In mathematics, the Ernst equation is an

integrable 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 ...

non-linear partial differential equation In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathem ...

, named after the American

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate caus ...

The Ernst equation

The equation reads:

\Re(u)(u_+u_r/r+u_) = (u_r)^2+(u_z)^2.

For its

Lax pair In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...

and other features see e.g. and references therein.

Usage

The ''Ernst equation'' is employed in order to produce the exact solutions of the

Einstein's 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 the

general theory of relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

References

Partial differential equations General relativity Integrable systems {{theoretical-physics-stub