radial coordinate — from formula anatomyThe radial coordinate is a spatial variable that measures distance from a chosen origin or axis in a coordinate system. In spherical coordinates, it is typically denoted r and ranges from 0 to infinity, while in cylindrical coordinates it is often ρ (or r) and measures perpendicular distance from a central axis. It is fundamental in describing systems with spherical or cylindrical symmetry, such as planetary orbits, atomic orbitals, and electromagnetic fields around point charges.
Radial coordinates have been used since ancient Greek astronomy, but their formal mathematical treatment in physics was developed by Leonhard Euler and others in the 18th century, particularly in the context of celestial mechanics and potential theory.
In a radial coordinate system, position is specified by the radial distance plus angular coordinates (e.g., θ and φ in spherical, z in cylindrical). The radial coordinate appears in the Laplacian operator, which governs many physical equations, and in radial distribution functions that describe how properties vary with distance from a center. For central forces, the radial coordinate often separates from angular variables in the Schrödinger equation, leading to radial wavefunctions.
Radial coordinates are essential in quantum mechanics for describing hydrogen-like atoms, in astrophysics for modeling stellar structure, and in engineering for analyzing stress in cylindrical or spherical vessels.
In general relativity, the radial coordinate does not always correspond to physical distance; for a Schwarzschild black hole, the radial coordinate r has a horizon at r = 2GM/c², where the coordinate becomes singular even though the curvature is finite.
| ds^2 | variable | spacetime interval — spacetime interval metre squared |
| dt | variable | coordinate time differential — time differential second |
| dr | variable | radial coordinate differential — radial coordinate differential metre |
| dΩ^2 | variable | solid angle element squared — solid angle element squared |
| ρ | variable | metric function of r and t — metric function |
| λ | variable | metric function of r and t — metric function |
| r | variable | radial coordinate — radial coordinate metre |
| v | variable | null coordinate (e.g., advanced time) — null coordinate metre |
| r | variable | radial coordinate — radial coordinate metre |
| ∂_v | operator | differentiates with respect to the null coordinate — partial derivative |
| ds^2 | variable | Spacetime interval squared — spacetime interval metre squared |
| \Phi(r) | variable | Redshift function — redshift function |
| b(r) | variable | Shape function — shape function metre |
| r | variable | Radial coordinate — radial coordinate metre |
| t | variable | Time coordinate — time coordinate second |
| d\Omega^2 | variable | Solid angle element squared — solid angle element |
| d | operator | Exterior derivative — differential operator |
| r | variable | radial coordinate of the throat — radial coordinate metre |
| z | variable | proper distance along the throat — proper distance metre |
| b | variable | shape function of the wormhole — shape function metre |
| d/dz | operator | derivative with respect to proper distance — ordinary derivative |
| d/dr | operator | derivative with respect to radial coordinate — ordinary derivative |
| ds^2 | variable | spacetime interval — spacetime interval metre squared |
| dt | variable | coordinate time differential — time differential second |
| dr | variable | radial coordinate differential — radial coordinate differential metre |
| dΩ^2 | variable | solid angle element squared — solid angle element squared |
| r | variable | radial coordinate — radial coordinate metre |
| t | variable | coordinate time — coordinate time second |
| G | constant | Newton's constant of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| M | constant | mass of the central object — mass kilogram |
| d | operator | infinitesimal change — differential |
| \Delta(r) | variable | Horizon function of the Kerr metric — horizon function square metre |
| r | variable | Radial coordinate (Boyer-Lindquist) — radial coordinate metre |
| M | variable | Black hole mass — mass kilogram |
| a | variable | Specific angular momentum (spin parameter) — specific angular momentum metre |
