Two Lorentz transformations based on relative space with absolute time and absolute space with relative time were presented here. Now we look at beginning with space and time both relative, in two different ways.
R-R Case 1
This case begins with: r´ = r – vt and t´ = t – r/v.
Consider then a linear function of these:
r´ = γ(r – vt) and t´ = γ(t – r/v)
along with the characteristic speed c such that r = ct and r´ = ct´. Combine this with r´ to get
ct´ = r´ = γ(r – vt) = γ(r – rv/c) = γ(ct – vt) and its inverse
ct = r = γ(r´ + vt´) = γ(r´ + r´v/c) = γ(ct´ + vt´). Multiply these together to get
c² tt´ = rr´ = γ² (rr´ – rr´v²/c²) = γ² (c²tt´ – v²tt´). Divide out rr´ or tt´ and get
γ² = 1 / (1 – v²/c²), which is the same as the Lorentz transformation.
Go back and combine t´ with the characteristic speed c to get
r´/c = t´ = γ(t – r/v) = γ(r/c – r/v) = γ(t – tc/v) and its inverse
r/c = t = γ(t´ + r´/v) = γ(r´/c + r´/v) = γ(t´ + t´c/v). Multiply these together to get
rr´/c² = tt´ = γ²(rr´/c² – rr´/v²) = γ²(tt´ – tt´c²/v²). Divide out rr´ or tt´ to get
γ² = 1 / (1 – c²/v²), which is the superluminal Lorentz transformation. Thus we have a contradiction.
R-R Case 2
This case begins with: r´ = r – tc²/v and t´ = t – rv/c².
Consider a linear function of these:
r´ = γ(r – tc²/v) and t´ = γ(t – rv/c²). Combine r´ with the characteristic speed c to get
ct´ = r´ = γ(r – tc²/v) = γ(r – rc) = γ(ct – tc²/v) and its inverse
ct = r = γ(r´ + t´c²/v) = γ(r´ + r´c) = γ(ct´ + t´c²/v). Multiply these together and get
c²tt´ = rr´ = γ²(rr´ – rr´c²/v²) = γ²(c²tt´ – tt´c4/v²), which simplifies to
1 = γ²(1 – c²/v²) or c² = γ²(c² – c4/v²) so that γ² = 1 / (1 – c²/v²),
which is the superluminal Lorentz transformation.
Alternatively, go back and combine t´ with the characteristic speed c to get
r´/c = t´ = γ(t – rv/c²) = γ(r/c – rv/c²) = γ(t – tv/c) and its inverse
r/c = t = γ(t´ + r´v/c²) = γ(r´/c + r´v/c²) = γ(t´ + t´v/c). Multiply these together and get
rr´/c² = tt´ = γ²(rr´/c² – rr´v²/c4) = γ²(tt´ – tt´v²/c²), which simplifies to
1/c² = γ²(1/c² – v²/c4) or 1 = γ²(1 – v²/c²) so that
γ² = 1 / (1 – v²/c²), which is the Lorentz transformation. Thus we have a contradiction.
Conclusion
Beginning with both space and time relative leads to a contradiction. We conclude that absolute and relative are jointly required.