Galilean transformation inconsistency with light sphere equation

[Prophylaxis]

Hi, I am a novice here and i am trying to solve the following problem:
1.Show that Galilean relativity transformation produces inconsistent classical physics result when used with equation of expansion of light sphere. (2 different frames)



2. x₂ = x₁ - vt₁
y₁ = y₂
z₁ = z₂
t₁ = t₂

expansion of light sphere: x₁² + y₁² + z₁² - ct₁² = 0
x₂² + y₂² + z₂² - ct₂² = 0



3. I have substituted the Galilean transformation equations in the later expansion of the light sphere equations:
x₁² + y₁² + z₁² - ct₁² = 0
x₂² + y₂² + z₂² - ct₂² = 0

x₁² + y₁² + z₁² - ct₁² = 0 = x₂² + y₂² + z₂² - ct₂²

x₁² + y₁² + z₁² - ct₁² = x₂² + y₁² + z₁² - ct₂²

x₁² - ct₁² = x₂² - ct₂²

x₁² - ct₁² = (x₁ - vt₁)² - ct₂²

x₁² - ct₁² = x₁² - 2vt₁ + v²t₁² - c²t₂²

So far I have not been able understand the inconsistency and it's bothering me. I do know about Lorentz contraction, but don't know where the inconsistency is? Is there something wrong with my thought process? Thanks in advance.

 

[PFStudent]

Hey,

An interesting problem, anyone have a solution to it?

Thanks,

-PFStudent