I'd like to find the lapse function and shift vector in 1+1 Minkowski as well as 1+1 de Sitter (flat foliation) for a region foliated this way:

enter image description here

The $y$-axis represents time while the x-axis represents space. The upper-most curve is the final spacelike boundary, and we take this region to be symmetric about the time axis.

The function which describes the final spacelike curve is given by

$$t = 1 - \sqrt{x^2+\alpha^2}$$

with $\alpha\in(0,1)$. In this plot I used $\alpha=0.6$ as an example.

The Minkowski metric is given by $ds^2=-dt^2+dx^2$ and the de Sitter metric is given by $ds^2=\eta^{-2}(-d\eta^2+dx^2)$ ($\eta$ is conformal time).

I've seen the unit normal written as $n_{\mu}=-N\partial_{\mu}t$ and this $t^\mu$ as $t^\mu=Nn^\mu+N^ae_a^\mu$ but it is not clear how this $t^\mu$ is related to the time here since it is multi-valued along a single spacelike surface.

As a start, I have already derived the unit normal vector:

$$n^\mu=\left(-\frac{\sqrt{x^2+\alpha^2}}{\alpha}, \frac{x}{\alpha}\right)$$

and the unit tangent vector is

$$e_a^\mu=\left(-\frac{x}{\sqrt{x^2+\alpha^2}}, 1\right)$$

Ultimately I'd like to use these expressions to find the relation between the full metric $g_{\mu\nu}$ and the induced metric $h_{\mu\nu}$ on the $d-1$ boundary as well as the induced metric $\sigma_{AB}$ on the $d-2$ boundary (the corners).


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