Skip to main content
SpringerLink
Log in

Aspects of GRMHD in high-energy astrophysics: geometrically thick disks and tori agglomerates around spinning black holes

  • Review Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

This work focuses on some key aspects of the general relativistic (GR)—magneto-hydrodynamic (MHD) applications in high-energy astrophysics. We discuss the relevance of the GRHD counterparts formulation exploring the geometrically thick disk models and constraints of the GRMHD shaping the physics of accreting configurations. Models of clusters of tori orbiting a central super-massive black hole (SMBH) are described. These orbiting tori aggregates form sets of geometrically thick, pressure supported, perfect fluid tori, associated to complex instability processes including tori collision emergence and empowering a wide range of activities related expectantly to the embedding matter environment of Active Galaxy Nuclei. Some notes are included on aggregates combined with proto-jets, represented by open cusped solutions associated to the geometrically thick tori. This exploration of some key concepts of the GRMHD formulation in its applications to High-Energy Astrophysics starts with the discussion of the initial data problem for a most general Einstein–Euler–Maxwell system addressing the problem with a relativistic geometric background. The system is then set in quasi linear hyperbolic form, and the reduction procedure is argumented. Then, considerations follow on the analysis of the stability problem for self-gravitating systems with determined symmetries considering the perturbations also of the geometry part on the quasi linear hyperbolic onset. Thus we focus on the ideal GRMHD and self-gravitating plasma ball. We conclude with the models of geometrically thick GRHD disks gravitating around a Kerr SMBH in their GRHD formulation and including in the force balance equation of the disks the influence of a toroidal magnetic field, determining its impact in tori topology and stability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. We adopt the geometrical units \(c=1=G\) and the \((-,+,+,+)\) signature, Latin indices run in \(\{0,1,2,3\}\). The four-velocity satisfy \(U^a U_a=-1\). The radius r has unit of mass [M], and the angular momentum units of \([M]^2\), the velocities \([U^t]=[U^r]=1\) and \([U^{\phi }]=[U^{\theta }]=[M]^{-1}\) with \([U^{\phi }/U^{t}]=[M]^{-1}\) and \([U_{\phi }/U_{t}]=[M]\). For the seek of convenience, we always consider the dimensionless energy and effective potential \([V_{eff}]=1\) and an angular momentum per unit of mass \([L]/[M]=[M]\).

  2. This also includes effects related to the disks energetics such as the mass-flux, the enthalpy-flux (evaluating also the temperature parameter), and the flux thickness. We assume polytropic fluids with pressure \(p=\kappa \rho ^{1+1/n}\). The mass-flux, enthalpy-flux and flux thickness have all form \({\mathscr {O}}(r_\times ,r_s,n)=q(n,\kappa )(W_s-W_{\times })^{d(n)}\), where \(q(n,\kappa )\) and d(n) are different functions of the polytropic index and constant: \(W_s\ge W_{\times }\) is the value of the equipotential surface, which is taken with respect to the asymptotic value, while \( W_{\times }\) is evaluated at the cusp \(r_{\times }\). Thus there is more specifically \(\mathrm {{{Enthalpy-flux}}}={\mathscr {D}}(n,\kappa ) (W_s-W)^{n+3/2}\), \(\mathrm {{{Mass-Flux}}}= {\mathscr {C}}(n,\kappa ) (W_s-W)^{n+1/2}\). The quantity \({\mathscr {L}}_{\times }/{\mathscr {L}}= {\mathscr {B}}/{\mathscr {A}} (W_s-W_{\times })/(\eta c^2)\) evaluates the fraction of energy produced inside the flow and not radiated through the surface (swallowed by the BH), \(({\mathscr {A}}, {\mathscr {B}},{\mathscr {C}})\) are constant depending on the polytropics. The efficiency is \(\eta \equiv {\mathscr {L}}/{\dot{M}}c^2\), \({\mathscr {L}}\) representing the total luminosity, the \({\dot{M}}\) is the total accretion rate where for a stationary flow \({\dot{M}}={\dot{M}}_\times \) that is the mass flow rate through the cusp (mass loss, accretion rates). Then \({\dot{M}}_\times \), the cusp luminosity \({\mathscr {L}}_\times \) (and the accretion efficiency \(\eta \)), measuring the rate the thermal-energy is carried at cusp, have the compact form \({\mathscr {P}}={\mathscr {O}}(r_\times ,r_s,n) r_\times (\varOmega _K(r_\times ))^{-1}\), where the relativistic frequency \(\varOmega \) reduces to the Keplerian one \(\varOmega _K\) at the edges of the accretion torus because the pressure forces vanish. There is \({\mathscr {L}}_{\times }={{\mathscr {B}}(n,\kappa ) r_{\times } (W_s-W_{\times })^{n+2}}/{\varOmega _K(r_{\times })}\), and accretion rate for the disk is \({\dot{m}}= {\dot{M}}/{\dot{M}}_{Edd}\), while \({\dot{M}}_{\times }={{\mathscr {A}}(n,\kappa ) r_{\times } (W_s-W_{\times })^{n+1}}/{\varOmega _K(r_{\times })}\) [13, 16, 77].

