6

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?

  • 4
    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
  • This is clear as mud to me. MBN's comment seems to show that this is in the context of Galilean relativity. In Galilean relativity there is no empirically testable homogeneity of time. Actually, GR doesn't even have a concept of homogeneity of time. Maybe he means that the laws of physics are position-invariant? That would make a lot more sense. –  May 28 '13 at 01:25

2 Answers2

3

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

Physicsworks
  • 1,291
  • 2
    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
  • 2
    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?

     –  Jul 01 '11 at 03:20
  • 2
    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 01 '11 at 13:25
  • @ Willie Wong I would like to know, can we do something in String Theory and then take some Newtonian approximations, and then play with equations, and then bang there comes out that space and time do have required properties and inertial frames exist? –  Jul 01 '11 at 17:18
  • 1
    @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? – Student Jul 01 '11 at 18:09
  • @Lakshya: depends on what you mean by "Newtonian approximation". If your approximation procedure contains in it some hidden assumptions about the homogeneity/isotropy conditions, then of course you will get such a space-time out in the end. – Willie Wong Jul 01 '11 at 18:30
  • @Anirbit: Not quite. My interpretation of Landau is more along the lines of there being a coordinate system/reference frame in which the (say) Poincare group is manifestly obvious. If you pick a bad coordinate system on Minkowski space, the underlying Lorentzian manifold still has all the symmetries. It is just hard to see because of the bad choice of coordinates. – Willie Wong Jul 01 '11 at 18:40
  • @Willie I think we need to be careful about whose symmetry we are talking of. No matter what reference frame one is sitting in the theory/lagrangian will have always have all the symmetries that you want - say the Poincare group. But this does not oblige that every solution of its equations of motion - the solution you are sitting in at any event - respects that that symmetry. But the space of all solutions will see the full symmetry group in some subtle way. – Student Jul 01 '11 at 21:50
  • @Willie In Einstein's theory I guess the analog is that one can get metrics with peculiar symmetries out of the lagrangian which always has the full diffeomorphism invariance of the manifold. – Student Jul 01 '11 at 21:51
0

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.

Helder Velez
  • 2,643
  • 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? :) –  Jul 02 '11 at 17:07
  • 1
    @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. – Student Jul 03 '11 at 07:17
  • 1
    @Anirbit - the computation of the energy of a system is reference dependent, and not all frames are equal. CMB reference is a special case of the Inertial_frame_of_reference. Newton , in this matter, was not worried with the properties of the light. May be we should; the Coriolis effect is evident only when we jump to a reference in the outside of the mobile Earth. We study the energy in the 'mobile' lab and we need artifacts to balance the equation of energy. Its mandatory. Physics did not ended with Einstein. We need to go forward. – Helder Velez Jul 03 '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? – Student Jul 03 '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. –  Jul 04 '11 at 10:39
  • @Lakshya now I see that you are seeking more insight. Good. I wish that someone can formulate a relevant Question. It will attract downvotes (both the Query and my Answer) by all those that think that Einstein relativity is good to deal with this, one of a kind, referential. In the meantime study the "Alfredo's papers"; you can find them. – Helder Velez Jul 04 '11 at 11:13
  • @Helder http://physics.stackexchange.com/questions/11849/cosmic-microwave-background-cmb-and-its-relation-to-inertial-frames –  Jul 05 '11 at 03:37
  • Actually, the first edition (Russian, of course) of "Mechanics" was published in 1940. The book was written in collaboration with Leonid Pyatigorskiy. After the strong critique by Vladimir Foсk (the first edition contained too many errors, not misprints, but physical ERRORS) the book was rewritten with Evgeniy Lifshitz and published in 1958. – Physicsworks Jul 10 '11 at 11:56