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As is widely know, the standard letter used for this constant is $c$.

So much for the speed of light according to the standard definition of terms in physics.

As is widely known, the standard letter used for this constant is $c$.

So much for the speed of light according to the standard definition of terms in physics. It is the same everywhere and it does not depend on any direction of travel.

The physics here is intermediate between special and general relativity (GR). It can all be treated using special relativity, but since coordinate transformations are involved (not just Lorentz transformations) it helps if one brings in a little GR as well.

In GR we assert that spacetime is a 4-dimensional pseudo-Riemannian manifold, with signature $(-1,1,1,1)$ or (equivalently) $(1,-1,-1,-1)$. This means that near any event there exists a coordinate system in which the line element takes the following form: $$ ds^2 = - A^2 dt^2 + dx^2 + dy^2 + dz^2 $$ where $A$ is a constant, and furthermore it is a universal constant because if the metric having this form appeared to have a different value of $A$ from one event to another, than one can rescale the coordinates to make it come out the same everywhere. Hence the constant $A$ earns a name, because it is a universal constant. It is called the speed of light. It gets this name because it is also found that light waves in empty space move in such a way that $ds = 0$ between events on the worldline, so their speed is given by $$ dx^2 + dy^2 + dz^2 = A^2 dt^2 $$ hence $$ v = \left( (\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2 \right)^{1/2} = A $$

Now if one chooses to adopt other systems of coordinates, then one can find coordinates say $T,X,Y,Z$ in which the worldline of a light ray can have $dX/dT = c/2$ when travelling in one direction, and $dX/dT = \infty$ while travelling in another direction. Quantities of this kind are called "the coordinate speed of light". They vary from one choice of coordinates to another, and do not tell us much of any relevance to physics.

So now, if we choose to use the unadorned phrase "speed of light" to mean "coordinate speed of light", without making it crystal clear that that is what we are doing, then we shall merely mislead people, as the video mentioned in the question clearly has mislead the questioner. The phrase "one way speed of light" will alert experts to the fact that something more technical and non-standard is being referred to, but that nuance will not be picked up in the context of popular presentations. It then appears that we are saying that light could really travel from Mars to Earth in the blink of an eye, crossing a spacelike interval. But light cannot cross a spacelike interval. So if one appears to be saying that light signals can cross a spacelike interval, without adding unambiguously that in fact this is not possible, then I think one is being misleading.

On Earth we have different time zones. For example, France is one hour ahead of England. This means one could set off on a journey from France to England, departing at noon (12:00) (French time) and, after an hour of travel, arrive in England at noon (English time). Does this mean you have travelled at infinite speed? Of course not. Is it a wonderful and amazing insight into the physics of relativity that challenges our ordinary perceptions about time? I don't think so.

The effect discussed in the video mentioned in the question is precisely this effect.

I'll unpack it algebraically in the following, which I hope will make it clear that this is all there is to it. The physics is intermediate between special and general relativity (GR). It can all be treated using special relativity, but since coordinate transformations are involved (not just Lorentz transformations) it helps if one brings in a little GR as well.

First let's present the standard approach. This first part will be a little technical for some readers, but you will be able to get the main point about how the speed of light is defined.

In GR we assert that spacetime is a 4-dimensional space of a certain kind, called "pseudo-Riemannian manifold, with signature $(-1,1,1,1)$ or (equivalently) $(1,-1,-1,-1)$". This means that near any event there exists a coordinate system in which to calculate the interval $ds$ between neighbouring events one can use the following equation: $$ ds^2 = - A^2 dt^2 + dx^2 + dy^2 + dz^2 $$ where $A$ is a constant, and furthermore it is a universal constant because if the metric having this form appeared to have a different value of $A$ from one event to another, than one can rescale the coordinates to make it come out the same everywhere. Hence the constant $A$ earns a name, because it is a universal constant. It is called the speed of light. It gets this name because it is also found that light waves in empty space move in such a way that $ds = 0$ between events on the worldline, so their speed is given by $$ dx^2 + dy^2 + dz^2 = A^2 dt^2 $$ hence $$ v = \left( (\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2 \right)^{1/2} = A $$

Now if one chooses to adopt other systems of coordinates, then one can find coordinates say $T,X,Y,Z$ in which the worldline of a light ray can have $dX/dT = c/2$ when travelling in one direction, and $dX/dT = \infty$ while travelling in another direction. Quantities of this kind are called "the coordinate speed of light". They vary from one choice of coordinates to another, and do not tell us much of any relevance to physics.

