Understanding maximally extended Schwarzschild solution

[JesseM]

Your right and I expressed myself very badly there. What I was trying to get at with the twins experiment, is that proper time can slow down in a real sense relative to other clocks with relative motion (or at a different gravitational potentials), as can be verified by bringing the clocks back together again.

Are you talking about a clock being objectively slowed relative to another at a particular moment? If so that doesn't make sense, at any given point on a twin's worldline we can pick different inertial frames which disagree about whether his clock is ticking slower or faster than his brother's clock at that moment, and yet every inertial frame will agree on the total time elapsed on each clock (the proper time) when they reunite. I posted an analogy with paths in 2D Euclidean geometry here if you're interested.

Given that proper time can vary relative to other clocks, I have to question why proper time is always given priority over coordinate time when they appear to be in conflict. For example if a particle is dropped into a black hole we could calculate that it takes for 10 minutes of the particle's proper time to arrive at the event horizon and a total of 15 minutes of its proper time to arrive at the central singularity, but is that what really happens? Does the particle actually arrive at the central singularity?

Yes, GR is a geometric theory, coordinate systems in curved spacetime have no more significance than coordinate systems on curved 2D surfaces like the surface of a sphere. No doubt we could come up with a coordinate system on a sphere where the coordinate distance between the equator and the North Pole is infinite, would this cause you to worry that a traveler might never actually reach the North Pole?

Speaking of coordinate systems, I was thinking a little more about the example of the North Pole being expanded into a line in a Mercator-like projection, and how you could show even in this coordinate system that the North Pole is "really" a geometric point. The basic idea is that every coordinate system on a surface is associated with a metric that defines a geometric, coordinate-independent notion of distance along a curve on that surface (ds^2 in spacetime, but just ordinary spatial distance when we're dealing with a metric on a 2D space). So, take any two points on the top edge of the map that are on the North Pole, draw a line between them, and calculate the length of this path using the metric. You should find that the length of the path is actually 0, which on a 2D spatial manifold shows that these points with different coordinate representations are actually the same geometric point.

Defining what it means for points with different coordinate representations to "really" be the same geometric point in a spacetime manifold is a little trickier, because different points on the worldline of a light beam are genuinely different geometrically, and yet the integral of ds^2 along a light beam worldline is always zero. But I'm sure physicists do have some definition. One guess I had about this is that two different points in a coordinate system could be defined to be the same geometric point if it is impossible to find a purely spacelike or purely timelike curve with nonzero ds^2 going between them, such that the curve has no sharp "kinks" in it (if you allow sharp kinks, it would be possible to find a nonzero timelike worldline connecting a single geometric point to itself--just draw two different timelike worldlines emanating from that point, wordlines which cross at some other point, then define a new closed worldline that travels 'up' the first worldline until it reaches the crossing point, then travels back 'down' the second worldline until it returns to the original point).

edit: actually I realized my hypothesized definition doesn't work, because I don't think you can find a purely spacelike or timelike path of nonzero length to connect two points on the worldline of a light ray, at least not if you're considering paths in only one space dimension, and with more space dimensions you can easily find a spacelike path connecting a single geometric point to itself, just consider a loop that stays within a single plane of simultaneity. So my proposal was obviously on the wrong track, but like I said, I'm sure physicists have some definition.

Next, I have to question why its OK for a particle (or observer) to travel backwards in coordinate time as long as its/his/her proper time is advancing?

For any particle whatsoever, even one in flat SR spacetime, if you define the direction of increasing proper time along its worldline (which itself is a matter of convention rather than a physical fact, as I said earlier), you can always find some coordinate system where proper time is increasing as coordinate time is decreasing. Do you see this as less problematic because you somehow think Schwarzschild coordinates are privileged over any other arbitrary coordinate system? Only coordinate-independent geometric facts are really physical, I keep saying this and yet you seem to keep ignoring it.

I am very aware that my point of view is not the textbook point of view, but I am asking if there is the possibility of other equally valid physical interpretations of the equations of General Relativity?

No, I don't think any physicist would see any validity in an "interpretation" that converted GR into a non-geometric theory where some particular coordinate system was privileged over others. If you want to advance such a crazy notion you should go to the independent research forum, not here.

 

[Hurkyl]

Going back to my first question, I was informed (through a different channel) that the universal property of the maximally extended Schwarzschild solution is that it is the (real) analytic continuation of the external Schwarzschild chart.

If I recall the theory correctly, that means every analytic extension of the exterior Schwarzschild chart is, indeed, a quotient of a subspace of the maximally extended one.


Of course, this classification wouldn't apply to smooth extensions.

 

[DrGreg]

Removing the Schwarzschild coordinate singularity

kev, and those who have been responding to him, might like to use the new Removing the Schwarzschild coordinate singularity[/color] thread I have created, as a place to further discuss the issues raised without diverting from the main topic of this thread.