Black holes

[Farsight]

You can be sure edd. RC referred to the standard example of cones where all of the curvature is at the vertex. You can take a vertical slice through a cone so it looks like this: V. Then flatten it out and the vertex is like where two flat regions meet.

 

[edd]

A conic section only looks like a V if it passes through the vertex.

I'm also not clear on your flattening out procedure.

My point is that on a polyhedron the curvature is not where two regions meet - that's an edge. Is it not at the vertices, in the same way it is at the vertex of a cone?

This is yet another reason for me to doubt that you know what you're talking about.

 

[Farsight]

A conic section only looks like a V if it passes through the vertex.

Sure, no problem.

I'm also not clear on your flattening out procedure.

You don't have to be. Take a slice through the two adjacent flat regions, and you've got a V.

My point is that on a polyhedron the curvature is not where two regions meet - that's an edge. Is it not at the vertices, in the same way it is at the vertex of a cone?

It doesn't matter. There's such a thing as the vertex of an angle. If you don't want to take a slice just note that two adjacent flat regions share a vertex.

This is yet another reason for me to doubt that you know what you're talking about.

I know what I'm talking about. And we all know that RC has been hoist with his own petard. He tried to ignore where his two flat regions meet, and then blundered into saying the standard example of cones where all of the curvature is at the vertex. And hey look, what can we see between those two adjacent flat regions? Why, it's a vertex!

Sheesh, this is getting boring.

 

[edd]

So if I take a sheet of paper and introduce a fold I've introduced some intrinsic curvature into the paper, is that right Farsight?

 

[sol invictus]

No I did nothing of the sort. I said a gravitational field was only flat in an infinitesimal region. Then RC tried to say I was wrong by likening it to a geodesic dome where finite regions are truly flat and the Riemann curvature is bundled into the place where two flat regions meet. It's garbage and you know it, and your howls of outraged accusation don't conceal that it's garbage.

You're lying, Farsight, and everyone knows it.

My point is that on a polyhedron the curvature is not where two regions meet - that's an edge. Is it not at the vertices, in the same way it is at the vertex of a cone?

Yes.

 

[Reality Check]

It's garbage RC. Go and look at a gravitational field. It isn't plated with little flat patches. It's curved.

Thats ignorant, Farsight. We all know what a gravitational field is without your reliance on cartoons. It is insane to state that it is plated with "little flat patches". It's based on a curved spacetime that is locally Minkowskian.

What GR states and you remain ignorant of is that GR uses a manifold which is locally Minkowskian. That is a requirement of the principle of equivalence. Try reading Einstein's papers or a text book sometime, Farsight ! Just kidding - I know you have read Einstein's papers. It is just you have little idea what they mean, e.g. that GR requires that spacetime is locally Minkowskian.

And each patch is infinitesimal in extent!

Wrong: Each Minkowskian patch is local in extent as Einstein stated!

No, it's because the local flatness is only an approximation. It isn't actually flat. It's just so nearly flat that you can't detect the curvature that's there.

That is what Einstein stated: That locally you cannot tell the difference between acceleration and a uniform gravitational field. Thus locally spacetime is Minkowskian. Therefore the manifold that you use to describe spacetime has to be locally Minkowskian. Duh !

And one more time for the ignorant: The geodesic dome is not a physically realistic spacetime. It is an analogy.

 

[Reality Check]

...snipped stuff that had nothing to do with my post...

Faersight: This is what I posted:

FYI, Farsight: Curvature for a cone is given as an example in GR classes because it is easy to visualize. The first thing to realize is that a cone is equivalent to a plane with a slice cut out of it so you can flatten out a cone into a plane by making a cut from the vertex.

Lecturers introduce the concept of parallel transport. Basically this is a manifold equipped with way of moving a tangent vector defined at one point along a path. This path can return to the original point, i.e. a closed path. The question then becomes: Is the transported vector the same as the original vector?

The answer is that it depends!
Consider the cone and a closed path that encloses the vertex. That means that the path has to jump across the slice in the equivalent plane representation. On each side of the slice the tangent vector will point at the vertex but it's direction will change. Thus the transported vector is different from the original vector for a closed path that encloses the vertex.

Consider the cone and a closed path that excludes the vertex. The transported and original vectors are the same.
Any change in the tangent vector is interpreted as curvature, e.g. the steeper the sides of the cone, the wider the slice and the larger the curvature.

Thus a cone has global curvature but is locally flat. Global means that you consider every single bit of the cone including the vertex. Local means a neighbourhood of a point. In the case of a cone it is any neighbourhood that does not include the vertex.

Try reading iwhat I wrote.
There is no mention of the cone example being a physically real spacetime. It is an interesting example of the differential geometry that is the basis of GR. It is an exercise that GR students go through in order to understand how GR works mathematically.

People who teach GR typically introduce parallel transport by looking at the problem of defining differentiation on a 2-sphere. Here the orientation of the original and transported tangent vector also depends on the path. They then go on to describe parallel transport on a cone because the example is more interesting and easier to visualize. A few lectures later they get to Einstein's field equations and examples of real spacetimes.

For example: Leonard Susskind's YouTube lecures on Modern Physics concentrating on General Relativity (recorded September 22, 2008 at Stanford University).

 

[edd]

I think Farsight's persistent reference to two flat regions plus silence on my previous question indicates he doesn't really understand the basics of the intrinsic curvature involved in GR.

