I emailed out exactly the same question to 5 experts. By now I received 3 respenses: =========== I do not understand what your question is. Read Wheeler and Taylor "Black Holes" (2000). > Dear Dr. , > > You must be very busy. Sorry for my disturbing. > I am a student of GR. I would be very very grateful if you could > answer > my question as follows which troubles me very much: > > ``Only when space is flat does one coordinate system exist which has > direct meaning of distances or angles, and if one set of coordinates > has 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 relativity claims 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, GR verified flat spacetime! > > FOR EXAMPLE: 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! > > 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. Only > when > the space is flat will the metric reduce to the Pythagoras theorem. > That is, only when the space is flat will the coordinates have direct > meaning of spatial distances and angles.'' > > Sincerely yours, ================== > You must be very busy with your new semester. > I am glad you replied to me while you was in my home country China. I'm still in Shanghai - I'm going home on September 20th. > I would be very very grateful if you could answer me one more question > as follows which troubles me a lot: > > ``Only when space is flat does one coordinate system exist which has > direct meaning of distances or angles, and if one set of coordinates > has 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 relativity claims 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, GR verified flat spacetime! > > FOR EXAMPLE: 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! > > 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. Only when > the space is flat will the metric reduce to the Pythagoras theorem. > That is, only when the space is flat will the coordinates have direct > meaning of spatial distances and angles.'' What's your question? The above stuff is nonsense, that's for sure. Best, ================= > General relativity claims curved spacetime. However, when testing GR > with data, I see that all textbooks consider Schwarzschild coordinates > to have direct meaning of distances or angles. No, that's not quite right. For large r -- r much greater than 2m -- the Schwarzschild spacetime is nearly flat, so a flat space interpretation of the coordinates is a good first approximation. But it is only an approximation. For example, the true radial distance between two points at constant angle and constant t but different r is not r'-r, but is obtained by integrating ds from r to r'. For large r and r', this is approximately equal to r'-r, but there are corrections -- the first is proportional to log(r'/r). For deflection of light, you will find a discussion about this issue in Bodenner and Will, "Deflection of light to second order: A tool for illustrating principles of general relativity," American Journal of Physics, Vol. 71, No. 8, pp. 770-773, August 2003. The authors work out the deflection of light to second order (where the non-Euclidean nature of the Schwarzschild coordinates becomes significant) in three different coordinate systems, and show how to get a consistent result by relating back to genuine invariants. It's true that elementary textbooks are often sloppy about this -- they only do the computation to first order, where it doesn't matter. Weinberg actually has a brief discussion on p. 191 of his text about the question of whether a prediction "really refers to an objective physical measurement or whether it has folded into it arbitrary subjective elements dependent on our choice of coordinate system." > 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. Only > when the space is flat will the metric reduce to the Pythagoras > theorem. That is, only when the space is flat will the coordinates > have direct meaning of spatial distances and angles. This is correct. For first order calculations in the Solar System, the curvature is small enough that one can get away with a "flat space" interpretation of coordinates, but for anything more accurate, or anything involving stronger fields (and therefore more curvature), you need to be very careful to not accidentally misinterpret the physical significance of your coordinates. If you look at how actual tests of general relativity, even in the Solar System, are done -- reading the details of the data analysis -- and not just the greatly simplified textbook descriptions, you will find that this is pretty well understood by the practitioners. ==================
Didn't you post the same thing in a recent thread? What do you think of the responses you have received?
