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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 ...
\Lambda.


Boyer–Lindquist coordinates

In those coordinates the local clocks and rulers are at constant \rm r and have no local orbital angular momentum \rm (L_z=0), 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 \rm G=M=c=k_e=1 the KNdS metric is g_= \rm -\frac g_= \rm -\frac g_= \rm -\frac g_= \rm\frac g_= \rm \frac with all the other metric tensor components g_=0, where \rm a is the black hole's spin parameter, \rm \mho its electric charge, and \rm \Lambda=3 H^2 the cosmological constant with \rm H 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 \rm A_ = \left\ The frame-dragging angular velocity is \omega = \frac= -\frac= \rm \frac and the local frame-dragging velocity relative to constant \rm \ 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 ...
) \nu = \sqrt = \rm \sqrt The escape velocity (the speed of light at the horizons) relative to the local corotating zero-angular momentum observer is = \sqrt = \rm \sqrt The conserved quantities in the equations of motion \ g_) where \rm \dot is the four velocity, \rm q is the test particle's specific charge and \rm F the Maxwell–Faraday tensor \rm _=\frac-\frac are the total energy =g_ +g_ + \rm q \ A_ 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 ...
=-g_ -g_ - \rm q \ A_ 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 ...
\tau 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 \rm \dot=dx/d\tau , \ \ddot=d^2x/d\tau^2.


Null coordinates

To get g_=0 coordinates we apply the transformation \rm dt=du-\frac \rm d \phi = d \varphi-\frac and get the metric coefficients g_=\rm -\frac g_=\rm \frac g_=g_ \ , \ \ g_=g_ \ , \ \ g_=g_ \ , \ \ g_=g_ and all the other g_=0, 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 ' ...
\rm A_=\left\ Defining \rm \bar=u-r ingoing lightlike worldlines give a 45^ 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 g^=0 and the ergospheres at g_, , g_=0. 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 \rm r, while the ergospheres' radii also depend on the polar angle \theta. 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 \rm r<0 in the antiverse behind the ring singularity, which is part of the probably unphysical extended solution of the metric. With a negative \Lambda (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 limitLeonard 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 \rm a= \mho =0 that would be the case when \Lambda=1/9).


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 \rm R=-4 \Lambda, 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 \rm K=\ \div \\text


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