Exact solutions with <i>w</i> modes

Mustapha Ishak; Luke Chamandy; Nicholas Neary; Kayll Lake

doi:10.1103/PhysRevD.64.024005

Access by Xinjiang University

Exact solutions with w modes

Mustapha Ishak, Luke Chamandy, Nicholas Neary, and Kayll Lake^*

Department of Physics, Queen’s University, Kingston, Ontario, Canada K7L 3N6

^*Email address: lake@astro.queensu.ca

PDF

Phys. Rev. D 64, 024005 – Published 6 June, 2001

DOI: https://doi.org/10.1103/PhysRevD.64.024005

Abstract

An explicit necessary condition for the internal trapping of null geodesics, along with the occurrence of resonance scattering of axial gravitational waves, is proposed for static spherically symmetric perfect fluid solutions of Einstein’s equations. Some exact inhomogeneous solutions which exhibit this trapping are given with special attention to boundary conditions and the physical acceptability of the space-times. In terms of the tenuity $(α = R / M$ at the boundary) all the examples given here lie in the narrow range $2.8 < α < 2.9.$ The tenuity can be raised to more interesting values by the addition of an envelope without altering the trapping.

References (40)

S. Chandrasekhar and V. Ferrari, Proc. R. Soc. London A434, 449 (1991); see also N. Andersson, Y. Kojima, and K. D. Kokkotas, Astrophys. J. 462, 855 (1996); and N. Andersson and K. D. Kokkotas, Mon. Not. R. Astron. Soc. 297, 493 (1998); an extensive review relevant to the background of this paper is given by K. D. Kokkotas and B. G. Schmidt, “Quasi-Normal Modes of Stars and Black Holes,” Living Reviews 1999-2 (http://www.livingreviews.org/Articles/Volume2/1999-2kokkotas/).
The idea of trapped waves in compact objects is not new. See B. Iyer and C. Vishveshwara, in A Random Walk in Relativity and Cosmology, edited by N. Dadhich, J. Krishna Rao, J. Narlikar, and C. Vishveshwara (Wiley Eastern Ltd., New Delhi, 1985), and references therein.
See Y. Kojima, Prog. Theor. Phys. 79, 665 (1988); and K. D. Kokkotas and B. Schutz, Mon. Not. R. Astron. Soc. 255, 119 (1992).
O. Benhar, E. Berti, and V. Ferrari, gr-qc/9901037.
Z. Andrade and R. Price, Phys. Rev. D 60, 104037 (1999).
K. Tominaga, M. Saijo, and K. Maeda, Phys. Rev. D 60, 024004 (1999).
M. Delgaty and K. Lake, Comput. Phys. Commun. 115, 395 (1998); this work (with corrections and updates) is being incorporated into an on-line database (http://grtensor.phy.queensu.ca/solutions/).
K. Lake and P. Musgrave, Gen. Relativ. Gravit. 26, 917 (1994).
G. F. R. Ellis, R. Maartens, and S. D. Nel, Mon. Not. R. Astron. Soc. 184, 439 (1978).
B. Schmidt, Kugelsymmetrische statische Materielösungen der Einsteinschen Feldgleichungen Diplomarbeit, Hamburg University.
Whereas at very high densities the adiabatic sound speed may not equal the actual propagation speed of the signal, we do not distinguish between these cases here and take the conservative view that $v_{s}^{2} >~ 1$ is acausal. See, for example, G. Caporaso and K. Brecher, Phys. Rev. D 20, 1823 (1979).
It is worth noting that the standard example for w modes, the Schwarzschild interior solution, fails essentially only at the last step. However, since the sound speed is (formally) infinite, it is physically unacceptable.
The stability of the Finch-Skea solution (Ref. [31]) cannot be estimated from their Table 1. These solutions should be specified by their parameters C and $β$ from which follow the pair ${(m, r}_{s})$ . That is, m and $r_{s}$ cannot be specified independently. In general, an exact solution can reasonably be expected to approximate only part of a full configuration. This means that the full model must be specified and a dynamical stability analysis is required for a reliable conclusion on stability.
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).
We use geometrical units throughout. For example, the tenuity $r_{Σ} / M$ is actually $c^{2} r_{Σ} / G M$ .
R. Tolman, Relativity Thermodynamics and Cosmology (Dover, New York, 1987).
J. Oppenheimer and G. Volkoff, Phys. Rev. 55, 374 (1939).
T. Baumgarte and A. Rendall, Class. Quantum Grav. 10, 327 (1993).
S. Hayward, Phys. Rev. D 53, 1938 (1996).
S. Sonego and M. Massar, Mon. Not. R. Astron. Soc. 281, 659 (1996).
In the Schwarzschild vacuum, for example, $B (r) = r / \sqrt{1 - 2 m / r}$ so that $B (r) \sim r$ in the Newtonian limit $r ≫ 2 m$ . This limit corresponds to straight-line motion. The trapping of null geodesics inside the Schwarzschild interior solution has been studied previously [e.g., F. de Felice, Nuovo Cimento Soc. Ital. Fis., B 63, 649 (1969); J. Jaffe, Mon. Not. R. Astron. Soc. 149, 395 (1970)]. The trapping of null geodesics is astrophysically relevant only to neutrinos and requires a detailed consideration of scattering.
Relation (9) returns the familiar $r < 3 m$ in the Schwarzschild vacuum. This field does not satisfy Eq. (2) since it is singular at the origin.
Equality in Eq. (9) is relevant to the study of “photon surfaces.” See C. M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42, 818 (2001).
S. Chandrasekhar and V. Ferrari, Proc. R. Soc. London A432, 247 (1991); apparently, Eq. (10) was first derived by K. S. Thorne and A. Campolattaro, Astrophys. J. 149, 591 (1967).
P. Musgrave and K. Lake, Class. Quantum Grav. 13, 1885 (1996).
Whereas $ρ_{Σ}$ is not required to be $0$ , models which are otherwise physically acceptable may give physically unreasonable values of $ρ_{Σ} .$ Uniform density models are, in this sense, the worst possible models.
As pointed out originally by Chandrasekhar and Ferrari [1], a Heaviside discontinuity in V at $Σ$ is of no physical consequence.
H. Buchdahl, Astrophys. J. 147, 310 (1967).
R. Tolman, Phys. Rev. 55, 364 (1939). A detailed study of the Tolman VII solution is provided in a separate analysis where it is shown that this solution exhibits trapped null orbits even for a tenuity $> 3$ .
J. Lattimer and M. Prakash, astro-ph/0002232.
M. Finch and J. Skea, Class. Quantum Grav. 6, 467 (1989).
Although the models considered in this paper are analytic, the limits quoted on $α$ are the result of detailed numerical studies.
The upper limit differs from the value given in [31] and may be traced to an error in their equation (6.14). It is worth noting that the sound speed fails to be monotone decreasing with r for $α >~ 2.679$ . The sound speed is subluminal throughout the solution for $α > 2.786$ .
H. Heintzmann, Z. Phys. 228, 489 (1969).
M. Durgapal, J. Phys. A 15, 2637 (1982).
M. Korkina, Sov. Phys. J. 24, 468 (1981).
The quantities which enter Eq. (9) can be invariantly defined.
H. Buchdahl, Phys. Rev. 116, 1027 (1959).
K. Rosquist, Phys. Rev. D 59, 044022 (1999), has recently shown an exact solution with trapping and $α = 5$ , but the solution is acausal in the region of interest. It is possible to construct “toy” models with multiple necks [see M. Karlovini, K. Rosquist, and L. Samuelsson, Class. Quantum Grav. 18, 817 (2001) and references therein]; recently Z. Andrade, Phys. Rev. D 63, 124002 (2001) has studied trapped and excited w modes in double uniform density models. All of these works treat, in the context of the present paper, unphysical models.
It is worth noting that if a solution represents only part of a more realistic configuration, then causality, but not stability, conditions need be applied to the solution.

