Schwarzschild metric and spherical symmetry

[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. :)