Laws of physics in noninertial frames

[bcrowell]

In a previous thread, the OP asked for answers at the "I" (i.e., undergraduate) level to the following question:

According to the postulates of Einstein theory, laws of physics are same in all inertial frame. What about non- inertial frames? Why they can't be same in non-inertial frame?

S/he received a variety of contradictory answers, and the discussion may have been hindered by attempts to present more sophisticated mathematical ideas as the appropriate level, resulting in some imprecision and confusion. I'm starting this thread so that we can have a discussion at a higher level of precision.

My answer to the OP's question is that the question states a common, historically-based misconception. The laws of physics *can* be the same in noninertial frames, if we write them in a certain way.

First some historical context. Einstein's original interpretation of relativity, when he created it a century ago, was in some ways hazy and incorrect, which is perfectly understandable since the theory was in its childhood. His original formulation of SR gives a prominent role to frames of reference, but it is now understood that the notion of a frame of reference is optional, and frames of reference play no foundational role. Einstein's original description of GR was that it generalized SR to noninertial frames. This is not the way modern relativists describe the distinction. The distinction is now understood as one between flat spacetime and curved spacetime. Unfortunately, many popularizations still state the distinction according to Einstein's original incorrect understanding.

The laws of physics can be stated in ways that do not require the introduction of the notion of a frame of reference. If one wants to adopt a frame of reference, then these statements of the laws of physics can be interpreted in terms of a particular frame, and it makes no difference whether the frame is inertial or noninertial.

Historically one of the most important examples was Maxwell's equations. One way of stating Maxwell's equations is as follows:

$\nabla_s F^{rs} = 4\pi J^r$

$\nabla_{[q} F_{rs]} = 0$

Here the electromagnetic field tensor F, the four-current J, and the covariant derivative $\nabla$ are all tensors, meaning that these equations remain valid under any diffeomorphism. The indices are abstract indices, so these equations make no reference to any coordinate chart. They can also be specialized to a particular coordinate system by replacing the abstract indices with concrete indices. The concrete-index versions are valid, for example, in Minkowski coordinates, which are the coordinates of an inertial observer. They are also valid in Rindler coordinates, which can be interpreted locally as the coordinates of a uniformly accelerated frame of reference.

Also on this topic: https://www.physicsforums.com/threads/distinction-between-special-and-general-relativity.827721/

 

[vanhees71]

I agree with all of that. It's a very modern view of "tensors". In physics we have a somewhat different definition, and that's where this kind of confusion results from.

Electromagnetism is an excellent example. When we learn electromagnetism, i.e., classical electrodynamics in terms of the Maxwell equations, (unfortunately) that's usually done in a fixed inertial frame and in terms of three-vector calculus. Then all quantities are expressed in terms of Cartesian tensors, i.e., multilinear forms with components taken with respect to a Cartesian right-handed basis, and one calls a quantity whose components transforms under rotations of the basis (represented by SO(3) matrices) like tensor components a tensor. One doesn't even distinguish between vectors and dual vectors and writes all indices of components and basis vectors as subscripts.

Then you introduce SR (which in my opinion one should have done before starting with E+M theory and then formulate E+M as a SR classical field theory from the very beginning) and you realize that with great advantage you introduce another kind of tensors, namely such whose components with respect to pseud-Cartesian basis vectors of four-dimensional Minkowski space transform with the $\mathrm{SO}(1,3)^{\uparrow}$ matrices (proper orthochronous Lorentz transformations). Then you have to distinguish between vectors and covectors, i.e., co- and contravariant tensor components, and much more becomes clear. It's also wise to distinguish between tensors and tensor fields, keeping the arguments and the transformation of the latter under good controll either.

In GR you must extend your mathematical toolbox by somewhat more general differentiable manifolds (the torsion free pseudo-Riemannian spaces). Then your tensors live in the tangent and co-tangent spaces at each point (and thus are tensor fields). They are tensors under any diffeomorphism mapping one set of generalized coordinates, defining a map of an open subset of the manifold, to another, defining another such map. This is usually called the "general covariance of GR".

 

[loislane]

The laws of physics *can* be the same in noninertial frames, if we write them in a certain way.

First some historical context. Einstein's original interpretation of relativity, when he created it a century ago, was in some ways hazy and incorrect, which is perfectly understandable since the theory was in its childhood. His original formulation of SR gives a prominent role to frames of reference, but it is now understood that the notion of a frame of reference is optional, and frames of reference play no foundational role. Einstein's original description of GR was that it generalized SR to noninertial frames. This is not the way modern relativists describe the distinction. The distinction is now understood as one between flat spacetime and curved spacetime. Unfortunately, many popularizations still state the distinction according to Einstein's original incorrect understanding.

The laws of physics can be stated in ways that do not require the introduction of the notion of a frame of reference. If one wants to adopt a frame of reference, then these statements of the laws of physics can be interpreted in terms of a particular frame, and it makes no difference whether the frame is inertial or noninertial.

Historically one of the most important examples was Maxwell's equations. One way of stating Maxwell's equations is as follows:

$\nabla_s F^{rs} = 4\pi J^r$

$\nabla_{[q} F_{rs]} = 0$

Here the electromagnetic field tensor F, the four-current J, and the covariant derivative $\nabla$ are all tensors, meaning that these equations remain valid under any diffeomorphism. The indices are abstract indices, so these equations make no reference to any coordinate chart. They can also be specialized to a particular coordinate system by replacing the abstract indices with concrete indices. The concrete-index versions are valid, for example, in Minkowski coordinates, which are the coordinates of an inertial observer. They are also valid in Rindler coordinates, which can be interpreted locally as the coordinates of a uniformly accelerated frame of reference.

I basically agree with the above. For instance when you say:"Einstein's original description of GR was that it generalized SR to noninertial frames.This is not the way modern relativists describe the distinction", that was exactly my point in the last paragraph of my post #5 in the other thread.

Just a few precisions: Certainly the distinction between inertial and noninertial frames can be problematic unless they are clearly defined which is not common, that is why I tried to restrict the discussion to something much clearer mathematicaly: a distinction between coordinates systems related to Minkowskian coordinates by transformations dictated by the tensor group of symmetries and those that are not. Such a mathematical clear distinction seems to have stirred a couple of emotional(in the sense of not rationally argumented) responses, and that's upsetting.
So when saying that "these equations remain valid under any diffeomorphism", you should specify that you mean under any diffeomorphism within the Minkowski group of symmetry(the Poincare group), otherwise you empty of mathematical content the difference between SR and GR, since GR is indeed defined so the EFE are valid under any diffeomorphism(not just those referred to the Poincare group). You seem to be ignoring the distinctions about tensors made by me and also by vanhees71.

The Rindler coordinates are a good example, since you are advocating leaving behind the concepts of inertial frames, let's not assign them a frame tag yet.
Rindler coordinates are those in which an observer undergoing continuous boosts(hyperbolic rotations) remains at rest. These singular coordinates are obviously related to Minkowski coordinates by transformations invariant under the Poincare group of symmetry, since the Rindler chart is a coordinate chart for Minkowski spacetime. In that sense these observers are undergoing inertial motion with respect to observer at rest in Minkowski(cartesian) coordenates, since they are just performing continuous compositions of boosts(that involve both pure rotations and boosts).
Whether you want to consider Rindler coordinates as an example of "inertial" or "noninertial" frames(since as you say you can interpret them as performing uniform acceleration) is a matter of interpretation but as you assert "The laws of physics *can* be the same in noninertial frames, if we write them in a certain way".

P.S:.I would have appreciated an specific analysis of the points that you and martinbn found wrong in my post instead of vague assesments about incorrect statements or not making sense without arguments . You at least have had the nice gesture of starting a new thread, but the other guy who claims in his profile to hold a math Phd just hits and runs.

 

[bcrowell]

Just a few precisions: Certainly the distinction between inertial and noninertial frames can be problematic unless they are clearly defined which is not common, that is why I tried to restrict the discussion to something much clearer mathematicaly: a distinction between coordinates systems related to Minkowskian coordinates by transformations dictated by the tensor group of symmetries and those that are not.

This just sounds to me like you're confused. In GR, Minkowski coordinates don't normally even exist, so it's not possible to define an inertial frame using the method that you're attempting to use. I don't know what idea you're trying to express by "the tensor group of symmetries."

So when saying that "these equations remain valid under any diffeomorphism", you should specify that you mean under any diffeomorphism within the Minkowski group of symmetry(the Poincare group),[...]

No, I meant what I said, and what I said was correct.

otherwise you empty of mathematical content the difference between SR and GR, since GR is indeed defined so the EFE are valid under any diffeomorphism(not just those referred to the Poincare group).

No, this is incorrect. You are referring to a historically based misconception. Modern relativists define the distinction between SR and GR as the distinction between flat spacetime and curved spacetime.

You seem to be ignoring the distinctions about tensors made by me and also by vanhees71.

The distinction vanhees71 made is fine. The distinction you're trying to make just indicates that you're confused.

These singular coordinates are obviously related to Minkowski coordinates by transformations invariant under the Poincare group of symmetry, since the Rindler chart is a coordinate chart for Minkowski spacetime.

You seem confused here as well. It doesn't make sense to talk about transformations that are invariant under a certain symmetry group. Other quantities, such as inner products, can be invariant under a transformation. That transformation might or might not be a member of the Poincare group.

In that sense these observers are undergoing inertial motion with respect to observer at rest in Minkowski(cartesian) coordenates, since they are just performing continuous compositions of boosts[...]

No. Such an observer is noninertial.

[...](that involve both pure rotations and boosts).

No. Here you seem confused by the fact that the composition of boosts along different axes can be considered as a boost plus a rotation. That is not the case here, since the boosts are along the same axis

P.S:.I would have appreciated an specific analysis of the points that you and martinbn found wrong in my post instead of vague assesments about incorrect statements or not making sense without arguments . You at least have had the nice gesture of starting a new thread, but the other guy who claims in his profile to hold a math Phd just hits and runs.

When a long post contains as many mistakes as that one, it's very time-consuming to go through and try to explain them all. That's not always going to be practical.

 

[PeterDonis]

Thread closed for moderation.