The Kerr–Newman–de–Sitter metric (KNdS) is one of the most general
stationary solutions of the
Einstein–Maxwell equations in
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
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
The Kerr–Newman metric describes the spacetime geometry around a mass which is electrically charged and rotating. It is a vacuum solution which generalizes the Kerr metric (which describes an uncharged, rotating mass) by additionally taking in ...
by taking into account the
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general rel ...
.
Boyer–Lindquist coordinates
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
Frame-dragging is an effect on spacetime, predicted by Albert Einstein's General relativity, general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary Field (physics), field is one that is ...
velocity relative to the fixed stars. In
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
and in
natural units
In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light may be set to 1, and it may then be omitted, equa ...
of
the KNdS metric is
with all the other metric tensor components
, where
is the black hole's spin parameter,
its electric charge, and
the cosmological constant with
as the time-independent
Hubble parameter
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther a galaxy is from the Earth, the faster ...
. 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
file:Kerr surfaces.svg, 300px, At the ergospheres (shown here in violet for the outer and red for the inner one), the temporal metric coefficient ''gtt'' becomes negative, i.e., acts like a purely spatial metric component. Consequently, timelike or ...
)
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
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
The
overdot
When used as a diacritic mark, the term dot refers to the glyphs "combining dot above" (, and "combining dot below" (
which may be combined with some letters of the extended Latin alphabets in use in
a variety of languages. Similar marks are ...
stands for differentiation by the testparticle's
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
or the photon's
affine parameter
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
, so
.
Null coordinates
To get
coordinates we apply the transformation
and get the metric coefficients
and all the other
, with the electromagnetic
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field \mathbf, a ' ...
Defining
ingoing lightlike worldlines give a
light cone on a
spacetime diagram
A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity. Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction witho ...
.
Horizons and ergospheres

The horizons are at
and the ergospheres at
.
This can be solved numerically or analytically. Like in the
Kerr
Kerr may refer to:
People
*Kerr (surname)
*Kerr (given name) Places
;United States
*Kerr Township, Champaign County, Illinois
*Kerr, Montana, A US census-designated place
*Kerr, Ohio, an unincorporated community
*Kerr County, Texas
Kerr Co ...
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 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
[Leonard Susskind]
Aspects of de Sitter Holography
timestamp 38:27: video of the online seminar on de Sitter space and Holography, Sept 14, 2021 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
).
Invariants
The
Ricci scalar
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
for the KNdS metric is
, and the
Kretschmann scalar
In the theory of pseudo-Riemannian manifold, Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic curvature invariant (general relativity), scalar invariant. It was introduc ...
is
See also
*
Kerr–Newman metric
The Kerr–Newman metric describes the spacetime geometry around a mass which is electrically charged and rotating. It is a vacuum solution which generalizes the Kerr metric (which describes an uncharged, rotating mass) by additionally taking in ...
*
De Sitter–Schwarzschild metric
*
de Sitter space
In mathematical physics, ''n''-dimensional de Sitter space (often denoted dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canonical Rie ...
*
de Sitter universe
A de Sitter universe is a cosmological solution to the Einstein field equations of general relativity, named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominat ...
*
Anti-de Sitter space
In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a symmetric_space, maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are na ...
*
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence (frequently abbreviated as AdS/CFT) is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) that are used ...
References
{{Relativity
Exact solutions in general relativity
Equations
Metric tensors