iSoul In the beginning is reality

Lorentz for space & time both relative?

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:

= γ(r – vt) and = γ(t – r/v)

along with the standard (characteristic) speed c such that r = ct and r´ = ct´. Combine this with 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 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:

= γ(r – tc²/v) and = γ(t – rv/c²). Combine 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´c4/v²), which simplifies to

1 = γ²(1 – c²/v²) or = γ²(c² – c4/v²) so that γ² = 1 / (1 – c²/v²),

which is the superluminal Lorentz transformation.

Alternatively, go back and combine 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/c4) = γ²(tt´ – tt´v²/c²), which simplifies to

1/ = 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.