Vanadium 50 said:
Let's take it one step back. (And pretend it's before the latest unit redefinition which makes things more complicated.) The question "is the speed of light isotropic" is the same as "is the permittivity of free space the same in all directions". That is a well-defined experimental question. The answer "it must be because of the definition of an inertial frame" is something I find unsatisfying. We have to look. And if we know that it's good to ε (no pun intended) we would like to make ε as small as possible.
Exactly, and that's why indeed isotropy is subject to experimental tests. Of course, it's true, you need a space-time model to measure time intervals and lengths. From the space-time model also the physical laws are constraint to some extent due to the symmetries implied by the assumed space-time model, and within this constraints you can build mathematical models to describe/predict real-world phenomena. Then you make observations and measurements in the real world using the underlying model to define quantitative measures for these observables. Nothing a priori ensures that everything turns out as predicted by the models, and you can check these (symmetry) assumptions by probing all kinds of consequences derived from them by building physical models.
Concerning the assumption of the existence inertial reference frames it's also clear that this is also subject to experimental test (and, sorry,
@Dale , when discussing this issue about experimental tests of space-time models I must use the "clocks-and-ruler definition" of a reference frame).
Of course, Newton simply assumed an absolute space and an absolute time (it's not even really a "space-time", because of the fiber-bundel structure of the Newtonian space-time model) and that was it for him. Nevertheless his view was already criticized, among other famously by his arch enemy Leibniz, who logically argued that motion cannot be absolute within Newton's own theoretical edifice, because all inertial frames are equivalent, and it's indeed necessary to define an inertial reference frame by realizing it by some reference point and three directions (realizable by rigid rods, which within Newton's physics of course exist) as well as a clock, which can be defined by a reference body assumed to move with constant velocity wrt. the (hopefully inertial) reference frame.
A naive starting point of course is, as done in any physics freshman lecture on day 1 (usually not expclitly ;-)), to just use your lab fixed at rest in Earth as an inertial reference frame, taking the ever present gravitational force of the Earth on all bodies into account as a homogeneous fource ##m \vec{g}##. As we all know, with this assumption you get very far.
It's of course clear that the Earth-fixed lab frame is for sure not an inertial reference frame. You may rather take the fixed stars as reference bodies defining an inertial frame, and then you expect that indeed the earth-fixed frame is even a rotating frame, both from the motion of the Earth around the Sun and its spin around its axis once per day. Then you develop the theory what to expect when using a non-inertial rotating reference frame and predict that the Foucault pendulum can be used to demonstrate the rotation of the Earth (wrt. the fixed-star reference frame), and as is well-known this indeed turns out to be right.
Then in the mid 19th century Maxwell developed his non-Galilei invariant electrodynamics, and many (if not all?) physicists thought that this finally fixes the reference frame for Newton's absolute space (and time). It was also theorized (including Maxwell himself) that Maxwell's electromagnetic waves are due to the vibrations of the aether, whose (global) rest frame defines Newton's absolute space. The history is known: From this it should be possible to empirically prove the existence of this absolute space and this preferred inertial aether rest frame. Then the null result of the Michelson Morley experiment, which was the first experiment being sensitive to order ##\mathcal{O}[(v/c)^2]##, showed that this idea is not correct and, even more famously, Einstein turned the argument around in 1905 and introduced a new space-time model (2 years later analyzed by Minkowski in its mathematical/geometrical structure and thus henceforth called "Minkowski space-time") valid for all of physics.
The up to now last step then was the development of GR by Einstein in his attempt to find a relativistic theory for the gravitational interaction, leading to a dynamical space-time picture. Here the important property is the equivalence of "inertial and gravitational mass", which finally in the mathematics boils down to the assumption that at any space-time point one can define a locally inertial reference frame. The extent of this local reference frame is determined by the homogeneity of the gravitational field as can be measured with test particles, and then the local inertial reference frames are defined in the neighborhood of the space-time point in question by a freely falling pointlike test body, which is then moving along a timelike geodesic of the curved Lorentzian spacetime, defining a time-like unit tangent vector (the four-velocity of the body) which then enables the construction for three space-like non-rotating unit vectors building together with the four-velocity of the body a free-falling non-rotating tetrad, defining a local inertial reference frame. It's of course only inertial to the extent defined by how accurately the gravitational field the test body is freely falling in can be regarded as homogeneous and to this accuracy the gravitational force can be regarded as equivalent to the inertial forces in a non-inertial reference frame being accerated relative to the free-falling tetrad just constructed to define the local inertial reference frame. Of course a "true gravitational field" can never be completely explained as equivalent to the inertial forces in a non-inertial local reference frame but there are always deviations from an exactly homogeneous gravitational field, leading to tidal forces measurable also in the local inertial reference frame.
