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

e_{(0)}=\frac{1}{\sqrt{2+(x^3)^2}}\left( x^3\partial_0-\partial_1+\partial_2\right)

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]

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]

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

{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

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

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.

References

↑ Ozsváth, I.; Schücking, E. (1962), "An anti-Mach metric" (PDF), Recent Developments in General Relativity: 339–350
↑ Pirani, F. A. E. (1957), "Invariant Formulation of Gravitational Radiation Theory", Phys. Rev., 105: 1089–1099, Bibcode:1957PhRv..105.1089P, doi:10.1103/PhysRev.105.1089

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.