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I am trying to use a certain parametrization on de-Sitter space $dS^n$ and I am getting both the wrong scalar curvature and metric determinant.

The formal definition of $dS^n$ in my work is $-x_1^2+x_2^2+...+x_{n+1}^2=1$ in $\mathcal{M}^{n+1}$ equipped with the induced metric $\eta$.

I work in different coordinates; the light-cone coordinates. Here $e_1=\frac{e_t+e_x}{\sqrt{2}}$ , $e_2=\frac{e_x-e_t}{\sqrt{2}}$ and same otherwise. The parametrization is

$\rho(x_2,...,x_{n+1})=\bigg(\frac{1-x_3^2-...-x_{n+1}^2}{2x_2},x_2,...,x_{n+1}\bigg), \qquad x_2>0.$

This clearly is a bijection onto half the $dS^n$, i.e. one connected component of $dS^n-H$ where $H$ is a $(n-1)$-dimensional coisotropic vector subspace defined through $x_2=x_1$.

When calculating the metric I get

$g_{22}=-\frac{1-x_3^2-...-x_{n+1}^2}{x_2^2}$

$g_{ii}=1,\qquad 2<i\leq n+1$

$g_{ij}=0,\qquad i\not=j$.

The determinant thus becomes $\sqrt{|det\:g|}=\frac{\sqrt{1-x_3^2-...-x_{n+1}^2}}{x_2}$. The sectional curvature becomes (regardless of 2d-span)

$\kappa=\frac{1}{1-x_3^2-...-x_{n+1}^2}$.

The determinant is supposed to be $\sqrt{g}=\frac{1}{x_2}$. Am I doing a conceptual/thought- error somewhere? Have redone the calculations and still get the same answer.

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  • $\begingroup$ Resolved; missing off-diagonal terms in the metric. $\endgroup$
    – Vertex
    Dec 21 '15 at 13:21

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