Schwarzschild metric and spherical symmetry

[grav-universe]

In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.

 

[bcrowell]

http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.2

 

[grav-universe]

http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.2

Sorry, I still don't understand.

 

[grav-universe]

Let's say that according to a distant observer, there is an observer on a planet. Now I can see that since the distant observer is in nearly flat space, the planet will be spherical since it has spherical symmetry, the gravity acts symmetrically in all directions and so forth and shapes the planet that way, say. The distant observer measures the radius of the planet to be r.

Okay, so let's say that the surface observer measures the radius to be r' = r / a, inferred by a ruler in the radial direction that is contracted by a factor of 'a' according to the distant observer. Let's also say that the surface observer measures the planet to be spherical. That would require that measurements along the surface are also 1 / a greater than the distant observer measures. dθ r is just the measurement along the circumference, so that would be factored by 1 / a in the metric also. Basically then, according to the distant observer, the surface observer's ruler is contracted by a factor of 'a' in any direction. I don't see anything about spherical symmetry according to the distant observer that would rule this out. So by what logic isn't it permitted, requiring that the distant observer and surface observer measure the same distances along the surface?

 

[Muphrid]

Under what transformation are the angular parts of the line element supposed to be invariant? I don't understand the "transformation" that you refer to in the first post.

 

[Mentz114]

In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.

Spherical symmetry means that the field can only depend only on r and no other spatial coordinates.

 

[grav-universe]

Under what transformation are the angular parts of the line element supposed to be invariant? I don't understand the "transformation" that you refer to in the first post.

Well, the local surface observer measures

v_r'^2 + v_t'^2 = c^2

where v_r' is the locally measured radial speed of a photon and v_t' is the locally measured tangent speed of a photon, whereby

(dr' / dt')^2 + (dθ'^2 r'^2 / dt')^2 = c^2

dr'^2 + dθ'^2 r'^2 = c^2 dt'^2

We assume that due to spherical symmetry, the quantity dθ' r' remains unchanged, so dθ' r' = dθ r. That is the distance measured in the tangent direction, which this transformation says will be measured the same by both the local observer and the distant observer. This is the part I'm asking about. From there, dr / dr' and dt' / dt are each derived to be sqrt(1 - 2m/r), giving

[dr / sqrt(1 - 2m/r)]^2 + dθ^2 r^2 = c^2 [dt sqrt(1 - 2m/r)]^2

dr^2 / (1 - 2m/r) + dθ^2 r^2 = c^2 dt^2 (1 - 2m/r)

which is the metric.

 

[grav-universe]

Spherical symmetry means that the field can only depend only on r and no other spatial coordinates.

Right, that I see easily see. Due to the spherical symmetry, all measurements can only vary by some function of r. But how does that translate to dθ' r' = dθ r?

 

[bcrowell]

Spherical symmetry means that the field can only depend only on r and no other spatial coordinates.

That doesn't work, because the definition needs to be coordinate-independent. See the link in #2 for a coordinate-independent definition.

 

[pervect]

In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.

Very loosely speaking, spherical symmetry means that the metric isn't changed by rotations. If you'd prefer a formal definition, try Wald, who says

A spacetime is said to be spherically symmetric if its isometry group contans a subgroup isomoprhic to the group SO(3), and the orbits of this subgroup (i.e the collection of points resulting from the action of the subgroup on a given point) are two-dimensional spheres.

Note that the only assumption being made here is the existence of the spherical symmetry. It's not an "assumption" to say that the line element can be put into the form you quote (which is the same line element from flat space-time), it's a consequence of the initial assumption of spherical symmetry.

The basic idea in very loose and informal terms is that if you take the set of points of constant distance from the black hole they must form a set which is isomorphic to a sphere. Thus we can use the same line element that we use for a sphere in flat space-time for the subset of the manifold of "points of constant r" without loss of generality.

Note that we are attempting to do our best to take advantage of the underlying symmetry to find the simplest possible solution. So we are deliberately restricting our coordinate choices to correspond to the underlying symmetry in an effort to find the set of equations that is easiest to solve. If you want to make a different coordinate choice after you've solved the equations, all you need to do is to provide a transformation of coordinates - for example, consider isotropic coordinates.

 

[grav-universe]

Very loosely speaking, spherical symmetry means that the metric isn't changed by rotations. If you'd prefer a formal definition, try Wald, who says



Note that the only assumption being made here is the existence of the spherical symmetry. It's not an "assumption" to say that the line element can be put into the form you quote (which is the same line element from flat space-time), it's a consequence of the initial assumption of spherical symmetry.

The basic idea in very loose and informal terms is that if you take the set of points of constant distance from the black hole they must form a set which is isomorphic to a sphere. Thus we can use the same line element that we use for a sphere in flat space-time for the subset of the manifold of "points of constant r" without loss of generality.

