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Feb23-12, 03:52 PM
Sci Advisor
PF Gold
P: 6,250
There are actually two separate issues here, and it may be easier to see how they are resolved if you keep them separate.

The first issue is the "when" issue, which is a matter of correctly understanding what the Schwarzschild spacetime geometry is telling you. This issue has nothing to do with entropy; you can ask the central question you are asking--when does an infalling object cross the horizon?--without bringing in entropy at all.

The best way to understand this, IMO, is to look at how Schwarzschild spacetime, and the worldline of an infalling object, look in Kruskal coordinates. Take a look at Figure 9 on this page:

The horizon (or at least the portion of it we're interested in) is the 45-degree line going up and to the right between regions I and II of the diagram. The worldline of the infalling object crosses that line, the horizon, at a definite event, which occurs at a definite, finite "time" according to the infalling object's clock. So there is a definite "when" that the object crosses the horizon from the object's own point of view.

Now take a look at this page from Wikipedia, and in particular the diagram in yellow part way down the page:

You will notice regions I and II again, but now two sets of dotted curves have been added. These are curves of constant Schwarzschild time coordinate t (the straight lines radiating out from the center) and constant Schwarzschild time coordinate r (the hyperbolas). Notice that in region I, the lines of constant t approach the horizon but never reach it; there is no finite value of t that corresponds to the actual horizon line (the future horizon, the part we're interested in, can be thought of as t = + infinity).

This means that (speaking somewhat loosely), from the point of view of an observer who remains outside the horizon forever, there is no finite "time" at which the infalling observer crosses the horizon. However, the reason for this is not that there is any mystery or strangeness about that event (the infalling observer crossing the horizon); it is simply that the most "natural" assignment of a time coordinate to events, for the observer outside the horizon, only covers region I; it does not cover region II. From the standpoint of the spacetime as a whole, this "time" coordinate is clearly limited (for one thing, all of the lines of constant Schwarzschild t intersect each other at the center of the diagram!).

The upshot of all this is that the definition of "now" depends on how you choose to draw coordinate lines on the spacetime. A "now" can in principle be any spacelike line (on the Kruskal diagram, this is any line whose slope is always less than 45 degrees, i.e., more horizontal than vertical). But *which* spacelike line (or set of lines) you choose makes a difference: there is no "Schwarzschild now" that passes through the event where the infalling observer crosses the horizon, but there are plenty of other perfectly good "now" lines that do.

I'll follow up with discussion of the second issue, entropy, in a separate post.