Graduate Schwarzschild Geometry: Evaluating Proper Distance

[kirkr]

TL;DR

Schwarzschild Geometry-proper distance.

Schwarzschild Geometry-proper distance. From what I have studied when the Schwarzschild line element is evaluated at constant time and at a constant radius , proper distance becomes a Euclidean distance on the surface of a sphere. What I don't understand is how to evaluate the integral associated with proper distance in terms of theta and phi. Could you provide an example calculation?

 

[strangerep]

What I don't understand is how to evaluate the integral associated with proper distance in terms of theta and phi. Could you provide an example calculation?

I'm not sure if this question is homework, or just homework-like, but the etiquette here at PF is that you should first show your attempted calculation (in latex), or at least list the formulas that are potentially relevant, rather than expecting others to spoon-feed you a whole calculation.

Now, when you say "the integral", over what (type of) path on the sphere do you wish to integrate? E.g., a segment of a great circle? Or some more complicated path?

 

[PeterDonis]

From what I have studied

What have you studied? Please give a reference.

 

[PeterDonis]

when the Schwarzschild line element is evaluated at constant time and at a constant radius , proper distance becomes a Euclidean distance on the surface of a sphere.

This doesn't make sense. The surface of a 2-sphere is not a Euclidean manifold, so there is no such thing as "a Euclidean distance" on it.

 

[Ibix]

Summary: Schwarzschild Geometry-proper distance.

From what I have studied when the Schwarzschild line element is evaluated at constant time and at a constant radius , proper distance becomes a Euclidean distance on the surface of a sphere.

No, but distance along a sphere at radial coordinate $r$ is the same as distance along a sphere of radius $r$ in a Euclidean space.

A useful way to think about it is to ask yourself how you would calculate distance along some path on a sphere in a normal Euclidean space. What integral do you do? And what does the Schwarzschild line element look like when $t$ and $r$ are constant?

 

[Nugatory]

What I don't understand is how to evaluate the integral associated with proper distance in terms of theta and phi. Could you provide an example calculation?

are you familiar with how the metric tensor is used to calculate distances in a manifold? If not, work through https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf to see how to set up the integral.

That’s the hard part - the integral itself is straightforward, and even trivial if you remember that you can choose your coordinates so that both points are on the “equator”

 

[kirkr]

Thanks. I will review the Preposterous Universe link that you sent. Kirk

 

[kirkr]

What have you studied? Please give a reference.

Hi Peter
Two of the sources that I have used are
Tensors, Relativity and Cosmology, Dalarson
Introduction to Cosmology, Narlikar

 

[PeterDonis]

Hi Peter
Two of the sources that I have used are
Tensors, Relativity and Cosmology, Dalarson
Introduction to Cosmology, Narlikar

These both seem focused more on cosmology, which does not make use of the Schwarzsschild geometry (FRW geometry, which is very different, is the important one in cosmology). That may be why they don't give a good understanding of the issue you are asking about.

 

[PeterDonis]

What I don't understand is how to evaluate the integral associated with proper distance in terms of theta and phi.

This works exactly the same as the corresponding integral on the surface of an ordinary 2-sphere. For example, it's the same as computing proper distances on the surface of (an idealized spherical) Earth in terms of latitude and longitude.