# Insights from the Boundary: black holes in a magnetic universe

Hari Kunduri

Following from the seminal work of Dain, a great deal is now known concerning geometric inequalities relating the area, charge, and angular momentum of axisymmetric black hole horizons in (possibly dynamical) spacetimes.  A key feature of these results is that they are quasi-local: they depend on spacetime only near the horizon itself and so are not sensitive to the asymptotic behaviour of the geometry.

For Einstein-Maxwell theory the celebrated uniqueness theorems tell us under certain conditions, that the Kerr-Newman (KN) family of solutions are the only stationary, axisymmetric and asymptotically flat black hole spacetimes. These are the model geometries that originally motivated the inequalities. However if we relax the condition of asymptotic flatness there are many other families of black hole solutions. While in general these will not contain event horizons (whose standard definitions require flat or AdS asymptotics) they still contain singularities and Killing horizons. In this paper we focussed on a particular family of solutions: the Melvin-Kerr-Newman (MKN) black hole.  Originally derived by Ernst four decades ago, it physically represents a charged, rotating black hole immersed in a background magnetic field.  This is a four-parameter family of solutions, with the extra parameter $B$ roughly corresponding to the strength of the background magnetic field; when $B=0$, we recover the usual KN solution.  The solution exhibits a number of physical features which may be of interest in astrophysics.

Ivan Booth

The non-flat asymptotic structure of the MKN geometry means that standard approaches to compute the ADM energy cannot be applied. Further there is no canonically normalized generator of time translation, leading to an ambiguity in computing the surface gravity, angular velocities, and gauge choice for the electric potential on the horizon.  While there are proposals for doing this (e.g. using the isolated horizon formalism or a more recent proposal by Gibbons, Pang, and Pope), the issue is unresolved. It is not even clear that there is a single “correct” definition. Thus despite the fact that the area $A$ and ‘horizon’ angular momentum $J$ and charge $Q$ can be meaningfully defined, the status of familiar results such as the Smarr relation and first law of black hole mechanics is unclear.

Nonetheless, we can show that the MKN horizon is a stable, axisymmetric marginally outer trapped surface. For any such geometry, there holds a universal bound $Q^4 + 4J^2 \leq R^4$ where $R = \sqrt{a/4\pi}$ is the areal radius.  Furthermore, there are bounds on the horizon angular momenta (not counting matter contributions) for MOTS in spacetimes satisfying the dominant energy condition: $|J| \leq R^2/2$.   Clearly, MKN horizons provide an interesting and explicit example to explore these bounds beyond the usual KN setting. In our recent article in Classical and Quantum Gravity, we have explored this issue in detail.  The relative complexity of the solution requires the use of both numerical and analytical methods.

Alberto Palomo-Lozano

The MKN family also admits an extreme limit for which the Killing horizon is degenerate. It should be noted that by uniqueness results for extreme electrovacuum horizon geometries (originally due to Hajicek and then within the isolated horizon formalism by Lewandowski and Pawlowski),   the metric and Maxwell field in a sufficiently small neighbourhood of any extreme horizon geometry must be that of extreme KN . This `rigidity’ result implies that, near the horizon, an extreme MKN solution is isometric to extreme KN; effectively, one of the extra parameters must ‘disappear’. In our work, we have found the explicit map between parameters of extreme MKN and KN (this map was also found independently by Bicak and Hejda).

The MKN family is naturally obtained by starting with a KN ‘seed’ solution and applying a Harrison transformation. It is natural to parameterize MKN by well-defined quantities such as $R$, $Q$, and $J$. We have found simple relations between $R$, $Q$ and $J$ and the corresponding quantities for KN seed. Interestingly, it turns out that a member of the MKN family is uniquely specified by fixing the magnetic strength parameter $B$ and $R$, $Q$, $J$.  Moreover the Harrison transformation preserves the degree of extremality.

In the near-extreme case, Gabach-Clement and Reiris have obtained ‘near-uniqueness’ results in the form of precise estimates that constrain the geometry of any non-extreme horizon to be ‘close’ (in a precise sense) to that of  geometry of extreme Kerr. In the spirit of these results we examined how MKN horizon geometries are constrained by extreme KN. Indeed, although the geometry of a non-extreme horizon can be quite different to KN for general parameters (in particular it can be ‘hour-glass shaped’) as one takes the extremality parameter to zero the geometry is tightly bound to approach extreme KN. We have checked this by comparing various geometric invariants (e.g. the ratio of lengths of great circles versus the equator) in the MKN phase space.

This research was done by members of the Memorial University Gravity Group which is led by faculty members Ivan and Hari. Alberto was a visiting post-doc and Matt an undergraduate student doing his honours research project. Memorial University is in St. John’s, Newfoundland and Labrador. St. John’s is the eastern-most city in North America and Newfoundland was the western-most limit of exploration for Europeans pre-Columbus (hence our title).

Image taken from I Booth et al 2015 Class. Quantum Grav. 32 235025. Copyright 2015 IOP Publishing Ltd.

Effect of Harrison transform on physical properties of MKN solutions – a) On the left the black dot marks the physical properties of the seed solution $Q= 0.2R$ and $J = 0.4R$ while the ref line records how they evolved for changing $B$. The dark purple surface around which it wraps is a surface of constant $\chi^2$ and the dashed line at the back shows the asymptotic value of the properties as $B \rightarrow \pm \infty$. b) On the right for the same values of $(Q,J)$, areal radius increase monotonically with the magnitude of $B$