# How to find a normal to an hypersurface?

I have to apply the Israel junction conditions in a region in which a hypersurface with O(3) symmetry separates two spacetime with Schwarzschild metric (with masses $$M_+$$, the exterior one, and $$M_-$$, the interior one). The hypersurface is: $$\Sigma_{\pm}=\{(t_{\pm}, r_{\pm}, \theta_{\pm}, \phi_{\pm})| F_{\pm}(t_{\pm},r_{\pm})=r_{\pm}-R(\tau(t_{\pm}))=0\}$$ where $$\tau$$ is the proper time on $$\Sigma$$. Now the question: the result given by the article for the normal unit vector is $$n_{\pm\mu}=\frac{\partial_{\mu}F_{\pm}}{\sqrt{|g^{(\pm)\mu\nu}\partial_{\mu}F_{\pm}\partial_{\nu}F_{\pm}|}}=(-\dot{R},\dot{t_{\pm}},0,0)$$ How can I find that $$n_{\pm\mu}=(-\dot{R},\dot{t_{\pm}},0,0)$$?

• The normal is the gradient with the denominator to normalize it... – Jan Bos Oct 3 '19 at 15:50
• Ok, but I need to understand the calculation to get the result $n_{\pm\mu}=(-\dot{R},\dot{t_{\pm}},0,0)$ in detail – Priuk Oct 3 '19 at 15:55

If your embedding of the hypersurface is given by a level-set function $$F$$, the normal vector to that hypersurface is given by the exterior derivative of that level-set function, ie

$$$$n = dF$$$$

Any vector tangent to your surface will be obtained by a path entirely within $$\Sigma$$. For a path $$\gamma \in \Sigma$$, the tangent vector to that path is $$\dot{\gamma}$$, but we also have that $$F(\gamma(\tau)) = 0$$ for every $$\tau$$, therefore

$$$$\dot{F}(\gamma(\tau)) = \dot{\gamma}(\tau) dF(\gamma(\tau)) = 0$$$$

This is the $$1$$-form $$dF$$ and the vector $$\dot{\gamma}$$. If we put this in coordinate form, this is simply (with $$u = \dot{\gamma}$$ our tangent vector)

$$$$g^{\mu\nu} u_\mu \partial_\mu F(\gamma(\tau)) = 0$$$$

In other words, this vector is indeed always orthogonal to any tangent vector.

To make it a unit vector, we simply divide it by its norm, as usual :

$$$$n = \frac{dF}{\|dF\|}$$$$

Or again, in coordinates,

$$$$n_\mu = \frac{\partial_\mu F}{\sqrt{g^{\mu\nu} \partial_\mu F \partial_\mu F}}$$$$

• Yes, but how can I find the result $n_{\pm \mu}=(-\dot{R}, \dot{t_{\pm}},0,0)$? – Priuk Oct 3 '19 at 13:23
• Is $R$ a specific quantity here? – Slereah Oct 3 '19 at 13:46
• It's the trajectory of the hypersurface (look at the definition I gave) – Priuk Oct 3 '19 at 13:55
• So can you help me? – Priuk Oct 4 '19 at 7:23