Constants and Scales

Constants enter a gravity calculation in two different ways. Some are standards values used to translate between units; others appear inside dimensionless ratios or curvature scales that diagnose a physical regime. This directory keeps those jobs separate, so an exact conversion is not mistaken for a measurement and a strong potential is not mistaken for a large local curvature.

Five complementary lookup cards

Record	Use it to answer	Distinction to preserve
Fundamental Constants Used in Gravity	Which values, units, precision, and uncertainty should enter an SI calculation?	Exact SI definitions, exact derived relations, measured values, and exact nominal astronomical conversions have different meanings.
Planck Scales	Which combinations of $c$, $\hbar$, and $G$ define the conventional quantum-gravity scales?	A dimensionful scale does not by itself establish new dynamics, discreteness, or a physical cutoff.
Solar-Mass Geometric Units	How do solar-mass ratios become geometric lengths, times, and frequency scales?	The IAU nominal solar mass parameter is an exact conversion factor; a mass in kilograms inherits the uncertainty of $G$.
Black-Hole Horizon and Ringdown Scales	Which radii, areas, rotation rates, thermal scales, and mode times characterize a stationary Kerr black hole?	Coordinate radius, intrinsic horizon size, surface gravity, temperature, and quasinormal frequencies are related but not interchangeable.
Compactness and Curvature Scales	Is a system weak-field, compact, strongly accelerated, or strongly curved?	Potential, acceleration, tidal curvature, and scalar invariants answer different questions and scale differently with size and mass.

The fundamental-constants card is numerical and standards-bound. It records the defining constants most often restored after a calculation in relativistic or natural units, the measured value of $G$ with its standard uncertainty, and nominal solar quantities that provide exact conversion factors. The significant digits and exactness labels are part of the lookup result; they should survive copying into code or a table.

The Planck and solar-mass cards build derived scales for two very different purposes. Planck units expose where quantum, relativistic, and gravitational dimensions meet, while retaining the distinction between reduced and unreduced mass conventions. Solar geometric units provide a clean conversion for relativistic systems without pretending that a nominal conversion is a new measurement of the Sun.

The horizon card specializes those scales to stationary Kerr geometry. It keeps the two horizon roots, intrinsic area, angular velocity, surface-gravity normalization, thermal conversion, and mode-dependent ringdown quantities explicit. Its limiting checks make the Schwarzschild and extremal cases easy to recognize without turning a stationary reference formula into a claim about an observed or evolving object.

The compactness card is analytic. It starts from the dimensionless compactness $GM/(Rc^2)$, then compares it with surface acceleration, mean density, tidal curvature, and the Schwarzschild Kretschmann invariant. This exposes an important black-hole limit: horizon compactness stays fixed as the mass grows, while horizon curvature falls. “Strong gravity” therefore needs a named diagnostic rather than an unspecified adjective.

A safe lookup sequence

First identify whether the task is a standards conversion, a dimensional estimate, a geometric specialization, or a physical diagnosis. For a conversion, retain the stated unit, exactness class, and uncertainty before rounding. For a derived scale, state the convention—especially factors of $2$, $2\pi$, and $\sqrt{8\pi}$. For a horizon quantity, fix the generator and time normalization before converting between inverse length, frequency, acceleration, and temperature. For a diagnosis, name the radius and observer or frame involved, then compare quantities with the same dimensions. In every case, keep $G$, $c$, and $\hbar$ visible until the unit system and physical meaning are clear.

These are compact lookup cards, not substitutes for derivations. Follow each card’s full-treatment links when a normalization, observer choice, or limiting argument matters to the conclusion.