Buchdahl Bound in Relativistic Stars

• Buchdahl bound is the compactness limit for static, spherically symmetric perfect fluids in general relativity.
• It is derived under strict assumptions such as isotropy, nonincreasing density, and regular center, marking the onset of infinite central pressure.
• Recent studies extend this bound through quasilocal energy methods, exploring charged, anisotropic, and higher-dimensional generalizations.

The Buchdahl bound is the classical compactness limit for a static, spherically symmetric relativistic star under restrictive but standard matter assumptions. In its familiar four-dimensional general-relativistic form, it states

$$\frac{2M}{R}\le \frac{8}{9}, \qquad\text{equivalently}\qquad \frac{M}{R}\le \frac{4}{9}, \qquad R\ge \frac{9}{4}M,$$

for a regular isotropic perfect fluid matched at a finite radius $R$ to an exterior Schwarzschild spacetime. In the surveyed literature, this bound functions simultaneously as a theorem about hydrostatic equilibrium, a criterion for central-pressure blow-up, a surface-potential limit $\Phi(R)\le 4/9$, and, in several recent reinterpretations, an energetic or virial threshold for the most compact horizonless object, often termed a Buchdahl star (Dadhich, 2019, Dadhich, 2022, Boehmer et al., 1 Jul 2025).

1. Classical statement and limiting configuration

The standard Buchdahl theorem applies to a static, spherically symmetric perfect-fluid configuration with isotropic pressure, nonnegative density and pressure, nonincreasing density outward, regular center, and smooth matching at the surface to the Schwarzschild vacuum. Under these hypotheses, the mass $M$ and areal radius $R$ satisfy

$\frac{2M}{R}\le \frac{8}{9}.$

Equivalent forms used in the literature include $M/R\le 4/9$ and $R\ge 9M/4$ (Dadhich, 2019, Dadhich, 2022, Reyes et al., 2023).

A central interpretation is that the inequality marks the onset of pressure pathologies in the incompressible reference solution. For the constant-density Schwarzschild interior solution, the central pressure diverges as the compactness approaches the limiting value. Several of the cited works use this divergence as the operational meaning of saturation: the bound is not merely algebraic, but the point beyond which no finite isotropic pressure can sustain static equilibrium (Lemos et al., 2015, Sharma et al., 2020).

The same limit is often written in terms of a surface potential. For a neutral object one has

$\Phi(R)=\frac{M}{R}\le \frac{4}{9},$

while a Schwarzschild black hole is characterized by $\Phi(R)=1/2$. This sharp separation between $R$0 and $R$1 underlies the recurrent distinction between the most compact horizonless object and a true horizon (Dadhich, 2022, Dadhich et al., 2023).

The literature surveyed here is not fully uniform on the associated redshift statement. Some works state that $R$2 implies surface redshift $R$3 (Dadhich, 2019, Boehmer et al., 1 Jul 2025), whereas one paper states a surface redshift bound $R$4 (Dadhich et al., 2023). This suggests that redshift conventions or auxiliary assumptions are not identical across these treatments.

2. Assumptions, proof strategy, and physical content

The proof architecture emphasized across the cited literature is highly structured. The essential ingredients are staticity, spherical symmetry, isotropy, regularity at the center, vanishing pressure at a finite boundary, and monotone density. In a standard metric representation,

$R$5

the Tolman–Oppenheimer–Volkoff equation is written as

$R$6

Within the classical Buchdahl argument, monotone decrease of density implies monotonicity of $R$7, and the isotropy condition yields the differential identity that drives the comparison inequality leading to $R$8 (Arrechea et al., 2024).

These assumptions are not incidental. The modern literature repeatedly stresses that Buchdahl’s theorem is a theorem for a specific class of matter models rather than a universal compactness ceiling. If isotropy is relaxed, if density is not nonincreasing, if charge is present, or if the gravitational sector differs from Einstein gravity, the original proof no longer applies in its classical form (Lake, 2016, Arrechea et al., 2024).

Within general relativity, the same bound is often presented as a stability statement. In the TOV framework for a constant-density star,

$R$9

the denominator becomes small as the Buchdahl threshold is approached, steepening the pressure gradient and driving $\Phi(R)\le 4/9$0 (Maier, 14 Apr 2026). A plausible implication is that the theorem can be read not only as a geometric inequality but also as a criterion for the breakdown of finite-pressure hydrostatic support.

3. Surface potential, Brown–York energy, and the Buchdahl star

A major recent theme is the reinterpretation of the Buchdahl bound in terms of quasilocal energy. In the Reissner–Nordström exterior, the Brown–York quasilocal energy inside radius $\Phi(R)\le 4/9$1 is written as

$\Phi(R)\le 4/9$2

while the non-gravitational matter energy is identified as

$\Phi(R)\le 4/9$3

The residual quantity

$\Phi(R)\le 4/9$4

is then interpreted as gravitational field energy outside the star. The compactness criterion is expressed as

$\Phi(R)\le 4/9$5

which yields

$\Phi(R)\le 4/9$6

For $\Phi(R)\le 4/9$7, this reproduces the classical Buchdahl inequality $\Phi(R)\le 4/9$8 (Dadhich, 2019).

