What is the Meaning of r in Schwartzschild Metric?

[jstrunk]

What is the meaning of r in the Schwartzschild metric?.
$$ds^2 = \frac{{dr^2 }}{{1 - \frac{{2GM}}{{c^2 r}}}} + r^2 (d\theta ^2 + \sin ^2 \theta d\varphi ^2 ) - c^2 \left( {1 - \frac{{2GM}}{{c^2 r}}} \right)dt^2$$
If you were to actually measure the radius, your observation would be affected by the factor
$$\frac{1}{{1 - \frac{{2GM}}{{c^2 r}}}}$$
So where would you get r from if its not the radius you would measure?
I am used to special relativity where you have two different observers that measure different
values but there are no different observers here. It seems like you are supposed to measure
some space to find r, then move a star there and measure it again to see how r is affected.
That hardly seems practical. Assuming there is already a star there, are you supposed to just
measure the circumference and divide by 2 pi to estimate what r would be if the star wasn't there?

 

[A.T.]

measure the circumference and divide by 2 pi

Yes.

 

[robphy]

$r=\sqrt{\frac{(\mbox{Area of a 2-sphere})}{4\pi} }$
Wald, p. 120 [via Google]

 

[wabbit]

Right. The key to this I think is in the expression $r^2 (d\theta ^2 + \sin ^2 \theta d\varphi ^2)$ which describes the geometry of a two-sphere (and the spherical symmetry of that metric). The coordinates can change but the area of that sphere or the circumference of its great circles are invariants.

 

[jstrunk]

Thank you for your help. I am now trying to understand Gravitational Redshift and I have a similar problem. The metric has tau (proper time) and t (time) which are not the same thing, even measured in a comoving frame. It seems like tau is now the real thing and t is some imaginary thing that doesn't really exist but is used to determine tau. This is a lot like what happened to r (radius) when I was looking at the advance of the perihelion of Mercury. The outcome depends on r but in fact there is no r and we can only supply and approximation of one by jiggery pokery.

Anyway, my point is I had developed the knack of thinking in Special Relativitistic terms by imagining myself comoving with one frame and applying the transformation equations to see how the world would look in another frame. I need to develop the corresponding skill for GR and I'm not getting it from any of the books I am using. Can someone recommend a book or web source that carefully develops this skill? Preferably with exercises and solutions.

 

[pervect]

Thank you for your help. I am now trying to understand Gravitational Redshift and I have a similar problem. The metric has tau (proper time) and t (time) which are not the same thing, even measured in a comoving frame. It seems like tau is now the real thing and t is some imaginary thing that doesn't really exist but is used to determine tau.

This is a good attitude - you can regard r and t as being arbitrary labels. Specifically, you can NOT say that the distance between two points with coordinates r' = r+$\epsilon$ and r=r is $\epsilon$, rather you say that the (proper) distance is given by the metric, so that $ds^2 = g_{rr} \Delta r^2$ with $\Delta r = r' - r$. This winds up implying that the (proper) distance between the points with coordinates $r+\epsilon$ and $r$ and the other coordinates the same is $\sqrt{g_{rr}} \epsilon$.

So you use the metric to convert coordinate changes into distances.

Similar things happen with proper time. Given two points with time coorditnates t' = t + \epsilon and t=t, you write the proper time $\tau$ as being proportional to $\sqrt{g_{tt}} \epsilon$.

Anyway, my point is I had developed the knack of thinking in Special Relativitistic terms by imagining myself comoving with one frame and applying the transformation equations to see how the world would look in another frame. I need to develop the corresponding skill for GR and I'm not getting it from any of the books I am using. Can someone recommend a book or web source that carefully develops this skill? Preferably with exercises and solutions.

If you happen to be familiar with tensor notation, there is an advanced concept called a "frame field basis" that will do something like what you want. But I suspect it's too advanced so I won't go into it except to say that it's there.

I would say that for starting to develop your intuition, the first step would be to regard the coordinates as "just labels", without any physical significance, and learn how to use the metric to compute the physical, "proper" distances and times between two nearby events.

The underlying philosophical concept is that rulers measure "proper" distances, and clocks measure "proper time", and that coordinates are just labels that don't measure anything in and of themselves, but are related to the physically measurable distances and times via the metric.

I'm not sure what the "best" text is to learn this, any textbook should explain the metric, though. I'm afraid my post assumed some familiarity with the metric already, if you're not familiar with the concept you'll have to learn what $g_{rr}$ and/or $g_{11}$ mean. It shouldn't be hard to find a text that mentions the notation, though.

 

[thedaybefore]

r is the speed of light times the amount of time it takes for the light to go from the center of mass to the measured location, where the light is assumed to be moving in a vacuum.

 

[Nugatory]

r is the speed of light times the amount of time it takes for the light to go from the center of mass to the measured location, where the light is assumed to be moving in a vacuum.

Not in Schwarzschild coordinates, which is what this thread is about.

 

[PeterDonis]

Not in Schwarzschild coordinates

And also not in a black hole spacetime in any coordinates, since it is impossible for light to go "from the center" to any other location.

 

[pervect]

Oh - something else I should have said. For the purpose of developing your intuition, you can create a new, local set of coordinates at a point, where in terms of the local coordinates, a small change in one of these re-defined local coordinates is the same as a small change in distance, though it only works in a limited area.

These "local coordinates" are basically what a frame field is about, minus all the math you'd need for a full treatment.

Recipies exist for creating such coordinates, i.e. Fermi Normal coordinates which are an example of coordinates which have this property (strictly speaking, it only happens at or around a specific point, though, the one you chose to create the coordinates around, typically regarded as an observer). So using Fermi Normal coordinates is a good way to support your physical intuition - if you can deal with the complexities.

Some of the unfortunate issues are that Fermi normal coordinates only cover a limited region of space-time, and typically there isn't any closed-form solution for equations to solve for them - you'll often wind up with series solutions.

For the Schwarzschld metric, because the metric is already diagonal, you can just offset and rescale the Schwarzschild coordinates to get your "local coordinates" by applying appropriate scale factors to make the diagonal metric a diagonal metric with unit magnitude coefficeints.