Mass Inflation in Black Holes: Poisson-Israel Theory Explained

Mass inflation is the phenomenon in which the internal mass parameter of a rotating or charged black hole grows without bound near the Cauchy horizon, driven by the infinite blueshift of infalling radiation. This causes spacetime curvature to reach Planckian scales on a spacelike surface just inside the black hole, where classical general relativity breaks down. The effect was first described by Eric Poisson and Werner Israel in 1990.

Key Takeaways

• Mass inflation was introduced by Eric Poisson and Werner Israel in their 1990 paper “Internal structure of black holes.”
• The inner mass parameter m₂(v) diverges near the Cauchy horizon because an outgoing shell’s energy Δm grows exponentially while the ingoing radiation tail δm decays only as an inverse power law.
• Before mass inflation was understood, some theorists believed wormholes to other universes might exist inside black holes; mass inflation closes off these hypothetical passages in realistic black holes.
• The Cauchy-horizon Killing surface gravity is given by κ₀ = (r₊ − r₋) / [2(r₋² + a²)], where r₊ and r₋ are the outer and inner horizon radii and a is the spin parameter.
• For gravitational radiation, the decay exponent p in the ingoing tail typically satisfies p ≥ 11.

What Did Poisson and Israel Discover About Black Hole Interiors?

Poisson and Israel’s 1990 paper “Internal structure of black holes” investigated the gravitational effects of the radiative tail produced by a rotating gravitational collapse. Their abstract states:

“The gravitational effects associated with the radiative tail produced by a gravitational collapse with rotation are investigated. It is shown that the infinite blueshift of the tail’s energy density occurring at the Cauchy horizon of the resulting black hole causes classically unbounded inflation of the effective internal gravitational-mass parameter of the hole. Since this effect is causally disconnected from any external observer, the black-hole external mass remains bounded. The mass inflation phenomenon causes the spacetime curvature to grow to Planckian scales on a spacelike hypersurface in the vicinity of the Cauchy horizon, beyond which the classical laws of general relativity break down. A consequence is that an observer’s trip to this hypersurface embraces all but the last Planck time of the entire black-hole classical history.”

The external mass of the black hole, the mass measured by a distant observer, remains bounded throughout this process. Only the internal mass parameter, which is causally disconnected from the outside universe, grows without limit.

How Did Later Work by Bonanno, Droz, Israel, and Morsink Extend Mass Inflation?

A follow-up 1994 paper, “Structure of the Inner Singularity of a Spherical Black Hole,” co-written by Poisson with Israel, describes the discovery as originating in Poisson’s doctoral dissertation on mass inflation, a concept distinct from cosmological inflation. The paper states:

“Prior to the understanding of mass inflation, it was believed that wormholes into other universes would be found inside some black holes. Mass inflation closes off these wormholes in realistic black holes. Poisson’s work on black hole interiors has been reviewed in popular science books as well as in the scientific press.”

This finding overturned an earlier expectation in general relativity: that the interior of a rotating or charged black hole might connect, via its inner horizon, to other regions of spacetime or other universes. Mass inflation shows that realistic, physically formed black holes instead develop a curvature singularity at the Cauchy horizon that seals off any such passage.

What Is the Equation for the Inner Mass Parameter?

The inner mass parameter m₂(v) diverges at the Cauchy horizon, producing a curvature singularity. Using the notation from the cited papers, the behavior is summarized as:

The terms in this equation are defined as follows:

• Δm — the mass-energy of the outgoing null shell S.
• M — the external gravitational mass, equivalent to the gravitational radius.
• δm — the mass-energy of the late-time radiation influx, known as the Price tail.
• h — a constant related to the gravitational source.
• v — the ingoing null coordinate.
• p — the decay rate exponent of the ingoing radiation; for gravitational radiation this is typically p ≥ 11.
• κ — the Cauchy-horizon Killing surface gravity.

The ingoing radiation tail δm decays according to an inverse power law, a behavior known as Price’s law. Because the outgoing shell contribution Δm grows exponentially as the shell approaches the Cauchy horizon while δm only decays as a power law, the total inner mass parameter m₂(v) diverges.

What Is the Cauchy-Horizon Killing Surface Gravity?

The Killing-surface gravity at the Cauchy horizon of a rotating (Kerr) black hole is given by:

κ₀ = (r₊ − r₋) / [2 (r₋² + a²)]

Here r₊ is the radius of the outer (event) horizon, r₋ is the radius of the inner (Cauchy) horizon, and a is the spin parameter, expressed in geometric units.

The inner mass parameter can be written explicitly as:

m₂(v) = M − h v^{-(p−1)} + v^{−p} e^{κ₀ v}

Inside the event horizon, the exponential term dominates as v approaches infinity, so Δm diverges and m₂(v) grows without bound near the Cauchy horizon. Outside the event horizon, where r is greater than r₊, the mass parameter stays bounded and at late times resembles an outer mass function m₁(v).

How Do Null Coordinates Work Near a Rotating Black Hole’s Horizons?

What Are the Ingoing and Outgoing Null Coordinates?

In Eddington-Finkelstein or related coordinate systems, the null coordinates are defined as v = t + r* for the ingoing direction and u = t − r* for the outgoing direction, where r* is the tortoise coordinate.

The tortoise coordinate encodes how coordinate time stretches near the horizon: geodesics take an infinite coordinate time to reach the horizon as measured from infinity. In flat, Minkowski space, v = t + r is simply equal to 2r, but in curved spacetime v = t + r* is not equal to 2r. The relations r* = (v − u) / 2 and t = (v + u) / 2 hold in both cases.

