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In general relativity, the van Stockum dust is an exact solution of the Einstein field equations in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is increasing with distance from this axis, the solution is rather artificial, but as one of the simplest known solutions in general relativity, it stands as a pedagogically important example.

This solution is named after Willem Jacob van Stockum, who rediscovered it in 1938 independently of a much earlier discovery by Cornelius Lanczos in 1924. It is currently recommended that the solution be referred to as the Lanczos–van Stockum dust.

Derivation

One way of obtaining this solution is to look for a cylindrically symmetric perfect fluid solution in which the fluid exhibits rigid rotation. That is, we demand that the world lines of the fluid particles form a timelike congruence having nonzero vorticity but vanishing expansion and shear. (In fact, since dust particles feel no forces, this will turn out to be a timelike geodesic congruence, but we won't need to assume this in advance.)

A simple ansatz corresponding to this demand is expressed by the following frame field, which contains two undetermined functions of ${\displaystyle r}$:

${\displaystyle {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=f(r)\,\partial _{z},\;{\vec {e}}_{2}=f(r)\,\partial _{r},\;{\vec {e}}_{3}={\frac {1}{r}}\,\partial _{\varphi }-h(r)\,\partial _{t}}$

To prevent misunderstanding, we should emphasize that taking the dual coframe

${\displaystyle \sigma ^{0}=dt+h(r)r\,d\varphi ,\;\sigma ^{1}={\frac {1}{f(r)}}\,dz,\;\sigma ^{2}={\frac {1}{f(r)}}\,dr,\;\sigma ^{3}=rd\varphi }$

gives the metric tensor in terms of the same two undetermined functions:

${\displaystyle g=-\sigma ^{0}\otimes \sigma ^{0}+\sigma ^{1}\otimes \sigma ^{1}+\sigma ^{2}\otimes \sigma ^{2}+\sigma ^{3}\otimes \sigma ^{3}}$

Multiplying out gives

${\displaystyle ds^{2}=-dt^{2}-2h(r)r\,dt\,d\varphi +(1-h(r)^{2})r^{2}\,d\varphi ^{2}+{\frac {dz^{2}+dr^{2}}{f(r)^{2}}}}$

${\displaystyle -\infty <t,z<\infty ,\;0<r<\infty ,\;-\pi <\varphi <\pi }$

We compute the Einstein tensor with respect to this frame, in terms of the two undetermined functions, and demand that the result have the form appropriate for a perfect fluid solution with the timelike unit vector ${\displaystyle {\vec {e}}_{0}}$ everywhere tangent to the world line of a fluid particle. That is, we demand that

${\displaystyle G^{{\hat {m}}{\hat {n}}}=8\pi \mu \operatorname {diag} (1,0,0,0)+8\pi p\operatorname {diag} (0,1,1,1)}$

This gives the conditions

${\displaystyle f^{\prime \prime }={\frac {(f^{\prime })^{2}}{f}}+{\frac {f^{\prime }}{r}},\;(h^{\prime })^{2}+{\frac {2h^{\prime }h}{r}}+{\frac {h^{2}}{r^{2}}}={\frac {4f^{\prime }}{rf}}}$

Solving for ${\displaystyle f}$ and then for ${\displaystyle h}$ gives the desired frame defining the van Stockum solution:

${\displaystyle {\stackrel{\to <!--\; \to \; -->}{e}}_{}}$Zdroj:https://en.wikipedia.org?pojem=Van_Stockum_dust
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