A map is not the territory. A chart is not the manifold.

**Farsight**'s argument with

**Vorpal**,

**sol invictus**,

**ben m** *et cetera* appears to involve a misunderstanding of the most basic facts of differential geometry. To understand what's going on here, it helps to understand the relationship between coordinate systems and differentiable manifolds.

(For a nice tutorial on that subject, I recommend

Jason J Sharples. Coordinate transformations and metric extension: a rebuttal to the relativistic claims of Stephen J. Crothers. *Progress in Physics*, 2010, Volume 1 (of the volumes for 2010), pages L1-L6. Online at http://www.ptep-online.com/index_fil...0/PP-20-L1.PDF

Stephen J Crothers is an advocate of Electric Universe pseudoscience who claims to have "definitive proof that black holes do not exist" and is really annoyed that so many people have found mistakes in that alleged proof.)

By definition, an n-dimensional

*topological manifold* is a paracompact Hausdorff space that's

*locally Euclidean*: Every point of the space is contained within an open set that's homeomorphic to an open subset of R

^{n}.

Those homeomorphisms are called

*charts*. Physicists generally refer to those charts as coordinate systems. As I'll explain below, the physicists' terminology is confusing, and that particular confusion may explain part of what we've seen in this thread.

By definition, a

*differentiable manifold* consists of a topological manifold plus an

*atlas*, which is a set of charts that cover the entire manifold and compose (in a certain way) to form sufficiently smooth diffeomorphisms.

The set of all possible charts (coordinate systems) for a manifold is called its complete atlas. The complete atlas contains infinitely many charts (coordinate systems). All of those charts are valid coordinate systems.

General relativity is a difficult subject, and there are many ways to misunderstand it. One of the more common mistakes is to treat some particular chart (coordinate system) as though it were somehow more correct than other charts (coordinate systems), and to treat the open set on which that particular chart (coordinate system) is defined as though it were the entire manifold.

With all but the simplest differentiable manifolds, there is no single chart (coordinate system) that's defined on the entire manifold. Consider, for example, the surface of the earth (which is a 2-sphere). There is no chart that covers the entire surface of the earth, because the 2-sphere is not homeomorphic to any open subset of 2-dimensional Euclidean space.

That example shows why it's confusing to refer to charts as coordinate systems. Spherical coordinates are usually thought of as a coordinate system that can deal with the entire 2-sphere, but spherical coordinates do not give us a chart that covers the entire 2-sphere because they are not bijective at the north pole: At zero inclination (90 degrees north latitude), all possible values for the azimuth (longitude) refer to the same point.

That's a simple example of a coordinate singularity. It's not a real singularity, because the vicinity of the north pole looks just like the vicinity of any other point on the 2-sphere. It's a mistake to think coordinate singularities imply any weird topology in the manifold itself. What a coordinate singularity does imply is that you'll have to switch to a different coordinate system if you want to calculate what's going on at the coordinate singularity and beyond.

With those basics in mind, let's look back at some of what's been said in this thread.

Originally Posted by

**Farsight**
And then sol said this?

*"For an external observer, yes. But for the person falling in, those time intervals are finite".*

The person falling in ends up with a stopped clock. It's stopped, so his "finite" intervals take forever. So they never ever happen.

No. We're talking about two different charts here: There's the chart (coordinate system) that maps the external observer's neighborhood onto an open subset of R

^{4}, and there's the chart (coordinate system) that maps the infalling person's neighborhood onto an open subset of R

^{4}. The external observer's chart stops at the event horizon; that's a coordinate singularity. The infalling observer's chart doesn't stop at the event horizon.

The external observer's coordinate singularity at the event horizon means the external observer calculates that the infalling person's clock stops at the event horizon. That's a mere artifact of the external observer's desire to calculate within a coordinate system that makes his own neighborhood look as Euclidean as possible. If the external observer were willing to calculate using a different coordinate system, such as one that makes the infalling person's spacetime neighborhood look as Euclidean as possible, then the external observer would realize that the infalling observer's clock does not stop.

Originally Posted by

**Farsight**
Sorry, I didn't look at it. The thing is, mathematics doesn't get this crucial point across. That's why people blithely switch to a different metric I suppose, and do that hop skip and a jump over the end of time without even noticing.

