# Buchdahl's theorem

**Buchdahl's theorem** is a result in general relativity, proved by H. A. Buchdahl in 1959, that places an upper limit on how compact a static star can be: for a static, spherically symmetric star made of an isotropic perfect fluid whose energy density does not increase outward, the mass-to-radius ratio cannot exceed 𝑀/𝑅 = 4/9, well before the black-hole limit of 1/2.<sup>[1](https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1027)</sup><sup> • </sup><sup>[2](https://arxiv.org/abs/gr-qc/0605097)</sup>

| Key fact | Detail |
|---|---|
| Original publication | H. A. Buchdahl, "General Relativistic Fluid Spheres", *Physical Review* **116**, 1027, published 15 November 1959 (received 16 June 1959)<sup>[1](https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1027)</sup> |
| The bound | 𝑀/𝑅 ≤ 4/9, equivalently 2𝑀/𝑅 ≤ 8/9 or 𝑅 ≥ (9/8)𝑅ₛ, where 𝑅ₛ = 2𝑀 is the Schwarzschild radius<sup>[2](https://arxiv.org/abs/gr-qc/0605097)</sup><sup> • </sup><sup>[3](https://arxiv.org/abs/2202.00043)</sup> |
| Assumptions | Isotropic pressure, energy density non-increasing outward, exterior matched to the Schwarzschild vacuum metric<sup>[2](https://arxiv.org/abs/gr-qc/0605097)</sup> |
| Saturation | Attained by the constant-density Schwarzschild interior solution in the limit of divergent central pressure<sup>[4](https://arxiv.org/abs/1505.03863)</sup> |
| Relation to black holes | The bound is weaker than the Schwarzschild static limit 𝑀/𝑅 = 1/2; no stable static spherical configuration exists for 4/9 < 𝑀/𝑅 < 1/2<sup>[5](https://www.sciencedirect.com/science/article/pii/S0003491625000673)</sup> |
| Equation of state | The derivation assumes nothing about the equation of state beyond ρ ≥ 0 and dρ/dr ≤ 0<sup>[6](https://www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2014.pdf)</sup> |

## Statement and assumptions

The theorem concerns a "regular sphere": a static, spherically symmetric distribution of matter with finite radius and finite central density, whose pressure and density are nowhere negative and whose density does not increase outward.<sup>[7](https://adsabs.harvard.edu/pdf/1966ApJ...146..275B)</sup> Under these conditions, Buchdahl showed that any such relativistic star must satisfy the inequality 𝑅ₛ/𝑅 < 8/9, that is, the radius must exceed 9/8 of the [Schwarzschild radius](https://www.edgepedia.org/schwarzschild-radius) of the same mass.<sup>[8](https://arxiv.org/html/2010.02859v2)</sup> In the common compactness notation 𝒜 ≡ 𝑀𝑀/(𝑅c²), the maximum is 𝒜 = 4/9 ≈ 0.444.<sup>[3](https://arxiv.org/abs/2202.00043)</sup>

The proof rests on hypotheses that are physical but restrictive: the pressure is isotropic (the tangential stress equals the radial stress), the energy density is non-increasing from center to surface, and the metric is matched at the boundary to the Schwarzschild vacuum exterior.<sup>[2](https://arxiv.org/abs/gr-qc/0605097)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/science/article/pii/S0003491625000673)</sup> Notably, the derivation assumes only ρ ≥ 0 and dρ/dr ≤ 0 and nothing about the equation of state, so the inequality also applies to hot stars satisfying these assumptions.<sup>[6](https://www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2014.pdf)</sup>

## Why the bound arises

The limit appears as a divergence of the central pressure. For the constant-density interior solution solved by [Karl Schwarzschild](https://www.edgepedia.org/karl-schwarzschild) in 1916, the central pressure is finite and positive only when 3√(1 − 𝒜) < 1, which corresponds to the upper mass bound 𝑀 < (4/9)𝑧ₘ; violating it would require infinite central pressure to hold the star up.<sup>[9](https://link.springer.com/article/10.1140/epjc/s10052-015-3824-8)</sup> As the compactness approaches the Buchdahl bound, the required central pressure grows without bound and diverges at 𝒜 = 4/9.<sup>[10](https://doi.org/10.1088/1361-6382/ae7a2e)</sup>

