This builds on the post Lorentz transformation derivations but given the round-trip light postulate (RTLP) here which states:
The mean round-trip speed of light in vacant space is a constant, c, which is independent of the motion of the emitting body.
From this empirical principle the round-trip Lorentz transformations may be derived, which are of the form: x′ = γ(x − vt) together with x = γ(x′ + vt′) so that xx′ = γ²(x − vt)(x′ + vt′), with v as the velocity of the primed frame relative to the unprimed frame.
I. Transformations with Velocity
Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with velocity v with respect to O in the positive x-direction. The origins of O and O′ initially coincide with each other. Let a light signal be emitted from the common origin and travel as a spherical wave front and be reflected back. Consider a point P on a spherical wavefront at a distance x and x′ from the origins of O and O′ respectively. According to the RTLP above, the round-trip speed of light, c, is the same for all inertial frames if both motions are included. Then for the point P we have:
x = ct and x′ = ct′ together so that xx′ = c²tt′ and t′x = tx′.
A. Length transformations
The length transformations are of the form: x′ = γ (x − vt) together with x = γ (x′ − vt′) so that xx′ = γ²(x − vt)(x′ − vt′). Then
xx′ = γ²(x − vt)(x′ + vt′) = γ²(xx′ − vtx′ + vxt′ − v²tt′)
= γ²(xx′ − v²xx′/c²) = γ²(xx′(1 − β²))
where β = v/c so that
γ² = 1/(1 − β²) or γ = 1/√(1 − β²).
B. Duration transformations
The duration transformations are of the form: t′ = γ (t − xv/c²) together with t = γ (t′ − x′v/c²) so that tt′ = γ²(t − xv/c²)(t′ − x′v/c²). Then
tt′ = γ²(t − xv/c²)(t′ + x′v/c²) = γ²(tt′ − xt′/v + tx′/v − xx′v²/c4)
= γ²(tt′ − tt′v²/c²) = γ²(tt′(1 − β²))
where β = v/c so that
γ² = 1/(1 − β²) or γ = 1/√(1 − β²).
Therefore, the Lorentz transformations with round-trip light for velocity are
ct′ = γ(ct − βx) and x′ = γ(x − βct) with γ = 1/√(1 − β²).
II. Transformations with Lenticity
Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with lenticity w with respect to O in the positive t-direction. The origins of O and O′ initially coincide with each other. Let a light signal be emitted from the common origin and travel as a spherical wave front and be reflected back. Consider a duration point P on a spherical wavefront at a distime t and t′ from the origins of O and O′ respectively. According to the RTLP above, the round-trip speed of light, c, is the same for all inertial frames if both motions are included. Then for the duration point P we have:
t = kx and t′ = kx′ together so that tt′ = k²xx′ and t′x = tx′.
A. Length transformations
The length transformations are of the form: x′ = γ (x − t/w) together with x = γ (x′ − t′/w) so that xx′ = γ²(x − t/w)(x′ − t′/w). Then
xx′ = γ²(x − t/w)(x′ + t′/w) = γ²(xx′ − tx′/w + xt′/w− tt′/w²)
= γ²(xx′ − k²xx′/w²) = γ²(xx′(1 − β²))
where β = k/w so that
γ² = 1/(1 − β²) or γ = 1/√(1 − β²).
B. Duration transformations
The duration transformations are of the form: t′ = γ (t − k²x/w) together with t = γ (t′ + k²x′/w) so that tt′ = γ²(t − k²x/w)(t′ + k²x′/w). Then
tt′ = γ²(t − k²x/w)(t′ + k²x′/w) = γ²(tt′ − xt′/w + tx′/w − k4xx′/w²)
= γ²(tt′ − k²tt′/w²) = γ²(tt′(1 − β²))
where β = k/w so that
γ² = 1/(1 − β²) or γ = 1/√(1 − β²).
Therefore, the Lorentz transformations with round-trip light for lenticity are
t′/k = γ(t/k − βx) and x′ = γ(x − βt/k) with γ = 1/√(1 − β²).
Note the duality by interchanging x and t.