Are Lorentz transformations exact and universal?
- Postulate 1 (the principle of relativity): All inertial frames of reference are equivalent.
- Postulate 2 (the invariance of the speed of light): The speed of light is the same for all observers and does not depend of the speed of the source.
Then the textbook discusses various physical events usually associated with light pulses. For such events one can use the Postulate 2 and derive the beautiful results of special relativity: the time dilation, the length contraction, the relativity of simultaneity, etc. All these results are summarized in the Lorentz transformations for the time and position of events. All this is fine and good.
The problems begin when one wants to extend the Lorentz transformations and other special-relativistic formulas to events associated not with light pulses, but with systems of interacting particles. How can we do that? There is no Postulate telling us anything about interacting particles. So, when we say that Lorentz transformations apply universally to all events and phenomena in the world, we make a huge leap of faith. Actually, we introduce into special relativity a third Postulate
- Postulate 3 (the universality of Lorentz transformations) The Lorentz transformations of special relativity apply to all events independent on their nature and involved interactions.
Only after this Postulate is introduced, we can notice the suspicious symmetry between spatial and temporal coordinates of events and how they transform to the moving frame of reference. Only then we can follow Einstein and Minkowski and boldly introduce the idea of space-time unification into one 4-dimensional continuum.
The question I would like to ask is this: What is the reason for introducing this third (hidden) Postulate? Yes, I know how well special relativity agrees with experiment. However, it could be possible that one can abandon Postulate 3 and still keep this perfect agreement with experiment. Actually, this is possible. One can construct a theory in which Postulates 1 and 2 are respected, but Postulate 3 is not. In this theory, therefore, there is no place for the 4D Minkowski space-time. In this theory, the Lorentz transformations are violated by a tiny amount that depends on the strength of interaction acting in the physical system. These violations are, in principle, observable. However, they are a few orders of magnitude less than the resolution of modern instruments. More about that in later posts.
posted by Eugene Stefanovich at 12:55 PM
6 Comments:
What about interpretations of Lorentz equations that are different to Einstein's? One can think of the Lorentz and Poincare one's, for example, which were not identical to Einstein's. Here are two new interpretations that I have just seen:
http://arxiv.org/abs/physics/0605199
http://www.metaresearch.org/cosmology/gravity/gps-twins.asp
There might very well be others that might provide new directions.
anonymous:
I briefly looked at these articles. It seems to me (please correct me if I am wrong) that their common thread is the rejection of the principle of relativity (= the equivalence of all inertial frames of reference). These articles seem to suggest that one can determine the absolute velocity of the reference frame (either with respect to distant stars or with respect to the aether), and that this velocity somehow affects the physical processes occuring in this reference frame.
There is nothing wrong with these ideas. However, I, personally, don't find them particularly attractive. For example, I would like to believe that if all distant stars and galaxies were somehow removed, then this wouldn't make any significant effect on physical processes here on Earth. I prefer to believe in the Galilei-Einstein principle of relativity. Of course, I don't have a proof of that. This is just my personal preference.
We are all entitled to our personal beliefs until these beliefs come into contradiction with reality (= experiment). So far, the principle of relativity was not disproved by any experiment or observation, and I would like to stick to it.
http://physicsmathforums.com/showthread.php?t=64
"http://arxiv.org/abs/gr-qc/0011064 on page 3 shows how the FitzGerald-Lorentz contraction formula for gamma can result from head-on pressure when a charge is moving in the Dirac sea vacuum, quoting C. F. Frank, 'On the Equations of Motion of Crystal Dislocations', Proceedings of the Physical Society of London, Society London A 62 (1949), 131–134. ...
"Mario also has the interesting paper http://arxiv.org/ftp/astro-ph/papers/0412/0412101.pdf which an interesting discussion of how you can get a heuristic understanding of general relativity:
" 'Friedwardt Winterberg (2002) presented a simple heuristic
derivation of the Schwarzschild metric. Even though it is not rigorous, it
does provide insight into eq. (2.1) in terms of Newtonian gravity (NG) and
special relativity. It is presented here with some minor additions.'
"You equate kinetic energy of a falling object with the gravitational potential energy gained, and that gives you a relationship between the square of the velocity, ie v^2, and gravity:
"v^2 = 2GM/r.
"You then stick that equivalence into the FitzGerald-Lorentz contraction of special relativity and you get the gravitational contraction factor,
"gamma = (1 - v^2 / c^2)^1/2 = [1 - 2GM/(rc^2)]^1/2
"Placed into the special relativity metric for contraction of distance by gravity, this gives you the Schwarzschild (1916) solution to the Einstein-Hilbert field equation.
"Plus, (see http://feynman137.tripod.com/ which is independent of Winterberg's earlier work) by using the binomial expansion on the gravitational contraction you get
"gamma = [1 - 2GM/(rc^2)]^1/2 = 1 – GM/(rc^2) + ...,
"So the spacetime around mass M is contracted approximately by the fraction GM/(rc^2), assuming that the gravitational field only operates in a single direction (like the contraction of special relativity). Because of the linkage of time and distance in spacetime, the contraction of distance automatically causes an identical-factor reduction in the speed of clocks locally, gravitational time-dilation.
"To see the contraction of Earth's radius, we need to multiply this fraction to Earth's radius, and to allow for the contraction to be spread over three orthagonal directions, which reduces the average contraction to 1/3 rd of that assuming only one dimension is contracted.
"Hence, contraction of Earth's radius = GM/(3c^2) = 1.5 millimetres,
"which agrees with Richard P. Feynman's finding in equation 42.3 of the Feynman Lectures on Physics, chapter 42, page 6.
"So I think general relativity can be understood in terms of very simple ideas. This contraction is what you would expect if exchange radiation (gauge bosons) cause forces by physical pressure acting at the fundamental particle level. Things get squeezed a bit by the spacetime fabric, gauge boson exchange radiation (or whatever the spacetime fabric)."
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Eugene,
I view your dream of a theory of possibly modifying the Lorentz transformation equations in the presence of systems of interacting particles as a hopelessly impossible task. Are you sure that there is no other way?
Eugene,
yes, I am sure there is no other way. I've put all my arguments in the book. If you think these arguments are invalid I would be happy to discuss your objections.
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