Wave hits beach-example: Does sweeping crest constitute a Frame of Reference?

Question

A long wave rolls in at 1 m/s and hits an almost parallel beach. It's 900k km long, and hits the beach at .1 micro degrees.

Basic math tells us the crest of the wave rides down the beach at a speed of 5.7*10^8m/s, or roughly twice the speed of light.

Let the moving crest define the origin (0,0,0) of a frame of reference, which we can call F-crab (from Frame-crest-running-along-beach).

Classical mechanics survive just fine in F-crab. The law of inertia still holds. F-crab should easily qualify as a 'classical inertial frame of reference' but not as an 'inertial frame of reference'.

Question: Does F-crab qualify as a 'frame of reference'?

The answer is relevant for wikipedia-entries like this one:

Particles with nonzero rest mass can be accelerated to approach c but can never reach it, REGARDLESS OF THE FRAME OF REFERENCE in which their speed is measured. (capitalized by me)

https://en.wikipedia.org/wiki/Speed_of_light

---

For the record, all measurements assume a stationary observer in the middle of the beach. The only physical movement is the wave at 1m/s.

I'm aware of posts like this: The reference frame of $c$

where the answer seems to be that F-crab isn't allowed, and so cannot exist.

This sounds weak, since F-crab is literally sweeping the beach right before our eyes.

Answer 1

You should be aware that different authors will define specific terms differently. Often the common definitions are equivalent in normal scenarios, but deviate from each other in edge cases.

The following is not an exhaustive list, but some common definitions

1. An inertial frame is a system of coordinates where objects experiencing no net force travel in straight coordinate lines at constant speed.

2. An inertial frame is one where the metric is $ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$

3. An inertial frame is a tetrad whose integral curves form a timelike congruence with no expansion, torsion, or proper acceleration.

Your frame would be a valid inertial frame under definition 1, but not under definitions 2 or 3.

In addition to definitions of inertial frames there are also definitions of reference frames that vary from author to author.

1. a reference frame is a system of coordinates covering some region of spacetime

2. a reference frame is an orthonormal tetrad with one timelike and three spacelike vectors.

Similarly, your frame would be valid as a general reference frame under definition 1, but not under 2.

Now, whether your frame is a valid frame or inertial frame is a matter of definitions and categorization. The physics is clear: no massive object can be at rest in your system whether you call it inertial, or a frame, or not. Furthermore, with the metric in your coordinates all of the other physics can be self consistently calculated, though the results may differ from inertial-frame calculations from authors using a different definition.

It is important to understand how a given author uses a term, particularly when trying to apply derived results by said author to edge cases like this one.

Answer 2

First, keep in mind that nothing is racing up the beach at twice the speed of light. You have arranged a wave in advance where points along the crest travel toward the beach at a few m/s. You have arranged things so successive points hit the beach in close succession. But these are independent hits that have been arranged in advance.

Note that if the wave hit the beach square on, the "speed" of the "point of intersection" would be infinite.

---

A set of Cartesian coordinates and clocks attached to the beach is an inertial frame of reference (ignoring the rotation of the Earth and such.) The beach is at rest in this frame. An object that experiences a total force of $0$ travels at a uniform speed in this frame. All massive objects travel at a speed less than $c$ in this frame, and light travels at $c$..

You could define a set of Cartesian coordinates and clocks that race up the beach at $2c$. No object can be at rest in this frame. It is moot whether you would name this an inertial frame. It isn't a useful one.