Here is an example.

Let $x,t$ be ordinary coordinates which can be used, for example, to describe the motion of things moving along a line between Earth and Mars, where we align the $x$ axis with this line (the line will stay still to good approximation during the few tens of minutes required for the motions we will discuss). Now define two other variables as follows: $$ X \equiv x, \;\;\;\;\;\; T \equiv t + x/c $$ These are definitions. The variables $X,T$ are a pair of quantities which I simply decided to define this way.

Now let's consider something moving along the $x$ axis. If its speed is $v$ then $dx/dt = v$ for motion in one direction, and $dx/dt = -v$ for motion in the other direction. We can track the motion also using the $X,T$ coordinates. We have $$ \frac{dX}{dt} = \frac{dx}{dt} = \pm v $$ and $$ \frac{dT}{dt} = 1 + \frac{1}{c} \frac{dx}{dt} = 1 \pm \frac{v}{c} $$ therefore $$ \frac{dX}{dT} = \frac{dX/dt}{dT/dt} = \frac{ \pm v }{1 \pm v/c}. $$ For example, in the case of a light pulse, where $v=c$, we shall find $$ \frac{dX}{dT} = \frac{c}{2} $$ in one direction and $$ \frac{dX}{dT} = \infty $$ in the other direction.

So is the light moving instantaneously from Mars to Earth? No: it is just like the different clock settings in France and England that I started with. The "clocks" indicated by $T$ have been arranged such that a clock on Mars is ahead of one on Earth. Amazing as it may seem to anyone who watched the Veritasium video, there really is no more to it than that. It is all based on a human decision to refer to the parameter $T$ as "time".

If we choose to use the unadorned phrase "speed of light" to mean "coordinate speed of light", without making it crystal clear that that is what we are doing, then we shall merely mislead people, as the video mentioned in the question clearly has mislead the questioner. The phrase "one way speed of light" will alert experts to the fact that something more technical and non-standard is being referred to, but that nuance will not be picked up in the context of popular presentations. It then appears that we are saying that light could really travel from Mars to Earth in the blink of an eye, crossing a spacelike interval. But light cannot cross a spacelike interval. So if one appears to be saying that light signals can cross a spacelike interval, without adding unambiguously that in fact this is not possible, then I think one is being misleading.

The question is,

1. "can the one way speed of light be instantaneous?"

2. "is taking the speed of light same in all directions an axiom of some sort?"

3. "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?"

My answer will be different from some others posted here, but this is not owing to a disagreement about the mathematics, it is a disagreement about terminology and what constitutes clear communication.

The physics here is intermediate between special and general relativity (GR). It can all be treated using special relativity, but since coordinate transformations are involved (not just Lorentz transformations) it helps if one brings in a little GR as well.

In GR we assert that spacetime is a 4-dimensional pseudo-Riemannian manifold, with signature $(-1,1,1,1)$ or (equivalently) $(1,-1,-1,-1)$. This means that near any event there exists a coordinate system in which the line element takes the following form: $$ ds^2 = - A^2 dt^2 + dx^2 + dy^2 + dz^2 $$ where $A$ is a constant, and furthermore it is a universal constant because if the metric having this form appeared to have a different value of $A$ from one event to another, than one can rescale the coordinates to make it come out the same everywhere. Hence the constant $A$ earns a name, because it is a universal constant. It is called the speed of light. It gets this name because it is also found that light waves in empty space move in such a way that $ds = 0$ between events on the worldline, so their speed is given by $$ dx^2 + dy^2 + dz^2 = A^2 dt^2 $$ hence $$ v = \left( (\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2 \right)^{1/2} = A $$

As is widely know, the standard letter used for this constant is $c$.

