Kerr–Newman–de–Sitter metric

The Kerr–Newman–de–Sitter metric (KNdS)[1][2] is the one of the most general stationary solutions of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass embedded in an expanding universe.

It generalizes the Kerr–Newman metric by taking into account the cosmological constant

In those coordinates the local clocks and rulers are at constant

and have no local orbital angular momentum

, therefore they are corotating with the frame-dragging velocity relative to the fixed stars.

In (+, −, −, −) signature and in natural units of

with all the other metric tensor components

is the black hole's spin parameter,

its electric charge, and

as the time-independent Hubble parameter.

The frame-dragging angular velocity is

and the local frame-dragging velocity relative to constant

positions (the speed of light at the ergosphere)

The escape velocity (the speed of light at the horizons) relative to the local corotating zero-angular momentum observer is

The conserved quantities in the equations of motion

is the test particle's specific charge and

and the covariant axial angular momentum

The overdot stands for differentiation by the testparticle's proper time

or the photon's affine parameter, so

coordinates we apply the transformation

, with the electromagnetic vector potential

ingoing lightlike worldlines give a

This can be solved numerically or analytically.

Like in the Kerr and Kerr–Newman metrics, the horizons have constant Boyer–Lindquist

, while the ergospheres' radii also depend on the polar angle

This gives 3 positive solutions each (including the black hole's inner and outer horizons and ergospheres as well as the cosmic ones) and a negative solution for the space at

in the antiverse[8][9] behind the ring singularity, which is part of the probably unphysical extended solution of the metric.

(the anti–de–Sitter variant with an attractive cosmological constant), there are no cosmic horizon and ergosphere, only the black hole-related ones.

In the Nariai limit[10] the black hole's outer horizon and ergosphere coincide with the cosmic ones (in the Schwarzschild–de–Sitter metric to which the KNdS reduces with

The Ricci scalar for the KNdS metric is