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In order to measure velocity, one needs a calibrated measuring stick and clock. But in order to calibrate a measuring stick you need a calibrated clock and velocity. And in order to calibrate a clock you need a calibrated measuring stick and velocity. Thus velocity, distance, and time are all defined in terms of each other. Is this circularity a problem that needs to be addressed by a fundamental theory of the universe? And is it addressed by any theories on the market?

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It is worth having a look at how the SI base units are defined (now, it has been different in the past and may be different in the future). The short version is that there is a degree of arbitrarity (is that a word?), but no circularity. –  dmckee Nov 27 '11 at 23:56
    
I suppose it works out because we are lucky in relying on the (well-founded) assumption that all caesium atoms are identical (ie we don't have to measure them first in order to calibrate the expected transition frequency). But ultimately this is an experimental assumption we often take for granted. Fundamentally I still see this as a theoretical circularity. –  user1247 Nov 28 '11 at 0:08
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""And in order to calibrate a clock you need a calibrated measuring stick and velocity."" This is plainly wrong. –  Georg Nov 28 '11 at 11:26
    
@dmckee: I think the word you are looking for is "arbirariness." –  AdamRedwine Nov 28 '11 at 12:23
    
@Georg: How else do you calibrate a clock, unless you take as an axiom that a comparison clock is true? –  user1247 Nov 28 '11 at 22:28

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You are getting confused because you are reading somewhere that the speed of light can meaninfully vary. Then if the speed of light changes, what does it change relative to? How can we tell if the speed of light just doubled, or if all the atoms in the universe suddenly shrunk by a factor of 2? Obviously you can't. There is no difference between these two statements, even philsophically.

There is no meaning to the statement that dimensional constants change. You are being confused by incompetent physicists that claim otherwise, because they want to sensationalize to the general lay public, which does not appreciate this well established fact. You asked this question because you realize that it is meaningless to speak about a dimensional constant changing, and this is causing cognitive dissonance. All physicists go through this cognitive dissonance, and it is fixed when you understand that dimensional constants cannot be meaningfully said to vary

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@user1247: Yes, for string landscape, only dimensionless quantities are different between vacua, it is meaningless to say a vacuum has a different speed of light. It is not completely trivial to sort this dimensional analysis stuff out, although basically everyone (who is competent) does it at some point. There was even a paper not so long ago with three opinions regarding how meaningful dimensional constants are! –  Ron Maimon Nov 28 '11 at 2:24
    
What dimensionless constants would need to change in order to effectively change the speed of light? Like in the classic story by Gamow, "Mr Tompkins in Wonderland", which explores the effects of relativity if the speed of light were 30 mph. –  user1247 May 11 '12 at 11:17
    
@user1247: It's difficult--- you need to keep atomic frequencies and atomic sizes fixed while changing the speed of light, which is the same as making atoms very big (while equally fast), or very fast (while equally big). One way to make the energy levels of atoms further apart is to increase the fine structure constant, but you also need to correct things so that you get the atoms ok again. It's extremely annoying to do. This is why Mr. Thomkins is not best phrased dimensionally, and also why it is a thought experiment that does not make good physics. –  Ron Maimon May 11 '12 at 14:34

My answer PSE-here is part of my answer. From there: "length, time, mass and charge units are deeply linked through the properties of atoms and speed of light."

Taking $c,\varepsilon,G$ as constants ("field constants" because they depend on space properties) the bulk constitutive equations of the world keep good as long as Mass, Charge, Length, Time units (M-L-Q-T) change by the same scale factor. c speed is an invariant (L/T) and it is imposed by space.
Double your atom's mass/size/time/charge and the world will behave as usual.
There are some people that (implicit or explicitly) ascribe a special character of absoluteness to the size of atoms around us. I'm sure that they can not offer a plausible reason to that beleif.
It is the relation [c]=[L]/[T] that allow us to untie the circular link, i.e. the units L and T are not independent.

