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.
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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

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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 $x^2 + y^2 - z^2 = a^2$.