Reference frame doubts about isotropy

Question

Landau & Lifshitz on p.5 in their "Mechanics" book states the following:

...a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous. This is called an inertial frame.

In Newtonian mechanics, ground frame is almost considered inertial, but we know that in ground frame, space is not isotropic (preferred direction due to gravity). So Landau's statement actually seems wrong if we consider Newtonian mechanics.

In General Relativity, ground frame is non-inertial and free fall frame is inertial in which we know that space is really isotropic - landau's statement holds true.

Even in newtonian mechanics, some inertial frames space can still be isotropic/homogeneous, but since we found even one example where it's not - i.e ground frame, then landau's statement can't be true in general for newtonian mechanics. i.e - when a statement is made, it must either be true in all cases or statement is false.

I think, Landau makes his statement in General Relativity, but nowhere, he mentions general relativity. The statement I copied is taken from his chapter of Galillean Relativity. What's your opinions on what he meant ? Should I also be thinking in terms of General relativity and not newtonian ?

Answer 1

You are quite correct that the ground frame is not inertial, or at least in General Relativity we would say it is an accelerating frame.

However in Newtonian mechanics we can say it is an inertial frame with gravitational forces acting. That is, we can say the ball accelerates downwards because there is a gravitational force acting on it not because the ground frame is non-inertial. By contrast Einstein would say the ground frame is non-inertial and the gravitational force is a fictitious force due to the non-inertial frame.

This is a slightly subtle point. Imagine starting with the observer floating in space far from any masses - we would all agree this is an inertial frame. Now tie the observer's feet to an extended spring so the spring starts to accelerate the observer. Clearly the frame is still inertial but we have an external force due to the spring. If we move the observer a small distance $y$ there would be a potential energy change of $Fy$, where $F$ is the tension in the spring, but this doesn't make the frame non-inertial. That PE change is the change in the PE of the spring.

In a similar way we could start floating in space then bring the Earth towards the observer until the surface of the Earth is just below the observer. Newton would say that just as in my previous example we have not done anything to the "space". We have just applied an external force because the Earth is now pulling on the observer just like the spring. So space itself is still an inertial frame but there are external forces acting. The potential energy change $mgy$ when we move the observer distance $y$ towards the Earth is the change in the energy stored in the Earth's gravitational field.

Answer 2

Landau's statement does not apply in General Relativity. But it is valid in both classical, in the sense of nonrelativistic mechanics and in Special Relativity.

Moreover even in General Relativity I know that there is a notion of "local inertial frame" but it is a bit complicated (for me, at least).

Answer 3

As far as I remember, the above-mentioned statement in L-L's book takes place in the first volume thus regarding classical mechanics.

Even if I dislike Landau-Lifschitz' approach, the requirements of homegeneity and isotropy of the space uniquely determine the inertial reference frames in classical mechanics.

That is because

(a) inertial forces are both homogeneous and isotropic only in those reference frames, as they vanish there;

(b) the remaining general dynamical laws are already homogeneous and isotropic in the inertial reference frames.

Homogeneity and isotropy of space in dynamics means the following in practice.

Consider an isolated (sufficiently far from the other bodies of the universe) mechanical system made of an arbitrary number of interacting bodies.

Prepare it with some initial conditions and record the dynamical evolution of the parties of the system with a fixed camera.

Next prepare a similar experiment with initial conditions translated (homogeneity) and/or rotated (isotropy) and perform the same action on the position of the camera.

Finally, compare the recorded movies: it turns out that you cannot distinguish them.

This happens only in inertial reference frames. If you try to do the same in non-inertial ones, you get different movies even if the dynamical systems are identical (including the interactions). That is because the apparent forces act differently in different places of the space (translated and/or rotated with respect to the initial ones).

Inertial reference frames posses also a further property: time homogeneity. The practical explanation is similar to the one above just considering translations in time.

Finally there is a further invariance property related to the boost translations, but it changes the (inertial) reference frame.

(As is well known, each of these (3+3+1+3=10 scalar) symmetries implies the existence of a corresponding (scalar) dynamically conserved quantity. But this is another story.)

Earth's frame is not inertial so that it is not a problem.

Passing to more modern formulations, like special and general relativity, the issue becomes much more delicate in view of the notion of rest frame with an observer which should deserve a deeper discussion.

Answer 4

Even though Rennie and Moretti's answers cover most of the issues connected to this question, I would like to address it more directly, eliminating possible misunderstandings from the beginning.

First, the volume's perspective on mechanics is classical mechanics. No general relativity anticipation was intended.

Second, the ground reference frame, i.e., the reference frame of a laboratory on Earth's surface, is approximately inertial, but not exactly. Deviations from the expected behavior do not depend directly on the presence of gravity but on Earth's motion. If the Earth were not rotating and not revolving around a star in a galaxy, but it was the only planet in the universe, it would behave like an inertial system because a body subject to Earth's gravity plus a compensating force would move with a uniform rectilinear motion. Deviations from the behavior in an inertial reference frame can be detected if the experiments' precision is high enough or the observation time is long enough.

Now, let's go to the Landau- Lifshitz approach. According to the modern view of mechanical phenomena, it is challenging to state Newton's laws satisfactorily.

The main difficulty is in the intertwined role played by the concepts of reference frame, mass, and force. Notice that in Newton's original perspective, the situation was different because the role of reference frames was not so important as after Special Relativity, and also because Newton had his concepts of absolute space and absolute time playing indirectly the role of inertial reference frames.

In Classical Mechanics, inertial reference frames play an important role as the only reference systems where accelerations depend only on real interactions between bodies. A possible way of characterizing an inertial frame is then to say that it coincides with a reference frame far enough from any other body, such that any single particle motion is on a straight line and uniform. The precise formulation may differ, but the essence is this one.

Such a definition may seem to exclude the surface of a planet, where the motion of particles on which only gravity acts is accelerated. However, this is not a real difficulty. Once we have one inertial reference frame where any external force is reduced below the observation threshold by the large distance from other bodies, we know that infinite other inertial frames differ only by the choice of the origin or by the relative uniform velocity. One of them may coincide with the center of a planet not interacting with other bodies.

Landau$Lifshitz's proposal is within the previously described approach. They only shift the emphasis from the characteristics of the motion of a test body to the symmetry conditions allowing them. Notice that from the physical point of view, any statement about space and time, to be amenable to observations must be translatable into a statement about the relative positions of physical systems and physical events. Hidden in their formulation is the asymptotic behavior of a body far enough from any interaction source. Again, once one has found at least one of Landau & Lifshitz's inertial reference frames, there is an infinite number of other inertial reference frames close enough to every possible source of interaction.