Note that we are attempting to do our best to take advantage of the underlying symmetry to find the simplest possible solution. So we are deliberately restricting our coordinate choices to correspond to the underlying symmetry in an effort to find the set of equations that is easiest to solve. If you want to make a different coordinate choice after you've solved the equations, all you need to do is to provide a transformation of coordinates - for example, consider isotropic coordinates.

I still don't see how it would be a consequence of the spherical symmetry, but I think I see something else in your reply. Are you really saying that it is not an assumption at all, but a given? That the rest of the metric is derived according to what would be required to keep the transformation of dθ r constant, in order to keep the derivation as simple as possible? And then, after deriving it this way, we can then find for other coordinate systems in which dθ r will vary?

 

[pervect]

I still don't see how it would be a consequence of the spherical symmetry, but I think I see something else in your reply. Are you really saying that it is not an assumption at all, but a given? That the rest of the metric is derived according to what would be required to keep the transformation of dθ r constant, in order to keep the derivation as simple as possible? And then, after deriving it this way, we can then find for other coordinate systems in which dθ r will vary?

Metrics exist in which the coefficient of d$\theta$ is different, for instance the isotropic solutions.

$$ds^2 = -\left(\frac{1-M/2R}{1+M/2R}\right)^2\,dt^2 + \left(1+M/2R\right)^4 \, \left[dR^2 + R^2 \,d\theta^2 + R^2 sin^2 \theta d \phi^2\right]$$

The point is that we know the solution must have spherically symmetric shells. Furthermore, once we have a metric, the shells must have an area. It's a coordinate choice to say that we can define a radial coordinate such that area = 4 pi r^2.

It's not the only coordinate choice possible, but it's a convenient one for getting a solution.

 

[DrGreg]

In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.

I think maybe the question you are asking is this. Given a static spherically symmetric spacetime, why do we assume the metric is of the form$$ds^2 = A(r)^2 \, dt^2 - B(r)^2 \, dr^2 - r^2 \, d\Omega^2$$where$$d\Omega^2 = d\theta^2 + \sin^2 \theta \, d\phi^2$$rather than the more general form$$ds^2 = C(R)^2 \, dt^2 - D(R)^2 \, dR^2 - E(R)^2 \, d\Omega^2 \mbox{ ?}$$The answer is that if we have a metric of the second form, we can perform a change of variables to get the first form, viz.$$\begin{align} r &= E(R) \\ R &= E^{-1}(r) \\ A(r) &= C(R) \\ B(r) &= \frac{D}{\left(\frac{dE}{dR}\right)}(R) \end{align}$$Also, as pervect indicates, its also possible to perform a different change of variables to get isotropic coordinates of the form$$ds^2 = F(\rho)^2 \, dt^2 - G(\rho)^2 \left( d\rho^2 + \rho^2 \, d\Omega^2 \right)$$

 

[grav-universe]

I think maybe the question you are asking is this. Given a static spherically symmetric spacetime, why do we assume the metric is of the form$$ds^2 = A(r)^2 \, dt^2 - B(r)^2 \, dr^2 - r^2 \, d\Omega^2$$where$$d\Omega^2 = d\theta^2 + \sin^2 \theta \, d\phi^2$$rather than the more general form$$ds^2 = C(R)^2 \, dt^2 - D(R)^2 \, dR^2 - E(R)^2 \, d\Omega^2 \mbox{ ?}$$The answer is that if we have a metric of the second form, we can perform a change of variables to get the first form, viz.$$\begin{align} r &= E(R) \\ R &= E^{-1}(r) \\ A(r) &= C(R) \\ B(r) &= \frac{D}{\left(\frac{dE}{dR}\right)}(R) \end{align}$$Also, as pervect indicates, its also possible to perform a different change of variables to get isotropic coordinates of the form$$ds^2 = F(\rho)^2 \, dt^2 - G(\rho)^2 \left( d\rho^2 + \rho^2 \, d\Omega^2 \right)$$

Looks interesting. Could you explain the mathematical manipulations involved with what you posted here in more detail please?

 

[grav-universe]

Metrics exist in which the coefficient of d$\theta$ is different, for instance the isotropic solutions.

$$ds^2 = -\left(\frac{1-M/2R}{1+M/2R}\right)^2\,dt^2 + \left(1+M/2R\right)^4 \, \left[dR^2 + R^2 \,d\theta^2 + R^2 sin^2 \theta d \phi^2\right]$$

The point is that we know the solution must have spherically symmetric shells. Furthermore, once we have a metric, the shells must have an area. It's a coordinate choice to say that we can define a radial coordinate such that area = 4 pi r^2.