Vern, for better understanding, I expose more my private communication: =============my email: Thanks for your message on my question. The question did bother me for about a year. I submitted my paper to over ten journals. The editors held my paper up for a while then rejected it, without much explanation except the words `your manuscript does not fit the prestigious jounal as bababa'. You are the first one who gave me explanation. But the question still bothers me. Because the second-order predictions of GR are hard to measure and are not verified, what was verified is the first-order predictions which are the same for two different theories. The first theory is the standard GR which deals with straight lines (geodesics) on curved spacetime. I simply proposed the theory which deals with curved lines on flat spacetime. Because I consider the metric form of GR to be the Lagrangian on flat spacetime, its first- order prediction is the same of GR. However, my theory does not have difficulty to explain galaxy dynamics and to quantize gravity and to model the universe. This flat universe model does not encounter such problems as horizon because I have varying speed of light. All my work is at xxxxxxxxxxxx Hopefully you could give me more explanations. Sincerely yours, =====================my more email: I found out that your answer is not satisfactory. You said that Bodenner and Will gave the invariant angle of light deflection which does not depend on the choice of coordinate system. But in the area of r approaching infinity, we have many inertial coordinate systems (flat spacetime Lorentz coordinate transformations) by which different angles are measured, due to length contraction! Because these inertial coordinate systems can be extended to the whole area near 2m, the assumption of curved spacetime leads to contradictory results: curved manifold requires SINGLE invariant angle while general covariance requires a set of MANY invariant angles!! I would be very very grateful if you can give me further explanation for the question. Sincerely yours, ================ > I found out that your answer is not satisfactory. > You said that Bodenner and Will gave the invariant angle of light > deflection which does not depend on the choice of coordinate system. > But in the area of r approaching infinity, we have many inertial > coordinate systems (flat spacetime Lorentz coordinate transformations) > by which different angles are measured, due to length contraction! > Because these inertial coordinate systems can be extended to the whole > area near 2m, the assumption of curved spacetime leads to > contradictory results: curved manifold requires SINGLE invariant > angle while general covariance requires a set of MANY invariant > angles!! Try reading the papers, instead of being so quick to jump to the conclusion that other people have made trivial mistakes. Will is the world's leading expert on experimental tests of relativity. Do you really think he forgot about Lorentz transformations? In this case, the Sun (or whatever object is deflecting the light) is static, and defines a preferred time-slicing. The deflection angle is independent of the spatial coordinate choice in that slicing. Of course, an observer in motion with respect to the Sun will observe a different deflection. That is irrelevant to the issue you originally raised. > You are the first one who gave me explanation. But the question still > bothers me. Because the second-order predictions of GR are hard to > measure and are not verified, what was verified is the first-order > predictions which are the same for two different theories. First: what are the PPN parameters for your theory. (If you don't know what this means, learn it! This is absolutely essential if you want to compare a new theory with observation. Try Will's book, _Theory and experiment in gravitational physics_.) Second: what does your theory predict for the gravitational radiation damping in a binary pulsar system? This is a key test of GR, in a strong field regime. If you can't reproduce the observed effect for PSR 1913+16, your model is dead. Third: what does your theory predict for the outcome of the Gravity Probe B measurement of frame dragging? Here you have a chance to make a prediction ahead of the announcement of the results. Fourth: from a quick glance at your papers, you seem to be predicting a value of the Shapiro time delay that differs from that of GR. Please note that the time delay has now been measured to a precision of two parts in 10^5, and agrees with the GR prediction. (See B. Bertotti, L. Iess, and P. Tortora, Nature 425:374,2003.) If you really disagree by as much as you state in your papers, your model is dead. There's nothing wrong with playing with new models, but you have not yet earned the right to be as arrogant as you appear. ==========my email Thanks for your message. It seems that you are a little angry with me. In fact, your suggestions are right. My idea was not studied deeply. I have my excuse. Firstly, I understand that I am not a genius. Secondly, I am jobless. Thirdly, I am not healthy enough due to the suffering of chinese cultural revolutionary. Fourthly, I was not supposed to study gravity at all. During several years, I tried to study galaxy patterns. Fortunately, I was able to put all regular galaxy patterns in single code! The ApJ referee, favoring my method, asked the second review of my paper to see if I could explore its physical meaning. Fortunately again, I found the meaning. But it is against the assumption of curved spacetime. ApJ editors no longer favored my paper. I realized that holy GR is the big story. After a little reseach, I found out that the spirit of GR is not as myterious as I thought. The full basis of GR is the Galileo equality principle of inertial mass and gravitational mass, plus the Einstein imagined principle of local common acceleration. The Einstein principle is definitely incorrect. I got a master degree in Defferential Geometry, I know what I am talking about. Einstein is lucky enough in that, without the Einstein principle, he can still assume, mathematically, that spacetime is curved. To conclude, the assumption of curved spacetime brings Einstein much more trouble than my paper brings to me. Rotation curves, galaxy dynamics, big bang, quantization. Anyone without religious prejudice must agree with me. Even there is great human resistance, the revolutionary era of new gravity physics must come. In fact, I am very thankful and happy to communicate with you. Best wish,
As you can see, our communication very possibly comes to the end. I need to read Bodenner and Will's book as the expert suggested. Because I live in a remote countryside, the
paper is not available right now. In the mean time the expert does not understand my paper at all and suggested ``arrogant`` (AMAZING!!) I answer his 4 suggestions: first, PPN directly assume r to be distance; Therefore, my theory can do the same. second, gravitational waves directly assume r to be distance, time; Therefore, my theory can do the same. third, I do not understand GRB much; My question is it still directly assume r to be distance? Fourth, he did not understand my paper at all; I suggest him read my paper carefully although I am not LEADING expert. For James R's question, the answer is at http://www.topology.org/sci/grav.html
cosmodel: Something I learned awhile ago may help you communicate. That something is that you should consider that most folks do not read past the second sentence. In a long post they read the first and last sentence and skip all in between. But I have read all of your post and still don't know what you're talking about.
