Hello, fellow truth seeker, Only when space is flat do its coordinates have direct meaning of distances or angles, and if one set of coordinates have direct meaning of distances or angles then the space must be flat. This is the famous Riemann theorem when he pineered the concept of curved space. General relativists claimed curved spacetime. However, when testing GR with data, I see that all textbooks consider Schwarzschild coordinates to have direct meaning of distances or angles. Therefore, I have strong feeling that they rape common human sense. If there is no scientific conspiracy here then there must be some relativist standing out to meet my challenge!! Hello, fellow truth seeker: Could you help me find such relativists to meet my challenge here? ============================ The following is from my previous post on sciforums: On a curved space, the sum of all angles of a triangle is not \pi, either greater or less than \pi. Two famous general relativity tests are about angles. You can check whatever textbook and see that they calculate angle by directly using the coordinate \phi. They indirectly assume the domain length of \phi is 2\pi. Therfore, the sum of all angles of a triangle is \pi. That is, they assume flat space!!!!!!!!!!!!!!!!!!! Any real mathematician can tell you that all coordinate systems on a curved space are curvilinear. All coordinates are merely parameters. Real angle and distance have to be calculated by employing the coefficients of the space metric. I TELL YOU that only when the space is flat will the metric reduces to the Pythagoras theorem. That is, only when the space is flat will the coordinates have direct meaning of spatial distances and angles.

Just about everybody has come around to accepting that the phenomena of relativity is real. But the phenomena does not require curved space-time. The Lorentz transforms are just as happy with classic space-time as with variable space-time. It is just easier with the curved space-time concept. If you want a preferred frame of reference, you have to double figure the transforms. This was thought to be redundant and unnecessary, so people generally accepted Einstein's view over the view of Lorentz.

cosmodel, You are being absurdly overly dramatic. No one is "raping common sense" and there is no scientific conspiracy. I am happy to answer any questions about general relativity that you may have. This assertion is silly and wrong. φ is simply a coordinate and the underlying geometry is completely independent of your coordinate choice. I could just as well use the coordinate φ' = φ / 2 π which runs from 0 to 1. This doesn't change anything about the geometry of spacetime, but it does immediately show that your argument is fallacious. Spacetime is locally flat which means you may always find local coordinates where the metric takes the standard Pythagorean form. Any mathematician will tell you this. Next question.

superluminal, A four dimensional manifold (spacetime) cannot look like a two dimensional manifold just because the dimensions don't match. What the local flatness means is that while the manifold may curve, it does so smoothly. In the case of a 2 dimensional manifold, local flatness means space looks like a flat plane locally. Spacetime looks like flat four dimensional Minkowski space locally. However, a two dimensional surface in spacetime would look locally like a flat plane provided the surface was smooth.

Ah. Of course. I was just reading up on manifolds since apparently the PoincarĂ© Conjecture is now officially proved. Spacetime is a 4-manifold, R4 yes?

Hi superluminal, Yeah, spacetime is a four manifold which looks locally like R4. Minkowski space is of course just R4 with the "weird" metric -1,1,1,1. And yeah, Poincare was finally proven! This Perelman fellow is quite a guy it seems. There is a really nice article in the New Yorker about Perelman which I highly recommend.

Monkey, You said \phi is coordinate, but relativists use it to be the real angle run by light, as in the famous test (1923?). You mean that the angle is coordinates dependent, no invariant meaning? You could choose what ever absurd coordinate X which is to be your angle? Secondly, local flat means infinitesimal area. But the Schwarzschild coordinates (including \phi ) are global ones covering even the absurd black hole area. You NAIVELY think no conspiracy. You heard of dictators? Now I present to you: You were Einstein, 27 years old, and published 3 famous papers. Then during several years (especialy 1923?), you was and you are considered god. You made a mistake on freely falling frames [If GR's calculation is correct (not the assumption of curved spacetime) then Galilei is not correct. The freely falling body near earth surface which has kinetic energy about c^2/3 per unit mass will not suffer gravity. That is, its acceleration is not 9.8m/s^2, but zero. I do not know if somebody saw the mistake some time ago. But I found it and submitted it to journals. They behave like thieves and reject my paper]. No body could not affend you the god in the early time. In modern time, so-called evil scientists (authorities, establishment) get millions and even billions of dollars (grants, funds, etc.) to study black holes, dark matters, dark energies, super strings, super knots, gravitons, big bangs, big explosiongs, etc. Now even you want to correct your fatal mistake, the establishment definitely does not accept your humbleness. They need paint and keep their own small-gods images on evergreen evy univ class rooms. Theoretically, black holes are proved wrong by Leonard Abrams, and observationally wrong by Rudolph Schild, Stanley Robertson and Darryl Leiter who is the final graduate student of mathematician coauthor of the god Einstein. You may check the story at http://gravity.myschoolclub.com The god's faith followers rape common sense for about 90 years!!!!

On a Riemannian manifold, any angle measured between two vectors at a point is a real angle, because at that point the manifold is locally flat.

Also, we know that in the Monty Hall problem both of the last 2 doors has an equal chance of winning. I mean, there are only two choices, and one of them must be the winner, so obviously each door has a 50% chance of winning. Come on people, stop using your ridiculously complicated Bayesian analysis techniques and use a little common sense!

Ah, the poverty of common sense. The problem with common sense is, of course, that it relates to the common, and is shaped by our experience with nature on just a few orders of magnitude. Even within the direct interaction realm, the heuristics of common sense often fail us. And once we leave the realm of direct human interaction with nature, it really stops working. Even though the Earth (and spacetime) is locally flat, its overall geometry is vastly different. Even though white light appears colourless, it is a mixture of all the visible colours. Even though the Monty Hall problem sets up a choice between two doors, the likelyhood of you winning the prize by a blind choice is not fifty/fifty, etc. etc. Trying to discern the nature of the universe with common sense has been tried time and time again. It doesn't work. We know this now, and appealing to common sense has become a warning sign in science. There's nothing wrong with conjecturing based on common sense, but the corroboration musn't rely on it.

Didn't Aristotle use common sense? Objects in motion tend towards rest. So how does an arrow fly? Well, the air pushed away by the front of the arrow rushes to the back of the arrow and shoves it forward, silly. Of course, Galileo disproved it which I suppose means that the scientific method did rape common sense. Alas, poor common sense.

The shape of physical objects has nothing to do with the geometry of space. You can in fact connect any two points on the earth's surface by a straight line if you drill a corresponding tunnel through the earth. So if you want to be logically consistent, then space must be overall flat (Euclidean) by definition. Thomas

Sure, but you can't do that with spacetime. Even stuff like wormholes would happen "on" spacetime. I'm not quite sure I get what you're trying to say. What logical inconsistencies follow by space not being overall flat?

What is your justification for your claim? The idea of a 'warped' space is exactly derived from the curved surfaces of physical objects (see for instance Hawking's webpage http://www.hawking.org.uk/lectures/warps.html ). The point is that a 'surface' is merely a mathematical abstraction but not a physical space (if you look at it from very close even the mathematical abstraction becomes invalid). Nobody forces you to stay on the 'surface' of a physical objects and that's why any two points on earth (or in the universe in general) can always be connected by a straight line (that's why people build tunnels through hills etc). The logical inconsistency that the shortest distance between any two objects in the universe is not a straight line (in the above indicated sense). Thomas