# Lorentz with round-trip light

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 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′ = 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′/+ xt′/wtt′/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.