|
|
| Line 1: |
Line 1: |
| 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>
| | Andrew Simcox is the title his parents gave him and he completely enjoys this title. Alaska is where I've always been living. I am an invoicing officer and I'll be promoted soon. It's not a typical thing but what she likes performing is to play domino but she doesn't have the time lately.<br><br>Feel free to visit my page ... [http://www.sirudang.com/siroo_Notice/2110 real psychic] |
| | |
| 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<sub>(0)</sub> is timelike, e<sub>(1)</sub>, e<sub>(2)</sub>, e<sub>(3)</sub> 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}}
| |
Andrew Simcox is the title his parents gave him and he completely enjoys this title. Alaska is where I've always been living. I am an invoicing officer and I'll be promoted soon. It's not a typical thing but what she likes performing is to play domino but she doesn't have the time lately.
Feel free to visit my page ... real psychic