  3. http://the-athena-x-ray-observatory.eu/.

  4. The global non-axis-symmetric hydrodynamic (HD) PPI implies also the formation of long-lasting, large-scale structures that may be also tracer for such tori in the in the gravitational wave emission—see for example [86].

  5. For example in [69], using three-dimensional GRMHD simulations it is studied the interaction between the PPI and the MRI considering an analytical magnetized equilibrium solution as initial condition. In the HD tori, the PPI selects the large-scale \(m = 1\) azimuthal mode as the fastest growing and non-linearly dominant mode. In different works it is practically shown that even a weak toroidal magnetic field can lead to MRI development which leads to the suppression of the large-scale modes. Notice also that the magneto-rotational instability in the disks is important because disks can be locally HD stable (according to Rayleigh criterion), but they are unstable for MHD local instability which is linear and independent by the field strength and orientation, and growing up on dynamical time scales. The torus (flow) is MHD turbulent due to the MRI.

  6. We note that, with particular symmetries of the background (static) if we set \(B^r=0\) from the Maxwell equations, we infer \( B^{\theta }\cot \theta =0 \) (with \(\ell =\)constant), that is satisfied for \(B^{\theta }=0\) or \(\theta =\pi /2\).

  7. The ratio \({\mathscr {M}}/\omega \) gives the comparison between the magnetic contribution to the fluid dynamics, through \( {\mathscr {M}}\), and the hydrodynamic contribution through its specific enthalpy \(\omega \).

  8. For the hydrodynamic case the squeezing function \(R_s\) for the Schwarzschild background increases monotonically with K, and at fixed K decreases with \(\ell ^2\). This therefore means that the toroidal surface is squeezed on the equatorial plane with decreasing “energy” K and increasing \(\ell ^2\)—[20]. However this trend is reversed, that is \(R_s=h/\lambda <1\) reaching a minimum values of \(R_s\approx 0.95\), for increasing values of K, the squeezing decreases, or decreases until it reaches a minimum and then increases. In the magnetic case, the influence of the magnetic field has to be evaluated through the parameter \({\mathscr {S}}\). In general, as for the case \({\mathscr {S}}=0\), the torus is thicker as the K parameter increases, and becomes thinner as fluid angular momentum increases. Furthermore \(R_s\) increases with \({\mathscr {S}}\). For a small region of values of \(\ell \), the torus becomes thinner with increasing of \({\mathscr {S}}\) and viceversa becomes thicker with decreasing \({\mathscr {S}}\).

  9. The Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism, also known as BSSNOK [121] consists in a modification of the ADM formalism of General Relativity (in Hamiltonian formalism not making possible long term and stable numerical simulations)—[120]. The modification introduced in the ADM equations by the BSSN formalism includes consists essentially in the introduction of the auxiliary variables (to be considered with the extrinsic curvature and the three metric). Such change has been used for (long term) analysis of linear and non-linear gravitational waves or black holes pair collision and generally different physics of spinning and double BH systems or neutron stars and merging of neutron stars, collapse of spinning stars leading to a BH solutions.

  10. Considering the BSSN formulation for the problem of weak hyperbolicity in the ADM equations, we also mention recent work [175] regarding the hyperbolicity of the \(3+1\) Valencia formulation of GRMHD when the solenoidal magnetic field constraint is enforced using hyperbolic divergence cleaning or presenting a vector potential formulations for GRMHD. Based on a dual-frame approach, the formulation for various fluid models is proved strongly hyperbolic, admitting consequently a well-posed initial value problem. The formulation with evolving vector potential is considered, and reducing to first-order it can be also strongly hyperbolic. The ideal GRMHD was consider in the Valencia flux-balance law form (but the equation were proved to be weakly hyperbolic therefore constituting an ill-posed initial value problem). Then GRMHD system in the hyperbolic divergence cleaning formulation and vector potential formulation results both strongly hyperbolic depending on sound speed (\(c_s\in ]0,1[\) or \(c_s\in ]0,1]\)).

