Solution of Einstein field equations
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 horizons are at
and the ergospheres at
.
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.
With a negative
(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
that would be the case when
).
The
Ricci scalar
for the KNdS metric is
, and the
Kretschmann scalar
is
See also
[
edit
]
References
[
edit
]
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