It is clear that you have not studied general relativity very deeply, either theoretically (otherwise you would have known about the role of coordinates and their relationship to physical observables) or observationally (otherwise you would have known that your prediction for Shapiro time delay disagreed with observation by three orders of magnitude). Yet despite this ignorance, you feel competent to claim > Anyone without religious prejudice must agree with me. Even there is > great human resistance, the revolutionary era of new gravity physics > must come. This is exactly the kind of arrogance I was complaining about. These are not the words of a physicist, but of a crank. I suggest that you approach physics with a little more modesty, and a little appreciation that maybe someone might know more about gravity than you. ======================my response: Wenberg said clearly that, Shapiro can not measure the ABSOLUTE delay shift. What Shapiro claimed is the difference between the observed temporal data pattern and the corresponding theoretical temporal curve based on GR!! The difference is small value as you found. However, You can propose a theory whose delay is much different as Shapiro's, but the corresponding difference between the observed temporal data pattern and the theoretical temporal curve based on your theory can be small value too! I have not finished my reply to your last email. So I continue. I answer your 4 suggestions: first, PPN directly assume r to be distance; Therefore, my theory can do the same. second, gravitational waves directly assume r, t to be distance, time; Therefore, my theory can do the same. third, I do not understand GRB much; My question is, does it still directly assume r to be distance? Fourth, my answer is the above paragraph. It is clearly that you did not take my papers seriously although I am not a LEADING expert. Stephen J Crothers distributed his lessons already at http://www.geocities.com/theometria/Kerr.pdf I am not Stephen. At least my English is very poor. I said that I am very grateful that you are the only one who gave me explanation. I have told you my opinion about GR. In fact my theory is the same as GR except that metric is considered to be the Lagrangian on flat spacetime. So all GR's successes are carried over to mine. AND WHAT IS MORE IMPORTANT, my idea can resolve difficulties which GR can not! GR is not bible and Einstein is not a god. Why are you so angry?
I finally found the so-called Bodenner and Will paper (Am.J.Phys. 71(8) 770, 2003) which is claimed by the expert to be calculating the deflection angle of light on curved spacetime. SHIT CLAIM!!!. The paper introduce Metric M (in my theory, called the Lagrangian L on flat spacetime (x,y,z) or (r, \theta, \phi) ). L(x, y, z) or L(r, \theta, \phi) ). The authors work at the assumption of r, \phi to be real distance, angle. Anyone in sense knows that it is impossible to have an angle \phi which is well defined over whole curved space. You can try, for example, if you can find it on a sphere surface. You can not!!! Therefore, the authors as all other GR textbook writers did, found the Lagrange's equations on flat spacetime and calculate the second order result. The authors' trick is that they calculated three coordinate systems: Schwarzschild, isotropic, harmonic. However, ALL THEIR CLAIM IS EQUIVALENT TO CHOOSE ONE COORDINATE SYSTEM TO BE CARTESIAN ONE ON FLAT SPACETIME WHILE OTHERS ARE curvilinear coordinate systems ON THE FLAT SPACETIME. I do not know if the leading experiments completely understand DIFFERENTIAL GEOMETRY. They smartly chose Schwarzschild coordinate to be the Cartesian one because Hillbert said that its radial coordinate has invariant meaning. The authors said that the paper is served in teaching general relativity for astute students. This course should have be given to Einstein. Then He the god would have withdrawn his assumption of curved spacetime and now NASA would not have put billions of bucks to the search of dark matters and black holes and big bangs and gravity knots. And 49 years old Stephen J Crothers would have a PhD degree and a job.
From http://wc3.worldcrossing.com/webx?14@@.1de091e6/18 Ike warned the american people about this in his farewell address ( http://www.americanrhetoric.com/speeches/dwightdeisenhowerfarewell.html ): Today, the solitary inventor, tinkering in his shop, has been overshadowed by task forces of scientists in laboratories and testing fields. In the same fashion, the free university, historically the fountainhead of free ideas and scientific discovery, has experienced a revolution in the conduct of research. Partly because of the huge costs involved, a government contract becomes virtually a substitute for intellectual curiosity. For every old blackboard there are now hundreds of new electronic computers. The prospect of domination of the nation's scholars by Federal employment, project allocations, and the power of money is ever present -- and is gravely to be regarded. Yet, in holding scientific research and discovery in respect, as we should, we must also be alert to the equal and opposite danger that public policy could itself become the captive of a scientific-technological elite. Note the last sentence regarding the "scientific-technological elite". Any genuine scientist who dares challenge this dangerous group is called a "crank" or a "psuedoscientist" - at the very least.
