Geometrized unit system
A unit system setting c and G to unity for relativity.
A geometrized unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum (c) and the gravitational constant (G) are used as defining constants. It is not a completely defined system; some systems, such as Stoney units and Planck units, set these two constants in addition to other constants to unity. This system is used in physics, especially in the special and general theories of relativity, where many equations appear simpler when expressed in geometrized units because all occurrences of G and c drop out.
In such a system, every time interval is interpreted as the distance light travels during that interval, so time takes on the dimension of length. Energy and momentum, as components of the four-momentum vector, and invariant mass, as the magnitude of that vector, all likewise acquire the dimension of length. A mass expressed in kilograms can be converted to an equivalent length in metres by multiplying by the conversion factor G/c². For instance, the Sun’s mass in SI units is equivalent to roughly half the Schwarzschild radius of a one-solar-mass black hole. All other conversion factors are derived by combining these two constants. The small numerical size of these conversion factors reflects that relativistic effects become noticeable only with large masses or high speeds. An alternative "rationalized" system of geometrized units is often used in particle physics and cosmology, where 4πG or 8πG are set to unity instead of G alone, making equations such as the Einstein field equations, the Einstein–Hilbert action, the Friedmann equations, and the Newtonian Poisson equation appear simpler and more natural. The definition of geometrized units as setting c, G, and the Boltzmann constant k to unity was formalized in the book *Gravitation* by Misner, Thorne, and Wheeler, with some authors referring to them as geometrodynamic units.
- field
- Physics (relativity, cosmology)
- known_for
- Setting c and G to unity; simplifying relativistic equations; converting mass to length via G/c²
Lore & Background
Geometrized units were defined in the book Gravitation by Misner, Thorne, and Wheeler such that the speed of light c, the gravitational constant G, and Boltzmann constant kB are all set to 1. Some authors refer to these units as geometrodynamic units. In geometrized units, every time interval is interpreted as the distance travelled by light during that given time interval—one second is interpreted as one light-second, so time has the geometrized units of length. Energy and momentum are interpreted as components of the four-momentum vector, and invariant mass is the magnitude of this vector, so in geometrized units these must all have the dimension of length.
Reader's Guide
The geometrized unit system is significant because it simplifies many equations in relativistic physics, making them more natural and easier to work with. For example, the Schwarzschild radius of a nonrotating uncharged black hole with mass m becomes rs = 2m. An alternative 'rationalized' system uses 4πG or 8πG instead, simplifying equations such as the Einstein field equations, the Einstein–Hilbert action, the Friedmann equations, and the Newtonian Poisson equation. Conversion factors between SI and geometrized units are derived from c and G; for instance, the Sun's mass of 2.0×10³⁰ kg is equivalent to 1.5 km. The small numerical size of these conversion factors reflects that relativistic effects are only noticeable when large masses or high speeds are considered. The system is not fully defined, as different variants (e.g., Stoney units, Planck units) set additional constants to unity.
Did You Know?
- In geometrized units, one second is interpreted as one light-second, giving time the dimension of length.
- The Sun's mass of 2.0×10³⁰ kg is equivalent to 1.5 km in geometrized units.
- An alternative 'rationalized' system uses 4πG or 8πG instead of G alone.
- The Schwarzschild radius of a nonrotating uncharged black hole simplifies to rs = 2m in geometrized units.
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