**Sharpness.** The inequality is sharp: stars of constant density can get arbitrarily close to saturating it, with the central pressure diverging in the limit in which equality is reached.<sup>[6](https://www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2014.pdf)</sup> Buchdahl himself noted that the bound was saturated by the Schwarzschild interior solution, the solution with constant energy density, when the central pressure blows up to infinity.<sup>[4](https://arxiv.org/abs/1505.03863)</sup>

## The compactness gap

Because the Buchdahl bound (4/9) is strictly weaker than the black-hole limit (𝑀/𝑅 = 1/2), general relativity predicts a gap: even though an object in the range 4/9 < 𝑀/𝑅 < 1/2 is not trapped behind a horizon, no stable static spherical configuration can occupy that range.<sup>[5](https://www.sciencedirect.com/science/article/pii/S0003491625000673)</sup> This result prevents the existence of exotic compact objects (ECOs), horizonless mimickers of black holes, with compactness arbitrarily close to that of a black hole, at least under the theorem's assumptions.<sup>[11](https://link.springer.com/article/10.1007/s41114-019-0020-4)</sup> [Boson](https://www.edgepedia.org/boson) stars, for example, are even less compact than the Buchdahl bound.<sup>[12](https://doi.org/10.1103/PhysRevD.99.104072)</sup>

## Extensions and generalizations

**Weakening the assumptions.** Several alternative derivations of the bound exist under different conditions for spherical, static compact stars with a Schwarzschild exterior.<sup>[5](https://www.sciencedirect.com/science/article/pii/S0003491625000673)</sup> Håkan Andréasson, a mathematician at [Chalmers University of Technology](https://www.edgepedia.org/chalmers-university-of-technology) working on kinetic theory and general relativity, later provided an 𝑅-independent bound on the compactness ratio 2𝑀(𝑅)/𝑅 under purely algebraic inequalities on the energy density and pressure, removing the monotonicity assumption on the density profile.<sup>[8](https://arxiv.org/html/2010.02859v2)</sup><sup> • </sup><sup>[13](https://research.chalmers.se/en/publication/94556)</sup> Work by Guven and Ó Murchadha showed that if the density is monotonically decreasing and the tangential stress does not exceed the radial stress, the 8/9 bound continues to hold through the bulk with the quasi-local mass; if the tangential stress exceeds the radial stress or the density is not monotonic, the 8/9 bound cannot be recovered, though 2𝑀/𝑅 remains strictly bounded away from unity.<sup>[14](https://www.researchgate.net/publication/1977813_Bounds_on_2mR_for_static_spherical_objects)</sup>

**Charged and anisotropic stars.** The bound has been extended to charged matter: a charged analog of the [Buchdahl limit](https://www.edgepedia.org/buchdahl-limit) reduces to 𝑀/𝑅 ≤ 4/9 when the charge is switched off, and permits 𝑀/𝑅 ≤ 2/3 for α² = 𝑄²/𝑀² = 1.<sup>[15](https://link.springer.com/article/10.1140/epjc/s10052-021-08894-3)</sup> For anisotropic matter, the same limit can be obtained by invoking the strong energy condition ρ ≥ 𝑝ᵣ + 2𝑝ₜ.<sup>[15](https://link.springer.com/article/10.1140/epjc/s10052-021-08894-3)</sup> Generalizations have also been developed in the presence of a cosmological constant<sup>[16](https://discovery.ucl.ac.uk/id/eprint/10225974/)</sup> and in pure Lovelock gravity, where the uniform-density limit 4/9 holds for spacetime dimension 𝑑 = 3𝑁 + 1.<sup>[17](https://arxiv.org/abs/1606.01330)</sup>

