Question about Spherical Metric and Approximations

In summary, The problem asks us to derive a metric given the coordinates of a colony of "eskimo mites" living at the north pole and the metric for usual spherical coordinates. The resulting metric is not diagonal to second order, but for small values of ##x## and ##y## it is flat and Euclidean. The attempt at a solution involves using the general relation for calculating the primed metric from the unprimed metric and finding expressions for the unprimed coordinates in terms of the primed coordinates. The mistake in the attempt at a solution was using an incorrect equation for ##\sin \theta##, but using the correct equation and a Taylor series approximation for ##\sin^2 \theta## leads to the correct result
  • #1
willballard137
2
0

Homework Statement



This is Problem 2 from Chapter 1, Section V of A. Zee's Einstein Gravity in a Nutshell. Zee asks us to imagine a colony of "eskimo mites" that live at the north pole. The geometers of the colony have measured the following metric of their world to second order (with the radius of the sphere equal to 1):

$$ ds^{2}=\left (1-\frac{y^{2}}{3} \right )dx^{2}+\left (1-\frac{x^{2}}{3} \right )dy^{2}+\frac{2}{3}xy\, dx\, dy+\cdots $$

Zee states: "For ## x,y \ll 1 ##, the space is flat and as Euclidean as it could be. But note that in the second order the metric is not diagonal."

The problem asks us to derive the above metric, given that we know (1) that the mite coordinates ## (x,y) ## are related to the usual spherical coordinates ## x = \theta \cos \phi ## and ## y=\theta \sin \phi ## and (2) that the metric for spherical coordinates is:

$$ ds^{2}=d\theta^{2}+\sin^{2}\theta\, d\phi^{2}. $$

Homework Equations



I believe that the most important equation allows us to calculate the metric in the primed coordinates ## (x,y) ## given the metric in the unprimed coordinates ## (\theta,\phi) ##:

$$ {g}'_{\rho\sigma}({x}')=g_{\mu\nu}(x)\frac{\partial x^{\mu} }{\partial {x}'^{\rho}}\frac{\partial x^{\nu} }{\partial {x}'^{\sigma}}. $$

I believe that I also need to find expressions for the unprimed coordinates as a function of the primed coordinates. In these equations, I assume that the radius of the sphere is 1.

$$ \sin \theta=\frac{\sqrt{x^{2}+y^{2}}}{r}=\sqrt{x^{2}+y^{2}} .$$

$$ \tan \phi=\frac{x}{y}.$$

The Attempt at a Solution


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So far, I have been unable to calculate ## {g}'_{xx} ##, which according to the problem statement should be (to second order) ## 1-\frac{y^{2}}{3} ##. According to our general relation for calculating the primed metric, I have:

$$ {g}'_{xx}={g}_{\theta\theta}\left (\frac{\partial\theta}{\partial x} \right )^{2}+g_{\phi\phi}\left (\frac{\partial\phi}{\partial x} \right )^{2} .$$

Then taking the (I hope) correct partial derivatives from above and substitute in some trigonometric identities.

$$ \frac{\partial\theta}{\partial x}=\frac{x}{\sqrt{x^{2}+y^{2}}}\cdot \frac{1}{\cos\theta}=\frac{x}{\sqrt{x^{2}+y^{2}}\sqrt{1-x^{2}-y^{2}}} .$$

$$ \frac{\partial\phi}{\partial x}=-\frac{y}{x^{2}}\cos^{2}\phi=-\frac{y}{x^{2}}\cdot \frac{x^{2}}{x^{2}+y^{2}}=-\frac{y}{x^{2}+y^{2}} $$

I can now plug these results back into my expression for ## {g}'_{xx} ##:

$$ {g}'_{xx}=\frac{x^{2}}{(x^{2}+y^{2})(1-x^{2}-y^{2})}+\sin^2\theta\frac{y^{2}}{(x^{2}+y^{2})^{2}}. $$

Simple trigonometry tells us that ## \sin^2 \theta = x^{2}+y^{2} ##. Thus I end up with the result, before simplification:

$$ {g}'_{xx}=\frac{x^{2}}{(x^{2}+y^{2})(1-x^{2}-y^{2})}+\frac{y^{2}}{(x^{2}+y^{2})}. $$

I have tried to simplify this to the correct second-order approximation, but cannot get there. It is very well possible that I am approaching this problem from the completely wrong direction. In the following chapter, which provides a brief introduction to Riemann Normal Coordinates, Zee makes use of Taylor Series Powers expansions. However, I was unsure of how to apply those to this problem.

Nonetheless, I take it that this problem emphasizes the fact that we can select coordinates such that the local metric -- for infinitesimal displacements -- is Euclidean to the zeroth order. Moreover, we can remove any first-order terms by careful selection of new coordinates. However, we cannot get rid of the second-order deviations.

I hope this provides enough information for some exterior guidance.
 
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  • #2
Welcome to PF!

I believe the mistake is with the equation ## \sin \theta=\sqrt{x^{2}+y^{2}} ##. This would be applicable if the "mite coordinates" ##x## and ##y## obeyed

##x = \sin\theta \cos \phi##
##y = \sin\theta \sin \phi##

But the mite coordinates are given to obey

##x = \theta \cos \phi##
##y = \theta \sin \phi##

If you use this second set of relations, then it seems to work out.
 
  • #3
Thank you for the suggestion. In addition to that change, I also used the Taylor series ## \sin ^2 \theta = \theta^2 + \frac{1}{3}\theta^4 + ... ## and it worked out.
 

1. What is a spherical metric?

A spherical metric is a way of measuring distances on a sphere using mathematical equations. It takes into account the curvature of the sphere and can be used to calculate distances, angles, and other geometric properties.

2. How is a spherical metric different from a Euclidean metric?

A spherical metric is different from a Euclidean metric because it takes into account the curvature of a sphere, whereas a Euclidean metric assumes a flat surface. This means that the calculations and measurements made using a spherical metric will be different from those made using a Euclidean metric.

3. What are some common approximations used in spherical metrics?

Some common approximations used in spherical metrics include the small angle approximation, the small distance approximation, and the small circle approximation. These approximations are used to simplify calculations and make them more manageable.

4. How is a spherical metric used in real-world applications?

A spherical metric is used in many real-world applications, such as navigation, astronomy, and geodesy. It is also used in the study of spherical objects, such as planets and other celestial bodies. Additionally, it is used in the development of maps and other geographic systems.

5. What are some limitations of using spherical metrics?

One limitation of using spherical metrics is that they are only applicable to spherical objects and cannot be used on flat surfaces. Another limitation is that they are only accurate for small distances and angles, and become less accurate for larger distances and angles. Additionally, spherical metrics can be more complex and difficult to work with compared to Euclidean metrics.

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