  11. We consider the method reviewed here a useful tool to establish uniqueness and stability properties although not applied in the formulation of numerical fully relativity code simulations. The first order formulations certainly represent a clear simplification in many contexts and are generally preferred in numerical computations as well as in the analysis of the motion of isolated bodies. Nevertheless, we mention that in [140] the local existence result for isolated dust (charged and uncharged) bodies based on a mixed order formulation has been provided. While, in [141, 142] the evolution equations for the EME system were first considered from the perspective from the Cauchy problem, where non-linear wave equations for the metric tensor describing the gravitational field adopting wave coordinates are obtained. However as consequence of this choice the evolution system is of mixed order (in [142] where Leray theory was implemented). More general discussions of the problem of the well-posedness of the evolution equations of the EME system and of MHD can be found in [143,144,145,146,147,148]. Then, the initial boundary value problem is a further relevant issue to be discussed, for example for the class of maximally dissipative boundary conditions used in [149], to show the well-posedness of the initial boundary problem for the vacuum Einstein field equations.

  12. Note, we saw in Sects. 2 and 3 how, in the case of gravitating toroids, using some symmetries of the background and the matter and fields adapted to the geometry background and the accretion orbiting disks, the evolution equations were automatically satisfied leading to the development of a constrain model for the constraining equations for the matter capable to provide a good approximation for the analysis of a variety of orbiting accreting disks.

  13. We mention here however that in many of the LSR spacetimes was naturally used the complex variable \(\psi =E+i B\) and \(\psi ^*=E-i B\), to decouple the equations with the appropriate symmetries and obtain linear equations in the fields—[156,157,158, 162].

  14. First simple method can be for example the Descartes criterion to determine the maximum number of positive and negative real roots of the characteristic polynomial and in particular simple cases one can make use of the so-called Routh–Hurwitz criterion to determine the number of roots with positive and negative real part of the polynomial by constructing the Routh associated matrix.

References

  1. Capellupo, D.M., Wafflard-Fernandez, G., Haggard, D.: Astrophys. J. 836(1), L8 (2017)

    Article  ADS  Google Scholar 

  2. McClintock, J.E., Shafee, R., Narayan, R., et al.: Astrophys. J. 652, 518 (2006)

    Article  ADS  Google Scholar 

  3. Daly, R.A.: Astrophys. J. 691, L72 (2009)

    Article  ADS  Google Scholar 

  4. Farr, W.M., Stevenson, S., Coleman Miller, M., et al.: Nature 548, 426 (2017)

    Article  ADS  Google Scholar 

  5. van Putten, M.H.P.M.: Astrophys. J. 810(1), 7 (2015)

    Article  ADS  Google Scholar 

  6. van Putten, M.H.P.M., Della Valle, M.: MNRAS 464(3), 3219 (2017)

    Article  ADS  Google Scholar 

  7. Pürrer, M., Hannam, M., Ohme, F.: Phys. Rev. D 93, 084042 (2016)

    Article  ADS  Google Scholar 

  8. Andrade-Santos, F., Bogdán, Á., Romani, R.W., et al.: Astrophys. J. 826, 91 (2016)

    Article  ADS  Google Scholar 

  9. Pugliese, D., Montani, G.: Phys. Rev. D 91(8), 083011 (2015)

    Article  ADS  Google Scholar 

  10. Pugliese, D., Stuchlik, Z.: Astrophys. J. 221, 25 (2015)

    Article  Google Scholar 

  11. Pugliese, D., Stuchlik, Z.: Astrophys. J. 223, 27 (2016)

    Article  Google Scholar 

  12. Pugliese, D., Stuchlik, Z.: Astrophys. J. 229, 40 (2017)

    Article  Google Scholar 

  13. Pugliese, D., Stuchlík, Z.: Class. Quant. Grav. 35(18), 185008 (2018a)

    Article  ADS  Google Scholar 

  14. Pugliese, D., Stuchlik, Z.: JHEAp 17, 1 (2018b)

    ADS  Google Scholar 

  15. Pugliese, D., Stuchlík, Z.: Class. Quant. Grav. 35(10), 105005 (2018c)

    Article  ADS  Google Scholar 

  16. Pugliese, D., Stuchlik, Z.: Eur. Phys. J. C 79(4), 288 (2019)

    Article  ADS  Google Scholar 

  17. Pugliese, D., Stuchlik, Z.: MNRAS 493(3), 4229–4255 (2020a)

    Article  ADS  Google Scholar 

  18. Pugliese, D., Stuchlik, Z.: Quantum Grav. 37, 195025 (2020)

  19. Pugliese, D., Stuchlik, Z.: 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, (2019). arXiv:1910.03925 [astro-ph.HE]

