Nonextremal

Nonextremal is an adjective used in physics and mathematics to describe systems that are not at an extremal limit. In physics, the term is most often applied to solutions or states that do not saturate a bound that defines an extremal, or limiting, case. Extremal cases are those in which a parameter reaches a maximal or minimal value allowed by the governing equations, often resulting in distinctive or simplified properties. Nonextremal thus refers to ordinary or generic states away from such a bound.

In black hole physics, extremal solutions saturate a charge, angular momentum, or other bound relative to the mass. Extremal black holes typically have zero Hawking temperature and a degenerate horizon. Nonextremal black holes do not saturate these bounds and generally possess nonzero Hawking temperature and a nondegenerate horizon. For example, a nonextremal Reissner-Nordström or Kerr-Newman black hole has Q^2 < M^2 or J^2 < M^4 (in suitable units), respectively, and exhibits thermodynamic properties that depend on the distance from the extremal limit. Nonextremal black holes may have one or two horizons and emit Hawking radiation, with entropy related to the horizon area.

Beyond gravity, nonextremal can describe thermal or generic states in systems where an extremal (zero-temperature or saturated) state is excluded. In string theory, nonextremal branes correspond to finite-temperature configurations, as opposed to extremal branes, which are at zero temperature and often preserve more symmetry.

Overall, nonextremal denotes the absence of saturation of a defining bound, marking a departure from special, extremal cases toward more generic behavior.