[c1] S. Chandrasekhar and V. Ferrari, Proc. R. Soc. London A434, 449 (1991); see also N. Andersson, Y. Kojima, and K. D. Kokkotas, Astrophys. J. 462, 855 (1996); and N. Andersson and K. D. Kokkotas, Mon. Not. R. Astron. Soc. 297, 493 (1998); an extensive review relevant to the background of this paper is given by K. D. Kokkotas and B. G. Schmidt, “Quasi-Normal Modes of Stars and Black Holes,” Living Reviews 1999-2 (http://www.livingreviews.org/Articles/Volume2/1999-2kokkotas/).

[c2] The idea of trapped waves in compact objects is not new. See B. Iyer and C. Vishveshwara, in A Random Walk in Relativity and Cosmology, edited by N. Dadhich, J. Krishna Rao, J. Narlikar, and C. Vishveshwara (Wiley Eastern Ltd., New Delhi, 1985), and references therein.

[c3] See Y. Kojima, Prog. Theor. Phys. 79, 665 (1988); and K. D. Kokkotas and B. Schutz, Mon. Not. R. Astron. Soc. 255, 119 (1992).

[c4] O. Benhar, E. Berti, and V. Ferrari, gr-qc/9901037.

[c5] Z. Andrade and R. Price, Phys. Rev. D 60, 104037 (1999).

[c6] K. Tominaga, M. Saijo, and K. Maeda, Phys. Rev. D 60, 024004 (1999).

[c7] M. Delgaty and K. Lake, Comput. Phys. Commun. 115, 395 (1998); this work (with corrections and updates) is being incorporated into an on-line database (http://grtensor.phy.queensu.ca/solutions/).

[c8] K. Lake and P. Musgrave, Gen. Relativ. Gravit. 26, 917 (1994).

[c9] G. F. R. Ellis, R. Maartens, and S. D. Nel, Mon. Not. R. Astron. Soc. 184, 439 (1978).

[c10] B. Schmidt, Kugelsymmetrische statische Materielösungen der Einsteinschen Feldgleichungen Diplomarbeit, Hamburg University.

