This is a re-do of the post *Lorentz for space & time both relative?* By making a few rearrangements, the contradictions disappear.

Case 1

This case begins with: *r´ = r – vt* or *t´ = t – rv/c²*.

Consider then a linear function of these: *r´* = *γ *(*r – vt*) or *t*´ = *γ* (*t – rv/c²*)

along with a standard 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 subluminal Lorentz transformation.

Alternatively, go back and combine *t´* with a standard 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/c ^{4}*) =

*γ²*(

*tt´ – tt´v²/c²*), which simplifies to

1/*c²* = *γ² *(1/c² – *v²/c ^{4}*) or 1 =

*γ²*(1 –

*v²/c²*) so that

*γ²* = 1 / (1 – *v²/c²*), which is again the subluminal Lorentz transformation.

Case 2

This case begins with: *r´ = r – tc²/v* or *t´* = *t – r/v*.

Consider a linear function of these: *r´* = *γ *(*r – tc²/v*) or *t´* = *γ *(*t – r/v*).

Combine *r´* with a standard 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´c ^{4}/v²*), which simplifies to

1 = *γ² *(1 – *c²/v²*) or *c²* = *γ² *(*c² – c ^{4}/v²*) so that

*γ²* = 1 / (1 – *c²/v²*), which is the superluminal Lorentz transformation.

Go back and combine *t´* with a standard 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 again the superluminal Lorentz transformation.

Conclusion

By beginning with the correct form of the non-relativistic transformation and its alternate, one may derive the Lorentz transformation and its alternate. Together both subluminal and superluminal velocities are covered.