Symmetric bowl associated with a line element

Hi! I have the following problem I don't really know where to start from:

A bowl with axial symmetry is built in flat Euclidean space ##R^3##, and has a radial profile giveb by ##z(r)##, where ##z## is the axis of symmetry and ##r## is the radial distance from the axis. What radial profile ##z(r)## is required for the bowl to have the same intrinsic geometry as the following line element:

$$ds^2=\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}dr^2+r^2d\phi^2$$

---- MY IDEA -----

The intrinsic geometry of this line element is described by:

$$g_{11}=\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}$$ and $$g_{22}=r^2$$

But how can I describe the intrinsic geometry of ##z(r)## to equal it to the latter?

Thank you so much for your help!

andrewkirk
Homework Helper
Gold Member
What radial profile z(r) is required for the bowl to have the same intrinsic geometry as the following line element:
Intrinsic geometry is a property of a manifold, like the bowl, not of a line element. So I am unable to understand that statement. What is the manifold (presumably 2D, ie a surface, not a line or line element) whose intrinsic geometry they are saying is the same as that of the bowl?

Also, in what coordinates is the formula for the line element written, and what is ##r_+##? I would guess it is Polar, but in that case ##r_+## is identical to ##r## and the formula is everywhere undefined.

nrqed
Homework Helper
Gold Member
Hi! I have the following problem I don't really know where to start from:

A bowl with axial symmetry is built in flat Euclidean space ##R^3##, and has a radial profile giveb by ##z(r)##, where ##z## is the axis of symmetry and ##r## is the radial distance from the axis. What radial profile ##z(r)## is required for the bowl to have the same intrinsic geometry as the following line element:

$$ds^2=\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}dr^2+r^2d\phi^2$$

---- MY IDEA -----

The intrinsic geometry of this line element is described by:

$$g_{11}=\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}$$ and $$g_{22}=r^2$$

But how can I describe the intrinsic geometry of ##z(r)## to equal it to the latter?

Thank you so much for your help!

You have to start with the metric in 3D in cylindrical coordinates (in terms therefore of ##r,\phi,z##). Now, substitute for z the function z(r) (and of course substitute also this function in dz). We must now impose that the resulting metric in terms of z(r), its derivative and ##r## and ##\phi## reproduces the metric that you wrote. That will give you a differential equation for z(r) in terms of r and the constant ##r_+##.