is not a generic feature of compact reflecting stars. In particular, we shall prove that charged reflecting stars can support charged massive scalar field configurations in their exterior spacetime regions. To this end, we solve analytically the characteristic Klein–Gordon wave equation for a linearized charged scalar field of mass mu , charge coupling constant q, and spherical harmonic index l in the background of a spherically symmetric compact reflecting star of mass M, electric charge Q, and radius R_{text {s}}gg M,Q. Interestingly, it is proved that the discrete set {R_{text {s}}(M,Q,mu ,q,l;n)}^{n=infty }_{n=1} of star radii that can support the charged massive scalar field configurations is determined by the characteristic zeroes of the confluent hypergeometric function. Following this simple observation, we derive a remarkably compact analytical formula for the discrete spectrum of star radii in the intermediate regime Mll R_{text {s}}ll 1/mu . The analytically derived resonance spectrum is confirmed by direct numerical computations.
Highlights
The no-hair conjecture for classical black-hole spacetimes has attracted much attention from physicists and mathematicians over the years
In a very interesting paper, Bhattacharjee and Sarkar [48] have recently extended the regime of validity of the no-hair theorem for horizonless spacetimes and proved that compact reflecting stars cannot support spatially regular configurations made of vector and tensor fields
It is well known that asymptotically flat black holes, which are characterized by compact event horizons with purely ingoing boundary conditions, cannot support spatially regular static configurations made of scalar, spinor, or vector fields [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
Summary
The no-hair conjecture for classical black-hole spacetimes has attracted much attention from physicists and mathematicians over the years. The elegant no-hair theorems presented in [4,5,6,7,8,9,10,11,12,13,14] have explicitly proved that asymptotically flat black holes cannot support physically acceptable (spatially regular) static configurations made of scalar, spinor, or vector fields1 [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. Below we shall prove that, for a spherically symmetric compact reflecting star of mass M and electric charge Q, there exists a discrete set {Rs(M, Q, μ, q, l; n)}nn==∞ 1 of star radii that can support an external spatially regular charged massive scalar field of proper mass μ, charge coupling constant q, and spherical harmonic index l.3,4 In particular, we shall explicitly show that the physical properties of the composed charged-reflecting-starlinearized-charged-massive-scalar-field configurations can be studied analytically in the intermediate radii regime M Rs 1/μ
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