[c11] Whereas at very high densities the adiabatic sound speed may not equal the actual propagation speed of the signal, we do not distinguish between these cases here and take the conservative view that $v_{s}^{2} >~ 1$ is acausal. See, for example, G. Caporaso and K. Brecher, Phys. Rev. D 20, 1823 (1979).

[c12] It is worth noting that the standard example for w modes, the Schwarzschild interior solution, fails essentially only at the last step. However, since the sound speed is (formally) infinite, it is physically unacceptable.

[c13] The stability of the Finch-Skea solution (Ref. [31]) cannot be estimated from their Table 1. These solutions should be specified by their parameters C and $β$ from which follow the pair ${(m, r}_{s})$ . That is, m and $r_{s}$ cannot be specified independently. In general, an exact solution can reasonably be expected to approximate only part of a full configuration. This means that the full model must be specified and a dynamical stability analysis is required for a reliable conclusion on stability.

[c14] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

[c15] We use geometrical units throughout. For example, the tenuity $r_{Σ} / M$ is actually $c^{2} r_{Σ} / G M$ .

[c16] R. Tolman, Relativity Thermodynamics and Cosmology (Dover, New York, 1987).

[c17] J. Oppenheimer and G. Volkoff, Phys. Rev. 55, 374 (1939).

[c18] T. Baumgarte and A. Rendall, Class. Quantum Grav. 10, 327 (1993).

[c19] S. Hayward, Phys. Rev. D 53, 1938 (1996).

[c20] S. Sonego and M. Massar, Mon. Not. R. Astron. Soc. 281, 659 (1996).

[c21] In the Schwarzschild vacuum, for example, $B (r) = r / \sqrt{1 - 2 m / r}$ so that $B (r) \sim r$ in the Newtonian limit $r ≫ 2 m$ . This limit corresponds to straight-line motion. The trapping of null geodesics inside the Schwarzschild interior solution has been studied previously [e.g., F. de Felice, Nuovo Cimento Soc. Ital. Fis., B 63, 649 (1969); J. Jaffe, Mon. Not. R. Astron. Soc. 149, 395 (1970)]. The trapping of null geodesics is astrophysically relevant only to neutrinos and requires a detailed consideration of scattering.

[c22] Relation (9) returns the familiar $r < 3 m$ in the Schwarzschild vacuum. This field does not satisfy Eq. (2) since it is singular at the origin.

[c23] Equality in Eq. (9) is relevant to the study of “photon surfaces.” See C. M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42, 818 (2001).

[c24] S. Chandrasekhar and V. Ferrari, Proc. R. Soc. London A432, 247 (1991); apparently, Eq. (10) was first derived by K. S. Thorne and A. Campolattaro, Astrophys. J. 149, 591 (1967).

[c25] P. Musgrave and K. Lake, Class. Quantum Grav. 13, 1885 (1996).

[c26] Whereas $ρ_{Σ}$ is not required to be $0$ , models which are otherwise physically acceptable may give physically unreasonable values of $ρ_{Σ} .$ Uniform density models are, in this sense, the worst possible models.

[c27] As pointed out originally by Chandrasekhar and Ferrari [1], a Heaviside discontinuity in V at $Σ$ is of no physical consequence.

[c28] H. Buchdahl, Astrophys. J. 147, 310 (1967).

[c29] R. Tolman, Phys. Rev. 55, 364 (1939). A detailed study of the Tolman VII solution is provided in a separate analysis where it is shown that this solution exhibits trapped null orbits even for a tenuity $> 3$ .

[c30] J. Lattimer and M. Prakash, astro-ph/0002232.

[c31] M. Finch and J. Skea, Class. Quantum Grav. 6, 467 (1989).

[c32] Although the models considered in this paper are analytic, the limits quoted on $α$ are the result of detailed numerical studies.

[c33] The upper limit differs from the value given in [31] and may be traced to an error in their equation (6.14). It is worth noting that the sound speed fails to be monotone decreasing with r for $α >~ 2.679$ . The sound speed is subluminal throughout the solution for $α > 2.786$ .

[c34] H. Heintzmann, Z. Phys. 228, 489 (1969).

[c35] M. Durgapal, J. Phys. A 15, 2637 (1982).

[c36] M. Korkina, Sov. Phys. J. 24, 468 (1981).

[c37] The quantities which enter Eq. (9) can be invariantly defined.

[c38] H. Buchdahl, Phys. Rev. 116, 1027 (1959).

[c39] K. Rosquist, Phys. Rev. D 59, 044022 (1999), has recently shown an exact solution with trapping and $α = 5$ , but the solution is acausal in the region of interest. It is possible to construct “toy” models with multiple necks [see M. Karlovini, K. Rosquist, and L. Samuelsson, Class. Quantum Grav. 18, 817 (2001) and references therein]; recently Z. Andrade, Phys. Rev. D 63, 124002 (2001) has studied trapped and excited w modes in double uniform density models. All of these works treat, in the context of the present paper, unphysical models.

[c40] It is worth noting that if a solution represents only part of a more realistic configuration, then causality, but not stability, conditions need be applied to the solution.

Physical Review D

Export citation

Export citation

Exact solutions with w modes

Abstract

References (40)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation