Technopedia

The technologist’s encyclopedia

Compact objects

Compact objects — white dwarfs, neutron stars, and black holes — are the end states of stellar evolution, where gravity has overwhelmed every pressure support except quantum degeneracy or, ultimately, nothing at all. They are the strong-field solutions of the Gravity topic: the regimes where Φ/c2\Phi/c^2 approaches 1 and general relativity is not a correction but the whole story.

Authoritative sources

White dwarfs: electron degeneracy

Definition. A white dwarf is supported by electron degeneracy pressure, giving a mass–radius relation R∝M−1/3R \propto M^{-1/3}: adding mass makes the star smaller, until the Chandrasekhar limit MCh≈1.44 M⊙M_{\rm Ch} \approx 1.44\,M_\odot, beyond which no electron pressure can hold it up.

Details
  1. Degeneracy pressure. Squeezing electrons into a smaller volume raises their Fermi momentum (Pauli exclusion); the resulting pressure is independent of temperature. Relativistic degeneracy softens the equation of state (p∝ρ4/3p \propto \rho^{4/3}), which caps the supported mass.
  2. Chandrasekhar limit. Above 1.44 M⊙1.44\,M_\odot, degeneracy fails and collapse proceeds — the trigger for Type Ia supernovae (accreting dwarfs) and core-collapse supernovae (iron cores reaching the limit).

Neutron stars and the TOV equation

Definition. A neutron star is supported by neutron degeneracy pressure plus nuclear forces; its structure follows the Tolman–Oppenheimer–Volkoff (TOV) equation, the general-relativistic generalization of hydrostatic equilibrium:

dpdr=−G(ρ+p/c2)(m(r)+4πr3p/c2)r2(1−2Gm(r)/(c2r)).\frac{dp}{dr} = -\frac{G\left(\rho + p/c^2\right)\left(m(r) + 4\pi r^3 p/c^2\right)} {r^2\left(1 - 2Gm(r)/(c^2 r)\right)}.
Details
  1. Pressure gravitates. The TOV equation adds p/c2p/c^2 to the density in both factors and multiplies the whole thing by the metric factor (1−2Gm/c2r)−1(1 - 2Gm/c^2r)^{-1}. Pressure, which helps in Newtonian mechanics, makes relativistic collapse worse — this is why there is a maximum neutron-star mass (∼2.3 M⊙\sim 2.3\,M_\odot observed, exact value equation-of-state dependent) and no stable object can be arbitrarily massive.
  2. Buchdahl bound. Even with exotic support, any static sphere of radius RR needs 2GM/(c2R)<8/92GM/(c^2R) < 8/9: get within ~11% of your Schwarzschild radius and no equation of state can save you.
  3. Observations. Radio pulsar timing weighs neutron stars to ∼10−4\sim 10^{-4} precision; NICER measures radii; gravitational-wave events like GW170817 constrain the equation of state through tidal deformability.

Black holes

Definition. When 2GM/(c2R)→12GM/(c^2R) \to 1, an event horizon forms at the Schwarzschild radius

rs=2GMc2≈3 km×MM⊙,r_s = \frac{2GM}{c^2} \approx 3\ \mathrm{km} \times \frac{M}{M_\odot},

a one-way causal boundary: the defining strong-field object of GR, fully described by the Kerr metric (mass + spin + charge).

Details
  1. No-hair theorem. A stationary black hole is completely characterized by mass, spin, and charge — an enormous information compression compared to the star that formed it.
  2. Kerr metric. Real astrophysical black holes spin; the Kerr solution’s frame dragging drags spacetime around the horizon and shrinks it to r+=GM/c2r_+ = GM/c^2 for an extremal hole.
  3. Observations. The Event Horizon Telescope imaged the shadows of M87* and Sgr A*; LIGO hears their mergers. Both probe the same geometry from complementary sides.

Relation to other classes

Compact objects are the strong-field stationary solutions of General relativity; their mergers are the loudest sources of Gravitational waves, and their abundance and formation history feed into Cosmological gravity through the matter budget of the universe.