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 (+, −, −, −) signature and in natural units of the KNdS metric is[3][4][5][6]
with all the other metric tensor components , where is the black hole's spin parameter, its electric charge, and [7] the cosmological constant with as the time-independent Hubble parameter. The electromagnetic 4-potential is
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
where is the four velocity, is the test particle's specific charge and the Maxwell–Faraday tensor
are the total energy
and the covariant axial angular momentum
The overdot stands for differentiation by the testparticle's proper time or the photon's affine parameter, so .
To get coordinates we apply the transformation
and get the metric coefficients
and all the other , with the electromagnetic vector potential
Defining ingoing lightlike worldlines give a light cone on a spacetime diagram.
The Ricci scalar for the KNdS metric is , and the Kretschmann scalar is