By "arc lenghts of world paths" you refer to the trajectory of a photon - or another phisical object in a gravitational field, yes?
World paths are there if you are moving or not. Stand on the surface of a planetary body that is not rotating and is in the middle of an empty universe... you need not have any spatial displacement, yet there
is a displacement, and you feel the effect on that displacement as gravity. The actual effect on the displacement is on the time component, which is what is creating the accelerated reference frame.
I understand what you say about treating time dilation and space curvature, essentially, only as space curvature.
I'm sure it makes the calculations easier.
But, as far as understanding the effects of mass goes, it seems to simplify the picture. Consider:
In euclidean geometry, time/space is absolute.
Py picturing time dilation and space curvature ONLY as space curvature, one is essentially assuming the observer's time is absolute and mass only influences distance. BUT we know gravity influences BOTH space and time.
Well, you have to remember that while we are treating the units as the same thing, the signature of the metric is absolutely non-euclidean (non-Riemannian) in nature. Even a flat manifold with a metric like that is not going to act like euclidean space.
More importantly, separating time is the biggest problem I see with anyone attempting this stuff. If you want to simplify, remove one (or two) of the spatial components. Remember that gravity is an accelerated reference frame, that requires time to be in play. The spatial aspects point you in the right direction, but curved space isn't gravity without the time aspect.
When you try to solve the equations of relativity without time, there is no accelerated reference frame, which is what we see as a force, so a test particle at rest (floating in mid air) would stay at rest in such a situation. If it was moving, it might follow a curved path based on the curvature of space, but what makes the earth press upwards against your feet is cause by time.
Universal time negates gravity (in general relativity).
So - you can only calculate ds if you know the other 4 terms of the equation. They should be hard to measure - unless one can calculate dx, dy and dz if one knows the mass and position of the interacting objects.
That part isn't that hard... you can learn it in a standard classical differential geometry course. The key is that
dx,
dy and
dz can be based on what ever coordinate system you want. Better yet, ignoring them altogether and working with this stuff in a coordinate free way will yield better information about the global geometry.
For example (short, as it is getting off topic), let say you have a curve on the surface of a 2-manifold like a sphere. The curve is parameterized by arc length, and you want to know something about it relative to the surface. It is a 2 dimensional surface, so you pick some easy coordinates for the surface at the start of the curve... but maybe those coordinates aren't as nice the further along you go, so you pick some others. And all the while, the surface doesn't care how you are parameterizing it.
And that is important when you start to deal with space-time of general relativity as the surface isn't regular (does have a nice fixed metric on the manifold), instead it has a metric that is effected by mass-energy.
But, when an object reaches the event horizon, it doesn't turn into a shell. It adds matter only in a part of the singularity, but increases the radius of the event horizon around the entire singularity. And, because the object is frozen in time, it doesn't deform further.
Thus, the singularity isn't an uniform sphere of matter - there are holes in it, empty spaces.
Well, are we talking about the growing of the black hole from within a star or the effect of a small object as if falls towards a fully formed black hole?
You are right... it isn't uniform, but what extent of effect are you worried about. It is less than the effect that you have on the earth by jumping up and down. It effects things, but not so much that you're talking about anything we would discuss here.
Now if you want to talk about another star or black hole falling straight into a non-spinning black hole, then sure, the geometry will be significantly altered... but that is beyond what we are discussing.
Frankly, until someone has a really good grasp of the simplest model, there is no point in attempting to discuss the more complex ones. And the more complex ones are really complex (and I didn't get that far along in this stuff).
Also - I often heard that a singularity has an infinitely small volume. How can this be reconciled with this model for a black hole?
Don't believe everything that you hear?
The example I gave earlier in the thread about a spherical shell about the earth having a larger volume than the surface area would normally have enclosed if it was empty (or in euclidean space) works just as well here. When taken to the extremes of a black hole, the distance to the center of a spherical shell about one could be greater than the distance around the circumference of the sphere.
The reason for learning differential geometry slowly, starting out with classical differential geometry of surfaces, learning about the first and second fundamental forms, is really to build up an intuition about how this stuff works before you discard many of those tools for newer ones that are more powerful and work in more exotic situations. Sadly most physicist learn their differential geometry at the same time they are learning relativity.
I mean you already have a better grasp of this stuff than some people with degrees in physics based on our conversation so far. But far too many physics students will never see any of this stuff.
Would you tell us Shaw, how we take the metric you've described, and calculate a satellite's trajectory with it? You've talked about geodesics before, and I do remember (more or less) what those are, but only in spatial manifolds. I'm unsure how we express things like the velocity of a satellite, where we're dealing with a space-time manifold.
The important aspect of finding a geodesic on any manifold is applying the metric to find the covariant derivative (which for a geodesic path has to equal zero).
So a satellite might have an initial momentum vector (which is a form of arc length parameterization which is both space and time dependent) and by knowing the covariant derivative (which is dependent on the metric, which in turn is dependent on the mass-energy of the planet, which is also effected by the distance from the planet) you can calculate the path. But the initial coordinate system might need changing too (unless you were starting with a satellite already in a stable orbit).
And it is the most messy process there is, which is why general relativity has not replaced Newtonian/Lagrangian mechanics for most applications.
Other than really simple examples, it wasn't until the wide availability of computers that people spent a lot of time applying general relativity to phenomena. It was just too painful to do by hand.
Honestly, just looking at my Riemannian Geometry book, I think it would take me a few days to get back to the point were I could set up the covariant derivative for some abstract manifold with a given metric... which I'd need to recall before being able to remember the steps for doing it in general relativity.
It is actually done
with general relativity in
The Geometry of Physics... but I can't seem to find my copy off hand.
I'm not sure if any of that was helpful to anyone... so I apologize for my shortcomings here.
