User:Quondum/sandbox/diffeo

"speed of light in vacuum" vs. "speed of light in a vacuum"
https://en.wikipedia.org/w/index.php?title=Wikipedia_talk:WikiProject_Physics&oldid=1136486979#%22speed_of_light_in_vacuum%22_vs._%22speed_of_light_in_a_vacuum%22

Quantities of confusion
This is primarily motivated by trying to bring WP into line with standards around angle-related quantities and units, which seem to confuse editors. This is an attempt to establish a framework for thinking about it, in which to clearly formulate the concepts. It is not to be used to motivate terminology for use in WP, only for terminology for use in discussing the concepts. It does this through extending SI for the purposes of discussion with extra base dimensions.

A ratio suffix '-ic'
Start by defining a few terms with a regular form for purposes of this page. In general, add 'ic' to indicate derivation of an intensive quantity from an extensive quantity by division by (or differentiation against) a given extensive quantity, and may be thought of as suggesting "measured against". There seem to be real-world examples of this.
 * lineic – quotient or derivative with respect to length, often called a linear density (introduced in ISO 31-0, defined in the IUPAC Gold Book)
 * areic – quotient or derivative with respect to area, e.g. 'areic volume' for measuring rainfall, 'areic mass' for area density (note: consider bivector implications) (introduced in ISO 31-0, defined in the IUPAC Gold Book)
 * volumic – quotient or derivative with respect to volume, e.g. 'volumic mass' for mass density (note: consider trivector implications) (introduced in ISO 31-0, defined in the IUPAC Gold Book)
 * massic – quotient or derivative with respect to mass, a.k.a. specific, as in 'massic heat capacity' (introduced in ISO 31-0, defined in the IUPAC Gold Book)
 * timeic – quotient or derivative with respect to elapsed time, e.g. 'timeic distance' for speed, 'timeic volume' for 'volume flow rate' (a trailing word 'rate' is commonly used instead, as in 'angular rate')
 * anguleic – quotient or derivative with respect to angle, e.g. 'anguleic work' is equal to torque (note: consider bivector implications) ['angle' should not do double duty as the quantity, so look for another, for example 'sweep']
 * phasic – quotient or derivative with respect to phase
 * quotient or derivative with respect to temperature?
 * quotient or derivative with respect to frequency?
 * a better form of '(amount-of-substance)ic' is needed ('molar' is the usual term, as in 'molar mass')

A positional derivation – 'moment'

 * moment of – the exterior product of displacement with a quantity, e.g. 'moment of momentum'

We should distinguish scalars, vectors, etc., in the systematic naming.

How should these combine? Presumably repeated application.

Keep in mind the distinct variants in Newtonian formulation and special relativity.

"Dimensionless" quantities and units
Keep angle and count as independent base quantities, with associated units: radian (rad) and count (c) respectively, to aid clarity of formulation.

Phase and plane angle are inherently distinct, and are distinguished here by a prime associated with phase. Thus, rad and rad′ would be distinguished coherent units in the SI extended to include distinct dimensions of angle and phase.

Moment-related quantities

 * "moment of energy–momentum" (relativistic, dust): ∑ x$μ$ ∧ p$ν$
 * "moment of mass" (Newtonian) – a vector; unit: kg⋅m: ∑ r m
 * "moment of momentum" (Newtonian) – a bivector; unit: N⋅m⋅s: ∑ r ∧ p
 * "centre of mass moment" (Newtonian) – a vector; unit: m: (∑ r m − t p)
 * massic moment of mass (centre of mass position) (Newtonian) – a vector; unit: m: (∑ r m) / (∑ m)
 * massic moment of momentum (centre of mass velocity) (Newtonian) – a vector; unit: m: (∑ r ∧ p) / (∑ m)
 * "moment of ??" (relativistic) – ??
 * "moment of power" (Newtonian) – a vector; unit: W⋅m: ∑ r E
 * "moment of force" (Newtonian) – a bivector; unit: N⋅m: ∑ r ∧ p (related to anguleic work (torque) through the circle constant)

Definitions:
 * d: exterior derivative
 * : partial derivative (with respect to observer's time, with observer's orthonormalized coordinates)
 * dust: "isolated gel blobs" (each particle with arbitrary four-momentum, but without internal stress or rotation in its rest frame

Geometric angle-related quantities

 * angle(?) – a bivector; unit: radian, degree, turn
 * anguleic work(?) (torque) – a bivector; unit: J/rad
 * timeic angle (angular velocity) – a bivector; unit: rad/s
 * rotation (to be renamed) – unit: c
 * timeic rotation (rotational frequency) – unit: c/s

Thoughts
In the Newtonian context (but not special relativity), angular velocity can be expressed
 * ω = $|r|$−2 r ∧ r

Without the scalar factor $|r|$−2, this is moment of velocity.

In the context of special relativity, the radius that is orthogonal to the worldline of the origin of the radius vector r might be used, suggesting that we might want the inverse square of (r ∧ v), where v is the four-velocity of the world line. In all, this could lead to some odd coherent names. What is currently called "angle" in the SI (but is similarly really a ratio of a bivector with some scalar), would get a new name under a system in which angle is dimensional.

Phase-related quantities

 * phase – a scalar; unit: radian′ (rad′)
 * timeic phase (angular frequency) – unit: rad′/s
 * phasic time – unit: s/rad′
 * lineic phase (angular wavevector) – unit: rad′/m
 * phasic length (angular wavelength) – vector quantity; unit: m/rad′

There are hyperbolic quantities that relate both phase and geometric angle in a similar way that time relates to distance. Information (unit: nat) is a similar logarithmic, though with out an obvious related analogue.