kev said:
Yes I do, and if the worldline is spacelike in Schwarzschild coordinates, then it is spacelike in Kruskal-Szekeres coordinates too.. precisely because it coordinate-independent.
And do you agree that the line A'-A'' in the diagram I posted above is timelike, not spacelike? Do you agree that a line of constant radial coordinate in Schwarzschild coordinates is timelike outside the horizon but spacelike inside, whereas a line of constant t-coordinate in Schwarzschild coordinates is spacelike outside the horizon but timelike inside? Finally, do you agree that a horizontal line of constant t coordinate in Kruskal coordinates is spacelike everywhere, and a vertical line in Kruskal coordinates is timelike everywhere? If you disagree with any of these you are simply confused, if you would actually do the math and check whether the line element ds^2 is positive or negative for a small increment on any of these lines, you'd see that all these statements are correct.
kev said:
The proper time of a falling observer is always positive as it falls in Schwarzschild coordinates and this is true above or below the horizon.
What do you mean by "always positive"? You can set the zero point of proper time anywhere you like. Perhaps you mean that proper time is always increasing with the t coordinate in Schwarzschild coordinates, but in this case you are mistaken, as long as we adopt the convention that there are no "local maxima" of proper time along a given worldline (meaning no points on a worldline where if you go in either direction away from a given point, the proper time decreases from its value at that point) then if the proper time is increasing with the t coordinate outside the horizon, it
must be decreasing with the t coordinate outside the horizon. The only way to have the proper time increase with t coordinate both inside and outside is to have such a "local maxima" on the object's worldline at the point it crosses the horizon (which is not represented in Schwarzschild coordinates, but geometrically it's a perfectly real point in the spacetime manifold, and it is represented in Kruskal-Szekeres coordinates).
kev said:
This means above and below the horizon the invariant interval is timelike but since timelike become spacelike below the horizon in Schwarzschild coordinates then the motion of the falling observer is spacelike below the horizon and by definition the observer is exceeding the speed of light.
No, you're completely confused here. First of all, just talking about the "invariant interval" ds^2 is meaningless without specifying what path you're calculating the invariant interval along--are you talking about a short interval along the worldline of the infalling particle? If so, then if you find it's timelike then it
is timelike, period, that's a coordinate-independent geometric fact and it has absolutely nothing to do with whether the t-coordinate of your chosen coordinate system is timelike or spacelike. You calculate the value of the ds^2 along a path by integrating the
line element along that path; the line element at every point is in turn determined by the metric, which is what encapsulates the actual geometric curvature of the spacetime. You can use any arbitrary coordinate system you like, including one where the t coordinate becomes spacelike or whatever, but the metric is
specifically adjusted to each possible coordinate system in such a way that the integral of ds^2 between two events along a given worldline will have the same value in every coordinate system, so that "path length" is a geometric fact which is
independent of your choice of coordinate system. Since the integral of ds^2 along a given path must be the same in ever possible coordinate system, then whether ds^2 is positive or negative must also be the same in every possible coordinate system, and
that is what determines whether the path is spacelike or timelike.
So the worldline is still timelike in the horizon, as I'm sure could be checked if you parametrized a path like A'-A'' in the diagram and then checked the value of the line element ds^2 for an infinitesimal variation in the parameter--it would have the same sign as the line element for an infinitesimal variation in the parameter for the path A-A' of the same particle outside the horizon. It is the line element, based on the metric, that tells you in a real geometric terms whether a worldline is spacelike or timelike--do you understand that this is the sole determinant of the geometry, that all other features of particular coordinate systems are irrelevant to the meaning of "spacelike" and "timelike"?
Here's the line element in Schwarzschild coordinates (copied from
here):
ds^2 = (1 - \frac{2m}{r}) c^2 dt^2 - \frac{1}{(1 - \frac{2m}{r})} dr^2 - r^2 ( d\theta^2 + r^2 sin^2 \theta d\phi^2 )
If you like, free feel to verify that for an infinitesimal increment along a line parallel to the t axis (meaning dr, d-theta and d-phi are all zero but dt is nonzero), if the line is at a value of r where r>2m (outside the event horizon), ds^2 will be positive (meaning the increment is timelike in a geometric sense), but if the line is at a value of r where r<2m (inside the event horizon), ds^2 will be negative (meaning it's spacelike). Likewise you could look at an infinitesimal increment along a line parallel to the r axis and verify that outside the horizon it's negative but inside the horizon it's positive.
Incidentally, if you look at the diagram below, also from MTW's
Gravitation, you can clearly see that the worldline of a particle lies inside the future light cone of events on its worldline both inside and outside the horizon, which is another way of thinking about the physical meaning of what it means for a worldline to be "timelike" (a physicist or mathematician could show that this definition is equivalent to the positive ds^2 notion, which is handy because in relativity light cones are very basic to the understanding of causality).
kev said:
A light ray at 45 degrees in a coordinate system does not indicate which way the light is going and this is difficult to determine because the proper time of a photon is zero. Proving that the path of a falling particle is timelike by comparing it to the supposed path of a photon whose direction you have not proven, proves nothing.
