# Eikonal Blog

## 2010.07.22

### Scripting user interfaces

Filed under: web tools — Tags: , , , , , — sandokan65 @ 13:32

Browser scripting:

### Volume of the ball

Filed under: geometry — Tags: , — sandokan65 @ 13:12
• The “volume” $\Omega_n :\equiv \hbox{Vol}(B^n)$ of the ball $B^n :\equiv \{x \in {\Bbb R}^n | s/t |x|<1\}$ is given by $\Omega_n = \frac{\pi^{n/2}}{\Gamma(1+\frac{n}2)}$.
• The “surface area” $\omega_n :\equiv \hbox{Area}({\Bbb S}^{n-1})$ of the sphere ${\Bbb S}^{n-1} :\equiv \{x \in {\Bbb R}^n | s/t |x|=1\} = \partial B^n$ is given by $\omega_n = (n+1) \Omega_{n+1}$.

Note: $\Omega_{n+2} = \frac{\pi}{n+1}\Omega_n$, so: $\Omega_{2n} = \frac{\pi^n}{(2n-1)!!}$ and $\Omega_{2n+1} = \frac{8 \pi^n}{3(2n)!!}$.

Examples:
$\begin{array}{ | c | c | c | c | c | c | c | c |} \hline n & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \Omega_n & \pi & \frac{4\pi}{3} & \frac{\pi^2}2 & \frac{\pi^2}3 & \frac{\pi^3}{10} & \frac{\pi^3}{18} & \frac{\pi^4}{70} \\ \hline \omega_{n-1} & 2\pi & 4\pi & 2\pi^2 & \frac{5\pi^2}{3} & \frac{3\pi^3}5 & \frac{7\pi^3}{18} & \frac{4\pi^4}{35} \\ \hline \end{array}$

Alzer’s inequalities (as cited in [1]):

• $a \Omega_{n+1}^{\frac{n}{n+1}} \le \Omega_n \le b \Omega_{n+1}^{\frac{n}{n+1}}$, where $a=\frac{2}{\sqrt{2}} = 1.128,37\cdots$, $b=\sqrt{e}=1.648,72\cdots$;
• $\sqrt{\frac{n+A}{2\pi}} \le \frac{\Omega_{n-1}}{\Omega_n} \le \sqrt{\frac{n+B}{2\pi}}$, where $A=\frac12$ and $B=\frac\pi2-1=0.570,79\cdots$;
• $\left(1+\frac1{n}\right)^\alpha \le \frac{\Omega_n^2}{\Omega_{n-1}\Omega_{n+1}} \le \left(1+\frac1{n}\right)^\beta$, where $\alpha=2-\frac{\ln(\pi)}{\ln(2)} = 0.348,50\cdots$ and $\beta=\frac12$.

Sources:

1. [1] “Topics in Special Functions” by G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen – http://arxiv.org/abs/0712.3856

### Library of metrics

Filed under: General relativity, geometry — Tags: , , — sandokan65 @ 09:42

## ADM decomposition

In $D=1+3$ dimensions the ADM (Arnowitt-Deser-Misner) split of the metric is given by ($i,j\in\overline{1,3}$, $\mu,\nu\in\overline{0,3}$):

$ds^2 = -N^2 dt^2 + g_{ij}^{(3)}(dx^i+N^i dt)(dx^j+N^j dt)$
where $N$ is the lapse variable, $N^i$ is the shift 3-vector.

Here:

• $R = K_{ij}K^{ij} - K^2 + R^{(3)} + 2\nabla_\mu (n^\mu \nabla_\nu n^\nu - n^\nu \nabla_\nu n^\mu)$ is the Ricci scalar,
• $K_{ij} :\equiv \frac1{2N} (\dot{g}_{ij}^{(3)} - \nabla_i^{(3)} N_j - \nabla_j^{(3)} N_i)$ is the extrinsic curvature, $K :\equiv g^{ij}K_{ij}$.

Source: “Unifying inflation with dark energy in modified F(R) Horava-Lifshitz gravity” by E. Elizalde, S. Nojiri, S. D. Odintsov, D. Saez-Gomez; arXiv.org > hep-th > arXiv:1006.3387 (http://arxiv4.library.cornell.edu/abs/1006.3387).

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