This exterior-energy viewpoint motivates the notion of a Buchdahl star as the limiting horizonless object saturating $\Phi(R)\le 4/9$9. In this interpretation, a Buchdahl star is the configuration for which gravitational energy is half of non-gravitational energy, whereas a black hole corresponds to equality of the two. One paper further relates this to the speed of a radially falling timelike particle via

$M$0

so that $M$1 for a Buchdahl star and $M$2 at a black-hole horizon (Dadhich, 2022).

This energetic picture is then connected to a virial-theorem analogy. In a Vlasov kinetic matter interpretation, internal gravitational energy is treated as kinetic energy, non-gravitational matter energy as potential energy, and the Buchdahl-star condition becomes

$M$3

A related geometric reformulation in the $M$4 covariant formalism defines

$M$5

and imposes the virial-type condition

$M$6

at the Buchdahl limit. In that framework, an accreting Buchdahl star remains in virial equilibrium only if it radiates the excess energy away as heat flux, appearing externally as Vaidya radiation (Dadhich et al., 2023).

4. Charged generalizations and sharp bounds

Charge modifies the compactness ceiling by reducing the effective attractive mass. In the exterior-potential formulation,

$M$7

and this leads to the charge bound

$M$8

The resulting compactness limit is therefore not a bound on $M$9 alone, but on the effective mass $R$0. One notable consequence stressed in the literature is that compact objects may satisfy $R$1 without being black holes (Dadhich, 2019).

The same structure appears in explicit charged stellar models. In a charged extension of the Schwarzschild interior using the Vaidya–Tikekar ansatz, the limiting infinite-central-pressure condition gives

$R$2

which yields

$R$3

This reduces to $R$4 in the neutral limit and again enforces $R$5 (Sharma et al., 2020).

The charged analogue of Buchdahl’s sharp inequality is the Buchdahl–Andréasson bound,

$R$6

equivalently,

$R$7

under the inequality

$R$8

In Andréasson’s proof, the equality case is a charged thin shell. However, Guilfoyle’s stars provide a regular charged-fluid realization of the same limiting compactness. For the subclass satisfying

$R$9

the infinite-central-pressure limit saturates the Buchdahl–Andréasson inequality. In the uncharged limit this reduces to the interior Schwarzschild constant-density condition, while in the extremal charged limit $\frac{2M}{R}\le \frac{8}{9}.$0 one obtains $\frac{2M}{R}\le \frac{8}{9}.$1, the quasiblack-hole limit (Lemos et al., 2015).

5. Anisotropy, evasion mechanisms, and super-Buchdahl configurations

The classical Buchdahl bound is not universal once isotropy is abandoned. A particularly direct demonstration is an infinite class of exact static anisotropic spheres satisfying regularity, positive monotone decreasing $\frac{2M}{R}\le \frac{8}{9}.$2, $\frac{2M}{R}\le \frac{8}{9}.$3, and $\frac{2M}{R}\le \frac{8}{9}.$4, a finite boundary $\frac{2M}{R}\le \frac{8}{9}.$5, the inequality $\frac{2M}{R}\le \frac{8}{9}.$6, and the anisotropy condition

$\frac{2M}{R}\le \frac{8}{9}.$7

For these solutions,

$\frac{2M}{R}\le \frac{8}{9}.$8

and the generalized TOV relation becomes

$\frac{2M}{R}\le \frac{8}{9}.$9

All standard energy conditions are satisfied except the dominant energy condition, whose violation arises from $M/R\le 4/9$0 by construction. Within the family

$M/R\le 4/9$1

the boundary exists only for $M/R\le 4/9$2, and for $M/R\le 4/9$3 the solutions break the classical Buchdahl bound while still satisfying the imposed regularity and monotonicity conditions. At the same time, they obey the Andréasson-type anisotropic compactness ceiling

$M/R\le 4/9$4

associated with the matter inequality above (Lake, 2016).

Anisotropy does not, however, always weaken the compactness limit. In model-dependent anisotropic exact solutions based on Karmarkar-class-I and Vaidya–Tikekar geometries, the derived upper bound can be lower than $M/R\le 4/9$5. For example, one studied model yields

$M/R\le 4/9$6

recovering $M/R\le 4/9$7 when $M/R\le 4/9$8, while another gives

$M/R\le 4/9$9

again reducing to $R\ge 9M/4$0 in the isotropic limit. In these constructions, increasing anisotropy lowers the maximum compactness (Sharma et al., 2021). This establishes that the effect of anisotropy is not sign-definite; it depends on the precise stress configuration and geometry.

