Ozsváth–Schücking metric

The Ozsváth–Schücking metric, or the Ozsváth–Schücking solution, is a vacuum solution of the Einstein field equations. The metric was published by István Ozsváth and Engelbert Schücking in 1962.[1] It is noteworthy among vacuum solutions for being the first known solution that is 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.[2]

With coordinates ${\displaystyle \{x^{0},x^{1},x^{2},x^{3}\}}$, define the following tetrad:

${\displaystyle e_{(0)}={\frac {1}{\sqrt {2+(x^{3})^{2}}}}\left(x^{3}\partial _{0}-\partial _{1}+\partial _{2}\right)}$

${\displaystyle e_{(1)}={\frac {1}{\sqrt {4+2(x^{3})^{2}}}}\left[\left(x^{3}-{\sqrt {2+(x^{3})^{2}}}\right)\partial _{0}+\left(1+(x^{3})^{2}-x^{3}{\sqrt {2+(x^{3})^{2}}}\right)\partial _{1}+\partial _{2}\right]}$

${\displaystyle e_{(2)}={\frac {1}{\sqrt {4+2(x^{3})^{2}}}}\left[\left(x^{3}+{\sqrt {2+(x^{3})^{2}}}\right)\partial _{0}+\left(1+(x^{3})^{2}+x^{3}{\sqrt {2+(x^{3})^{2}}}\right)\partial _{1}+\partial _{2}\right]}$

${\displaystyle e_{(3)}=\partial _{3}}$

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

${\displaystyle {d\tau }^{2}=-(dx^{0})^{2}+4(x^{3})(dx^{0})(dx^{2})-2(dx^{1})(dx^{2})-2(x^{3})^{2}(dx^{2})^{2}-(dx^{3})^{2}.}$

The Riemann tensor has only one algebraically independent, nonzero component

${\displaystyle R_{0202}=-1,}$

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

${\displaystyle d\tau ^{2}=[(x^{2}-y^{2})\cos(2u)+2xy\sin(2u)]du^{2}-2dudv-dx^{2}-dy^{2}}$

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