It's not the only coordinate choice possible, but it's a convenient one for getting a solution.

I had always thought of the Schwarzschild metric as the "true" solution. We could switch to other coordinate systems to make things easier in some way to find an answer to a problem, a "psuedo" coordinate system, but then we must switch that answer back to Schwarzschild to get the actual result. In SR, we can switch between coordinate systems because synchronization is not absolute, but we do not generally change ruler lengths and tick rates of clocks between two frames because it would not be "natural", not representing actual length contraction and time dilation, even though with different synchronizations, different length contractions and time dilations would be measured, but we would only transport identical clocks and rulers from the first frame to the second and then leave them "as is" and measure accordingly after the clocks have been synchronized some particular way while leaving their tick rates alone.

In GR, all hovering observers are in static space, and while there is time dilation between radiuses, all hovering observers agree that all clocks at a particular radius are synchronized to read the same, so there are no synchronization issues between radiuses. Well, if all local clocks are synchronized using the Einstein synchronization method, granted, but all GR coordinate systems apply that I think. So what we are really changing is distance measurements as far as I can tell. But as far as tangent distances being the same in Schwarzschild as a coordinate choice, if local observers at r were to measure the circumference around r by physically laying very small rulers end to end around the circumference, then transport the very same set of rulers to the radius s, the local observers there would physically measure the same distance 2 pi in relation to s around the circumference, so having the same ruler length in the tangent direction, or they will not. By comparing the local measurements at all radiuses this way, there would be only one function that relates them, and that would give us the length contraction of a ruler in the tangent direction at a particular radius. So we could not set that value to any coordinate choice we want in order to derive the metric if we want to find the "natural" length contraction of rulers in the tangent direction, could we?

 

[PAllen]

I had always thought of the Schwarzschild metric as the "true" solution. We could switch to other coordinate systems to make things easier in some way to find an answer to a problem, a "psuedo" coordinate system, but then we must switch that answer back to Schwarzschild to get the actual result. In SR, we can switch between coordinate systems because synchronization is not absolute, but we do not generally change ruler lengths and tick rates of clocks between two frames because it would not be "natural", not representing actual length contraction and time dilation, even though with different synchronizations, different length contractions and time dilations would be measured, but we would only transport identical clocks and rulers from the first frame to the second and then leave them "as is" and measure accordingly after the clocks have been synchronized some particular way while leaving their tick rates alone.

In GR, all hovering observers are in static space, and while there is time dilation between radiuses, all hovering observers agree that all clocks at a particular radius are synchronized to read the same, so there are no synchronization issues between radiuses. Well, if all local clocks are synchronized using the Einstein synchronization method, granted, but all GR coordinate systems apply that I think. So what we are really changing is distance measurements as far as I can tell. But as far as tangent distances being the same in Schwarzschild as a coordinate choice, if local observers at r were to measure the circumference around r by physically laying very small rulers end to end around the circumference, then transport the very same set of rulers to the radius s, the local observers there would physically measure the same distance 2 pi in relation to s around the circumference, so having the same ruler length in the tangent direction, or they will not. By comparing the local measurements at all radiuses this way, there would be only one function that relates them, and that would give us the length contraction of a ruler in the tangent direction at a particular radius. So we could not set that value to any coordinate choice we want in order to derive the metric if we want to find the "natural" length contraction of rulers in the tangent direction, could we?

This is the opposite of what GR is about. There is no preferred coordinate system, and all physical observables are coordinate independent. The modern preference is coordinate independent operations whenever feasible.

 

[grav-universe]

Oh wait, here's a surprisingly simple mathematical manipulation that would easily get rid of the function tied to the angle.

v_r'^2 + v_t'^2 = c^2

(dr' / dt')^2 + (dθ' r' / dt')^2 = c^2

dr'^2 + dθ'^2 r'^2 = c^2 dt'^2

f(r) c^2 dt^2 - g(r) dr^2 - h(r) dθ^2 r^2 = 0

where all three functions are time independent and angle independent, leaving them only as functions of r. From there we can gain

[f(r) / h(r)] c^2 dt^2 - [g(r) / h(r)] dr^2 - dθ^2 r^2 = 0

A(r) = f(r) / h(r) and B(r) = g(r) / h(r)

A(r) c^2 dt^2 - B(r) dr^2 - dθ^2 r^2 = 0

I don't know how I didn't see that before. But then, even after solving for A(r) and B(r) and gaining the actual metric by plugging in those values, we still have the problem of not knowing what h(r) really is to begin with. The overall metric must still work out using A(r) and B(r), and if we leave the metric as is, we have Schwarzschild, but how do we know if the tangent length contraction is really unity or something else?

