General Relativity Basics: The Principle of Equivalence

[PeterDonis]

We may be talking in different contexts, but I want to make sure you realize: no event has a unique proper time.

I agree.

I see how our concepts differ about events. You think of events remaining in place, while you progress forward in time. I think of events drifting from the future into the present toward the past, while I remain in the present.

I don't have a problem with either of these points of view; however, I'm not sure the first one accurately captures the one I've been implicitly using. The point of view I've been implicitly using is that *nothing moves*: spacetime, all of it, just *is*, as a four-dimensional geometric object. When we make statements about events, worldlines, etc., we're making geometric statements about geometric objects within this overall geometric object.

True, the worldlines of the falling objects do cross the line x = c t, but always at a point t>0. That event of crossing will remain forever in the future for the observer on the rocket. It will always be something that has not happened yet.

Yes, that's one way of putting it. However, that way of putting it tempts you, as I said before, to say things like...

Whether something that happens in the future actually "exists" is a metaphysics question

...which is *not* justified, in my opinion. The event of crossing is only "forever in the future" for observers on the rocket; it does *not* remain forever in the future for inertial observers. That's not a metaphysical question; it's a direct logical consequence of the construction of the spacetime, as a geometric object.

Furthermore, it's a logical consequence that is accessible to the observers on the rocket; even though they can't themselves "see" the event of the free-falling observer crossing the horizon, they can tell that there *must* be such an event (and further events after it along the free-falling worldline). How? By integrating the proper time along the free-falling worldline (using *their* metric), and realizing that, even as their "accelerated" coordinate time goes to infinity, the worldline of the free-falling observer only contains a finite amount of proper time (because the integral converges to a finite value as Rindler coordinate time t goes to infinity). But worldlines can't just stop at a finite proper time; more precisely, the Rindler observers can find no physical *reason*, even in their frame, why the free-falling worldline would just stop at a finite proper time. It's not just that there's no catastrophic event there, no explosion, no laser blast blowing the free-falling observer to bits, no "wall" for the observer to run into. Even if there were such an event, the debris from it would have to go *somewhere*--there would be other worldlines to the future of the catastrophic event. In other words, physically, the free-falling worldline *has to have a future* past the last point the rocket observers can see; there must be a further portion of the free-falling worldline that they can't observe.

 

[Passionflower]

We may be talking in different contexts, but I want to make sure you realize: no event has a unique proper time.

I agree.

Peter I am not sure what you agree on. As far as I understand it an event is a point in spacetime, points don't have proper time or any other time for that matter. Time is a timelike interval between two events. Proper time is such an interval for a real clock traversing a path in spacetime.

 

[PeterDonis]

Peter I am not sure what you agree on. As far as I understand it an event is a point in spacetime, points don't have proper time or any other time for that matter. Time is a timelike interval between two events. Proper time is such an interval for a real clock traversing a path in spacetime.

The term "proper time" can be used that way, yes, but it can also be used to denote the single *value* of the "proper time" used to parametrize a worldline, at a given event on that worldline. But since any event will lie on multiple worldlines, each of which could assign a different value of their "proper time" parameter to the event, no event can have a single, unique value of the "proper time" parameter. That's what I was agreeing to, and I'm pretty sure that's what JDoolin meant (he'll correct me, I'm sure, if he meant something else).

 

[yuiop]

But worldlines can't just stop at a finite proper time; more precisely, the Rindler observers can find no physical *reason*, even in their frame, why the free-falling worldline would just stop at a finite proper time. It's not just that there's no catastrophic event there, no explosion, no laser blast blowing the free-falling observer to bits, no "wall" for the observer to run into.

Would you agree that this exposes a limitation of using Rindler coordinates as an analogy of what happens in the case of real black hole, because we know in that case, there is a real singularity at the centre of the black hole (where the worldline of a free falling observer does stop at a finite proper time) and yet there is no indication of this real singularity in Rindler coordinates. In Rindler coordinates the free falling observer continues to fall for infinite proper time after crossing the horizon so if we use Rindler coordinates to analyse what happens in the case of a real black hole we would be forced to conclude that the free falling observer never arrives at the real central singularity in finite coordinate or proper time.

 

[PeterDonis]

Would you agree that this exposes a limitation of using Rindler coordinates as an analogy of what happens in the case of real black hole, because we know in that case, there is a real singularity at the centre of the black hole (where the worldline of a free falling observer does stop at a finite proper time) and yet there is no indication of this real singularity in Rindler coordinates.

Yes; in fact, the case is even stronger than you suggest, because this...

In Rindler coordinates the free falling observer continues to fall for infinite proper time after crossing the horizon so if we use Rindler coordinates to analyse what happens in the case of a real black hole we would be forced to conclude that the free falling observer never arrives at the real central singularity in finite coordinate or proper time.

is false as it stands; the correct way to state it is that Rindler coordinates do not even *cover* the portion of spacetime "below" the horizon--in other words, we can't say that "In Rindler coordinates the free falling observer continues to fall...after crossing the horizon", because we can't even *assign* coordinates at all, not even infinite ones, to any events below the horizon in Rindler coordinates, even though we can show that the events themselves must be there.

Also, strictly speaking, we wouldn't use "Rindler coordinates" in the case of a real black hole; we'd use Schwarzschild coordinates, which are analogous to Rindler coordinates. But those coordinates have exactly the same limitation: they don't cover the portion of spacetime at or below the horizon, so they can't be used to analyze what happens there.

(Technically, the "exterior Schwarzschild coordinates", with r > 2M, are the ones analogous to Rindler coordinates; there are also "interior Schwarzschild coordinates", with r < 2M, which *do* cover the portion below the horizon, but do *not* cover the portion above. Popular books often obscure this by using the term "Schwarzschild coordinates", without qualification, to refer to both sets of coordinates as though they were one single coordinate system, but they're not; they're two separate ones, which can't be connected into one because they are both singular at the horizon, r = 2M, which acts as an impassable "barrier", so to speak, between them.)

 

[Passionflower]

In this context for an interesting lecture about 'Rindler and Schwarzschild' see:

Lecture 12 of Leonard Susskind's Modern Physics concentrating on General Relativity. Recorded December 9, 2008 at Stanford University: http://academicearth.org/lectures/modern-physics-general-relativity-12