So much for the speed of light according to the standard definition of terms in physics.

Now if one chooses to adopt other systems of coordinates, then one can find coordinates say $T,X,Y,Z$ in which the worldline of a light ray can have $dX/dT = c/2$ when travelling in one direction, and $dX/dT = \infty$ while travelling in another direction. Quantities of this kind are called "the coordinate speed of light". They vary from one choice of coordinates to another, and do not tell us much of any relevance to physics.

So now, if we choose to use the unadorned phrase "speed of light" to mean "coordinate speed of light", without making it crystal clear that that is what we are doing, then we shall merely mislead people, as the video mentioned in the question clearly has mislead the questioner. It then appears that we are saying that light could really travel from Mars to Earth in the blink of an eye, crossing a spacelike interval. But light cannot cross a spacelike interval. So if one appears to be saying that light signals can cross a spacelike interval, without adding unambiguously that in fact this is not possible, then I think one is being misleading.

The answer to the three questions listed at the start is, then: "if someone asserts that light can move between different locations instantaneously then beware: they may be adopting some non-standard way of dissecting spacetime using coordinate systems, and they may be using the terminology "speed of light" in a misleading way".

The question is,

1. "can the one way speed of light be instantaneous?"

2. "is taking the speed of light same in all directions an axiom of some sort?"

3. "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?"

My answer will be different from some others posted here, but this is not owing to a disagreement about the mathematics, it is a disagreement about terminology and what constitutes clear communication.

The physics here is intermediate between special and general relativity (GR). It can all be treated using special relativity, but since coordinate transformations are involved (not just Lorentz transformations) it helps if one brings in a little GR as well.

In GR we assert that spacetime is a 4-dimensional pseudo-Riemannian manifold, with signature $(-1,1,1,1)$ or (equivalently) $(1,-1,-1,-1)$. This means that near any event there exists a coordinate system in which the line element takes the following form: $$ ds^2 = - A^2 dt^2 + dx^2 + dy^2 + dz^2 $$ where $A$ is a constant, and furthermore it is a universal constant because if the metric having this form appeared to have a different value of $A$ from one event to another, than one can rescale the coordinates to make it come out the same everywhere. Hence the constant $A$ earns a name, because it is a universal constant. It is called the speed of light. It gets this name because it is also found that light waves in empty space move in such a way that $ds = 0$ between events on the worldline, so their speed is given by $$ dx^2 + dy^2 + dz^2 = A^2 dt^2 $$ hence $$ v = \left( (\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2 \right)^{1/2} = A $$

As is widely know, the standard letter used for this constant is $c$.

So much for the speed of light according to the standard definition of terms in physics.

Now if one chooses to adopt other systems of coordinates, then one can find coordinates say $T,X,Y,Z$ in which the worldline of a light ray can have $dX/dT = c/2$ when travelling in one direction, and $dX/dT = \infty$ while travelling in another direction. Quantities of this kind are called "the coordinate speed of light". They vary from one choice of coordinates to another, and do not tell us much of any relevance to physics.

So now, if we choose to use the unadorned phrase "speed of light" to mean "coordinate speed of light", without making it crystal clear that that is what we are doing, then we shall merely mislead people, as the video mentioned in the question clearly has mislead the questioner. The phrase "one way speed of light" will alert experts to the fact that something more technical and non-standard is being referred to, but that nuance will not be picked up in the context of popular presentations. It then appears that we are saying that light could really travel from Mars to Earth in the blink of an eye, crossing a spacelike interval. But light cannot cross a spacelike interval. So if one appears to be saying that light signals can cross a spacelike interval, without adding unambiguously that in fact this is not possible, then I think one is being misleading.

The answer to the three questions listed at the start is, then: "if someone asserts that light can move between different locations instantaneously then beware: they may be adopting some non-standard way of dissecting spacetime using coordinate systems, and they may be using the terminology "speed of light" in a misleading way".