Manifolds, tensors and fields Split

Quarkhead, note this from Wolfram MathWorld:

"A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in

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). To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly "flat" on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré."

But note that space isn't curved where a gravitational field is. See Baez for that. Note this: "similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial". Space isn't curved, instead it's inhomogeneous. If you were to plot this inhomogeneity using for example light clock rates in an equatorial slice through the Earth and surrounding space, you would notice that the inhomogeneity was not some straight-line non-uniformity, but instead exhibited a curvature, like so:

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GNUFDL image by Johnstone, see http://commons.wikimedia.org/wiki/File:Spacetime_curvature.png


Call me a pedant, but IMHO it's important to understand that it's the inhomogeneity of space that's curved, and space isn't.

 

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Farmoron, I never once mentioned space, spacetime, curvature etc in any of my posts here. One wonders at the motive for your irrelevant (and rather unwelcome) intervention.

Note well - I am attempting, however poorly, to explain the mathematics of differential geometry. Some of which is applicable to GR, some of which may not be. If you have a mathematical point to make in this thread, make it. Otherwise do the other thing

 

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Quite. A manifold is a topological "space", and here you have a thread entitled Manifolds, tensors and fields, and you don't say what a manifold is. Your opening post needs work. Lots of it I'm afraid. Because you do not really understand something unless you can explain it to your grandmother.

Your opening line was as there seems to be a prevailing interest in GR. I made a contribution that was relevant to that.

See above, I'll make points concerning general relativity and gravity as appropriate. You may find my expertise in this area somewhat threatening, but you shouldn't. It's unlikely that I'll say anything that isn't factually correct, and the feedback I give ought to help you to improve your presentation. And of course help you to avoid becoming bogged down with abstraction and "lost in maths". By the way a tensor is akin to a matrix, and according to Einstein, a field is a state of space.

 

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Agreed

 

...though of course there may be the odd typo which I shall graciously and promptly correct with all due humility. It is of course nice to know that you've read my post carefully.

 

Then perhaps you'd like to summarise them for the rest of us?

 

I formed the view from previous thread that you don't understand gravity. But perhaps I was wrong. Please do continue with your presentation.

 

Farsight - let me just make a couple of (maybe 3) points:

1. I "messed" with your on-screen name. This was extremely impolite of me, and very probably illegal on this forum. I apologize

2.. Given "world enough and time" (as the quote goes) I may get on to mention curvature in the abstract setting I am trying to establish here. At NO point had I intended to mention its "cause" - this is the province of those that apply differential geometry to physics and engineering (say). I never claimed to be, nor wanted to be, a physicist

3. In spite of all your reading of the "mere" words that Einstein wrote, he and his friend Marcel Grossman understood everything contained (or to be contained) in this thread and much more than I do. They (he) were not "lost in mere mathematics" - it was central to his programme

If you want to contribute to this thread on its own terms, you will be welcome. But please do not interject with irrelevant and self-serving remarks

 

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You have me there. I cannot.
My point was that rpenner and quarkhead on the evidence available, are far more reputable then yourself.
My apologies for butting in.

 

You are such a moron.

You don't even understand the difference between a physical space and a mathematical space, even though you are citing fucking Wolfram.

You need to take the time to read the shit you post. It will waste our time less and make you seem like less of the half-wit that you so clearly are.

No, you spouted you standard lies. Nothing in your link has anything to do with a supposed impossibility of curved space in GR. It is certainly possible to have curved space in GR.

No, you will just add your lies.

You can't follow the math here, so you have no expertise. You have lies. Only lies.

 

Manifolds, tensors and fields!

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Apology accepted. But do resist the urge to be abusive. It does you no credit.

Noted. How about if you press on with your presentation instead of getting bogged down with rpenner?

Agreed.

I won't. At some juncture when you mention curvature, do look again at my post #11. Meanwhile, carry on.