Minkowski metric in spherical polar coordinates

Is there any flaw in the above approach that I took?No, your approach is fine. I just thought it would be more efficient to calculate ##ds^2## directly because the spherical coordinates are given in terms of the Cartesian coordinates.
  • #1
spaghetti3451
1,344
33

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?
 
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  • #2
failexam said:

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:

[itex]\frac{\partial x}{\partial r}[/itex], [itex]\frac{\partial y}{\partial r}[/itex], [itex]\frac{\partial z}{\partial r}[/itex]
[itex]\frac{\partial x}{\partial \theta}[/itex], [itex]\frac{\partial y}{\partial \theta}[/itex], [itex]\frac{\partial z}{\partial \theta}[/itex]
[itex]\frac{\partial x}{\partial \phi}[/itex], [itex]\frac{\partial y}{\partial \phi}[/itex], [itex]\frac{\partial z}{\partial \phi}[/itex]
 
  • #3
Isn't that in part (c)?

Shouldn't I do (b) first?
 
  • #4
failexam said:
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.
 
  • #5
(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?
 
  • #7
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
Chestermiller said:
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.
 
  • Like
Likes Chestermiller

1. What is the Minkowski metric in spherical polar coordinates?

The Minkowski metric in spherical polar coordinates is a mathematical representation of the spacetime interval in special relativity. It takes into account both space and time coordinates in a curved coordinate system.

2. How is the Minkowski metric different from other metrics?

The Minkowski metric is unique because it combines both spatial and temporal coordinates, whereas other metrics may only consider one or the other. It also takes into account the curvature of spacetime, which is a key concept in Einstein's theory of general relativity.

3. What are the equations for the Minkowski metric in spherical polar coordinates?

The Minkowski metric in spherical polar coordinates is represented by the equation ds² = -dt² + dr² + r²dθ² + r²sin²θdφ². This equation takes into account the time coordinate (t) and the three spatial coordinates (r, θ, φ).

4. How is the Minkowski metric used in physics?

The Minkowski metric is a fundamental tool in the study of special relativity and general relativity. It is used to calculate the spacetime interval between two events, which is a crucial concept in understanding the behavior of particles and objects in spacetime.

5. Can the Minkowski metric be applied to other coordinate systems?

Yes, the Minkowski metric can be applied to other coordinate systems, such as Cartesian coordinates or cylindrical coordinates. However, the equations will differ depending on the coordinate system, but the concept of combining both space and time coordinates remains the same.

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