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In a loose sense (you need to add a mass constant, to speak of momentum instead of velocity), the circularity is addressed by the three constants $\hbar, G, c$ and their respective theories, quantum mechanics, general relativity, and special relativity.

As say by David, $c$ solves the space vs time.

For speed vs distance, it is $\hbar$ what is needed, and in fact the lack of this constant delayed the publication of Newton's Principia, who -after suggestions of two friends- did a lot of rewritting for the theorem of angular momentum, aka equal areas at equal times. Note that the area in an orbit, at fixed $\delta t$, is proportional to speed cross times distance.

I can not argue so easily for speed against time, but if it can be reduced to an issue of spacetime geometry, then $G$ controls it, but again there is the issue that you have no considered mass in your circles. Here it could be more about acceleration or about force.

Another way to see all of this is that in natural units $\hbar=1, c=1$ everything can be written in energy, say $eV$, and then $G$ provides you a natural calibrated stick, Planck mass.

So the second point of your question is yes, it is addressed.

As for the first question, ¿is the circularity a problem for a fundamental theory?, amazingly it seems not, as it is proved by the fact that Newton work was seen as a valid fundamental theory for centuries.

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No, and no. For several reasons:

  1. You do not need a calibrated measuring stick and clock to measure distance and time. You can pick up a random piece of wood and call that your length unit, and measure time using any convenient repetitive physical process, such as your pulse. (Of course a pulse does not make a particularly accurate clock, but just pretend for a minute.) The issue of calibration only arises when you need to make different measuring devices agree with each other.
  2. If you do need to calibrate, say, a measuring stick, the easiest way is to just compare it with another measuring stick. Or to calibrate a clock, you compare it with another clock. You do not need a calibrated measuring stick and velocity. There are many extremely reliable natural clocks that can be used for this purpose. For example, our current time standard is defined in terms of the oscillation period of the EM radiation emitted by a hyperfine transition of the cesium atom. As long as you're sufficiently careful when you measure it, this provides an easily available, naturally pre-calibrated clock.
  3. We also have a natural pre-calibrated velocity, namely the speed of light. Again, as long as you're sufficiently careful when you measure it, this provides an easily available, naturally pre-calibrated velocity. In combination with the definition of the time scale using the cesium HF transition, we use this to define our unit of length.
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Your answer would be satisfactory to an experimental perspective, but it doesn't seem to address the point from a theoretical one. Theoretically, without relying on assumed physical constants (after all the speed of light etc could conceivably be a function of space and/or time), there is indeed a circularity issue. But this isn't necessarily an unsolvable problem. –  user1247 Nov 28 '11 at 0:17
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@user1247: it is a consequence of special and general relativity that the speed of light is a unique universal constant. Now, if you're asking about whether a circularity problem exists without relying on existing physical knowledge, then you should probably take this question elsewhere. You could see whether the people at Philosophy would handle it. –  David Z Nov 28 '11 at 0:22
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@user1247: It is meaningless to say that the speed of light varies with space and time without an independent definition of a unit of time and space. If the length of everything doubles tomorrow and the speed of light halves, and everything happens four times as fast, nothing changes. The thing you are being confused by is that your intuition says that it must be meaningful to speak about a varying speed of light, despite the fact that careful thinking about units shows you that it is not. –  Ron Maimon Nov 28 '11 at 0:37
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@user1247: Can you provide some references? What I know of string theory assumes a universally unique speed of light, to the extent that such a thing can be defined. And again, as far as physics is concerned, there is no problem with any circularity that may exist in the definitions of velocity, distance, and time. This is widely recognized throughout the scientific community. So if you do view it as a problem, it's a sign that you may be considering something which is not science. –  David Z Nov 28 '11 at 1:16
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@user1247: I am indeed familiar with string theory, although I'm not an expert in the field. However, I can say with some confidence that variation in the speed of light, whether from one point in spacetime to another or from one theory to another, is not associated with any sort of string theory or any related theory. –  David Z Nov 28 '11 at 1:37

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