  20. Pugliese, D., Montani, G.: Mon. Not. R. Astron. Soc. 476(4), 4346 (2018)

    Article  ADS  Google Scholar 

  21. Pugliese, D., Montani, G., Bernardini, M.G.: MNRAS 428(2), 952 (2013)

    Article  ADS  Google Scholar 

  22. Pugliese, D., Montani, G.: Europhys. Lett. 101, 19001 (2013)

    Article  ADS  Google Scholar 

  23. Shafee, R., McKinney, J.C., Narayan, R., et al.: Astrophys. J. 687, L25 (2008)

    Article  ADS  Google Scholar 

  24. Fragile, P.C., Blaes et al: Astrophys. J. 668, 417–429 (2007)

    Article  ADS  Google Scholar 

  25. De Villiers, J.-P., Hawley, J.F.: Astrophys. J. 577, 866 (2002)

    Article  ADS  Google Scholar 

  26. Porth, O., Olivares, H., Mizuno, Y., et al.: Comput. Astron. Cosm. 4, 1 (2017)

  27. Abramowicz, M.A., Fragile, P.C.: Liv. Rev. Relativ. 16, 1 (2013)

    Article  ADS  Google Scholar 

  28. Alho, A., Mena, F.C., Valiente Kroon, J.A.: Adv. Theor. Math. Phys. 21, 857–899 (2017)

    Article  MathSciNet  Google Scholar 

  29. Lubbe, C., Valiente Kroon, J.A.: Ann. Phys. 328, 1 (2013)

    Article  ADS  Google Scholar 

  30. Pugliese, D., Kroon, J.A.V.: Gen. Relativ. Gravit. 44, 2785 (2012)

    Article  ADS  Google Scholar 

  31. Pugliese, D., Kroon, J.A.V.: Gen. Relativ. Gravit. 48(6), 74 (2016)

    Article  ADS  Google Scholar 

  32. Komissarov, S.S.: MNRAS 368, 993 (2006)

    Article  ADS  Google Scholar 

  33. Montero, P.J., Zanotti, O., Font, J.A., Rezzolla, L.: MNRAS 378, 1101 (2007)

    Article  ADS  Google Scholar 

  34. Rezzolla, L., Zanotti, O.: Relativistic Hydrodynamics. Oxford University Press, Oxford (2013)

    Book  MATH  Google Scholar 

  35. Shakura, N.I.: Sov. Astron. 16, 756 (1973)

    ADS  Google Scholar 

  36. Shakura, N.I., Sunyaev, R.A.: Astron. Astrophys. 24, 337 (1973)

    ADS  Google Scholar 

  37. Page, Don N., Thorne, Kip S.: Astrophys. J. 191, 499–506 (1974)

    Article  ADS  Google Scholar 

  38. Kozłowski, M., Jaroszyński, M., Abramowicz, M.A.: Astron. Astrophys. 63, 209 (1998)

    ADS  Google Scholar 

  39. Alig, C., Schartmann, M., Burkert, A., Dolagapj, K.: Astrophys. J. 771, 119 (2013)

    Article  ADS  Google Scholar 

  40. Blanchard, P.K., et al.: Astrophys. J. 843, 106 (2011)

    Article  ADS  Google Scholar 

  41. Nixon, N., King, A., Price, D., Frank, J.: Astrophys. J. 757, L24 (2012)

    Article  ADS  Google Scholar 

  42. Lovelace, R.V.E., Chou, T.: Astrophys. J. 468, L25 (1996)

    Article  ADS  Google Scholar 

  43. Carmona-Loaiza, J.M., Colpi, M., Dotti, M., Valdarnini, R.: MNRAS 453(2), 1608 (2015)

    Article  ADS  Google Scholar 

  44. Dyda, S., Lovelace, R.V.E., Ustyugova, G.V., et al.: MNRAS 446, 613 (2015)

    Article  ADS  Google Scholar 

  45. Volonteri, M., Haardt, F., Madau, P.: Astrophys. J. 582, 559 (2003)

    Article  ADS  Google Scholar 

  46. Aly, H., Dehnen, W., Nixon, C., King, A.: MNRAS 449(1), 65 (2015)

    Article  ADS  Google Scholar 

  47. King, A., Nixon, C.: Astrophys. J. 857(1), L7 (2018)

    Article  ADS  Google Scholar 

  48. Abramowicz, M.A., Jaroszyński, M., Sikora, M.: Astron. Asrophys. 63, 221 (1978)

