Metric of a globally negatively curved space

[BOAS]

Homework Statement

I think I have managed to do the first three parts of this problem ok, but I am struggling with part 4.
[/B]
A 2D negatively curved surface can be described in 3D Euclidean Cartesian coordinates by the equation:

##x^2 + y^2 + z^2 = −a^2##.​

1) Find the 2D line element for points in the 2D space (x,y):

##dl^2 = h_{ij}dx^i dx^j##​

2) Find the metric coefficients for the line element.

3) Write down the transformation of the Cartesian coordinates to polar coordinates, and compute the transformation matrix:

##\frac{\partial x^{' a}}{\partial x^{b}}## , with ##x^{'a} = (x, y)## and ##x^a = (\rho, \phi)##.​

4) Find the expression for the metric under the transformation to polar coordinates.

5) Explain two advantages of this transformation

Homework Equations

The Attempt at a Solution

[/B]
1) In Euclidean 3-space we have ##ds^2 = dx^2 + dy^2 + dz^2##. Using the surface as a constraint equation and differentiating the line element:

##2x dx + 2y dy + 2z dz = 0##

Solving for ##dz##, ##dz = \frac{- x dx - y dy}{z} = \frac{- x dx - y dy}{\sqrt{- a^2 - x^2 - y^2}}##

and so ##ds^2 = dx^2 + dy^2 - \frac{(x dx + y dy)^2}{a^2 + x^2 + y^2}##

2) Multiplying this out and reading off the coefficients

##h_{xx} = 1 - \frac{x^2}{a^2 + x^2 + y^2}##

##h_{yx} = h_{xy} = - \frac{xy}{a^2 + x^2 + y^2}##

##h_{yy} = 1 - \frac{y^2}{a^2 + x^2 + y^2}##

3) ##x = \rho \cos \phi##, ##y = \rho \sin \phi##

##X^{'a} = (x, y)##, ##X^a = (\rho, \phi)##

##\begin{pmatrix}
\frac{\partial X^{'1}}{\partial X^1} & \frac{\partial X^{'1}}{\partial X^2} \\ \frac{\partial X^{'2}}{\partial X^1} & \frac{\partial X^{'2}}{\partial X^2}
\end{pmatrix} =
\begin{pmatrix}
\frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \phi} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \phi}
\end{pmatrix} =
\begin{pmatrix}
\cos \phi & - \rho \sin \phi \\ \sin \phi & \rho \cos \phi
\end{pmatrix}
##

4) This is where I am having troubles. I am confused about how this all fits together to actually perform this transformation.

Some guidance would be really appreciated!

Thanks in advance!

 

[Orodruin]

A 2D negatively curved surface can be described in 3D Euclidean Cartesian coordinates by the equation:

x^2 + y^2 + z^2 = −a^2.

Is this correctly quoted? That looks like the empty set to me.

 

[BOAS]

Is this correctly quoted? That looks like the empty set to me.

It is correctly quoted

 

[Orodruin]

Well, then it is wrong. The set ##x^2 + y^2 + z^2 = -a^2## is the empty set (assuming ##a## is real, if it is imaginary it is just a sphere).

 

[BOAS]

Well, then it is wrong. The set ##x^2 + y^2 + z^2 = -a^2## is the empty set (assuming ##a## is real, if it is imaginary it is just a sphere).

Ah, you're right. When dealing with negatively curved spaces we do say ##a \rightarrow ia##

 

[Orodruin]

Then it is a sphere.

 

[Orodruin]

What you are probably looking for is the one-sheet hyperboloid ##x^2 + y^2 - z^2 = a^2##.