 

[grav-universe]

This is the opposite of what GR is about. There is no preferred coordinate system, and all physical observables are coordinate independent. The modern preference is coordinate independent operations whenever feasible.

Okay, well, this is what I want to know. Since the distant observer is in very nearly flat space-time, I want to apply his coordinate system the same way as we would in Euclidean space. According to his Euclidean coordinate system, there is a gravitating body at a point in space and hovering observers at radiuses r and s from that point. Observers at r physically lay rulers around the circumference at r and measure C_r. The same set of rulers are then transported to s and observers there do the same thing, measuring C_s. What I want to know is what will the ratio (C_r / C_s) (s / r) be? And how do you determine that?

 

[PAllen]

Okay, well, this is what I want to know. Since the distant observer is in very nearly flat space-time, I want to apply his coordinate system the same way as we would in Euclidean space. According to his Euclidean coordinate system, there is a gravitating body at a point in space and hovering observers at radiuses r and s from that point. Observers at r physically lay rulers around the circumference at r and measure C_r. The same set of rulers are then transported to s and observers there do the same thing, measuring C_s. What I want to know is what will the ratio (C_r / C_s) (s / r) be? And how do you determine that?

There is no global inertial coordinates in GR. Depending on which feature of an inertial coordinates system you want to try to extend globally, you will get different global coordinate systems. How do you pick which to prefer? In SR, this problem does not exist - all 'reasonable' ways of building a global frame produce the same result. In GR, each 'feature' you want to emphasize produces different global coordinates.

For example, why favor hovering observers? They have proper acceleration, which is not at all like a global inertial frame. As for rulers, if one end is under different proper acceleration than the other, no material consistent with relativity will be unafffected, so the result depends on the material.

IMO, SC coordinates are one of less informative - they have a coordinate singularity that has no physical meaning; they are anisotropic, while no observer actually sees local anisotropy.

 

[PAllen]

Okay, well, this is what I want to know. Since the distant observer is in very nearly flat space-time, I want to apply his coordinate system the same way as we would in Euclidean space. According to his Euclidean coordinate system, there is a gravitating body at a point in space and hovering observers at radiuses r and s from that point. Observers at r physically lay rulers around the circumference at r and measure C_r. The same set of rulers are then transported to s and observers there do the same thing, measuring C_s. What I want to know is what will the ratio (C_r / C_s) (s / r) be? And how do you determine that?

As for your question, in SC coordinates, r and s are defined by the result of such a measurement. You cannot measure to the center, as that is a singularity. So r and s are defined as C_r/2π and C_s/2π. If you then want to measure the distance between r and s, it depends on who measures it and how. Some ways of measuring it will produce the proper distance integrated using SC coordinates along r for dt=0. For any given way of measuring distance between shells, any coordinates will produce the same answer for the distance - because this is a function of the measurement method, not the coordinates used.

 

[DrGreg]

Looks interesting. Could you explain the mathematical manipulations involved with what you posted here in more detail please?

It all follows almost directly from equating the first two equations for ds² in post #13, term by term.

$r \, d\Omega = E(R) \, d\Omega$ gives you $r = E(R)$.

$R = E^{-1}(r)$ is the inverse (assuming, of course, that E is invertible).

$A(r) \, dt = C(R) \, dt$ gives you $A(r) = C(R) = C(E^{-1}(r))$.

$B(r) \, dr = D(R) \, dR$ gives you
$$B(r) = D(R) \frac{dR}{dr} = \frac{D(R)}{\left( \frac{dr(R)}{dR} \right)} = \left. \frac{D}{\left(\frac{dE}{dR}\right)}\right|_{R = E^{-1}(r)}$$

 

[pervect]

Okay, well, this is what I want to know. Since the distant observer is in very nearly flat space-time, I want to apply his coordinate system the same way as we would in Euclidean space.

As others have remarked, you can't do this literally. And it's not really clear what features of the coordinate system you want to preserve, i.e. if we try to come "as close as we can", it's not clear what features you consider important, and what features you don't.

According to his Euclidean coordinate system, there is a gravitating body at a point in space and hovering observers at radiuses r and s from that point. Observers at r physically lay rulers around the circumference at r and measure C_r. The same set of rulers are then transported to s and observers there do the same thing, measuring C_s. What I want to know is what will the ratio (C_r / C_s) (s / r) be? And how do you determine that?

None of what you ask for - measuring distances - requires you to use a particular coordinate system. But, there is one thing you do have to specify.

Do you recall from special relativity that space and time intermix for different observers? I'm afraid I don't know your background.

In order to split space-time into space + time, you need to specify some notion of simultaneity, because of the fact that, speakig loosely, distances depend on one's state of motion in relativity - i.e. length contraction.

The most common notion is the shared notion of simultaneity of static observers, one who hover at constant distances and are all stationary relative to one another.