    ADS  Google Scholar 

  49. Jaroszynski, M., Abramowicz, M.A., Paczynski, B.: Acta Astronom. 30, 1 (1980)

    ADS  Google Scholar 

  50. Stuchlík, Z., Slaný, P., Hledík, S.: Astron. Astrophys. 363, 425 (2000)

    ADS  Google Scholar 

  51. Rezzolla, L., Zanotti, O., Font, J.A.: Astron. Astrophys. 412, 603 (2003)

    Article  ADS  Google Scholar 

  52. Slaný, P., Stuchlík, Z.: Class. Quantum Grav. 22, 3623 (2005)

    Article  ADS  Google Scholar 

  53. Stuchlik, Z.: Mod. Phys. Lett. A 20, 561–75 (2005)

    Article  ADS  Google Scholar 

  54. Rees, M.J., Phinney, E.S., Begelman, M.C., Biford, R.D.: Nature 295, 17 (1982)

    Article  ADS  Google Scholar 

  55. Abramowicz, M.A., Straub, O.: Accretion Discs Scholarped. 9(8), 2408 (2014)

    Article  ADS  Google Scholar 

  56. Narayan, R., Mahadevan, R., Quataert, E.: (1998) arXiv:astro-ph/9803141

  57. Hawley, J.F.: Mon. Not. R. Astron. Soc. 225, 677 (1987)

    Article  ADS  Google Scholar 

  58. Hawley, J.F.: Astrophys. J. 356, 580 (1990)

    Article  ADS  Google Scholar 

  59. Hawley, J.F.: Astrophys. J. 381, 496 (1991)

    Article  ADS  Google Scholar 

  60. Hawley, J.F., Smarr, L.L., Wilson, J.R.: Astrophys. J. 277, 296 (1984)

    Article  ADS  Google Scholar 

  61. Balbus, S.A.: Nature 470, 475 (2011)

    Article  ADS  Google Scholar 

  62. Paczyński, B.: Acta Astron. 30, 4 (1980)

    Google Scholar 

  63. Paczyński, B., Wiita, P.: Astron. Astrophys. 88, 23 (1980)

    ADS  Google Scholar 

  64. Font, J.A., Daigne, F.: Astrophys. J. 581, L23–L26 (2002)

    Article  ADS  Google Scholar 

  65. Font, J.A.: Liv. Rev. Relat. 6, 4 (2003)

    Article  Google Scholar 

  66. Lei, Q., Abramowicz, M.A., Fragile, et al.: Astron. Astrophys. 498, 471 (2008)

  67. Boyer, R.H.: Proc. Camb. Philos. Soc. 61, 527 (1965)

    Article  ADS  Google Scholar 

  68. Blaes, O.M.: Mon. Not. R. Astron. Soc. 227, 975 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  69. Bugli, M., Guilet, J., Müller, E., Del Zanna, L., Bucciantini, N., Montero, P.J.: MNRAS 475, 108 (2018)

    Article  ADS  Google Scholar 

  70. Zanotti, O., Pugliese, D.: Gen. Relat. Gravit. 47(4), 44 (2015)

    Article  ADS  Google Scholar 

  71. Bekenstein, J., Oron, D.: Phys. Rev. D 18(1809), 1–71819 (1978)

    Google Scholar 

  72. Bekenstein, J., Oron, D.: Phys. Rev. D 19(2827), 1–72837 (1979)

    ADS  Google Scholar 

  73. Abramowicz, M.A.: Acta Astron. 21, 81 (1971)

    ADS  Google Scholar 

  74. Chakrabarti, S.K.: Mon. Not. R. Astron. Soc. 250, 7 (1991)

    Article  ADS  Google Scholar 

  75. Chakrabarti, S.K.: Mon. Not. R. Astron. Soc. 245, 747 (1990)

    ADS  Google Scholar 

  76. Igumenshchev, I.V., Abramowicz, M.A.: Astrophys. J. Suppl. 130, 463 (2000)

    Article  ADS  Google Scholar 

  77. Abramowicz, M.A.: Astron. Soc. Jpn. 37(4), 727–734 (1985)

    ADS  Google Scholar 

  78. Frank, J., King, A., Raine, D.: Accretion Power in Astrophysics. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  79. Volonteri, M., Sikora, M., Lasota, J.-P.: Astrophys. J. 667, 704 (2007)

