Schwarzschild Tangherlini Metric

In general relativity , the higher-dimensional generalization of the Schwarzschild metric is referred to as the Schwarzschild-Tangherlini metric (after Karl Schwarzschild , Frank R. Tangherlini ). The general shape of the line element (in Weinberg's signing convention) is

${\displaystyle ds^{2}=-{\Big [}1-{\Big (}{\frac {a}{r}}{\Big )}^{d-3}{\Big ]}dt^{2}+{\Big [}1-{\Big (}{\frac {a}{r}}{\Big )}^{d-3}{\Big ]}^{-1}dr^{2}+r^{2}d\Omega _{d-2}^{2},}$

where was set and denotes the number of dimensions of space-time. In "ordinary" spacetime would be . The standard metric on the -dimensional unit sphere is denoted by, which is inductively defined by ${\displaystyle c=1}$ ${\displaystyle d}$ ${\displaystyle d=4}$ ${\displaystyle d\Omega _{d-2}^{2}}$ ${\displaystyle d-2}$ ${\displaystyle S^{d-2}}$

${\displaystyle d\Omega _{1}^{2}=d\varphi ^{2},\quad d\Omega _{i+1}^{2}=d\theta _{i}^{2}+\sin ^{2}\theta _{i}d\Omega _{i}^{2}\;(i\geq 1),}$

where the coordinate takes values ​​between and , while the coordinates take values ​​between and . For example ${\displaystyle \varphi }$ ${\displaystyle 0}$ ${\displaystyle 2\pi }$ ${\displaystyle \theta _{i}}$ ${\displaystyle 0}$ ${\displaystyle \pi }$ ${\displaystyle d=6}$

${\displaystyle d\Omega _{4}^{2}=d\theta _{3}^{2}+\sin ^{2}\theta _{3}d\theta _{2}^{2}+\sin ^{2}\theta _{3}\sin ^{2}\theta _{2}d\theta _{1}^{2}+\sin ^{2}\theta _{3}\sin ^{2}\theta _{2}\sin ^{2}\theta _{1}d\varphi ^{2}.}$

The interesting result for is that in this metric there are no stable, bound orbits of massive particles that do exist for . One sees this by looking at the movement in the equatorial plane and introducing the coordinate . The equation results from the Lagrange density by introducing the conservation quantities (" energy ") and (" angular momentum ") ${\displaystyle d\geq 5}$ ${\displaystyle d=4}$ ${\displaystyle \theta _{1}=\theta _{2}=\ldots ={\frac {\pi }{2}}}$ ${\displaystyle u(\varphi )={\frac {a}{r}}}$ ${\displaystyle E}$ ${\displaystyle l}$

${\displaystyle {\frac {1}{2}}\left({\frac {du}{d\varphi }}\right)^{2}+{\frac {1}{2}}u^{2}-{\frac {1}{2}}u^{d-1}-{\frac {a^{2}}{l^{2}}}u^{d-3}={\frac {a^{2}}{l^{2}}}(E-1),}$

where the last three terms on the left represent an effective potential . If you sketch the course over , you can see immediately that there is a maximum of one or exactly one ( ) extreme point. Thus every particle trajectory is either unbounded or leads to the singularity . ${\displaystyle u}$ ${\displaystyle d\geq 5}$ ${\displaystyle d\geq 6}$ ${\displaystyle u=\infty }$

literature

• Tangherlini, FR, "Schwarzschild field in n dimensions and the dimensionality of space problem", Nuovo Cim. 27: 636-651 (1963)