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′* = *γ*²(*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′* = *c*²*tt′* 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′* = *γ*²(*x − vt*)(*x′ − vt′*). Then

*xx′* = *γ*²(*x − **v**t*)(*x′ + **v**t′*) = *γ*²(*xx′* − *vtx′* + *v**xt′ *− *v*²*tt′*)

= *γ*²(*xx′* − *v*²*xx′*/*c*²) = *γ*²(*xx′*(1 − *β*²))

where *β = v*/*c* so that

* γ*² = 1/(1 − *β*²) or *γ* = 1/√(1 − *β*²).

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′* = *γ*²(*t − xv/c²*)(*t′ − x′v/c²*). Then

*tt′* = *γ*²(*t − x**v*/*c*²)(*t′ + x′**v*/*c*²) = *γ*²(*tt′* − *xt′*/*v* + *tx′*/*v* − *xx′**v*²/*c*^{4})

= *γ*²(*tt′* − *tt′**v*²/*c*²) = *γ*²(*tt′*(1 − *β*²))

where *β = v*/*c* so that

* γ*² = 1/(1 − *β*²) or *γ* = 1/√(1 − *β*²).

Therefore, the Lorentz transformations with round-trip light for velocity are

*ct′ = γ*(*ct − βx*) and *x′* = γ(*x − βct*) with *γ* = 1/√(1 − *β*²).

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 an instant *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 instant *P* we have:

*t = kx* and *t′ = kx′* together so that *tt′* = *k*²*xx′* 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′* = *γ*²(*x − t*/*w*)(*x′ − t′*/*w*). Then

*xx′* = *γ*²(*x − t*/*w*)(*x′ + t′*/*w*) = *γ*²(*xx′* − *tx′*/*w *+ *xt′*/*w*− *tt′*/*w*²)

= *γ*²(*xx′* − *k*²*xx′*/*w*²) = *γ*²(*xx′*(1 − *β*²))

where *β = k*/*w* so that

* γ*² = 1/(1 − *β*²) or *γ* = 1/√(1 − *β*²).

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′* = *γ*²(*t − k²x*/*w*)(*t′ + k²x′*/*w*). Then

*tt′* = *γ*²(*t − k²x*/*w*)(*t′ + k²x′*/*w*) = *γ*²(*tt′* − *xt′*/*w* + *tx′*/*w* − *k ^{4}xx′*/

*w*²)

= *γ*²(*tt′* − *k²tt′*/*w*²) = *γ*²(*tt′*(1 − *β*²))

where *β = k*/*w* so that

* γ*² = 1/(1 − *β*²) or *γ* = 1/√(1 − *β*²).

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 − *β*²).

Note the duality by interchanging *x* and *t*.