    Article  ADS  Google Scholar 

  80. Volonteri, M.: Astrophys. J. 663, L5 (2007)

    Article  ADS  Google Scholar 

  81. Volonteri, M.: Astron. Astrophys. 18, 279 (2010)

    ADS  Google Scholar 

  82. Li., L. X. 2012 MNRAS, 424, 1461

  83. Oka, T., Tsujimoto, S., et al.: Nat. Astron. 1, 709 (2017)

    Article  ADS  Google Scholar 

  84. Kawakatu, N., Ohsuga, K.: MNRAS 417(4), 2562–2570 (2011)

    Article  ADS  Google Scholar 

  85. Allen, S.W., Dunn, R.J.H., Fabian, A.C., et al.: MNRAS 1(372), 21 (2006)

    Article  ADS  Google Scholar 

  86. Kiuchi, K., Shibata, M., Montero, P.J., Font, J.A.: Phys. Rev. Lett. 106, 251102 (2011)

    Article  ADS  Google Scholar 

  87. Adamek, K., Stuchlik, Z.: Class. Quantum Grav. 30, 205007 (2013)

    Article  ADS  Google Scholar 

  88. Hamersky, J., Karas, V.: Astron. Astrophys. 32, 555 (2013)

    Google Scholar 

  89. Karas, V., Kopácek, O., Kunneriath, D., Hamersky, J.: Acta Polytech. 54(6), 398 (2014)

    Article  ADS  Google Scholar 

  90. Slany, P., Kovar, J., Stuchlik, Z., Karas, V.: Astrophys. J. Suppl. 205, 3 (2013)

    Article  ADS  Google Scholar 

  91. Kovar, J., Slany, P., Stuchlik, P., et al.: Phys. Rev. D 84(8), 084002 (2011)

    Article  ADS  Google Scholar 

  92. Fragile, P.C., Sadowski, A.: MNRAS 467, 1838 (2017)

    Article  ADS  Google Scholar 

  93. Gimeno-Soler, S., Font, J.A.: Astron. Astrophys. 607, A68 (2017)

    Article  ADS  Google Scholar 

  94. Cruz-Osorio, A., Gimeno-Soler, S., Font, J.A.: MNRAS 492, 5730 (2020)

    Article  ADS  Google Scholar 

  95. Del Zanna, L., Zanotti, O., Bucciantini, N., Londrillo, P.: Astron. Astrophys. 473, 11 (2007)

    Article  ADS  Google Scholar 

  96. Wielgus, M., Fragile, P.C., Wang, Z., Wilson, J.: MNRAS 447, 359 (2015)

    Article  Google Scholar 

  97. Das, U., Begelman, M.C., Lesur, G.: MNRAS 473, 2791 (2017)

    Article  ADS  Google Scholar 

  98. Parker, E.N.: Astrophys. J. 122, 293 (1955)

    Article  ADS  MathSciNet  Google Scholar 

  99. Parker, E.N.: Astrophys. J. 160, 383 (1970)

    Article  ADS  Google Scholar 

  100. Yoshizawa, A., Itoh, S.I., Itoh, K.: Plasma & Fluid Turbulence: Theory and Modelling. CRC Press, London (2003)

    Book  MATH  Google Scholar 

  101. Trova, A., Karas, V., Slany, P., Kovar, J.: Astrophys. J. Suppl. 226(1), 12 (2016)

    Article  ADS  Google Scholar 

  102. Trova, A., Schroven, K., Hackmann, E., et al.: Phys. Rev. D 97(10), 104019 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  103. Kovár, J., Slany et al: Phys. Rev. D 93(12), 124055 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  104. Kovář, J., Slaný et al: Phys. Rev. D 90(4), 044029 (2014)

    Article  ADS  Google Scholar 

  105. Schroven, K., Trova, A., Hackmann, E., Lämmerzahl, C.: Phys. Rev. D 98(2), 023017 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  106. Bardeen, J.M., Petterson, J.A.: Astrophys. J. 195, L65 (1975)