What Is the Tortoise Coordinate for a Kerr Black Hole?

For Kerr spacetime, the radial differential is dr* = (r² + a²) / Δ · dr, where Δ = r² − 2Mr + a² = (r − r₊)(r − r₋).

An explicit form of the tortoise coordinate is:

r*(r) = r + [2 M r₊ / (r₊ − r₋)] ln |(r − r₊) / (2 M)| − [2 M r₋ / (r₊ − r₋)] ln |(r − r₋) / (2 M)|

This expression reduces to the static Schwarzschild form when the spin parameter a equals zero. Typical coordinate limits are as follows: with v = t + r*, v approaches infinity at large radii, v approaches negative infinity at r = r₊, v approaches infinity at r = r₋, and v is finite at r = 0. The opposite signs apply for the outgoing coordinate u.

For a charged, rotating Kerr-Newman black hole, the same transformations apply with Δ = r² − 2Mr + a² + Q² and horizon radii r₊, r₋ = M ± sqrt(M² − a² − Q²).

How Does Radiation Scattering Drive Mass Inflation?

Perturbations falling toward a black hole are scattered by its external potential barrier. Part of the resulting late-time radiation, the Price tail, is backscattered into the hole and contributes to the ingoing flux term δm in the mass inflation equation. Researchers reference effective scattering peaks in this process, sometimes associated with a radius r₀ or with the coordinate value v = 0.

Inside the event horizon, radiation that crosses the horizon is scattered again by an inner gravitational potential barrier. These multiple scatterings feed the ingoing tail that ultimately drives mass inflation near the Cauchy horizon.

These scattering and peak features are discussed in the papers archived at arXiv:gr-qc/9411050 and arXiv:gr-qc/9805008, which include supporting notes and figures.

How Does Mass Inflation Affect Spacetime Curvature?

Mass inflation replaces the external mass parameter M with the divergent inner mass parameter m₂(v) inside curvature invariants such as the Kretschmann scalar. For Kerr spacetime, one form of this scalar incorporating m₂(v) is:

K = R_{abcd} R^{abcd} = 48 m₂(v)² (r² − a² cos²θ) [(r² + a² cos²θ)² − 16 r² a² cos²θ] / (r² + a² cos²θ)⁶

Here a = J / (m c), often written simply as a = J / m in geometric units, and θ is the polar angle measured from the rotation axis.

Replacing M with the diverging m₂(v) produces a curvature invariant that grows without bound at the Cauchy horizon. Kerr geometry contains two relevant singular loci: one associated with the inner horizon at r = r₋, driven by mass inflation, and the separate ring singularity located at r = 0, θ = π/2.

Frequently Asked Questions

What is mass inflation in the context of black holes?

Mass inflation is the unbounded growth of a black hole’s internal mass parameter near its Cauchy horizon, caused by the infinite blueshift of infalling radiation. Eric Poisson and Werner Israel described this effect in 1990. It causes spacetime curvature to reach Planckian scales inside the black hole while the mass measured by an external observer remains unaffected.

Does mass inflation affect what an outside observer sees?

No. Mass inflation is causally disconnected from any external observer, so the black hole’s external gravitational mass remains bounded throughout the process. The effect only influences the internal structure of the black hole beyond the event horizon.

Did scientists once think black holes contained wormholes to other universes?

Yes. Before mass inflation was understood, some theorists believed wormholes into other universes might exist inside certain black holes. Poisson and Israel’s work on mass inflation showed that in realistic, physically formed black holes, these wormholes are closed off by a curvature singularity at the Cauchy horizon.

What causes the inner mass parameter to diverge?

The inner mass parameter diverges because the outgoing shell contribution grows exponentially as it approaches the Cauchy horizon, while the ingoing radiation tail only decays according to an inverse power law known as Price’s law. This mismatch in growth versus decay rates causes the total inner mass parameter to become unbounded.

What is the Price tail?

The Price tail refers to the late-time radiation influx that decays according to an inverse power law as a black hole settles down after formation. This tail contributes the term δm to the mass inflation equation and is partly responsible for driving the divergence of the inner mass parameter near the Cauchy horizon.

Where can I read the original papers on mass inflation?

Eric Poisson and Werner Israel’s foundational 1990 paper is titled “Internal structure of black holes.” A related follow-up paper, “Structure of the Inner Singularity of a Spherical Black Hole,” by A. Bonanno, S. Droz, W. Israel, and S. M. Morsink, is archived at arXiv:gr-qc/9403019. A. Bonanno’s paper “Mass Inflation in a Rotating Charged Black Hole” is archived at arXiv:gr-qc/9507047.

Sources and Further Reading

• J. S. F. Chan, “The Universality of Mass Inflation Inside Black Holes”; see related discussion in Black Holes Really Exist.
• A. Bonanno, S. Droz, W. Israel, S. M. Morsink, “Structure of the Inner Singularity of a Spherical Black Hole”: arXiv:gr-qc/9403019.
• A. Bonanno, “Mass Inflation in a Rotating Charged Black Hole”: arXiv:gr-qc/9507047.
• Eric Poisson and Werner Israel, “Internal structure of black holes” (1990).
• Eric Poisson’s doctoral dissertation, University of Waterloo: http://uwspace.uwaterloo.ca/handle/10012/225
• Scattering and peak feature discussion: arXiv:gr-qc/9411050
• Discussion thread: What Is Mass Inflation? — Physics Forums