1.8 Another Symmetry

1.8  Another Symmetry
 
We saw in previous sections that Maxwell’s equations are invariant under Lorentz transformations, as well as translations and spatial rotations.
- We also saw that this was of no consequence.  Nature has a absolute reference system.  

...by reciprocity we have vij = ?vji
- A key relation for future analysis.

If acceleration were relative (like position and velocity), we would expect the cyclic symmetry vij + vjk + vki = 0, which is a linear function of all three components. Indeed, this is the Galilean composition formula. However, since acceleration is absolute, it's to be expected that the actual relation is non-linear in each of the three components.
- In sum, position and velocity are relative; acceleration is absolute. 
So saieth mathpages.

... the relativistic composition law for velocities accounts for the increasing inertia of an accelerating object. This leads to the view that inertia itself is, in some sense, a consequence of the non-linearity of velocity compositions.
- 
1. Velocity compositions are linear.
2. Inertia is the effect of aether.

These are the Lorentz transformations for velocity v in the x direction.  The y and z coordinates are unaffected, so we have y' = y and z' = z.  From this it follows that the quantity t^2 - x^2 - y^2 - z^2 is invariant under a general Lorentz transformation, so we have arrived at the full Minkowski spacetime metric.
-Further analysis of Lorentz transforms is of interest to mathematicians, but has no application to physics.