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Galilean transformation expanded

The Galilean transformation is typically presented for motion in direction of the x-axis, with the other axes unchanged:

x´ = xvt, y´ = y, z´ = z, and t´ = t,

where v is the relative velocity of the observers. This is incompatible with the Lorentz transformation, but more than that, it is inconsistent with the two-way (round-trip) speed of light in a vacuum.

The Lorentz transformation can be made compatible with the round-trip speed of light if light is considered to travel instantaneously to its observer, which is usually the final leg. The speed of light for the other part of the round trip can be inferred so that their harmonic mean equals c, which is the most that is known (see One-way speed of light).

That is, if the speed of light in the Lorentz transformation is allowed to approach infinity, then the transformation will approach the Galilean transformation. Here the Galilean transformation arises as a limit of the Lorentz transformation by the speed of light approaching infinity, rather than usual the relative velocity approaching zero.

The Lorentz transformation for motion in the direction of the x-axis is:

x´ = γ (xvt), y´ = y, z´ = z, and t´ = γ (tvx/c²), with γ = (1 – v²/c²)–1/2,

where γ (gamma) is the Lorentz factor. As c → ∞, γ → 1 and t´ → t. This can only be the case for one part of the light trip, which we’re taking as the last part of the trip.

Why the last part? Because that’s what is observed, directly or reflected in a mirror. And in everyday conversation the place where something is observed to be is spoken of as where it is now. Even with the convention of a constant speed of light, one has to be very pedantic to keep correcting others and oneself by saying that where something is seen to be is in fact where it was in the past.

For a round trip, the speed of light for the part not directly observed can be inferred from the empirical result that the round trip speed equals the constant, c:

x´ = γ (xvt), y´ = y, z´ = z, and t´ = γ (t – 4vx/c²), with γ = (1 – 4v²/c²)–1/2.

The speed of light for the unobserved part is inferred from the necessity that the harmonic mean equals c:

(1/c1 + 1/c2) = 2/c,

where c1c/2 as c2 → ∞. This harmonic mean of speeds is the arithmetic mean of paces. What is actually measured is the pace of light from the independent length traversed in the dependent time.

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