The „one way“ speed of light from a source to a detector cannot be measured independently of a convention as to how synchronize the clocks at the source and the detector. To synchronize these clocks one needs to know the one way speed of light, since it is the “greatest available speed” and no instantaneous transfer of signal is possible. So, there is a circular reasoning.
What can however be experimentally measured is the round-trip speed (or "two-way" speed of light) from the source to the detector and back again. Measured round trip speed of light is always equal to constant c.
A. Einstein‘s synchronization is a clock synchronization convention that assumes, that velocity of light in all direction is c or isotropic. It synchronizes distant clocks in such a way that the one-way speed of light becomes equal to the two-way speed of light.
H. Reichenbach's (or Reichenbach - Grünbaum) synchrony convention is self - consistent and admits that speed of light is different in different directions, while measured „round-trip“ speed of light is equal to c. For example, speed of light in one direction can be infinitely large and in the other infinitely close to c/2.
As soon as definition of simultaneity depends on the clock's synchronization scheme and is conventional, any one - way velocity of everything also depends on the same scheme and is conventional.
Therefore, considering aberration - in an inertial reference frame, in addition to the tilt angle of the telescope, one needs to know the speed of the laboratory. But clocks synchronization scheme would affect one – way speed of this laboratory. Hence, as soon as one determines the speed of the laboratory using Einstein-synchronized clocks, the tilt angle of the telescope will indicate that the speed of light is equal exactly to constant c.
Method Nr. 2 is known as slow clock transport and is equivalent to Einstein synchrony convention. Measured by means of this synchronization method one way speed of light will be equal exactly to constant c.
The first experimental determination of the speed of light was made by Ole Christensen Rømer. It may seem that this experiment measures the time for light to traverse part of the Earth's orbit and thus determines its one-way speed. However, the Australian physicist Leo Karlov showed that Rømer actually measured the speed of light by implicitly making the assumption of the equality of the speeds of light back and forth.
It is also not possible to "instantly" synchronize clocks by means of rigid rod, since absolutely rigid bodies do not exist and signal cannot move inside the rod faster than light.
Many experiments that attempt to directly probe the one-way speed of light have been proposed, but none have succeeded in doing so.
For example, from the center of a room, using identical catapults, one throws two identical clocks at the same distance. But, in a moving frame, these clocks will slow down to varying degrees. Even if one way speed of light is anisotropic, due to this discrepancy, the speed of light measured using these clocks will be exactly equal to constant c.
S. Marinov once proposed synchronization of clocks by means of a chain (or a conveyor belt). (See: S. J. Pokhovnik, „The empty ghosts of Michelson and Morley: A critique of the Marinov coupled-mirrors experiment“). But one has to keep in mind, that in a moving laboratory opposite sides of the chain would Lorentz – contract at different magnitude. It would lead to “desynchronization” of clocks and measured in this way the speed of light will be exactly equal to constant c.
R. W. Wood has considered a modification of Fizeau’s method for determining the speed of light, in which two toothed wheels are mounted at the ends of a long axle, and light is sent in one direction only (S. Marinov, M.D. Farid Ahmet employed this method). However, a stress – free relativistic twist of the axle would be an additional compensating factor. Measured by this apparatus one way speed of light will also be equal precisely to c (Herbert E. Ives, “ Theory of Double Fizeau Toothed Wheel”).
Max Jammer's „Concepts of Simultaneity“ presents a comprehensive, accessible account of the historical development of the concept as well as critique of many proposed experiments to measure the one way speed of light.
In rotating frames, even in Special Relativity, the non-transitivity of Einstein synchronisation diminishes its usefulness. If clock 1 and clock 2 (are equidistant from the center of the ring) on a rim of rotating ring are not synchronised directly, but by using a chain of intermediate clocks, the synchronisation depends on the path chosen. Synchronisation around the circumference of a rotating disk gives a non vanishing time difference that depends on the direction used. If one synchronizes clocks 1 and 2 by means of a flash of light from the center of the ring, measured by means of these clocks one – way speeds of light will be different clockwise and counterclockwise, but still satisfying Reichenbach‘s synchrony condition.
Lorentz‘s theory assumes that the speed of light is isotropic only in the preferred frame (Ether). The introduction of length contraction and time dilation for all phenomena in a "preferred" frame of reference, which plays the role of Lorentz's immobile aether, leads to the complete Lorentz transformation. Because the same mathematical formalism occurs in both, it is not possible to distinguish between LET and SR by experiment (see: Simulation of kinematical effects of the Special Relativity by means of classical mechanics in water environment)
Even though Reichenbach’s synchronization is more "universal", undoubtedly, from the practical point of view the Einstein – synchronization is the most convenient in inertial frames. The Lorentz transformation is defined such that the one-way speed of light will be measured to be independent of the inertial frame chosen.
This made me ask this question is taking the speed of light same in all directions an axiom of some sort?
Yes, although it is called a postulate rather than an axiom. This is Einstein's famous second postulate:
Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence
$${\rm velocity}=\frac{{\rm light\ path}}{{\rm time\ interval}} $$
where time interval is to be taken in the sense of the definition in § 1.
A. Einstein, 1905, "On the Electrodynamics of Moving Bodies"
https://www.fourmilab.ch/etexts/einstein/specrel/www/
This postulate is simply assumed to be true and the consequences are explored in his paper. The subsequent verification of many of the rather strange consequences is then taken to be strong empirical support justifying the postulate. This is the heart of the scientific method.
So are all of our physics theories based on the assumption and what would happen if light turns out to be moving at different speeds in different direction? Will that enable transfer of information faster than the speed of light and is there any way for us knowing that the transfer happens faster than the speed of light?