 

[sol invictus]

I think Farsight's persistent reference to two flat regions plus silence on my previous question indicates he doesn't really understand the basics of the intrinsic curvature involved in GR.

You "think" that??

I know you're a nice person and you're giving every benefit of the doubt, but...

 

[edd]

I have a tendency to understate, that's all.

 

[Reality Check]

You can be sure edd. RC referred to the standard example of cones where all of the curvature is at the vertex. You can take a vertical slice through a cone so it looks like this: V. Then flatten it out and the vertex is like where two flat regions meet.

There are no "two flat regions" and so they can never meet.
See for example: Leonard Susskind's YouTube lecures on Modern Physics concentrating on General Relativity (recorded September 22, 2008 at Stanford University). I am not sure which lecture discusses the cone example though.

The point is that you can change a cone to a plane with a slice out of it by adding a cut from the vertex outward. This is easily illustrated with a sheet of paper (but neglects the fact that the cone and plane extend to infinity). Looking at the plane makes it obvious that the cone has no intrinsic curvature for any path that does not enclose the vertex. It is locally flat. Any path that encloses the vertex has intrinsic curvature (a cone is globally curved).

And remember, Farsight - this is all intrinsic. The cone is not embedded in a higher space that allows extrinsic curvature to be measured.

 

[Reality Check]

I think Farsight's persistent reference to two flat regions plus silence on my previous question indicates he doesn't really understand the basics of the intrinsic curvature involved in GR.

We already know that Farsight doesn't really understand the basics of the intrinsic curvature involved in GR.

But his "two flat regions" is a failure of visualization about how to construct (or de-construct) a cone, e.g. take a sheet of paper (one intrinsically flat region), cut a slice out of it (still one intrinsically flat region) and join the sides of the slice.

 

[edd]

We already know that Farsight doesn't really understand the basics of the intrinsic curvature involved in GR.

But his "two flat regions" is a failure of visualization about how to construct (or de-construct) a cone, e.g. take a sheet of paper (one intrinsically flat region), cut a slice out of it (still one intrinsically flat region) and join the sides of the slice.

Quite. It sounds like a remarkable failure of intuitive understanding (for a self-professed expert) as well as the ability to handle the mathematics more precisely.

 

[ctamblyn]

@Farsight:

Given your lack of response to my earlier post (and other similar ones), you are evidently going to continue to dodge the empirical evidence issue forever. I've literally lost count of the number of times you've been asked for whatever empirical evidence you have which contradicts MTW GR while supporting your alternative FGR, and you haven't yet shown us anything that qualifies (link to where you did if I'm wrong).

 

[Farsight]

So if I take a sheet of paper and introduce a fold I've introduced some intrinsic curvature into the paper, is that right Farsight?

Yes. Say it's a 180-degree fold. Get a microscope and look closely at the folded region. The paper there is sharply curved like this: U. If you had a very long strip of paper and you made a large number of very small very slight "folds" or creases of say 0.00001 degrees apiece, then between them the paper is flat, but because they're there the strip isn't flat any more.

 

[edd]

Yes. Say it's a 180-degree fold. Get a microscope and look closely at the folded region. The paper there is sharply curved like this: U. If you had a very long strip of paper and you made a large number of very small very slight "folds" or creases of say 0.00001 degrees apiece, then between them the paper is flat, but because they're there the strip isn't flat any more.

OK, you simply don't understand the nature of the curvature we're talking about then. The relevant curvature to this discussion is intrinsic rather than extrinsic.

 

[Farsight]

You're lying, Farsight, and everyone knows it.

I'm not lying. You're only saying that to try to cover up RC's faux pas. It's only infinitesimal regions that are locally flat. When you say non-infinitesimal regions are flat, then shuffle along a little and repeat ad infinitum, you end up with no curvature. I suggest you try it for yourself with a strip of paper.

 

[Farsight]

I think Farsight's persistent reference to two flat regions plus silence on my previous question indicates he doesn't really understand the basics of the intrinsic curvature involved in GR.

I understand it edd. It's really simple. The curvature you can see on the bowling-ball depictions is Riemann curvature. Without that, the grid is a flat level plane, and there's no gravitational field in it. Things don't fall down. With it, the force of gravity at any one location is proportional to the slope of the grid.

When you're in an accelerating rocket it's like tilting a flat level grid so that it's sloping. Then you experience something similar to gravity. In the cabin of your rocket, things appear to fall down.

The principal of equivalence relates a small region of the bowling-ball curved grid to the tilted grid. It's such a small region that you can't measure any local curvature. But if it's absolutely utterly flat, and if all other small regions are similary absolutely utterly flat, the grid can't be curved. It's all very simple and there's nothing mysterious about it.

 

[Farsight]

Given your lack of response to my earlier post (and other similar ones), you are evidently going to continue to dodge the empirical evidence issue forever. I've literally lost count of the number of times you've been asked for whatever empirical evidence you have which contradicts MTW GR while supporting your alternative FGR, and you haven't yet shown us anything that qualifies (link to where you did if I'm wrong).

And I've lost count of the number of times I've given you empirical evidence for the varying speed of light that can't vary to less than zero. That knocks KS coordinates on the head and demonstrates that the black hole is indeed a frozen star. Now stop dismissing what Einstein said, and stop pretending that the varying speed of light is my idea. It isn't, it's Einstein's.

 

[edd]

You don't understand it. You think you do, but you don't. You posted earlier about the cross section of the paper being shaped like a U. That's blatantly and clearly (to anyone who knows what they're talking about) intrinsically flat.