The Origins of Causality Violations in Force-free Simulations of Black Hole Magnetospheres

Abstract

Recent simulations of force-free, degenerate electrodynamic (ffde) black hole magnetospheres indicate that the fast mode radiated from (or near) the event horizon can modify the global potential difference in the poloidal direction orthogonal to the magnetic field, V, in a black hole magnetosphere. By contrast, in MHD (meaning perfect magnetohydrodynamics hereafter), a combination of Alfvén and fast waves are required to alter V in a magnetosphere. This distinction is significant because it changes the causal boundary surface for evolving V from the event horizon (the inner fast critical surface in ffde) to the inner Alfvén critical surface. Second, there is a fundamental contradiction in a wave that alters V coming from near the horizon. The background fields in ffde satisfy the "ingoing wave condition" near the horizon (which arises from the requirement that all matter is ingoing at the event horizon), yet outgoing waves are radiated from this region in the simulation. The resolution of these two issues is important to our understanding of causality in black hole magnetospheres and ffde as a faithful representation of tenuous MHD near a black hole. Studying the properties of the waves in the simulations are useful tools to this end. It is shown that the regularity of the stress-energy tensor in a freely falling frame requires that the outgoing (as viewed globally) waves near the event horizon are redshifted away and are ineffectual at changing V. It is also concluded that waves in massless MHD (ffde) are extremely inaccurate depictions of waves in a tenuous MHD plasma, near the event horizon, as a consequence of black hole gravity. Any analysis based on ffde near the event horizon is seriously flawed.

General relativistic magnetohydrodynamic simulations of monopole magnetospheres of black holes

“…Fig. 3 also shows that the Lorentz factor of the ingoing wind remains very low and, thus, there is no reason for the breakdown of the FFDE approximation at the event horizon suggested in Punsly & Bini (2004) and Macdonald & Thorne (1982) .…”

“…A very relevant example can be found in the theory of magnetically dominated pulsar winds where particle inertia becomes an important factor as a result of acceleration by the electromagnetic field (Mestel 1999, section 13.2.3) Other potential examples in the theory of black hole magnetospheres are discussed by Punsly (2001, sections 8 and 9). Recently, Punsly & Bini (2004) argued that FFDE is deficient near the event horizon where ‘the plasma attains an infinite inertia in a global sense’. Macdonald & Thorne (1982) also anticipated a breakdown of the FFDE approximation near the event horizon, although no detailed explanations were given.…”

Fast-Wave Polarization, Charge Horizons, and the Time Evolution of Force-free Magnetospheres

“…The 2-D solutions should reveal any dispersive effects of the fast waves that result from boundary conditions as is the case in a vacuum waveguide or a plasma-filled waveguide (Punsly 2001). The axisymmetric 2-D fast wave discontinuities in an ffde black hole magnetospheres were determined in Punsly and Bini (2004). The oscillatory 2-D fast waves in a black hole magnetosphere are linear combinations of spin-weighted spheroidal harmonics convolved with a radial function that is a solution of an extremely complicated differential equation (Teukolsky 1973;Punsly 2001).…”

“…A constant K-S azimuthal coordinate, dφ = 0 transforms to the constraint dφ = −(a/∆)dr in Boyer-Lindquist (B-L, hereafter) coordinates, where ∆ ≡ r 2 − 2Mr + a 2 . Note that this implies a strong toroidal magnetic field near the event horizon in B-L coordinates (Punsly and Bini 2004). The dφ = 0 condition defines the particle trajectories that are restricted to the field lines, a requirement of ffde that was derived in section 2.…”

Astrophysics in 2004

“…'s), followed by a "no" in the sense that you never seem to get a positive outward energy flux at the horizon (Punsly 2004reacting to Levinson 2004.…”