iSoul In the beginning is reality

Subluminal and superluminal Lorentz transformations

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: = 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 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 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/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 again the subluminal Lorentz transformation.

Case 2

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

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

Combine 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´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.

Go back and combine 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.

Post Navigation