|Feb20-13, 07:49 AM||#18|
Is Poincare wrong about no preferred geometry?
I think people are getting confused in this discussion. Poincare's philosophical views did not agree with the standard modern interpretation of GR. From wikipedia:
Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics. He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.
This is what Poincare meant by no preferred geometry.
|Feb20-13, 07:59 AM||#19|
|Feb20-13, 08:14 AM||#20|
Poincare's point about measuring the geometry of the universe was that to do so you have to assume your ruler is straight. But his point was that instead of straight ruler and bent geometry, one could also imagine bent rulers and straight geometry. I believe the ability to have flat and curved spacetime versions of Newton, Nordstrom, and Einstein gravity illustrate the spirit of his point.
Naty1's quote from Thorne is relevant: "Isn't it conceiveable that spacetime is actually flat but the clocks and rulers we use to measure it are actually rubbery?...yes...Both viewpoints give precisely the same predictions for any measurements performed....Some problems are solved most easily and quickly using the curved spacetime paradigm; others, using the flat spacetime...Black hole problems, for example, are most amenable to curved spacetime techniques; gravitational wave problems (for, example computing the waves produced when two neutron stars orbit each other) are most amenable to flat spacetime techniques....the laws that underlie the two paradigms are mathematically equivalent....That is why physicsts were not content with Einstein's curved spacetime paradigm and have developed the flat spacetime paradigm as a supplement to it..."
|Feb20-13, 09:29 AM||#21|
If a flat and hyperbolic geometry can be constructed to describe the same physical event and it is only convention or convenience as to which to select, then can we use the angles made by distant stars to show that we inhabit a flat (euclidean) geometry? Is it impossible to discover the shape of our spacial geometry?
And is it more useful to just consider the interactions of massive bodies by the most convenient mathematical model and not interpret the model used as a representation of the nature(shape) of space?
|Feb20-13, 11:39 AM||#22|
Here is a quote that covers the general idea.
Poincare Science and Hypothesis“The straight line is a line such that a figure of which this line is a part can move without the mutual distances of its points varying, and in such a way that all the points in this straight line remain fixed" ? Now, this is a property which in either Euclidean or non-
Euclidean space belongs to the straight line, and belongs to it alone. But how can we
ascertain by experiment if it belongs to any particular concrete object ? Distances
must be measured, and how shall we know that any concrete magnitude which I have
measured with my material instrument really represents the abstract distance? We
have only removed the difficulty a little farther. In reality, the property that I have just
enunciated is not a property of the straight line
(75) alone; it is a property of the straight line and of distance. For it to serve as an
absolute criterion, we must be able to show, not only that it does not also belong to
any other line than the straight line and to distance, but also that it does not belong to
any other line than the straight line, and to any other magnitude than distance. Now,
that is not true, and if we are not convinced by these considerations, I challenge any
one to give me a concrete experiment which can be interpreted in the Euclidean
system, and which cannot be interpreted in the system of Lobatschewsky. As I am
well aware that this challenge will never be accepted, I may conclude that no
experiment will ever be in contradiction with Euclid's postulate; but, on the other
hand, no experiment will ever be in contradiction with Lobatschewsky's postulate.
5. But it is not sufficient that the Euclidean (or non-Euclidean) geometry can ever be
directly contradicted by experiment. Nor could it happen that it can only agree with
experiment by a violation of the principle of sufficient reason, and of that of the
relativity of space. “
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