Opposite velocities and lenticities

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Two opposite velocities — or lenticities — are invariant over time and space.

The standard Galileian transformation in the space-time domain is

x'=x-vt,\; y'=y,\; z'=z

Velocity u transforms as

u'=\frac{\mathrm{d} x'}{\mathrm{d} t}= \frac{\mathrm{d} x}{\mathrm{d} t}-\frac{\mathrm{d} vt}{\mathrm{d} t}=u-v

Velocity is not invariant relative to a single inertial observation, but it is relative to observations with opposite relative velocities:

u'+u'=(u-v)+(u+v)

That is

u'=u

Harmonic velocities are opposites and so are Galileian invariant in space-time.

Similarly, the standard Galileian transformation in the time-space domain is

t'=t-wx,\; t_{2}'=t_{2},\; t_{3}'=t_{3}

Lenticity n transforms as

n'=\frac{\mathrm{d} t'}{\mathrm{d} x}= \frac{\mathrm{d} t}{\mathrm{d} x}-\frac{\mathrm{d} wx}{\mathrm{d} x}=n-w

Lenticity is not invariant relative to a single inertial observation, but it is relative to observations with opposite relative lenticities:

n'+n'=(n-w)+(n+w)

That is

n'=n

Harmonic lenticities are opposites and so are Galileian invariant in time-space. Notably, the pace of light reflected a given distance and back is invariant.

The time-space domain may be represented to a limited extent with the reciprocals of the Galileian transformation:

\frac{1}{x'}=\frac{1}{x}-\frac{1}{vt},\; \frac{1}{y'}=\frac{1}{y},\; \frac{1}{z'}=\frac{1}{z}

Velocity u transforms over distance as

(u')^{-1}=\left (\frac{\mathrm{d} x'}{\mathrm{d} t} \right )^{-1}= \left (\frac{\mathrm{d} x}{\mathrm{d} t} \right )^{-1}- \left (\frac{\mathrm{d} vt}{\mathrm{d} t} \right )^{-1}= (u)^{-1}-(v)^{-1}

Velocity is invariant relative to observations with opposite relative velocities:

(u')^{-1}+(u')^{-1}=(u)^{-1}-(v)^{-1}+(u)^{-1}+(v)^{-1}

That is

(u')^{-1}=(u)^{-1}