Homework Help: Minkowski metric in spherical polar coordinates

1. Mar 4, 2016

spaghetti3451

1. The problem statement, all variables and given/known data

Consider Minkowski space in the usual Cartesian coordinates $x^{\mu}=(t,x,y,z)$. The line element is

$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}$

in these coordinates. Consider a new coordinate system $x^{\mu'}$ which differs from these Cartesian coordinates. The Cartesian coordinates $x^{\mu}$ can be written as a function of these new coordinates $x^{\mu}=x^{\mu}(x^{\mu'})$.

(a) Take a point $x^{\mu'}$ in this new coordinate system, and imagine displacing it by an infinitesimal amount to $x^{\mu'}+dx^{\mu'}$. We want to understand how the $x^{\mu}$ coordinates change to first order in this displacement $dx^{\mu'}$. Argue that

$dx^{\mu}=\frac{\partial x^{\mu}}{\partial x^{\mu'}}dx^{\mu'}$.

(Hint: Taylor expand $x^{\mu}(x^{\mu'}+dx^{\mu'})$.)

(b) The sixteen quantities $\frac{\partial x^{\mu}}{\partial x^{\mu'}}$ are referred to as the Jacobian matrix; we will require this matrix to be invertible. Show that the inverse of this matrix is $\frac{\partial x^{\mu'}}{\partial x^{\mu}}$. (Hint: Use the chain rule.)

(c) Consider spherical coordinates, $x^{\mu'}=(t,r,\theta,\phi)$ which are related to the Cartesian
coordinates by

$(t,x,y,z)=(t,r\ \text{sin}\ \theta\ \text{cos}\ \phi,r\ \text{sin}\ \theta\ \text{sin}\ \phi,r\ \text{cos}\ \theta)$.

Compute the matrix $\frac{\partial x^{\mu}}{\partial x^{\mu'}}$. Is this matrix invertible everywhere? Compute the displacements $dx^{\mu}$ in this coordinate system (i.e. write them as functions of $x^{\mu'}$ and the infinitesimal displacements $dx^{\mu'}$).

(d) Compute the line element $ds^{2}$ in this coordinate system.

2. Relevant equations

3. The attempt at a solution

(a) By Taylor expansion,

$x^{\mu}(x^{\mu'}+dx^{\mu'}) = x^{\mu}(x^{\mu'}) + \frac{\partial x^{\mu}}{\partial x^{\mu'}}dx^{\mu'}$

$x^{\mu}(x^{\mu'}+dx^{\mu'}) - x^{\mu}(x^{\mu'}) = \frac{\partial x^{\mu}}{\partial x^{\mu'}}dx^{\mu'}$

$dx^{\mu}=\frac{\partial x^{\mu}}{\partial x^{\mu'}}dx^{\mu'}$

Am I correct so far?

Last edited: Mar 4, 2016
2. Mar 4, 2016

stevendaryl

Staff Emeritus
Yes, but you should go ahead and calculate the nine components:

$\frac{\partial x}{\partial r}$, $\frac{\partial y}{\partial r}$, $\frac{\partial z}{\partial r}$
$\frac{\partial x}{\partial \theta}$, $\frac{\partial y}{\partial \theta}$, $\frac{\partial z}{\partial \theta}$
$\frac{\partial x}{\partial \phi}$, $\frac{\partial y}{\partial \phi}$, $\frac{\partial z}{\partial \phi}$

3. Mar 4, 2016

spaghetti3451

Isn't that in part (c)?

Shouldn't I do (b) first?

4. Mar 4, 2016

stevendaryl

Staff Emeritus
Yeah, I guess you should, even though part c doesn't actually depend on part b.

5. Mar 4, 2016

spaghetti3451

(b) Via the chain rule,

$\frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\mu'}}{\partial x^{\nu}}=\delta_{\nu}^{\mu}$,

where we are using the summation convention only over $\mu'$.

Therefore, the inverse of the matrix $\frac{\partial x^{\mu}}{\partial x^{\mu'}}$ is the matrix $\frac{\partial x^{\mu'}}{\partial x^{\nu}}$.

Is this correct?

6. Mar 4, 2016

stevendaryl

Staff Emeritus
Yes.

7. Mar 4, 2016

Staff: Mentor

What's wrong with taking the differentials of x, y, and z (expressed in terms of the spherical coordinates in post #1), evaluating their differentials (in terms of the spherical coordinates and their differentials), and then taking the sum of their squares? This should give the Minkowski metric in spherical coordinates, correct?

Chet

8. Mar 5, 2016

spaghetti3451

I know that this is a correct and shorter approach, but I'm trying to follow the instructions of the question.