With this much set, distances are defined (loosely speaking again) as the shortest distance between two points in a surface of constant time, said surface being defined by the agreed-upon notion of simultaneity of the hoovering observers, for example.

In general you have to solve the geodesic equation, but for the simple examples the path you need to integrate over is obvious.

All you need to do then is to construct the proper curve, and measure its length using the metric.

I'll go into the math more later on, at the moment I don't have the time.

Meanwhile, as far as coordinate systems go, consider the equivalent problem on the Earth. You can't define a coordinate system or make a map of the Earth on a flat sheet of paper that preserves all distances. Distances on the surface of the Earth are defined, and you can make a flat map match up with any particular local neighborhood you want it to - but you can't cover the whole globe with a flat map that preserves distances.

 

[pervect]

OK - the way you integrate the length of a curve in general coordinates isn't much different from the way you'd do it, in for example, polar or spherical coordinates.

[1] It's easiest if you parametirize the curve, say for example r(s), theta(s), phi(s). THe derivatives of the curve will be the tangents to it, i.e. dr/ds, dthet/ds, dphi/ds.

You also have some line element, let say it's A(r)dr^2 + B(r)dtheta^2 + C(ri)D(theta)*dphi^2

For spherical coordinates A(r) would be 1, B(r)=r^2, C(r)=r^2, D(theta)=sin^2(theta) for example

Then you can find the length of the curve segment ds by taking

length^2 = A(r)*(dr/ds)^2 ds^2 + B(r)*(dtheta/ds)^2*ds^2 + C(r)D(theta) (dphi/ds)^2 ds^2

where dr/ds, dtheta/ds and dphi/ds are the tangents to the parameterized curve in [1].

so you just integrate sqrt(length^2) over ds.

And that's all there is to it.

Making sure that the curve is the shortest one mathematically is a bit trickier - you have to know about the geodesic equation. But you can often guess the geodesic correctly by inspection, for instance, you know to measure the circumference around great circles (r=phi=constant), or (theta=0,r=constant), and you know to measure radial distances radially (theta=phi=constant).

 

[jarod765]

Another way to think about it is the space-time can be foliated by submanifolds with a metric equivalent to a 2-sphere metric.

 

[grav-universe]

Okay, let me ask a different question then. I understand that distances can't actually be physically and directly measured in curved space using only the distant observer's own ruler, but only inferred. So I suppose one could say that the inferred distance depends upon the coordinate system and how distances relate according to the metric associated with that coordinate system. I am looking for a particular coordinate system in that case.

Let's say that in flat space-time, there are three distant stationary observers that are equidistant from each other. They can triangulate to find the location of a point at the center of their positions, the center of the triangle. They also locate that position relative to the fixed stars. They chart their own positions in relation to each other and the fixed stars by drawing a two dimensional map with their triangle lying upon the plane of the map.

Okay, so one of the distant observers, call her observer A, sends a probe to a distance r from that point at the center of the triangle, directly along the radial line between the point and observer A. In flat space-time, the distance r can be directly measured using observer A's own ruler. The position of the probe is also charted on the map.

But now, at some point and in some manner, a small spherical planet is moved so that its center lies directly upon the point at the center of the triangle. The distant observers can still triangulate the position of the point and the center of the planet and chart this position on another map, along with their own positions in relation to each other and the fixed stars. They lay this map on top of the first map and all of the positions almost perfectly coincide. The distant observers are not infinitely remote but at such a great distance that the difference between the maps is infinitesimal, so insignificant.

Okay, so now, the distant observers will employ the Schwarzschild coordinate system in relation to the gravity of the planet. Observer A sends a probe as she did before to a distance r from the center of the planet, the distance r that will correspond to her current coordinate system. She infers what that distance r will be in relation to herself and the other two distant observers, and to the fixed stars, and charts that position upon the second map as well, according to her current coordinate system, where her coordinate system says r will lie. She then places the second map over the first map again and compares positions. Will the charted position of the first probe at r in the flat space-time of the first map match the charted position of the second probe at r in the curved space-time of the second map? If not, what coordinate system would be required so that the positions correspond?

 

[grav-universe]

I think I may have figured out the answer to my question. As distant observers are so remote as to approach flat space-time, that space is nearly Euclidean and the position of the inferred distance r is the same in any coordinate system, so the charted positions on the maps for any coordinate system would always match according to the distant observers. However, the coordinate system they choose refers to what is measured locally at r, depending upon how they parameterize the curvature of the space in close proximity to the planet.

