Mass and angular momentum, left ambiguous by Einstein, get defined

In support of your idea, engineers routinely use FEA simulation software that models dynamic systems without any explicit representation of angular momentum or rotational inertia. Only linear displacement/velocity/acceleration, "translational" mass, and elasticity. Such software can still model rotating systems and all the complex effects that come with rotation. So at least classically, I don't think angular momentum is a fundamentally essential concept.

The CMB is assumed to be what it is stated to be. If you look at the various theoretical predictions for the CMB, you will see that the theoretical predictions have been quite inaccurate. In absolute magnitude, the temperature that would be expected is quite small. However, the error in those predictions relative to each other is quite high (way too high, if I recall correctly this error is on the order of +/- 50%). I have done laboratory experiments which had similar errors and it was considered a failed experiment. The predicted value from theory was supposed to be less than a couple of percent to give some credence that the theory was actually applicable.

Based on the evidence so far provided, I have some concerns that what the CMB is supposed to represent is not actually what it represents.

Once you have experimental or observational anomalies that conflict with theory, one has to look at that theory as either being wrong or incomplete. After 40 years of watching these things, I conclude that we are still very ignorant of the nature of our universe.

Unfortunately, there are inherent infrastructure problems in scientific investigation and we often see politics and dogma interfering. This works to limit our understanding. I see this as a repeat of what was happening in the late 19th century. We didn't learn from our mistakes then and so we are doomed to repeat those mistakes now.

Isn't the CMB far more uniform than we would expect it to be if the speed of light was not the same in all directions?

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tl;dr: coincident-events first, then labels (coordinates). [Einstein 1916, p.117 [2] although I remembered to look there only after writing all of the below]. One-way speed of light arguments are in danger of being coordinates-first, and thus insufficiently general for physics.[3]

The key word in your comment is

> directly

But why do we care? We have an abundance of indirect evidence, premised on direct tests of coordinate-independent features of our best most-fundamental theory. The two important features of (general) relativity are pointwise local Lorentz covariance -- where c is the only free parameter of the Lorentz group -- and the minimal coupling. Special relativity's Minkowski space is in this view a special static time-orientable spacetime in which we have global Poincaré invariance (c again is the only free parameter of the Poincaré group; the Lorentz group is a subgroup of the Poincaré group -- the latter includes all the spacetime translations, and in the Minkowski case the space-translation and time-translation symmetries all commute). When we go blithely parallel-transporting null vectors, this is what matters.

We can certainly write down an f(c) theory. Dicke did this in in his superb 1957 "Gravitation without a Principle of Equivalence" >https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29....? and so have several others (See Ellis or Magueijo for a review https://link.springer.com/article/10.1007/s10714-007-0396-4> corresponding with https://arxiv.org/abs/astro-ph/0703751> resp. https://iopscience.iop.org/article/10.1088/0034-4885/66/11/R...> open access, but corresponds with https://arxiv.org/abs/astro-ph/0305457>).

It is far from silly to write down a theory where c varies in spacetime. It is the foundation of several alternative-to-cosmic-inflation decaying-bimetric theories of the very early universe, where c eventually stabilizes to its value in our local spacetime having been a different (typically much much much -- ~30 orders of magnitude -- higher) value during the formation of primordial matter density variations. The faster speed of light allows for distant reaches of the early universe to reach the same temperature with uniformity up to the small fluctuations in the cosmic microwave background.

Of course we run into the same point you've been working in this thread: it's hard to discover the exact function on c in the early universe. We have to rely on indirect evidence, and strong gravitational lensing is useful there. SVOM https://svom.cnes.fr/en/SVOM/GP_mission.htm> is looking for Lorentz-invariance-violation (LIV)-induced modifications to the photon dispersion relation in vacuum, and is a particularly good platform for test of a Taylor-series expansion like E^2=p^2 c^2 ( 1 +- \sum_{n=1}^{\infty} a_n ), since GRBs at least somewhat escape the problem that the lowest order terms dominate at small energies, and they are distributed across the sky and at different redshifts. We are also now better equipped to study light echos (oh for a galactic supernova!) and detailed strong galactic lensing studies. Spoiler: the constraints on a spacetime-translational variation of c grow tighter with every observation. However, to fully rule out a sharp phase-change in c, we will need practical ~ 10^-15 Hz gravitational-wave astronomy. LIGO is most sensitive around 10^2 Hz; eLISA would be around 10^-2 Hz. (I am fairly sure the authors of most modern variable-speed-of-light early cosmologies knew as they were writing that they probably could hide in that hard-to-explore space through a few generations of gravitational wave observatories. One might say the same about a wide variety of recent cosmic inflation theories, too.)

In a general dynamical spacetime the notion of a two-way path is tricky. Even in Minkowski space, for a two-way signal, the return detection arrives at a different, later, point in spacetime than the outbound signal, even if the spacelike coordinates are always (0,0,0) [this is somewhat reminiscent of the twin paradox]. Outbound-and-return are two future-directed null geodesics. Your argument in this setting is equivalent to saying that we are somehow in trouble because the "outbound" and "return" null geodesics may have, without rescaling, different affinely-parametrized lengths.