    Article  ADS  Google Scholar 

  107. Sochora, V., Karas, V., Svoboda, J., Dovciak, M.: MNRAS 418, 276–283 (2011)

    Article  ADS  Google Scholar 

  108. Karas, V., Sochora, V.: Astrophys. J. 725(2), 1507–1515 (2010)

    Article  ADS  Google Scholar 

  109. Schee, J., Stuchlik, Z.: Gen. Relativ. Gravit. 41, 1795 (2009)

    Article  ADS  Google Scholar 

  110. Schee, J., Stuchlik, Z.: JCAP 1304, 005 (2013)

    Article  ADS  Google Scholar 

  111. Stuchlik, Z., Schee, J.: Class. Quant. Grav. 30, 075012 (2013)

    Article  ADS  Google Scholar 

  112. Stuchlik, Z., Schee, J.: Class. Quant. Grav. 27, 215017 (2010)

    Article  ADS  Google Scholar 

  113. Stuchlik, Z., Hledik, S., Truparova, K.: Class. Quant. Grav. 28, 155017 (2011)

    Article  ADS  Google Scholar 

  114. Stuchlik, Z., Schee, J.: Class. Quant. Grav. 29, 065002 (2012)

    Article  ADS  Google Scholar 

  115. Stuchlik, Z., Slany, P., Kovar, J.: Class. Quant. Grav. 26, 215013 (2009)

    Article  ADS  Google Scholar 

  116. Stuchlik, Z., Kovar, J.: Int. J. Mod. Phys. D 17, 2089–105 (2008)

    Article  ADS  Google Scholar 

  117. Stuchlik, Z., Kološ, M., Kovář, J., Slaný, P., Tursunov, A.: Univ 6, 26 (2020)

    Article  ADS  Google Scholar 

  118. Stuchlik, Z., Pugliese, D., Schee, J., Kucáková, H.: Eur. Phys. J. C 75(9), 451 (2015)

    Article  ADS  Google Scholar 

  119. Shibata, M., Sekiguchi, Y.: Phys. Rev. D 72, 044014 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  120. Baumgarte, T.W., Shapiro, S.L.: Phys. Rev. D 59, 024007 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  121. Nakamura, T., Oohara, K., Kojima, Y.: Prog. Theor. Phys. Suppl. 90, 1 (1987)

    Article  ADS  Google Scholar 

  122. Baumgarte, T.W., Shapiro, S.L.: Astrophys. J. 585, 921 (2003)

    Article  ADS  Google Scholar 

  123. Palenzuela, C., Garrett, D., Lehner, L., Liebling, S.: Phys. Rev. D 82, 044045 (2010)

    Article  ADS  Google Scholar 

  124. Anile, A.M.: 1989 Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  125. Lichnerowicz, A.: Relativistic Hydrodynamics and Magnetohydrodynamics. Benjamin, New York (1967)

    MATH  Google Scholar 

  126. Disconzi, M.M.: Nonlinearity 27, 1915 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  127. Radice, D., Rezzolla, L., Galeazzi, F.: Class. Quant. Grav. 31, 075012 (2014)

    Article  ADS  Google Scholar 

  128. Anton, L., Zanotti, O., Miralles, J.A., et al.: Astrophys. J. 637, 296 (2006)

    Article  ADS  Google Scholar 

  129. Friedrich, H.: Phys. Rev. D 57, 2317 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  130. Friedrich, H., Rendall, A.D.: Lect. Notes Phys. 540, 127 (2000)

    Article  ADS  Google Scholar 

  131. Friedrich, H.: Class. Quantum Grav. 13, 1451 (1996)

    Article  ADS  Google Scholar 

  132. Reula, O.: Liv. Rev. Rel. 3, 1 (1998)

    Google Scholar 

  133. Friedrich, H.: J. Diff. Geom. 34, 275 (1991)

    Google Scholar 

  134. Ellis, G.F.R., van Elst, H. (1998) NATO Adv. Study Inst. Ser. C. Math. Phys. Sci. 541, 1

  135. Marklund, M., Clarkson, C.: Mon. Not. R. Astron. Soc. 358, 892 (2005)

    Article  ADS  Google Scholar 

  136. Etienne, Z.B., Liu, Y.T., Shapiro, S.L.: Phys. Rev. D 82, 084031 (2010)

    Article  ADS  Google Scholar 

  137. Gundlach, C., Martín-García, J.M.: Class. Quantum Grav. 23, S387 (2006)

    Article  ADS  Google Scholar 

  138. Alcubierre, M.: Introduction to \(3+1\) Numerical Relativity. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  139. Rendall, A.D.: Partial Differential Equations in General Relativity. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  140. Choquet-Bruhat, Y., Friedrich, H.: Class. Quantum Grav. 23, 5941 (2006)

    Article  ADS  Google Scholar 

  141. Choquet-Bruhat, Y.: C. R. Acad. Sci. Paris 261, 354 (1965)

    Google Scholar 

  142. Choquet-Bruhat, Y.: General Relativity and the Einstein Equations. Oxford University Press, Oxford (2008)

    Book  MATH  Google Scholar 

  143. van Putten, M.H.P.M.: Commun. Math. Phys. 141, 63 (1991)

    Article  ADS  Google Scholar 

  144. Friedrichs, K.O.: Comm. Pure Appl. Math. 28, 749 (1974)

    ADS  Google Scholar 

  145. Renardy, M.: J. Math. Fluid Mech. (2011)

  146. van Putten, M.H.P.M.: J. Math. Phys. 43, 6195 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  147. Zenginoglu, A.: Ideal Magnetohydrodynamics in Curved Spacetime. Master thesis, University of Vienna (2003)

