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2013-05-09T11:21:36Z

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The '''Ozsváth–Schücking metric''', or the '''Ozsváth–Schücking solution''', is a [[vacuum solution (general relativity)|vacuum solution]] of the [[Einstein field equations]]. The metric was published by István Ozsváth and [[Engelbert Schücking]] in 1962.<ref>{{citation|first1=I.|last1=Ozsváth|first2=E.|last2=Schücking|title=An anti-Mach metric|journal=Recent Developments in General Relativity|pages=339–350|year=1962|url=http://web.mit.edu/jwk/www/docs/Ozsvath-Schucking%201962%20-%20Anti-Mach%20Metric.pdf}}</ref> It is noteworthy among vacuum solutions for being the first known solution that is [[stationary spacetime|stationary]], globally defined, and singularity-free but nevertheless not isometric to the [[Minkowski metric]]. This stands in contradiction to a claimed strong Mach principle, which would forbid a vacuum solution from being anything but Minkowski without singularities, where the singularities are to be construed as mass as in the [[Schwarzschild metric]].<ref>{{citation|first1=F. A. E.|last1=Pirani|title=Invariant Formulation of Gravitational Radiation Theory|journal=Phys. Rev.|volume=105|pages=1089–1099|year=1957|doi=10.1103/PhysRev.105.1089|bibcode = 1957PhRv..105.1089P }}</ref>

With coordinates <math>\{x^0,x^1,x^2,x^3\}</math>, define the following [[tetrad (general relativity)|tetrad]]:

:<math>e_{(0)}=\frac{1}{\sqrt{2+(x^3)^2}}\left( x^3\partial_0-\partial_1+\partial_2\right)</math>
:<math>e_{(1)}=\frac{1}{\sqrt{4+2(x^3)^2}}\left[ \left(x^3-\sqrt{2+(x^3)^2}\right)\partial_0+\left(1+(x^3)^2-x^3\sqrt{2+(x^3)^2}\right)\partial_1+\partial_2\right]</math>
:<math>e_{(2)}=\frac{1}{\sqrt{4+2(x^3)^2}}\left[ \left(x^3+\sqrt{2+(x^3)^2}\right)\partial_0+\left(1+(x^3)^2+x^3\sqrt{2+(x^3)^2}\right)\partial_1+\partial_2\right]</math>
:<math>e_{(3)}=\partial_3</math>

It is straightforward to verify that e(0) is timelike, e(1), e(2), e(3) are spacelike, that they are all [[orthogonal]], and that there are no singularities. The corresponding proper time is

:<math>{d \tau}^{2} = -(dx^0)^2 +4(x^3)(dx^0)(dx^2)-2(dx^1)(dx^2)-2(x^3)(dx^2)^2-(dx^3)^2.</math>

The [[Riemann tensor]] has only one algebraically independent, nonzero component

:<math>R_{0202}=-1,</math>

which shows that the spacetime is [[Ricci flat]] but not [[conformally flat]]. That is sufficient to conclude that it is a vacuum solution distinct from Minkowski spacetime. Under a suitable coordinate transformation, the metric can be rewritten as

:<math>
d\tau^2 = [(x^2 - y^2) \cos (2u) + 2xy \sin(2u)] du^2 - 2dudv - dx^2 - dy^2
</math>

and is therefore an example of a [[pp-wave spacetime]].

==References==
<references/>

{{DEFAULTSORT:Ozsvath-Schucking metric}}
[[Category:Exact solutions in general relativity]]

{{relativity-stub}}

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