So locally, r for one coordinate system is not r for another coordinate system, meaning that a local hovering observer at r in Schwarzschild coordinates is not the same local hovering observer at r for another coordinate system. So if the distant observer A sends a probe to a distance r using Schwarzschild coordinates, that position would correspond on that map to the same position for the distance r that was charted in flat space-time. But if that distant observer then changes the coordinate system to Eddington's isotropic coordinates, say, then the probe is no longer at a distance r, but somewhere else according to her new coordinate system. But if observer A were to send a third probe to the distance r according to her new coordinate system, that would still correspond to the position of the first probe on the first map and the position of the second probe on the second map, even though the positions of the second and third probes, when charted together on the same map, are now different in relation to each other on both of the last two maps. Hopefully I explained that well enough. Does that make sense to you guys?

 

[pervect]

Okay, let me ask a different question then. I understand that distances can't actually be physically and directly measured in curved space using only the distant observer's own ruler, but only inferred. So I suppose one could say that the inferred distance depends upon the coordinate system and how distances relate according to the metric associated with that coordinate system. I am looking for a particular coordinate system in that case.

I suspect you've missed my point, probably because of an insufficient background in special relativity, which one really needs to learn (and learn well) before one tries to tackle general relativity.

The reason I suspect you've missed the point is you are still worried about coordinate systems in spite of my previous remarks.

Distances don't depend directly on the choice of coordinates, except insofar as the coordinates impose some sort of notion of simultaneity.

For instance, one can calculate distances in polar coordinates, or we can calculate them in cartesian coordinates. Or any other sort of coordinates, really.

I hope this is obvious, though I'm getting the niggling feeling that you might NOT know the details of how to compute distances in polar coordinates, and thus out of a sense of caution based on ignorance think that the coordinates do matter.

Coordinates are just a choice of labels - they don't really matter to anything physical.

I'll get back to simltaneity later. For now, I'll assume that you use the more-or-less usual notion of simutlaneity, that of static observers use, in which case you will find that your spatial slices (the slices of space-time of constant time are curved). This isn't any more - or less - scary and complicated than measuring distances on the curved surface of the Earth.

You might have seen the pictures of how the spatial slices are curved, they look like this:

http://en.wikipedia.org/wiki/File:Flamm.jpg

http://en.wikipedia.org/wiki/File:Flamm.jpg

[I'm not sure why the image isn't displaying? I'll put in the link...]

Note that there is no coordinate system for the surface of the Earth in which you measure distances by just subtracting coordinates. If that's your ultimate goal, it's doomed to fail in GR on the above surface (called a Flamm's paraboloid), just as it is doomed to fail on the surface of the Earth. To deal with measuring distances you'll need to learn techniques that are more complicated mathematically than just subtracting coordinates.

Going back to the notion of simultaneity:

The notion of simultaneity is important to defining distances. Length contraction in SR is the cannonical example.

See any of the mega threads about length contraction, and/or the simultaneity of light flashes, to get more details on how length contraction and the relativity of simultaneity intertwine.

Fundamentally, the Lorentz interval is what different observers agree on. They specifically do not agree on "distances" or times, in general. They DO agree on the Lorentz interval.

In GR specifying the notion of simultaneity is trickier than it is in SR. Definining a coordinate system, which defines surfaces of constant time by a constant "t" coordinate, is one way of defining the notion of simultaneity, but perhaps not the only one. But fundamentally it's not the coordinate choice that matters, what matters is the notion of "now", the notion of "simultaneity".

To wrap it up - distance is just the length of some curve. The devil is in the details - what curve, exactly?

 

[DrGreg]

http://en.wikipedia.org/wiki/File:Flamm.jpg

http://en.wikipedia.org/wiki/File:Flamm.jpg

[I'm not sure why the image isn't displaying? I'll put in the link...]

Attribution: AllenMcC. Wikimedia Commons CC-BY-SA-3.0

(It wasn't working because the URL you used was the Wikipedia description page, not the actual picture http://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Flamm.jpg/320px-Flamm.jpg)

 

[grav-universe]

I suspect you've missed my point, probably because of an insufficient background in special relativity, which one really needs to learn (and learn well) before one tries to tackle general relativity.

The reason I suspect you've missed the point is you are still worried about coordinate systems in spite of my previous remarks.

Distances don't depend directly on the choice of coordinates, except insofar as the coordinates impose some sort of notion of simultaneity.

For instance, one can calculate distances in polar coordinates, or we can calculate them in cartesian coordinates. Or any other sort of coordinates, really.

I hope this is obvious, though I'm getting the niggling feeling that you might NOT know the details of how to compute distances in polar coordinates, and thus out of a sense of caution based on ignorance think that the coordinates do matter.

Coordinates are just a choice of labels - they don't really matter to anything physical.

