In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
, the Newman–Janis algorithm (NJA) is a
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...
technique for finding
exact solutions
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
to the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
. In 1964, Newman and Janis showed that the
Kerr metric
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of ...
could be obtained from the
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...
by means of a coordinate transformation and allowing the radial coordinate to take on complex values. Originally, no clear reason for why the algorithm works was known.
In 1998, Drake and Szekeres gave a detailed explanation of the success of the algorithm and proved the uniqueness of certain solutions. In particular, the only perfect fluid solution generated by NJA is the Kerr metric and the only
Petrov type D solution is the
Kerr–Newman metric
The Kerr–Newman metric is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating ma ...
.
The algorithm works well on ''ƒ''(''R'') and Einstein–Maxwell–Dilaton theories, but doesn't return expected results on
Braneworld and
Born–Infield theories.
See also
*
Birkhoff's theorem (relativity)
References
Algorithms
Exact solutions in general relativity
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