**Geometric interpretation.** A 2025 analysis provides a model-independent geometric framework for the limit and shows that the Buchdahl bound is equivalent to a relativistic version of the Newtonian virial theorem; it also characterizes the bound as the limiting case of a static, spherical, thermodynamically stable configuration, with thermodynamic stability implying that the average density is monotonic and non-increasing from center to surface.<sup>[5](https://www.sciencedirect.com/science/article/pii/S0003491625000673)</sup>

## Buchdahl's own assessment

In a 1966 follow-up paper, Buchdahl observed that some of the inequalities in the 1959 work were too weak: in particular, when the parameter δ is small, the right-hand side of one formula does not tend to unity in the [Newtonian limit](https://www.edgepedia.org/newtonian-limit), and another becomes effectively empty for sufficiently small δ.<sup>[7](https://adsabs.harvard.edu/pdf/1966ApJ...146..275B)</sup> The 1959 paper also went beyond the compactness bound, deriving a minimal theorem for the central pressure corresponding to a classical result, considering inequalities involving proper and potential energy, and addressing whether the total radiation emitted during symmetric gravitational contraction can exceed the initial energy 𝐅0 of the dispersed matter.<sup>[1](https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1027)</sup>

## References

1. [H. A. Buchdahl (1959). General Relativistic Fluid Spheres. Physical Review 116, 1027.](https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1027)
2. [On the Buchdahl inequality for spherically symmetric static shells (arXiv:gr-qc/0605097)](https://arxiv.org/abs/gr-qc/0605097)
3. [Compactness bounds in General Relativity (arXiv:2202.00043)](https://arxiv.org/abs/2202.00043)
4. [Sharp bounds on the radius of relativistic charged spheres: Guilfoyle's stars saturate the Buchdahl-Andréasson bound (arXiv:1505.03863)](https://arxiv.org/abs/1505.03863)
5. [The Buchdahl bound denotes the geometrical Virial theorem. Annals of Physics (2025).](https://www.sciencedirect.com/science/article/pii/S0003491625000673)
6. [Black Holes lecture notes, H. S. Reall, University of Cambridge (2014).](https://www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2014.pdf)
7. [H. A. Buchdahl (1966). General Relativistic Fluid Spheres. Astrophysical Journal 146, 275.](https://adsabs.harvard.edu/pdf/1966ApJ...146..275B)
8. [E. Chávez Nambo and O. Sarbach (2021). Static spherical perfect fluid stars with finite radius in general relativity: a review. arXiv:2010.02859.](https://arxiv.org/html/2010.02859v2)
9. [Compact objects from gravitational collapse: an analytical toy model. European Physical Journal C (2015).](https://link.springer.com/article/10.1140/epjc/s10052-015-3824-8)
10. [Black-Hole mimickers in GR and λ-gravity. Classical and Quantum Gravity (2025).](https://doi.org/10.1088/1361-6382/ae7a2e)
11. [Testing the nature of dark compact objects: a status report. Living Reviews in Relativity (2019).](https://link.springer.com/article/10.1007/s41114-019-0020-4)
12. [Anisotropic stars as ultracompact objects in general relativity. Physical Review D 99, 104072 (2019).](https://doi.org/10.1103/PhysRevD.99.104072)
13. [H. Andréasson (2007). Sharp bounds on 2m/r of general spherically symmetric static objects. Chalmers preprint.](https://research.chalmers.se/en/publication/94556)
14. [J. Guven and N. Ó Murchadha (1999). Bounds on 2m/R for static spherical objects. Physical Review D 60, 084020.](https://www.researchgate.net/publication/1977813_Bounds_on_2mR_for_static_spherical_objects)
15. [An electromagnetic extension of the Schwarzschild interior solution and the corresponding Buchdahl limit. European Physical Journal C (2021).](https://link.springer.com/article/10.1140/epjc/s10052-021-08894-3)
16. [Buchdahl stars and bounds with cosmological constant. UCL Discovery.](https://discovery.ucl.ac.uk/id/eprint/10225974/)
17. [Buchdahl compactness limit for a pure Lovelock static fluid star (arXiv:1606.01330)](https://arxiv.org/abs/1606.01330)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Exact solutions and spacetime metrics › Interior and localized solutions › Schwarzschild interior solution*

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