# Opposite velocities and lenticities

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

The standard Galileian transformation in the space-time domain is

$x'=x-vt,\;&space;y'=y,\;&space;z'=z$

Velocity u transforms as

$u'=\frac{\mathrm{d}&space;x'}{\mathrm{d}&space;t}=&space;\frac{\mathrm{d}&space;x}{\mathrm{d}&space;t}-\frac{\mathrm{d}&space;vt}{\mathrm{d}&space;t}=u-v$

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

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

Lenticity n transforms as

$n'=\frac{\mathrm{d}&space;t'}{\mathrm{d}&space;x}=&space;\frac{\mathrm{d}&space;t}{\mathrm{d}&space;x}-\frac{\mathrm{d}&space;wx}{\mathrm{d}&space;x}=n-w$

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:

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

Velocity u transforms over distance as

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

Velocity is invariant relative to observations with opposite relative velocities:

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

That is

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