# What is Laplace operator of Schwarzschild-Spherical coordinates? [closed]

This is the Laplace operator of Spherical coordinates: What is the Laplace operator of Schwarzschild-Spherical coordinates?

$$d{\bf{l}}=(1-\frac{r_s}{r})^{-1/2}dr\hat{\bf{r}}+rd\theta{\bf\hat{\theta}}+\sin{\theta}rd\phi{\bf\hat{\phi}}$$

• I've linked the appropriate Wiki article under "Laplace operator". There's even an explicit formula to calculate it from the metric there - what is your question? Oct 26, 2014 at 20:09
• Hi Achmed. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. Oct 26, 2014 at 20:44
• en.wikipedia.org/wiki/… Oct 26, 2014 at 21:29
• for the generalisation of the laplacian operator in riemannian manifolds and derivation of its form in local coordinates you can also check macs.hw.ac.uk/~hg94/pdst11/pdst11_sphere.pdf Oct 27, 2014 at 18:24

It seems you want to calculate the laplacian in the spacial surface of $t$ constant. If the global metric is defined as:
$$ds^2 = (1-\frac{r_s}{r}) \, dt^2 + (1-\frac{r_s}{r})^{-1} \, dr^2 + r^2 \, d\theta^2 + r^2 \, \sin^2(\theta) \, d\phi^2$$
The metric on the surface $t$ constant is $$ds^2 = (1-\frac{r_s}{r})^{-1} \, dr^2 + r^2 \, d\theta^2 + r^2 \, \sin^2(\theta) \, d\phi^2$$
So your metric written in matrix form is: $$[g_{ij}]=\begin{pmatrix} (1-\frac{r_s}{r})^{-1} && 0 && 0 \\ 0 && r^2 && 0 \\ 0 && 0 && r^2 \, \sin^2(\theta) \end{pmatrix}$$ The laplacian operator can be written in terms of this matrix elements as other people pointed in the comments. The general formula can be found in wikipedia as $$\nabla^2 \phi = \sum_{ij} \frac{1}{\sqrt{|\det(g)|}} \partial_i \left(g^{ij} \sqrt{|\det(g)|} \partial_j \phi \right)$$ Where $g^{ij}$ is the inverse of the matrix of the metric $g_{ij}$. So $$[g^{ij}]=\begin{pmatrix} (1-\frac{r_s}{r}) && 0 && 0 \\ 0 && \frac{1}{r^2} && 0 \\ 0 && 0 && \frac{1}{ r^2 \, \sin^2(\theta)} \end{pmatrix}$$ Now you can use the formula above for get the laplacian you want.