The motivation for the Lorentz transformation was rooted both in experiment (the most important reason) and some conceptual challenges with Maxwell’s equations for electrodynamics. From a conceptual standpoint, it was desirable (for reasons I’ve since forgotten) that Maxwell’s equations in one frame (let us call this the “rest” frame) that yield:

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x^2}+\frac{\partial^2 \vec{\Psi}}{\partial y^2}+\frac{\partial^2 \vec{\Psi}}{\partial z^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t^2}=0

will have a comparable form in a frame moving with relative velocity to the rest frame which is described with prime coordinates (x’, y’, z’, t’) instead of (x,y,z,t):

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x'^2}+\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=0

But as I mentioned, there is both a conceptually deduced and experimentally deduced (such as the Hafele Keating experiment) that suggests there really is a true rest frame where V = 0 because of the integration of acceleration over all time that would yield:

\LARGE v = \int a(t) = 0

and I described how one can conduct and experiment in principle where one can determine’s one’s absolute speed relative to the Aether. The Lorentz transformations apply to the Aether from of relativity, btw, but instead of time dilation we have clock slowing, and clock slowing based on absolute speed is indicated experimentally.

However to create a transformation that preserves the form or Maxwell’s equations such as:

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x'^2}+\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=0

required the genius of Lorentz to accomplish. The transformation entailed doing the following using the chain rule on multiple varialbles on the d’Alamberitan above. I’ll elaborate it in brutal detail in order to make the idea more accessible:

Consdier that going from the rest frame to a moving frame, we have a change of coordinages from the unprimed to the primed system. For example, the unprimed coordinates in the x dimension are related the primed coordinates according to the chain rule for multiple variables:

\Large \frac{\partial^2 \vec{\Psi}}{\partial x^2} = \frac{\partial}{\partial x} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{x} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{x} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{x} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{x}\right]

I’m elaborating this in brutal detail to show the genius that Lorentz must have had find a transformation that would make Maxwell’s equations invariant! The above equation could be expanded in even more detail for the second derivative, but for brevity, I’ll spare the reader and keep it in that “simpler” form. Doing the same for all the dimensions the d’Alambertian in primed coordinates would be (gasp!):

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x^2}+\frac{\partial^2 \vec{\Psi}}{\partial y^2}+\frac{\partial^2 \vec{\Psi}}{\partial z^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t^2} =

\Large \frac{\partial}{\partial x} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{x} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{x} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{x} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{x}\right]

\Large + \frac{\partial}{\partial y} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{y} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{y} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{y} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{y}\right]

\Large + \frac{\partial}{\partial z} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{z} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{z} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{z} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{z}\right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{t} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{t} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{t} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{t}\right]

=0

Lorentz searched for a transformation to reduce the above monstrosity to:

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x'^2}+\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=0

and his solution contributed to him getting the Nobel prize. The above will not be solve by the Galilean transformation, so he had to concoct his own transformation. One way to make the above monstrosity manageable is to hyothesize the following 16 relations which can be substituted into the above monstrosity:

[for the x’ dimension]

\large \frac{\partial x'}{\partial x} = \gamma

\large \frac{\partial x'}{\partial y} = 0

\large \frac{\partial x'}{\partial z} = 0

\large \frac{\partial x'}{\partial t} = -\gamma V

[for the y’ dimension]

\large \frac{\partial y'}{\partial x} = 0

\large \frac{\partial y'}{\partial y} = 1

\large \frac{\partial y'}{\partial z} = 0

\large \frac{\partial y'}{\partial t} = 0

[for the z’ dimension]

\large \frac{\partial z'}{\partial x} = 0

\large \frac{\partial z'}{\partial y} = 0

\large \frac{\partial z'}{\partial z} = 1

\large \frac{\partial z'}{\partial t} = 0

[for the t’ prime dimension]

\large \frac{\partial t'}{\partial x} = -\gamma \frac{V}{c^2}

\large \frac{\partial t'}{\partial y} = 0

\large \frac{\partial z'}{\partial z} = 0

\large \frac{\partial z'}{\partial t} = \gamma

Hypothesizing the above 16 relations greatly reduces the nastiness of the above monstrosity to something more manageable which I cover in the next comment. How Lorenztz came up with that hypothesis is testament to his genius.