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Reinhard Meinel

Publications and source records attributed to Reinhard Meinel.

At least 19 recordsLinked to original sources
arXiv2023

Multipole moments of a charged rotating disc of dust in general relativity

The gravitational and electromagnetic multipole moments of the charged rotating disc of dust, which is an axisymmetric, stationary solution of the Einstein-Maxwell equations in terms of a post-Newtonian expansion, are calculated and discussed. It turns out that the individual mass, angular momentum, electric and magnetic moments are ordered in the sense that higher moments have a lower absolute value. There is an interesting conjecture stating that the absolute values of all higher multipole moments of a uniformly rotating perfect fluid body are always greater than those of the corresponding Kerr spacetime, which we generalize to include charged bodies. We find that for the charged rotating disc of dust the conjecture holds (within the limits of accuracy of the post-Newtonian expansion).

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arXiv2023

Geometry of charged rotating discs of dust in Einstein-Maxwell theory

Within the framework of Einstein-Maxwell theory geometric properties of charged rotating discs of dust, using a post-Newtonian expansion up to tenth order, are discussed. Investigating the disc's proper radius and the proper circumference allows us to address questions related to the Ehrenfest paradox. In the Newtonian limit there is an agreement with a rotating disc from special relativity. The charged rotating disc of dust also possesses material-like properties. A fundamental geometric property of the disc is its Gaussian curvature. The result obtained for the charged rotating disc of dust is checked by additionally calculating the Gaussian curvature of the analytic limiting cases (charged rotating) Maclaurin disc, electrically counterpoised dust-disc and uncharged rotating disc of dust. We find that by increasing the disc's specific charge there occurs a transition from negative to positive curvature.

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arXiv2020

Bäcklund transforms of the extreme Kerr near-horizon geometry

We apply the method of Bäcklund transformations to generate a new hierarchy of exact solutions to the vacuum Einstein equations starting from the extreme Kerr near-horizon geometry. Solutions with extreme Kerr near-horizon asymptotics containing an arbitrary number of free parameters are included.

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arXiv2018

A continuous Riemann-Hilbert problem for colliding plane gravitational waves

We present the foundations of a new solution technique for the characteristic initial value problem (IVP) of colliding plane gravitational waves. It has extensive similarities to the approach of Alekseev and Griffiths in 2001, but we use an inverse scattering method with a Riemann-Hilbert problem (RHP), which allows for a transformation to a continuous RHP with a solution given in terms of integral equations for non-singular functions. Ambiguities in this procedure lead to the construction of a family of spacetimes containing the solution to the IVP. Therefore the described technique also serves as an interesting solution generating method. The procedure is exemplified by extending the Szekeres class of colliding wave spacetimes with 2 additional real parameters. The obtained solution seems to feature a limiting case of a new type of impulsive waves, which are circularly polarised.

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arXiv2016

Gyromagnetic factor of rotating disks of electrically charged dust in general relativity

We calculated the dimensionless gyromagnetic ratio ("$g$-factor") of self-gravitating, uniformly rotating disks of dust with a constant specific charge $ε$. These disk solutions to the Einstein-Maxwell equations depend on $ε$ and a "relativity parameter" $γ$ ($0<γ\le 1$) up to a scaling parameter. Accordingly, the $g$-factor is a function $g=g(γ,ε)$. The Newtonian limit is characterized by $γ\ll 1$, whereas $γ\to 1$ leads to a black-hole limit. The $g$-factor, for all $ε$, approaches the values $g=1$ as $γ\to 0$ and $g=2$ as $γ\to 1$.

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arXiv2015

A note on circular geodesics in the equatorial plane of an extreme Kerr-Newman black hole

We examine the behaviour of circular geodesics describing orbits of neutral test particles around an extreme Kerr-Newman black hole. It is well known that the radial Boyer-Lindquist coordinates of the prograde photon orbit $r=r_{\rm ph}$, marginally bound orbit $r=r_{\rm mb}$ and innermost stable orbit $r=r_{\rm ms}$ of the extreme Kerr black hole all coincide with the event horizon's value $r=r_+$. We find that for the extreme Kerr-Newman black hole with mass $M$, angular momentum $J$ and electric charge $Q=\pm\sqrt{M^2-J^2/M^2}$ ($|J|\le M^2$) the coordinate equalities $r_{\rm ph}=r_+$, $r_{\rm mb}=r_+$ and $r_{\rm ms}=r_+$ hold if and only if $|J|$ is greater than or equal to $M^2/2$, $M^2/\sqrt{3}$ and $M^2/\sqrt{2}$, respectively.

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arXiv2015

On the black hole limit of rotating discs of charged dust

Investigating the rigidly rotating disc of dust with constant specific charge, we find that it leads to an extreme Kerr-Newman black hole in the ultra-relativistic limit. A necessary and sufficient condition for a black hole limit is, that the electric potential in the co-rotating frame is constant on the disc. In that case certain other relations follow. These relations are reviewed with a highly accurate post-Newtonian expansion. Remarkably it is possible to survey the leading order behaviour close to the black hole limit with the post-Newtonian expansion. We find that the disc solution close to that limit can be approximated very well by a "hyper\-extreme" Kerr-Newman solution with the same gravitational mass, angular momentum and charge.

