Working in Natural Units

Natural unit systems set a chosen set of fundamental physical constants to the dimensionless value 1. This removes those constants from equations and makes quantities of different “everyday” dimensions share a scale — a mass expressed as an energy, a length as a time, and so on.

unxt provides four natural unit systems built in. They are realized numerically: the base units are chosen so the named constants evaluate to 1.0, while the full dimensional structure is preserved (so unxt’s dimension checking keeps working). This tutorial works through each one on a small physics problem.

>>> import numpy as np
>>> import unxt as u
>>> from astropy import constants as const, units as apu
>>> from unxt.unitsystems import unitsystem, hep, geometrized, atomic, planck

High-energy physics: masses as energies

In particle physics one sets \(\hbar = c = 1\) and measures everything in powers of an energy, conventionally the GeV. The built-in hep system uses exactly this scale, so its mass unit is \(1\,\mathrm{GeV}/c^2\):

>>> bool(np.isclose((1 * hep["mass"]).to_value("kg"),
...                 (1 * apu.GeV / const.c**2).to_value("kg")))
True

A proton then weighs about 0.938 in these units — the familiar \(m_p \approx 0.938\ \mathrm{GeV}/c^2\):

>>> round(float(const.m_p / (1 * hep["mass"])), 3)
0.938

The energy scale is configurable through HEPUSysFlag. A larger energy gives a smaller length and time (since \(\hbar = c = 1\)):

>>> from unxt.unitsystems import HEPUSysFlag
>>> unitsystem(HEPUSysFlag, energy="TeV")["time"] == hep["time"] / 1000
True

Geometrized units: masses as lengths

In general relativity one sets \(c = G = 1\), turning masses and times into lengths. The natural length scale for a gravitating body is its gravitational radius \(r_g = G M / c^2\). Building a geometrized system at the Sun’s gravitational radius makes the mass unit exactly one solar mass:

>>> from unxt.unitsystems import GeometrizedUSysFlag
>>> r_g = const.G * const.M_sun / const.c**2   # the Sun's gravitational radius
>>> apu.Unit(r_g)
Unit("1476... m")

>>> geo_sun = unitsystem(GeometrizedUSysFlag, length=apu.Unit(r_g))
>>> bool(np.isclose((1 * geo_sun["mass"]).to_value("kg"),
...                 const.M_sun.to_value("kg")))
True

By construction, \(c\) and \(G\) are both 1 in any geometrized system:

>>> bool(np.isclose(const.c.decompose(geometrized).value, 1.0))
True
>>> bool(np.isclose(const.G.decompose(geometrized).value, 1.0))
True

Atomic units: the scale of the atom

Atomic (Hartree) units set \(m_e = \hbar = e = 4\pi\varepsilon_0 = 1\). The length unit is the Bohr radius — about half an ångström:

>>> round(float((1 * atomic["length"]).to_value("Angstrom")), 4)
0.5292

Unlike the other systems, atomic units carry an electric-charge base dimension (the elementary charge):

>>> [str(d) for d in atomic.base_dimensions]
['length', 'mass', 'time', 'electrical charge']

Planck units

Planck units set \(\hbar = c = G = k_B = 1\) and are fully determined — there is no free scale. The base units are the Planck length, mass, time, and temperature:

>>> [str(d) for d in planck.base_dimensions]
['length', 'mass', 'time', 'temperature']

>>> for name in ("c", "hbar", "G", "k_B"):
...     assert np.isclose(getattr(const, name).decompose(planck).value, 1.0)

A note on semantics

unxt’s natural unit systems are numeric: they choose base units so the named constants equal 1. They do not (yet) add equivalences that let you convert directly between, say, a mass and an energy — a Quantity in MeV is still an energy, not a mass. Decomposing into a natural unit system (as above) is the supported way to obtain natural-unit values.