Minkowski metric in spherical polar coordinates

[spaghetti3451]

Homework Statement

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.

Homework Equations

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?

 

[stevendaryl]

Homework Statement

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.

Homework Equations

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?

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}

 

[spaghetti3451]

Isn't that in part (c)?

Shouldn't I do (b) first?

 

[stevendaryl]

Isn't that in part (c)?

Shouldn't I do (b) first?

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

 

[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?

 

[stevendaryl]

Yes.

 

[Chestermiller]

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

 

[spaghetti3451]

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

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