Yes, all of our physics theories are based on this assumption, but the assumption itself is simply a convention. The nice thing about conventions is that there is no "wrong" or "right" convention. This specific convention is known as the Einstein synchronization convention, and it is what the second postulate above referred to by "time interval is to be taken in the sense of the definition in § 1". From the same paper in section 1:
Let a ray of light start at the “A time” $t_{\rm A}$from A towards B, let it at the “B time” $t_{\rm B}$ be reflected at B in the direction of A, and arrive again at A at the “A time” $t'_{\rm A}$.
In accordance with definition the two clocks synchronize if $$t_{\rm B}-t_{\rm A}=t'_{\rm A}-t_{\rm B}$$
A. Einstein, 1905, "On the Electrodynamics of Moving Bodies" https://www.fourmilab.ch/etexts/einstein/specrel/www/
If we define $\Delta t_A= t'_A-t_A$ then with a little rearranging this becomes $t_B=\frac{1}{2}(t_A+t'_A)=t_A+\frac{1}{2}\Delta t_A$. This is a convention about what it means to synchronize two clocks. But it is not the only possible convention. In fact, Reichenbach extensively studied an alternative convention where $t_B=t_A+ \epsilon \Delta t_A$ where $0 \le \epsilon \le 1$. Einstein's convention is recovered for $\epsilon = \frac{1}{2}$ and the Veritasium video seemed oddly excited about $\epsilon = 1$.
Note that the choice of Reichenbach's $\epsilon$ directly determines the one way speed of light, without changing the two way speed of light. For Einstein's convention the one way speed of light is isotropic and equal to the two way speed of light, and for any other value the one way speed of light is anisotropic but in a very specific way that is sometimes called "conspiratorial anisotropy". It is anisotropic, but in a way that does not affect any physical measurement. Instead this synchronization convention causes other things like anisotropic time dilation and even anisotropic stress-free torsion which conspire to hide the anisotropic one way speed of light from having any experimental effects.
This is important because it implies two things. First, there is no way to determine by experiment the true value, there simply is no true value, this is not a fact of nature but a description of our coordinate system's synchronization convention, nature doesn't care about it. Second, you are free to select any value of $\epsilon$ and no experiment will contradict you.
This means that $\epsilon=\frac{1}{2}$ is a convention, just like the charge on an electron being negative is a convention and just like the right-hand rule is a convention. No physical prediction would change if we changed any of those conventions. However, in the case of $\epsilon=\frac{1}{2}$ a lot of calculations and formulas become very messy if you use a different convention. Since there is no point in making things unnecessarily messy, it is a pretty strong convention.
Finally, regarding FTL information transfer. If we use $\epsilon \ne \frac{1}{2}$ then there is some direction where information can travel faster than $c$. However, since in that direction light also travels faster than $c$ the information still does not travel faster than light. It is important to remember that under the $\epsilon \ne \frac{1}{2}$ convention the quantity $c$ is no longer the one way speed of light, so faster than light and faster than $c$ are no longer equivalent.
Best Answer
Indeed, as you suggest, the form of basic expressions for energy and momentum are different under the assumption that the one-way speed of light (OWSOL) is anisotropic. I will use Anderson's notation where the anisotropy of the OWSOL is defined by a parameter, $\kappa$, defined such that the OWSOL in the positive $x$ direction is $c/(1-\kappa)$ and the OWSOL in the negative $x$ direction is $c/(1+\kappa)$. Anderson's notation is related to Reichenbach's by $\kappa = 2 \epsilon-1$ with the standard isotropic convention for $\kappa=0$. I will also use units where $c=1$ for convenience.
If we start from a standard inertial frame using Einstein synchronization $(t,x,y,z)$ then we can transform to the Anderson anisotropic frame with $$t=T+ \kappa X$$$$(x,y,z)=(X,Y,Z)$$ Note that positions are unchanged in this frame, and all that is changed is the synchronization convention. This leads to the metric $$ds^2=-(dT+\kappa \ dX)^2+dX^2+dY^2+dZ^2$$ Note that this is the standard metric plus additional terms that depend on the OWSOL via $\kappa$. In particular, if you expand this expression there is a cross term $-2\kappa \ dT dX$ leads to changes in many of the standard physics equations. However, importantly, because this is merely a coordinate change, there is no change in the underlying physics and no experimental measurement is changed with this convention.
To derive the formula for KE in this metric, we first write the Lagrangian of a free particle in this metric: $$\mathcal L=m \sqrt{\left(\dot T +\kappa \ \dot X\right)^2-\dot X^2 - \dot Y^2 - \dot Z^2}$$
From this Lagrangian we can find the conserved quantities through the usual Lagrangian-based approach. This gives the conserved energy and the conserved momentum: $$E=m\frac{1+\kappa \ V_X}{\sqrt{\left(1+\kappa \ V_X\right)^2-V^2}}$$ $$ \vec p = \frac{m}{\sqrt{\left(1+\kappa \ V_X\right)^2-V^2}}\left( - \kappa - \kappa^2 \ V_X+V_X, V_Y, V_Z\right) $$ where $V$ is the speed in the Anderson frame and $V_X$ is the component of the velocity in the $X$ direction.
For the energy, if we set $(V_X,V_Y,V_Z)=(V,0,0)$ and do a Taylor series expansion around $V=0$ we get $$E\approx m+\frac{1}{2}m V^2 -\kappa m V^3$$ where we see the first term is the usual mass-energy, the second term is the usual kinetic-energy, and the third term is an additional kinetic-energy term that depends on both the velocity and the OWSOL.
Now, according to this formula if the OWSOL is anisotropic ($\kappa \ne 0$) then energy will also be anisotropic in the following sense. If a given amount of potential energy is converted to kinetic energy, then the resulting speed will depend on the direction. Energy is still conserved, but it is anisotropic in relation to the anisotropy of the OWSOL.