I'll get back to simltaneity later. For now, I'll assume that you use the more-or-less usual notion of simutlaneity, that of static observers use, in which case you will find that your spatial slices (the slices of space-time of constant time are curved). This isn't any more - or less - scary and complicated than measuring distances on the curved surface of the Earth.

You might have seen the pictures of how the spatial slices are curved, they look like this:

http://en.wikipedia.org/wiki/File:Flamm.jpg

http://en.wikipedia.org/wiki/File:Flamm.jpg

[I'm not sure why the image isn't displaying? I'll put in the link...]

Note that there is no coordinate system for the surface of the Earth in which you measure distances by just subtracting coordinates. If that's your ultimate goal, it's doomed to fail in GR on the above surface (called a Flamm's paraboloid), just as it is doomed to fail on the surface of the Earth. To deal with measuring distances you'll need to learn techniques that are more complicated mathematically than just subtracting coordinates.

Going back to the notion of simultaneity:

The notion of simultaneity is important to defining distances. Length contraction in SR is the cannonical example.

See any of the mega threads about length contraction, and/or the simultaneity of light flashes, to get more details on how length contraction and the relativity of simultaneity intertwine.

Fundamentally, the Lorentz interval is what different observers agree on. They specifically do not agree on "distances" or times, in general. They DO agree on the Lorentz interval.

In GR specifying the notion of simultaneity is trickier than it is in SR. Definining a coordinate system, which defines surfaces of constant time by a constant "t" coordinate, is one way of defining the notion of simultaneity, but perhaps not the only one. But fundamentally it's not the coordinate choice that matters, what matters is the notion of "now", the notion of "simultaneity".

To wrap it up - distance is just the length of some curve. The devil is in the details - what curve, exactly?

In SR, simultaneity determines what lengths we measure in moving frames because we must use clocks as well as rulers to measure the lengths of objects in motion. But for what I am considering in GR, all observers being considered are static hovering observers, all applying the Einstein simultaneity convention, so there are no simultaneity issues between them, only gravitational time dilation and length contraction.

But I think I have already figured out what I needed to know in post #26. I was mostly looking for a definition of r. As far as I can tell, space-time for distant observers is nearly flat, and they can just apply their usual definition of r to the position of a hovering observer in curved space, inferred by simply extending their near Euclidean coordinate system. r wouldn't make sense if it were anything else. But depending upon how they apply the parameters for the curved space stemming from the origin at the center of the gravitating mass, their coordinate system will vary, and the curvature at r will vary, so r in one coordinate system, corresponding to one set of hovering observers, will be different than r in another coordinate system, which corresponds to a different set of hovering observers. Basically, although the space is flat for the distant observers, they are re-scaling the local space at r.

It appears to me, then, that different coordinate systems are just re-scaling of the spatial dimensions from the origin outward, much in the same way that different synchronizations in SR re-scales the measured dimensions. We are allowed to do that in SR because synchronization is not absolute. I'm still not clear yet how this is allowed in GR, but I can see that it is feasible as long as the distances are only inferred, not absolute. So from this perspective, I can see that Schwarzschild coordinates scale distances such that small tangent distances are measured locally at any r the same as the distant observer infers for the tangent distances at r. And with Eddington's isotropic coordinates, distances are scaled locally such that the ratio of the tangent distance measured locally to the tangent distance measured by a distant observer is the same ratio between local and distant observers for a small radial distance measured, so that if the local hovering observer measures the same radial and tangent speeds of something, such as photons, measuring it isotropically, the distant observer will also, although at a different speed that depends upon the ratio of the local to distant distances measured and the ratio for the gravitational time dilation.

I'm still looking for an absolute way to measure ratios of distances, but using only inferred distance, I suppose that perhaps I will not find one. I could try the ruler distances, the distances found by physically laying rulers end to end along a path, as would be integrated using the length contraction for each local ruler, which must be the same regardless of the coordinate system used, but I'm assuming I will not find one that way either, that the ruler distance in any coordinate system is the same. But I want to know why, something tangible that will give a concrete reason different coordinate systems are allowed, the reasoning behind it that is as easily comprehendable as changing synchronizations in SR, where I can clearly see how the results can vary.

 

[PAllen]

In SR, simultaneity determines what lengths we measure in moving frames because we must use clocks as well as rulers to measure the lengths of objects in motion. But for what I am considering in GR, all observers being considered are static hovering observers, all applying the Einstein simultaneity convention, so there are no simultaneity issues between them, only gravitational time dilation and length contraction.

There are many things worth responding to in your post, but I'll start with this. Your family of static observers are no more privileged in GR than an arbitrary inertial frame in SR. Just as in SR, a different family of observers will make different measurements. In many ways, a family of inertial observers (free fall) is more analogous to flat spacetime than the hovering observers (in fact, a family of free fall observers will measure flat 3-space, while the hovering observers will not ). The family of free fallers may also use Einstein simultaneity convention, resulting in different slicing of spacetime and different proper distances than the hovering observers - even though both use the same simultaneity convention - just as in SR.

