General Relativity Basics: The Principle of Equivalence

[PeterDonis]

They are both legitimate ways of explaining what is going on. The light coming from the Rindler Horizon can't ascend from the horizon, and the light and objects falling toward the rindler horizon approach it asymptotically, but never reach it. Objects beyond the rindler horizon, mathematically also move toward the rindler horizon, but their clocks are mathematically going backwards.

In the view of the rocket, the event at (x=0,t=0) is an event that cannot be crossed because nothing can happen there.

If you use the term "horizon" to refer only to the single *event* at x = 0, t = 0, then I can see why you're describing things this way. Obviously a free-falling observer who drops off the rocket at event x = s_0, t = 0 will never pass through the event x = 0, t = 0. However, that observer *will* cross the line x = t (at x = s_0, t = s_0), so if the term "horizon" refers to that entire line (which is the standard usage, and also makes more sense--see below), it is simply false to say that "objects falling toward the rindler horizon approach it asymptotically, but never reach it". The objects' worldlines *do* reach the horizon, and pass beyond it. That's the fact. It is also a fact that observers in the rocket will never *see* that portion of a free-falling object's worldline, but that doesn't mean that portion of the worldline doesn't exist.

Why does it make more sense to use the term "horizon" to refer to the *line* x = t, instead of just the single *event* x = 0, t = 0? (Actually, if we include the other two space dimensions, the horizon is a null *surface* which includes the line x = t.) Because the hyperbolas along which the Rindler observers move, x^{2} - t^{2} = s_{0}^{2}, asymptote to the line, *not* the single point. Put another way, it's not just the single event x = 0, t = 0 that can't send light signals to any of the Rindler observers; it's *any* event on the line x = t (or in the region "above" it--i.e., with t > x--so the line x = t is the *boundary* of the region, which is what "horizon" means in standard usage). The only reason to single out the particular event x = 0, t = 0 is that it is the "pivot point", so to speak, of all the lines of simultaneity for the Rindler observers as they move along their hyperbolas. But since, as you note, nothing actually *happens* at that event, singling it out tempts you to ignore real stuff that *does* happen, like free-falling observers crossing the line x = t.

 

[Mentz114]

What PeterDonis says above is correct. At the risk of repeating this, have a look at Greg Egan's artricle, which has useful diagrams.

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

 

[JDoolin]

If you use the term "horizon" to refer only to the single *event* at x = 0, t = 0, then I can see why you're describing things this way. Obviously a free-falling observer who drops off the rocket at event x = s_0, t = 0 will never pass through the event x = 0, t = 0. However, that observer *will* cross the line x = t (at x = s_0, t = s_0), so if the term "horizon" refers to that entire line (which is the standard usage, and also makes more sense--see below), it is simply false to say that "objects falling toward the rindler horizon approach it asymptotically, but never reach it". The objects' worldlines *do* reach the horizon, and pass beyond it. That's the fact. It is also a fact that observers in the rocket will never *see* that portion of a free-falling object's worldline, but that doesn't mean that portion of the worldline doesn't exist.

Why does it make more sense to use the term "horizon" to refer to the *line* x = t, instead of just the single *event* x = 0, t = 0? (Actually, if we include the other two space dimensions, the horizon is a null *surface* which includes the line x = t.) Because the hyperbolas along which the Rindler observers move, x^{2} - t^{2} = s_{0}^{2}, asymptote to the line, *not* the single point. Put another way, it's not just the single event x = 0, t = 0 that can't send light signals to any of the Rindler observers; it's *any* event on the line x = t (or in the region "above" it--i.e., with t > x--so the line x = t is the *boundary* of the region, which is what "horizon" means in standard usage). The only reason to single out the particular event x = 0, t = 0 is that it is the "pivot point", so to speak, of all the lines of simultaneity for the Rindler observers as they move along their hyperbolas. But since, as you note, nothing actually *happens* at that event, singling it out tempts you to ignore real stuff that *does* happen, like free-falling observers crossing the line x = t.

There are two different things we're talking about here. The events, which never cross the line x=ct (or x=-ct for that matter), and the objects, which never cross the point x=0.

 

[PeterDonis]

There are two different things we're talking about here. The events, which never cross the line x=ct (or x=-ct for that matter), and the objects, which never cross the point x=0.

Huh? Events don't "cross" anything--they're single points in spacetime. Objects travel on worldlines, which are ordered sequences of events (the usual ordering/parametrization of the worldline is given by the "proper time" according to the object traveling on that worldline; each event on the worldline has its own unique proper time). The point where a free-falling object crosses the horizon, the line x = ct, is an event just like any other on that object's worldline.

What I've just given are the standard definitions of "event" and the other terms. If you're using different definitions in the statement I just quoted above, please state them explicitly, because I don't understand the distinction you're making.

 

[JDoolin]

Huh? Events don't "cross" anything--they're single points in spacetime. Objects travel on worldlines, which are ordered sequences of events (the usual ordering/parametrization of the worldline is given by the "proper time" according to the object traveling on that worldline; each event on the worldline has its own unique proper time). The point where a free-falling object crosses the horizon, the line x = ct, is an event just like any other on that object's worldline.

What I've just given are the standard definitions of "event" and the other terms. If you're using different definitions in the statement I just quoted above, please state them explicitly, because I don't understand the distinction you're making.

We may be talking in different contexts, but I want to make sure you realize: no event has a unique proper time. As you say, "proper time" is determined by an object traveling on "that world line" but since there is no unique worldline through any event, there can be no unique proper time at that event.. Under the context we have been discussing there are two proper times assigned to each event. We can assign a one proper time to each event according to the reading on the rocket's clocks. We could also assign a proper time by using clocks stationary in the inertial reference frame. But the proper time of an event is not uniquely, since it depends on the prior motion of the clocks arriving at that event.

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.

The events also move according to the rules of Special Relativity--Lorentz Transformations. It is under this context, the people aboard the rocket would model events approaching the line x=c t and x=-c t from either side, but not crossing.

The objects' worldlines *do* reach the horizon, and pass beyond it. That's the fact. It is also a fact that observers in the rocket will never *see* that portion of a free-falling object's worldline, but that doesn't mean that portion of the worldline doesn't exist.

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. (Whether something that happens in the future actually "exists" is a metaphysics question, but in this case of perfectly predictable motion, the worldlines can be modeled, of course.)

 

[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