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 standard (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 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 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 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.

Alternatively, go back and combine *t´* with the 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 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.