Relaxing monotone density provides a second route beyond the classical theorem. Bilayered isotropic stars with outward-increasing density,

$R\ge 9M/4$1

can exceed $R\ge 9M/4$2 while preserving positive density. For $R\ge 9M/4$3, the cited toy model yields a new ceiling

$R\ge 9M/4$4

If negative core density is allowed, then

$R\ge 9M/4$5

becomes possible with finite regular pressure, at the cost of energy-condition violation. Thin-shell models provide a parallel mechanism: for a shell with Minkowski interior, the dominant energy condition gives

$R\ge 9M/4$6

while other anisotropic bounds quoted in the same work include $R\ge 9M/4$7 and $R\ge 9M/4$8 (Arrechea et al., 2024).

6. Extensions beyond four-dimensional general relativity and current directions

The Buchdahl problem has been generalized extensively beyond standard four-dimensional Einstein gravity. In five-dimensional Einstein–Gauss–Bonnet gravity, the compactness limit depends qualitatively on the sign of the Gauss–Bonnet coupling $R\ge 9M/4$9. For $\Phi(R)=\frac{M}{R}\le \frac{4}{9},$0, the bound becomes structure-dependent through the central density, with

$\Phi(R)=\frac{M}{R}\le \frac{4}{9},$1

and the corresponding inequality

$\Phi(R)=\frac{M}{R}\le \frac{4}{9},$2

In this regime, stable stars can exist arbitrarily close to the event horizon. For $\Phi(R)=\frac{M}{R}\le \frac{4}{9},$3, the bound is more restrictive than the five-dimensional GR result $\Phi(R)=\frac{M}{R}\le \frac{4}{9},$4 (Wright, 2015).

In Eddington-inspired Born–Infeld gravity, the bound depends on the $\Phi(R)=\frac{M}{R}\le \frac{4}{9},$5-energy condition

$\Phi(R)=\frac{M}{R}\le \frac{4}{9},$6

and takes the form

$\Phi(R)=\frac{M}{R}\le \frac{4}{9},$7

with

$\Phi(R)=\frac{M}{R}\le \frac{4}{9},$8

For $\Phi(R)=\frac{M}{R}\le \frac{4}{9},$9, one has $\Phi(R)=1/2$0, so the star must be less compact than in GR. The same analysis yields a possible observational constraint $\Phi(R)=1/2$1 if a neutron star with mass around $\Phi(R)=1/2$2 is observed, and identifies the center-regularity condition

$\Phi(R)=1/2$3

as a Hagedorn-like equation of state (Feng et al., 2018).

Einstein-æther theory provides another non-GR deformation. For $\Phi(R)=1/2$4, the derived Buchdahl-type inequality is

$\Phi(R)=1/2$5

with the standard GR value $\Phi(R)=1/2$6 recovered at $\Phi(R)=1/2$7. After conversion to the asymptotic Schwarzschild mass, the physical compactness limit is lower than in GR, consistent with the outward shift of the throat, photon ring, and ISCO (Hsu et al., 2024).

Higher-dimensional generalizations also exist within ordinary Einstein gravity. In $\Phi(R)=1/2$8 dimensions, under the Andréasson condition

$\Phi(R)=1/2$9

one obtains

$R$00

while for $R$01 and $R$02,

$R$03

These inequalities imply corresponding redshift bounds, including $R$04 for the Andréasson condition and $R$05 in the Buchdahl/isotropic case (Wright, 2015).

Two current directions are especially notable. First, the presence of a cosmological constant leads to several inequivalent-looking generalized bounds. In Schwarzschild–de Sitter language,

$R$06

gives

$R$07

whereas the traditional interior-solution route yields a different small-$R$08 correction proportional to $R$09. This discrepancy has motivated a proposed new inequality,

$R$10

which reproduces $R$11 at $R$12 and expands as $R$13 for small $R$14 (Boehmer et al., 1 Jul 2025).

Second, semiclassical and interacting-vacuum analyses suggest that the classical limit need not remain decisive near the Buchdahl radius. For conformal quantum fields on a constant-density star approaching

$R$15

the renormalized stress tensor diverges faster than the classical source: $R$16 whereas the classical pressure diverges only as $R$17. The same study finds null-energy-condition violation near the inner light ring, indicating that quantum backreaction cannot be ignored arbitrarily close to $R$18 (Reyes et al., 2023). In a different direction, interacting-vacuum extensions of the TOV system introduce a dynamical vacuum contribution $R$19 and can keep the central pressure finite beyond the classical threshold for suitable couplings, thereby making the compactness limit model-dependent rather than purely geometric (Maier, 14 Apr 2026).

Taken together, these developments support a precise but non-universal reading of the Buchdahl bound. It remains the sharp compactness theorem for regular isotropic, monotone-density perfect fluids in classical four-dimensional general relativity, yet many nearby theories and matter models replace it with modified inequalities, alternative saturation mechanisms, or genuinely super-Buchdahl horizonless configurations.