Efficiency of the Keplerian accretion in braneworld Kerr-Newman spacetimes and mining instability of some naked singularity spacetimes

We show that the braneworld rotating Kerr-Newman black hole and naked singularity spacetimes with both positive and negative braneworld tidal charge parameters can be separated into 14 classes according to the properties of circular geodesics governing the Keplerian accretion. We determine the efficiency of the Keplerian accretion disks for all braneworld Kerr-Newman spacetimes. We demonstrate the occurrence of an infinitely deep gravitational potential in Kerr-Newman naked singularity spacetimes having the braneworld dimensionless tidal charge $b\in (1/4,1)$ and the dimensionless spin $a\in (2\sqrt{b}-\sqrt{b(4b-1)}$, $2\sqrt{b}+\sqrt{b(4b-1)})$, implying unbound efficiency of the Keplerian accretion and the possibility of extracting the whole naked singularity mass. Therefore, we call them braneworld “mining-unstable” Kerr-Newman naked singularity spacetimes. Fundamental restriction on the relevance of the extraordinary—but fully classical—phenomenon of the mining instability is given by validity of the assumption of geodesic motion of the accreting matter.

Figure 1

Example of the behavior of the Kretschmann scalar $K$ for $a=0.8$ and $b=-0.8$ to illustrate its similarity to the Kerr-Newman case.

Figure 2

(Upper left panel) Polar slice through the braneworld Kerr-Newman spacetime in the Cartesian Kerr-Schild coordinates. The dimensionless spin parameter $a$ is fixed at 1 and the braneworld parameter $b$ is appropriately chosen to demonstrate its influence on the ergosphere. (Upper right panel) Polar slice through the braneworld Kerr-Newman spacetime in the Cartesian Kerr-Schild coordinates. The braneworld parameter $b$ is fixed at 0.9 and the spin parameter $a$ is appropriately chosen to demonstrate its influence on the ergosphere. (Lower left panel) Causality violation region. (Lower right panel) Ergosphere and causality violation region.

Figure 3

Contour plot for radii of the boundary of the causality violation region in the equatorial plane.

Figure 4

Braneworld Kerr-Newman black holes and naked singularities can be divided into ten distinguish classes according to the properties of the circular photon geodesics. Curve $a_{\mathrm{ph}}(4b/3,b)$ (the solid line), given by (88), plays the main role in the classification. The corresponding regions of the $b\text{−}a$ plane are denoted by I–X; the numbers in brackets denote the number of circular photon orbits in the respective classes. The first number determines the number of stable circular photon geodesics, while the second number determines the number of unstable circular photon geodesics.

Figure 5

Mapping of existence of the marginally stable circular geodesics in the parameter space of the braneworld Kerr-Newman spacetimes.

Figure 6

Classification of the braneworld Kerr-Newman spacetimes according to the properties of circular geodesics relevant to the Keplerian accretion. The parameter space $b\text{−}a$ is separated by curves governing the extrema of the functions determining the photon circular orbits (the solid lines) and the marginally stable orbits (the dashed lines). Point ($\sqrt{0.5}$, 0.5) is an intersection of the dashed line and the curve separating black holes from naked singularities ($b=1-a^2$). These two curves are tangent at the common point.

Figure 7

Classification of the braneworld Kerr-Newman spacetimes according to the properties of circular geodesics relevant to the Keplerian accretion. The parameter space $b\text{−}a$ is separated by curves governing the extrema of the functions determining the photon circular orbits (the solid lines) and the marginally stable orbits (the dashed lines). Detailed structure for small values of spin $a$ and $b\sim 1$.

Figure 8

$E$ and $L$ for Kerr black holes and naked singularities.

Figure 9

$L$, $E$, and effective potential for class I.

Figure 10

$L$, $E$, and effective potential for class II.

Figure 11

$L$, $E$, and effective potential for class IIIa.

Figure 12

Energy measured by the LNRF observers (upper sign family orbits only) in the mining-unstable Kerr-Newman spacetimes of class IIIa. The energy diverges at the radius of the stable photon circular orbit.

Figure 13

Absolute value of covariant energy $|E|$ divided by LNRF energy (for the upper sign family orbits) in the mining-unstable Kerr-Newman spacetimes of class IIIa. The fraction is defined down to the radius of the stable circular photon orbit, and at this point it has a finite value.

Figure 14

$L$, $E$, and effective potential for class IIIb.

Figure 15

$L$, $E$, and effective potential for class IVa.

Figure 16

$L$, $E$, and effective potential for class IVb.

Figure 17

$L$, $E$, and effective potential for class Va.

Figure 18

$L$, $E$, and effective potential for class Vb.

Figure 19

$L$, $E$, and effective potential for class Vc.

Figure 20

$L$, $E$, and effective potential for class VI.

Figure 21

$L$, $E$, and effective potential for class VII.

Figure 22

$L$, $E$, and effective potential for class VIII.

Figure 23

$L$, $E$, and effective potential for class IX.

Figure 24

$L$, $E$, and effective potential for class X.

Figure 25

The energetic efficiency $\eta (a,b)$ of the Keplerian accretion following the upper family circular geodesics is given depending on the spacetime dimensionless tidal charge parameter $b$ and the spin parameter $a$. The 3D diagram reflects the position of the special class of the mining-unstable Kerr-Newman spacetimes of class IIIa in the plane of the spacetime parameters. Because of the complex character of the efficiency function, we also give the characteristic $a=\mathrm{const}$ sections in the $\eta \text{−}b$ plane.

Figure 26

Energy efficiency of the Keplerian accretion following the lower family circular geodesics is given depending on the spacetime dimensionless tidal charge parameter $b$ for characteristic values of the spin parameter $a=\mathrm{const}$.