In SR what we care about is that the signal is Lorentz invariant at each spacetime point where it could be sampled, even as sender and receiver/reflector are moving ultrarelativistically or are ultraboosted. Given Lorentz-invariance we can determine the three relevant points on the manifold. (Poincaré invariance means we can do this same test at any time or place in the flat space universe). Your complaint is that this is not a direct one-way measurement. OK, it's not. So what? We can in principle directly test Lorentz-invariance at any point (e.g. we can have a sparse gas with a well-understood (as in at the Standard Model of Particle Physics level) low extinction coefficient). If we have no flat space violation of Lorentz-invariance, we must have symmetry of light travel time for constant light-like separation.

In a dynamical general spacetime (Lorentz invariance -> local Lorentz invariance (LLI)), we can readily move the intended recipient of a one-way light pulse outside the reach of the light pulse itself; nature already does this for us in at least a couple of ways (metric expansion and astrophysical black holes). We can also have different delays on each arm of a two-way measurement, e.g. through Shapiro delay, around a spinning mass, or in the presence of a gravitational wave. However, at each point in the (vacuum part of the) curved spacetime [a] light obeys the massless wave equation and [b] local Lorentz invariance demands that the fraction of the wave at X propagates to a neighbouring point X' at c, and that X' must be drawn only from certain available neighbouring points.

So, really, it's not so much "what is the one-way velocity of light?" but rather "how much spacetime does a pulse of light traverse between two spacetime points?". Or in other words, we are looking for an affine parametrization on a curve of zero interval and that extremizes the length between two points on the manifold m and is constructed by parallel-propagating a tangent vector on m and in its own direction.

Consequently, I think the issue at the core of your points about measuring one-way speed of light is how to best label, with coordinates, two particular coincidence-points in a Lorentzian spacetime, rather than choosing two labels[1] (via your favourite synchronization scheme, for example) which are then used as the basis for a parametrization of a null curve. Then the question you ask is: "how can we know that the spacetime is Lorentzian?" or alternatively, "how do we know there is not some vicious additional gravitational field and vacuum polarization which makes the spacetime only seem Lorentzian?" for which experimentalists have generated a century worth of answers. The answer of how to best label two points in a Lorentzian manifold is \mu : it really depends on what and how you want to calculate. The physics here are that the two points (in your null-curve-parametrizing / one-way-light-travel-time experiment) are timelike-separated.

>> If we shine a light on a detector, we can measure the time the light takes to reach the detector. Are you saying that the light hits the detector instantly, and then goes back to the source, or that the detector is the source and the light comes back to it after hitting the source instantly?

> If you shine a light on a detector, you measure the time it takes to travel both legs.

reflector, surely, rather than detector? Your director can record a timestamp locally while you and your flashlight are across the room. You don't need two legs.

The main objection upthread is that while your flashlight's "on" switch can also record a timestamp within the flashlight, synchronizing the flashlight's time to the detector's time so that one can compare the timestamps after the experiment is done (or days later, or weeks later, again and again) is not feasible within the framework of Special Relativity. Therefore if one has confidence in one's local timestamps, one can compare them when measuring the round trip of the flashlight beam reflected off a mirror across the room back onto a detector mounted right beside the flashlight's bulb, inside the flashlight.

However, nature provides several gravitationally-driven systems which can provide a "preferred foliation", or a naturally-produced easily-compared timestamp. Examples include fancy sundials (including observations of millisecond pulsars), lunisolar tides, and the cosmological scale factor. These can be recovered independently in a (probably pretty large) flashlight and the detector looking for the light from the flashlight. Timestamps can then be compared with greater precision than available via Special Relativity synchronization methods.

Indeed, this is something that has been worked on in lunar-ranging experiments over the past decades. Can we use detailed knowledge of the solar system to predict a signal aimed Earthward by a 21st century space probe on or very near the moon as well as a signal launched from Earth to a reflector on the moon that is picked up by a detector on Earth? It turns out we can, to good precision!

> Final note

"c" is the free parameter of the Lorentz group, which is the fundamental isometry group of Special Relativity (and quantum electrodynamics and the Standard Model of particle physics). "c" is also a constant in the Einstein Field Equations of General Relativity, and there is a mathematical connection between that and Special Relativity.

In both theories, unless interfered with by interaction with matter, massless objects are constrained to move, obligatorily, from one spacetime point to another point from a restricted set of neighbours. These points are on the surface of a null cone.

Light is experimentally massless, so null curves are often called lightlike curves, null cones are usually called light cones, and so on. However, electromagnetic radiation is not the only massless species in the cosmos.

> very specific environment

An excellent approximation of vacuum (very long mean free paths for electromagnetic radiation at all frequencies) dominates the observable universe, so the permeability and permittivity of free space is a good choice except in very specific environments (planet-bound electrical engineering laboratories and copper wires are out-volumed by intergalactic, or even interplanetary, space by a lot). In those matter-rich environments, the mean free path of light is low. The constant c remains the same in such matter-rich environments, but electromagnetic radiation passing through them does not propagate at c.

> the value that we put as c

is exactly 1. https://en.wikipedia.org/wiki/Geometrized_unit_system>