Spacetime is homogeneous and isotropic

[BookWei]

I read the Special Theory of Relativity in Jackson's textbook, Classical Electrodynamics 3rd edition.
Consider the wave front reaches a point ##(x,y,z)## in the frame ##K## at a time t given by the equation,
$$c^{2}t^{2}-(x^{2}+y^{2}+z^{2})=0 --- (1)$$
Similarly, in the frame ##K^{'}## the wave front is specified by
$$c^{2}(t')^{2}-[(x')^{2}+(y')^{2}+(z')^{2}]=0 --- (2)$$
With the assumption that spacetime is homogeneous and isotropic, the connection between
the two sets of coordinates is linear.
The quadratic forms (1) and (2) are then related by
$$c^{2}(t')^{2}-[(x')^{2}+(y')^{2}+(z')^{2}]=(\lambda)^{2}[c^{2}t^{2}-(x^{2}+y^{2}+z^{2})]$$
where ##\lambda=\lambda(v)## is a possible change of scale between frames.
Why do we need to assume the spacetime are homogeneous and isotropic?
Will the special relativity fail if we ignore those two assumptions?
Many thanks!

 

[Nugatory]

Why do we need to assume the spacetime are homogeneous and isotropic?

We assume that because it is consistent with all of our observations of how the universe we live in works,

 

[sweet springs]

If not isotropic, ##\lambda=\lambda(\mathbf{v})\neq \lambda(|\mathbf{v}|)## depending on which direction we are going
If not homogeneous, ##\lambda=\lambda(x,y,z,t)## depending on when and where we are.

 

[Sorcerer]

If spacetime isn't isotropic, how can the principle of relativity hold? (even Galileo's version?). How can the laws of physics be the same in every inertial frame if the laws of physics depend on direction? Surely there would exist inertial frames with different spatial orientations, moving in different directions. If spacetime depended on direction, then how can the laws of physics be independent of direction? And if they are not independent of direction, then how can the principle of relativity hold in all inertial frames?

I would ask the same question regarding homogeneity. If the laws of physics depend on location, then clearly the principle of relativity cannot hold, since different inertial frames would presumably be in different locations.

 

[Ibix]

If we don't assume isotropy and homogeneity shouldn't we just end up with a direction-dependent one-way speed of light? Since that's unmeasurable, it really boils down to a different "natural" clock synchronisation and no more. It just makes the maths more complex but yields the same measurables.

Or am I oversimplifying?

 

[Sorcerer]

If we don't assume isotropy and homogeneity shouldn't we just end up with a direction-dependent one-way speed of light? Since that's unmeasurable, it really boils down to a different "natural" clock synchronisation and no more. It just makes the maths more complex but yields the same measurables.

Or am I oversimplifying?

Why wouldn't that also affect the two-way speed of light? And why wouldn't it affect the other laws of physics and measurements as well, in different locations, and in different positions in space and time?

 

[Ibix]

Why wouldn't that also affect the two-way speed of light?

I think it does in general. But inhomogeneities that affect the one way speed of light but not the two way speed are permitted, I think. If you choose to use non-orthogonal coordinates you are implicitly assuming such an inhomogeneity, as I understand it