Landau's ambiguous statement about the existence of inertial frames

Landau writes "It is found, however, that a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous."

Does he mean that we can prove the existence of an inertial frame or does he want to say that it is assumed by doing enough number of experiments?

Can we start with some axioms and definitions of properties of space and time and then deduce the existence of such a frame in which space is homogeneous and isotropic and time is homogeneous?

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Can you give an exact citation? Is it in Classical Mechanics? Where exactly? Context would help a lot in evaluating this statement. – Ted Bunn Jun 30 '11 at 18:58
It is in "Mechanics" on page five, of the edition I looked at, in the part about Galileo's principle of relativity. – MBN Jun 30 '11 at 19:17

2 Answers

I believe this is just a restatement of the first Newton's law.

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I think that's right, and I think Landau means the statement as an empirical fact (derived from experiments), rather than a mathematical theorem. It's not a mathematical theorem in any sense I can think of, and I think Landau's too smart to claim it is. – Ted Bunn Jun 30 '11 at 21:58

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Perhaps the equivalence of this statement and the first law can be stated as a theorem – MBN Jun 30 '11 at 22:50
Hi, thanks for your replies. But, see, I didn't post this question to know the opinions of whether one thinks it is can be shown or it is assumed experimentally. I personally believe that it is taken as a granted fact and we assume (in classical mechanics) that the frame fixed to this universe as a whole is inertial. As Landau is just a starting book for Classical Mechanics and this statement does appear to be ambiguous, I want to know if using some advanced physics (string theory?) can we theoretically show the existence of such a frame in which space and time have the required properties? – Lakshya Bhardwaj Jul 1 '11 at 3:20

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No. Those are postulated. Newtonian mechanics can be constructed on any Galilean manifold. In particular, Newtonian mechanics can be constructed with a spatially inhomogeneous geometry (take any Riemannian manifold for space and cross it against $\mathbb{R}$ for time). Even if you consider relativistic mechanics, the situation is not better. In general relativity already homogeneity and isotropy of space-time is abandoned (except in cosmological models); string theory won't make it better. – Willie Wong Jul 1 '11 at 13:25

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@Willie Great to hear from you again. I was wondering if this statement is basically about the symmetry group - Poincare group lets say. You can always be sitting in a reference frame where the symmetry group is not the full Poincare group - if you are looking at the world by sitting on a merry-go-round..but you can always do a Lorentz transformation to such a frame where the symmetry group is the full Poincare group. Is that what is being said? – user6818 Jul 1 '11 at 18:09
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The edition Mechanics was 1st published at 1960 and was written earlier than that. Landau died in 1968 aged 60.
Just 4 years before, in 1964, the CMB was discovered and the properties of it was unknown for years after.
It appears that Landau's saying was vindicated with a referential with the properties of the CMB.

The Earth (we, and the labs) is moving in relation to the CMB and the universe appear to us to be non isotropic. From the perspective of CMB, the referential of light, i.e. where light propagates equally in all directions, the universe is isotropic.

In addition every observer in the universe can use (share) a common length and time base using the CMB properties for the calibration purposes.

This special referential is not attached to any observer as with the Einstein ones.

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So, did Landau sense the existence of something like CMB 4 years before its discovery? And can we arrive at all properties of CMB with something more fundamentally theoretical? :) – Lakshya Bhardwaj Jul 2 '11 at 17:07
@Helder I have little clue as to what you are trying to say. Just to point out that the existence of CMB does not provide a notion of any fixed/special reference frame (..in the Newtonian sense..) One still has the foundational ideas of Einstein that physical parameters like length, mass.time intervals and concepts like simultaneity of events are dependent on the observer's frame of reference. The CMB frame is as good or as bad as any other frame. – user6818 Jul 3 '11 at 7:17

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– Helder Velez Jul 3 '11 at 12:58
@Helder I simply don't get what you are trying to say. Many things are fuzzy about your statement. Can you precisely define what would you call the CMB reference? Why is that an inertial frame? (..how are you defining it?..definitions can differ depending on what approximation you are making about gravity..) Why are you picking that out? Can you give a technical reference to what you are saying? May be a paper or a book reference? – user6818 Jul 3 '11 at 19:25
@Anirbit I think Helder has given this great idea on which we can think of finding some kind of universal inertial frame. See, at the start of universe, no direction had any kind of reservation as that Big Bang should favour that direction over others. So, it seems pretty reasonable to beleive that as a whole the universe is isotropic in nature although we still can't say its homogeneous. Moreover, CMB is the purest form of remains of early universe free from fluctuations and it should (acc to aesthetic appeal of CMB) form the basis of next theory which involves inertial & non-inertial frames. – Lakshya Bhardwaj Jul 4 '11 at 10:39
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