  148. Choquet-Bruhat, Y., York, J.W.: Lect. Notes Phys. 592, 29 (2002)

    Article  ADS  Google Scholar 

  149. Friedrich, H., Nagy, G.: Commun. Math. Phys. 201, 619 (1999)

    Article  ADS  Google Scholar 

  150. Reula, O.: Phys. Rev. D 60, 083507 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  151. Clarkson, C.: Phys. Rev. D 76, 104034 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  152. Horst, E.: Commun. Math. Phys. 126, 613–633 (1990)

    Article  ADS  Google Scholar 

  153. Laskyand, P.D., Lun, A.W.C.: Phys. Rev. D 75, 104010 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  154. Viana, R.L., Clemente, R.A., Lopes, S.R.: Plasma Phys. Control. Fusion 39, 197 (1997)

    Article  ADS  Google Scholar 

  155. Guo, Y., Tahvildar-Zadeh, A.S.: Contemp. Math. 238, 151–161 (1999)

    Article  Google Scholar 

  156. Betschart, G., Clarkson, C.A.: Class. Quantum Grav. 21, 5587 (2004)

    Article  ADS  Google Scholar 

  157. Burston, R.B.: Class. Quantum Grav. 25, 075004 (2008b)

    Article  ADS  MathSciNet  Google Scholar 

  158. Burston, R.B., Lun, A.W.C.: Class. Quant. Grav. 25, 075003 (2008)

    Article  ADS  Google Scholar 

  159. Stewart, J.M., Walker, M.: Proc. R. Soc. Lond. A 431, 49 (1974)

    ADS  Google Scholar 

  160. Barrow, J.D., Maartens, R., Tsagas, C.G.: Phys. Rep. 449, 131 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  161. Tsagas, C.G.: Class. Quantum Grav. 22, 393 (2005)

    Article  ADS  Google Scholar 

  162. Burston, R.B.: Class. Quantum Grav. 25, 075002 (2008a)

    Article  ADS  MathSciNet  Google Scholar 

  163. Mościbrodzka, M., Falcke, H., Shiokawa, H.: Astron. Astrophys. 586, A38 (2016)

    Article  ADS  Google Scholar 

  164. White, C.J., Dexter, J., Blaes, O., et al.: Astrophys. J. (2020)

  165. Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al.: Astrophys. J. 875, L5 (2019)

  166. Porth, O., Chatterjee, K., Narayan, R., et al.: Astrophys. J. Suppl. (2019)

  167. Kerr, R.P.: Phys. Rev. Lett. 11, 237 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  168. Ripperda, B., Bacchini, F., Philippov, A.A.: Astrophys. J. 900, 100 (2020)

    Article  ADS  Google Scholar 

  169. Tomei, N., Del Zanna, L., Bugli, M., et al.: MNRAS 491, 2346 (2020)

    ADS  Google Scholar 

  170. Del Zanna, L., Bucciantini, N.: MNRAS 479, 657 (2018)

    ADS  Google Scholar 

  171. Bugli, M., Del Zanna, L., Bucciantini, N.: MNRAS 440, L41 (2014)

    Article  ADS  Google Scholar 

  172. Stergioulas, N.: J. Phys. Conf. Ser. 283, 012036 (2011)

    Article  Google Scholar 

  173. Tsokaros, A., et al.: Phys. Rev. D 99, 041501 (2019)

    Article  ADS  Google Scholar 

  174. Uryu, K., et al.: Phys. Rev. D 100, 123019 (2019)

    Article  ADS  Google Scholar 

  175. Hilditch, D., Schoepe, A.: Phys. Rev. D 99, 104034 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  176. Korobkin, O., et al.: Phys. Rev. D 83, 043007 (2011)

    Article  ADS  Google Scholar 

  177. Mewes, V., et al.: Phys. Rev. D 93, 064055 (2016)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Centre of Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo náměstí 13, 74601, Opava, Czech Republic

    D. Pugliese

  2. ENEA-R.C. Frascati, UTFUS-MAG, Via Enrico Fermi 45, Frascati, Rome, 00044, Italy

    G. Montani

  3. Physics Department, “Sapienza” University of Rome, P.le Aldo Moro 5, Rome, 00185, Italy

    G. Montani

Authors
  1. D. Pugliese
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Montani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Pugliese.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pugliese, D., Montani, G. Aspects of GRMHD in high-energy astrophysics: geometrically thick disks and tori agglomerates around spinning black holes. Gen Relativ Gravit 53, 51 (2021). https://doi.org/10.1007/s10714-021-02820-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-021-02820-4

Keywords

Search

Navigation