Huh? I wasn't talking about the "direction" of a photon, and timelike vs. spacelike has nothing to do with the direction that proper time is increasing on a given path, it's only about the set of points in spacetime that the path passes through. Again, in GR it's always true that a path which remains inside the light cones (future or past) of every point along the path is a timelike one.
kev said:
I don't atribute special holy significance to Schwarzschild coordinates but I do not think it is an unfortunate choice of coordinates as it often described. If something is coordinate-independent geometric fact in any given coordinate system then it true in any other coordinate system. Kruskal-Szekeres coordinates do not have any higher claim to the truth over the supposedly misguided Schwarzschild coordinates. They have equally valid.
I wonder why you attribute holy significance to the quirks of Kruskal-Szekeres coordinates?
Er, I don't. You don't seem to understand that timelike vs. spacelike is a geometric fact which is the same in all coordinate systems (once you've figured out the correct form of the metric in that coordinate system, you can use it to calculate the integral of ds^2 along a given path and your answer will be a coordinate-independent one), and that physicists have
checked this already and verified that Kruskal-Szekeres happen to have the nice property that worldlines which are timelike in the coordinate-independent sense always have a slope closer to the vertical than 45 degrees when represented in these coordinates. Of course it's the geometry that's fundamental, but because of this "nice" property of Kruskal-Szekeres coordinates they are less likely to
mislead you about the basic geometrical issues, as you have clearly been mislead by thinking about the "less nice" Schwarzschild coordinates.
kev said:
It means nothing get younger, not even twins in relativity. In SR and GR, the proper time can slow down relative to another clock, but never reverse. None of the twins ever ends up younger than started in any thought experiment.
Then like I said, if you want to avoid a "local maxima" in proper time which would correspond to time decreasing in both directions from the maxima, you
must define proper time in such a way that if proper time is increasing with Schwarzschild t coordinate on a given worldline outside the horizon, then it is decreasing with Schwarzschild t coordinate on the same worldline outside the horizon.
kev said:
Yep, it does not make sense, if objects really do continue to fall below the horizon but I agree that is that conventional wisdom.
You're being a crackpot if you keep talking as though these issues are a matter of "conventional wisdom" as opposed to objectively right or wrong and probably not hard to check mathematically for anyone well-versed in GR.
kev said:
Take a look at the attached world maps in various projections. The top chart of the first diagram shows a view of the globe (Orthographic azimuthal projection) from both sides and the blue lines represent the path of a polar orbiting satellite projected onto the surface of the earth. The lower chart of the first image is a Mercator type projection of the world map that flattens the globe into a flat rectangle. The satellite path is also shown on this projection. In the second image the Mercator projection is squeezed in the middle to produce an unusual but still still valid chart of the Earth. Notice the large white empty triangular spaces. There is no need to fill those empty spaces with a parallel virtual Earth to make the geodesics complete.
As on a
previous thread, you talk as though it's a matter of arbitrary fancy whether we think of different points in a coordinate map as being "the same" or "different", ignoring this thing called the
metric which tells you how distances between points
actually work in a geometric sense. The globe analogy is actually helpful in thinking about the meaning of the metric, though. Do you agree that on a globe, there is a geometric notion of distance along a given path, and that path with the shortest distance between any two points is a section of the
great circle that contains both points? Do you agree that the apparent length of a given path in a given coordinate representation will not in general match up well with the actual geometric length? For example, if you have two horizontal lines of apparent equal length in a Mercator projection, one at the equator and one near the pole, they will not actually have equal geometric length if you map the corresponding points on the globe, the one at the equator will have a greater geometric length. Well, the metric in Mercator coordinates would tell you the line element at every point, and it would tell you that a given increment of the horizontal coordinate near the equator corresponds to a larger increment of ds^2 then the same sized increment of the horizontal coordinate near the pole, and if you integrated ds^2 along a path using the right formula for the line element in Mercator coordinates, you'd get back the actual geometric length of that path on the surface of the globe. And if you can calculate the coordinate-independent distance along all possible paths, then you can define how "nearby" an arbitrary pair of points are by looking at the length of the shortest possible path between them.
Even if the coordinate system does something weird like stretching out a single geometric point on the globe into a line in the coordinate representation (which is exactly what happens with the set of events at the event horizon in Schwarzschild coordinates), you could figure out that this set of different coordinate points is "really" a single geometric point by nothing that the shortest path between such points would have a geometric length of zero. Similarly, in the case of your triangular Earth diagram with empty white spaces, if you define the metric on each triangular slice and then identify points on the diagonals of each slice as in the right-hand version of your diagram, you could show geometrically using the metric on either slice that all the points along the diagonal are really a single geometric point (the North Pole). In GR I think defining what it means for different coordinate points to be the "same" geometrically would be a bit more complicated, since you can have lightlike paths between distinct points which have ds^2 = 0 along the path, but just from reading physicists talking about coordinate representations vs. geometry I'm sure that physicists and mathematicians do have
some well-defined notion of what it means for points in spacetime with distinct coordinates to be the "same point" geometrically. And from reading physicists talk about the Kruskal diagram, they make clear that different points along the diagonal representing the event horizon actually
are geometrically distinct points in the spacetime manifold, so this is unlike your triangular mapping of the Earth where all the points on the diagonal are really just the single geometric point of the North Pole, and thus the blue line can really be a single continuous line geometrically even though it appears to cross the diagonal at two different places.