You need to let go of the idea that there is anything privileged about hovering observers or SC coordinates. Note also that if there is more than one significant gravitating body, there is no family of static hovering observers at all. In short, you will get nowhere understanding GR by focusing on static observers in SC geometry.

 

[pervect]

In SR, simultaneity determines what lengths we measure in moving frames because we must use clocks as well as rulers to measure the lengths of objects in motion.

I"m really not sure what you're trying to say here. But it sounds like you do know that distances depend on the state of motion, even though I don't quite follow your remarks here.

But for what I am considering in GR, all observers being considered are static hovering observers, all applying the Einstein simultaneity convention, so there are no simultaneity issues between them, only gravitational time dilation and length contraction.

I'll interpret this as saying that you are interested in what distances the static observers will measure, which is legitimate question.

But I think I have already figured out what I needed to know in post #26. I was mostly looking for a definition of r.

Unfortunately, I can't make heads or tails of your post #26. This is a bad sign.

To measure a distance in GR, you can imagine setting up a chain of observers along the curve you are going to measure the distance along. In your case, static observers. Each (static) observer measures the distance to the next in the chain, using his local clocks and rulers. Then you add up all these local measurements and call it "the distance".

This is the standard procedure used in , for instance, cosmology.

You are correct in noting that there is a scaling factor between local times and distances and coordinate times an distances. I'm getting the impression from what little I can follow from #26 that you don't realize that the distance is measured by adding up all the local distances. Perhaps you are adding up coordinate deltas? But coordinate deltas are not distances!

In other words, if you want to measure the length of a curve, in principle, you can imagine a bunch of observers, each with a local ruler, on the curve - (there really isn't any such thing as a remote ruler!), all of whom measures the distance to the next observer in the chain.

When the observers are all in a straight line from the source to the destination, the sum of all these local measurements is the distance, the length of the shortest curve connecting the two points. The result is independent of any particular coordinate system you choose, just as points that are six inches apart are six inches apart regardless of whether you use rectangular coordinates or polar coordinates.

 

[pervect]

I was thinking about this some more, and I came up with what is probably the simplest explanation I can think of, based mainly on a reference which is both very good and available on the internet for a more detailed exposition than I care to write.

Exploring Black holes - chapter 2

Nothing is more distressing on first contact with the idea of curved spacetime
than the fear that every simple means of measurement has lost its
power in this unfamiliar context. One thinks of oneself as confronted with
the task of measuring the shape of a gigantic and fantastically sculptured
iceberg as one stands with a meterstick in a tossing rowboat on the surface
of a heaving ocean.

Were it the rowboat itself whose shape were to be measured, the procedure
would be simple enough (Figure 1). Draw it up on shore, turn it
upside down, and lightly drive in nails at strategic points here and there
on the surface. The measurement of distances from nail to nail would
record and reveal the shape of the surface. Using only the table of these
distances between each nail and other nearby nails, someone else can
reconstruct the shape of the rowboat.

There's a little diagram , a sketch of the "rowboat", with all the distances measured. The distances are measured locally - they are the distances one measures/ would measure with a physical ruler transported to that location.

Now, we can imagine applying this process , exactly as described, to the r, theta plane of the Schwarzschild geometry - i.e. we will take a slice of constant Schwarzschild time, and of constant phi.

When we go through this process to reconstruct the "shape" of this particular slice of space-time, we get the well known result of Flamm's paraboliod. (This particular result isn't in Taylor's exposition, but it's in the wikipedia).

And that's really all there is to it!

We know that this curved spatial geometry models all the distances is the r-theta plane measured by local observers with there little, local rulers. We might need one more thing - that is the mapping of events from this diagram to the r, theta plane of Schwarzschild space-time.

The mapping is pretty simple - look at Flamm's parabaloid in a cyindrical coordinate system, i.e. r, theta, and z. r and theta will correspond to the Schwarzshild r and theta coordinates. z doesn't correspond to anything physically measurable, but it makes all the distances work out.

The distances measured between any two events on the curved surface of Flamm's paraboloid will be the same distances that observers measure with their local rulers.

I.e.there is a 1:1 corresondence between points on the Flamm's paraboliod, and points on the r-theta space-time slice through the black hole, and this 1:1 correspondence preserves the distances between points. So if you want to get the distance in the Schwarzschild space-time, you can just measure it on the paraboloid.

 

[grav-universe]

Thank you very much for your help, guys. I now have a particular GR coordinate system in mind and I have started a new thread "shrinking event horizon to point singularity" for responses to this new subject. Thanks again. :)