I agree that different people will disagree about where the event horizon is located. But a zero or very low speed of light is no artefact. Your coordinate system is. It isn't something real that actually exists. It's an artefact of measurement. And when *all processes slow and stop*, you can't measure anything. You can't see anything, and you can't even think, because all processes have stopped. So you don't have a coordinate system. Your proper time is being measured on a stopped clock. Just because you're stopped too doesn't mean that *nothing even slightly unusual happens*. It means nothing happens.

No. What it means is that you're trying to use a chart (coordinate system) outside the open set on which that chart is defined. That's a mistake. If you want to understand what's going on at the event horizon itself, you have to switch to a different chart (coordinate system) whose domain includes the event horizon.

Originally Posted by

**Farsight**
At the event horizon it isn't *nothing unusual happens*, it's *nothing happens*. There are no more events. The light isn't moving any more. You can't measure space, you can't measure time, you can't see, or think, and your reference frame and coordinate system have utterly collapsed.

The highlighted part is almost correct, but not for the reason

**Farsight** gave. The reason you can't use the external observer's chart (coordinate system) to measure space or time at the event horizon is that the event horizon lies outside the open set on which that chart (coordinate system) is defined. If you use a chart (coordinate system) whose domain includes the event horizon, you'll realize that nothing unusual happens at the event horizon.

Originally Posted by

**Farsight**
No. I've already told you twice that the maths doesn't show you this. I've given you a good explanation, now deal with the argument instead of trying to play the inscrutable mathematics card.

**Farsight** is telling us untrue things about the mathematics because he doesn't understand the mathematics. Indeed, he refers to the math as "inscrutable".

The mathematics of general relativity is fairly advanced. That's why people who have little or no training in mathematics generally know better than to argue about general relativity. There are, of course, exceptions.

Originally Posted by

**Farsight**
I've told you before, don't try hiding behind the maths, not with me.

We're not hiding behind the math. We're trying to explain the math.

Originally Posted by

**Farsight**
It isn't. Remember what sol said: *all processes slow and stop on the horizon as viewed from outside*. Now imagine that you're a light wave bouncing back and forth inside his parallel-mirror light clock. At the event horizon, you stop forever.

This isn't my idea, this is the original frozen star idea, in line with Schwarzschild and Einstein. Typical textbooks do that hop skip and a jump over the Schwarzschild event-horizon singularity and write it off as a coordinate artefact, missing the whole point that when light stops you don't have any coordinates any more.

As

**sol invictus** said, all processes appear to slow, never reaching the event horizon,

*as viewed from outside* by an external observer who's calculating with a chart (coordinate system) that does not include the event horizon. The reason for that is simple: The external observer's chart (coordinate system) contains a coordinate singularity at the event horizon.

That coordinate singularity doesn't imply that light actually stops at the event horizon. In reality, nothing at all unusual happens as light passes through the event horizon.

Originally Posted by

**Farsight**
If it's "straightforwardly wrong", try explaining it. Or try addressing my straightforward argument. When you can't do either, stop digging, and yield.

**Farsight**'s argument is indeed straightforwardly wrong, because it's based on

**Farsight**'s apparent misunderstanding of the most basic definitions of differential geometry.

Originally Posted by

**Farsight**
I'm the one who's saying it's more than just some coordinate artefact. People who swallow Kruskal-Szekeres coordinates treat the event-horizon singularity like window dressing to be disregarded.

The event-horizon singularity is a coordinate singularity. The event horizon lies outside of the open set on which the usual Schwarzschild chart (coordinate system) is defined.

That means we can't understand what happens at the event horizon without using a chart (coordinate system) other than the one

**Farsight** insists we use. So long as

**Farsight** insists upon using a particular mathematical formalism that becomes completely useless at the event horizon, he won't understand what happens at the event horizon and beyond.

Originally Posted by

**Farsight**
Everybody can see you're ducking the argument. Let me take a look at the other posts, then I'll nail you to the floor. Hurry up and start backing down, because it's going to be embarrassing.

Originally Posted by

**Farsight**
No it isn't. I've said this and I'll say it again. Gravitational time dilation goes infinite at the event horizon, so your clock has stopped and you've stopped too. Light has stopped, so you can't measure anything, and *there is no metric*.

No matter how many times

**Farsight** says it, it isn't true.

**Farsight** is saying the well-known Schwarzschild coordinate singularity at the event horizon is a genuine spacetime singularity. He's wrong.