If we see inertial frames from a basic point of view (precisely more basic axiom from which I can at least derive the law of free body as in landau mechanics first chapter) that inertial frames are ones in which space is homogenous and isotropic and time being homogenous then my question is how to determine from this point of view,which frame is inertial?And why can't accelerated frames be inertial and more precisely why can't accelerated frames have homogenous and isotrpic space and time as homogenous?


Landau proves on p. 5 (3rd ed.) that in an inertial frame, the velocity of a free particle is constant. So if we have inertial frame A and a frame B that is accelerated relative to A, then B cannot be inertial. For a particle whose velocity was constant in A would not have a constant velocity in B.

He proves this using the Lagrangian formalism, but I think it can be made more obvious by considering a particle that is at rest in A, in the case where B is initially at rest relative to A. Then in B, the particle will be initially at rest and then accelerate in some direction. This acceleration picks out a direction in space, which violates the isotropy of space.

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  • how to determine from this point of view, which frame is inertial?

One really doesn't do that. There is no (commonly used) way to determine directly whether "space is uniform and isotropic"; it isn't something operationally defined. In physics, this homogeneity and isotropy is just a code language (or possibly a bad terminology) for "Lagrangian does not depend on position, or rotation angles, or time".

Inertial frame is an idealization that is never fully verified to exist in practice. What one can do, is to state some acceptable deviation from the inertial behaviour for the given frame and space-time extent and measure motions of apparently free bodies in this extent and check if the actual deviations from rectilinear motion are within the acceptable deviations. If so, such frame in such space-time context can be idealized as inertial.

  • And why can't accelerated frames be inertial

They can, if the acceleration has negligible impact. For example, frame centered in the Earth but not rotating with it is, for the motions on the Earth, inertial with great accuracy. It is also an accelerated frame, since its origin orbits around the Sun. This acceleration is so small that it is neglected and that's why the frame is said to be inertial.

  • can't accelerated frames have homogeneous and isotropic space and time as homogeneous?

If the acceleration has significant effects, such as the inertial force or Coriolis force, then the system cannot be described by a Lagrangian that ignores this. The proper Lagrangian for a massive particle will include this acceleration in a way that will make the Lagrangian dependent on position of the particle or time, which will make "space not homogeneous" or "space not isotropic" or "time not homogeneous".

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    $\begingroup$ There is no way to determine directly whether "space is uniform and isotropic"; it isn't something operationally defined. Sure it is. If I release a test particle at rest, with no forces acting on it, and it accelerates in some direction, then space is either inhomogeneous or anisotropic. The question is about the logical development of Landau. Rejecting Landau's logical framework doesn't answer the question. $\endgroup$ – user4552 May 21 '19 at 13:57
  • $\begingroup$ Landau assumes that inertial frame exists and can be chosen, but all he does in making the choice is that he discusses how things look like only in this preferred subset of frames. He does not specify how a frame should be tested for inertiality in practice, because that is of no importance to his argument. He does not claim that this is possible in a practical sense. There are two problems with your method. First, one has to make sure the particle is subject to zero net force. Second, one has to test all points of spacetime in the studied frame.These things are just not done in practice. $\endgroup$ – Ján Lalinský May 21 '19 at 16:24

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