Anyone can be mistaken.
Some people are mistaken more often than others.
When it comes to general relativity, people who literally don't know the first thing about the relevant mathematics generally make more mistakes than those who have spent years learning that math.
Then if you say gravitational time dilation goes infinite at the event horizon, you're left with a zero speed of light at that location. It's all horribly simple.
Yes, it's horribly simple.
Those who understand the math realize that the chart (aka coordinate patch/system)
Farsight is using does not include the event horizon. Those who understand the math know that the time dilation implied by that particular chart increases without bound as you approach the event horizon, but never becomes infinite because the chart does not include the event horizon.
The observer travelling with the clock gets to a place where the speed of light is zero. So his light-clock does stop ticking, and because the speed of light goes to zero, he doesn't observe it.
Those who understand the math realize that
Farsight is pulling a bait and switch here.
Up to now, he was using a chart that corresponds to an observer who is so far away that the gravitational pull of the central mass is negligible. That is the chart in which the infalling observer's clock slows forever as the infalling observer approaches (but never reaches) the event horizon. Contrary to
Farsight's repeated claims, that faraway observer never actually observes any stoppage of the infalling observer's clock, and that faraway observer never actually observes the infalling observer reaching the event horizon.
In the above, however,
Farsight begins to talk about what the infalling observer observes, which means he should be using a chart that corresponds to that infalling observer. That's an altogether different chart. In that chart, the infalling observer's clock doesn't slow down at all, and the infalling observer passes through the event horizon with no fanfare whatsoever.
Reading your lack of counter argument which you attempt to cover up with abuse, everybody else is thoroughly convinced that you're a posturing quack.
Farsight reads minds about as well as he reads charts.
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 Rn.
Let's not forget that such is a mathematical abstraction.
General relativity cannot be understood without understanding the mathematical abstraction known as a spacetime manifold. To understand spacetime, you must understand manifolds.
Those homeomorphisms are called charts.
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.
OK, but let's not forget that you cannot step outside and point up to the clear night sky and say "look, that's a coordinate system". A coordinate system is an artefact of measurement.
It would be more accurate to say each chart (aka coordinate patch/system) is a convention for measuring. Along the 3-kilometer stretch of Massachusetts Avenue that runs between MIT and Harvard, pedestrians often speak of walking a certain distance north (toward Harvard) or south (toward MIT). That convention works just fine (unless some out-of-towner is navigating by compass).
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.
It's not a mistake. The CMBR dipole anisotropy allows us to gauge our motion through the space of the universe, which when coupled with time gives us our entire spacetime manifold. Knowing this allows one to essentially "step back" out of any one particular coordinate system to see the big picture, as it were.
It's a mistake.
Throughout this thread,
Farsight has been making incorrect claims about one particular mathematical abstraction of spacetime: the Schwarzschild manifold, which is the unique maximal solution of Einstein's field equations that describes an asymptotically flat, mostly static, spherically symmetric gravitational field around a fixed mass M.
In that particular mathematical abstraction of spacetime, there are no stars apart from that mass M, and there is no CMBR. It's true, however, that the spherical symmetry implies a distinguished "center" for the spatial components of the Schwarzschild manifold as observed from afar (although that "center" corresponds to a genuine singularity, so it does not correspond to any world line in the manifold). The existence of that spatial "center" is entirely consistent with the existence of infinitely many charts, all of them correct.
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.
I have a globe in my study. It gives me a map of the surface of the earth.
Farsight appears to be demonstrating ignorance of the basic definitions.
By definition, a chart is a homeomorphism
WP from an open subset
WP of the manifold to an open subset of n-dimensional Euclidean space
WP. Although the entire 2-sphere (such as
Farsight's globe) is indeed an open subset of that 2-sphere, there is no homeomorphism from that entire 2-sphere onto an open subset of the 2-dimensional Euclidean plane.
If
Farsight is claiming that his globe gives him a chart, then he is claiming to know how to flatten the entire surface of his globe onto a flat floor without punching any holes or opening any seams or tears in the globe and without placing any two distinct points of the globe above the same point on the floor. It can't be done; that's a mathematical theorem.
If
Farsight is not claiming that his globe gives him a chart, then his remark was utterly irrelevant.
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 not an example of a coordinate singularity when you're looking at a globe, which is a good representation of what's actually there. The Mercator projection isn't so good; in essence your map is not a good representation of the territory.
Farsight appears to believe his globe gives him a chart (aka coordinate system). He's wrong: The 2-sphere is not homeomorphic to any open subset of the 2-dimensional Euclidean plane.
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.
If you can. This thread is discussing whether you can actually do it, and that discussion involves the nature of the territory.
We know we can do it. It was done a long time ago.
Georges Lemaître
did it in 1938.
Eddington and Finkelstein did it.
Kruskal and Szekeres did it.
Farsight has posted more of the same since I wrote the above, but everything he wrote in his more recent post is answered by the above and by other contributors to this thread.