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arXiv2013

A physical derivation of the Kerr-Newman black hole solution

According to the no-hair theorem, the Kerr-Newman black hole solution represents the most general asymptotically flat, stationary (electro-) vacuum black hole solution in general relativity. The procedure described here shows how this solution can indeed be constructed as the unique solution to the corresponding boundary value problem of the axially symmetric Einstein-Maxwell equations in a straightforward manner.

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arXiv2013

Post-Newtonian expansion of a rigidly rotating disc of dust with a constant specific charge

We present an algorithm for obtaining the post-Newtonian expansion of the asymptotically flat solution to the Einstein-Maxwell equations describing a rigidly rotating disc of dust with a constant specific charge. Explicit analytic expressions are calculated up to the eighth order. The results are used for a physical discussion of the extreme relativistic limiting cases. We identify strong evidence for a transition to an extreme Kerr-Newman black hole.

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arXiv2012

Black holes and quasiblack holes in Einstein-Maxwell theory

Continuous sequences of asymptotically flat solutions to the Einstein-Maxwell equations describing regular equilibrium configurations of ordinary matter can reach a black hole limit. For a distant observer, the spacetime becomes more and more indistinguishable from the metric of an extreme Kerr-Newman black hole outside the horizon when approaching the limit. From an internal perspective, a still regular but non-asymptotically flat spacetime with the extreme Kerr-Newman near-horizon geometry at spatial infinity forms at the limit. Interesting special cases are sequences of Papapetrou-Majumdar distributions of electrically counterpoised dust leading to extreme Reissner-Nordstrom black holes and sequences of rotating uncharged fluid bodies leading to extreme Kerr black holes.

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arXiv2011

Constructive proof of the Kerr-Newman black hole uniqueness including the extreme case

A new proof of the uniqueness of the Kerr-Newman black hole solutions amongst asymptotically flat, stationary and axisymmetric electro-vacuum spacetimes surrounding a connected Killing horizon is given by means of an explicit construction of the corresponding complex Ernst potentials on the axis of symmetry. This construction, which makes use of the inverse scattering method, also works in the case of a degenerate horizon.

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arXiv2011

On the black hole limit of electrically counterpoised dust configurations

By means of a simple scaling transformation any asymptotically flat Papapetrou-Majumdar solution of the Einstein-Maxwell equations corresponding to a localized regular distribution of electrically counterpoised dust can be reformulated as a one-parameter family of solutions admitting a black hole limit. In the limit, a characteristic separation of spacetimes occurs: From the exterior point of view, the extreme Reissner-Nordstrom metric outside the event horizon is formed. From the interior point of view, a regular, non-asymptotically flat (and in general non-spherically symmetric) spacetime with the extreme Reissner-Nordstrom near-horizon geometry at spatial infinity results.

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arXiv2010

On the black hole limit of rotating discs and rings

Solutions to Einstein's field equations describing rotating fluid bodies in equilibrium permit parametric (i.e. quasi-stationary) transitions to the extreme Kerr solution (outside the horizon). This has been shown analytically for discs of dust and numerically for ring solutions with various equations of state. From the exterior point of view, this transition can be interpreted as a (quasi) black hole limit. All gravitational multipole moments assume precisely the values of an extremal Kerr black hole in the limit. In the present paper, the way in which the black hole limit is approached is investigated in more detail by means of a parametric Taylor series expansion of the exact solution describing a rigidly rotating disc of dust. Combined with numerical calculations for ring solutions our results indicate an interesting universal behaviour of the multipole moments near the black hole limit.

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arXiv2006

Quasi-stationary routes to the Kerr black hole

Quasi-stationary (i.e. parametric) transitions from rotating equilibrium configurations of fluid bodies to rotating black holes are discussed. For the idealized model of a rotating disc of dust, analytical results derived by means of the "inverse scattering method" are available. They are generalized by numerical results for rotating fluid rings with various equations of state. It can be shown rigorously that a black hole limit of a fluid body in equilibrium occurs if and only if the gravitational mass becomes equal to twice the product of angular velocity and angular momentum. Therefore, any quasi-stationary route from fluid bodies to black holes passes through the extreme Kerr solution.

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arXiv2006

The Ernst equation and ergosurfaces

We show that analytic solutions $\mcE$ of the Ernst equation with non-empty zero-level-set of $\Re \mcE$ lead to smooth ergosurfaces in space-time. In fact, the space-time metric is smooth near a "Ernst ergosurface" $E_f$ if and only if $\mcE$ is smooth near $E_f$ and does not have zeros of infinite order there.

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arXiv2006

On the black hole limit of rotating fluid bodies in equilibrium

Recently, it was shown that the extreme Kerr black hole is the only candidate for a (Kerr) black hole limit of stationary and axisymmetric, uniformly rotating perfect fluid bodies with a zero temperature equation of state. In this paper, necessary and sufficient conditions for reaching the black hole limit are presented.

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