Cauchy horizon
Light-like boundary of a Cauchy problem's domain.
In physics, a Cauchy horizon is a light-like surface marking the edge of where a Cauchy problem—a specific type of boundary value problem for partial differential equations—remains valid. On one side of this horizon, closed space-like geodesics exist; on the other, closed time-like geodesics are found. The term honors Augustin-Louis Cauchy. The simplest example is the inner horizon of a Reissner–Nordström black hole.
Under the averaged weak energy condition (AWEC), Cauchy horizons are inherently unstable and highly sensitive to time-dependent disturbances. Even a tiny perturbation would cause proper time to contract and energy density to rise exponentially for an observer moving toward the horizon. That observer would witness the entire future history of the universe flash by, until they hit a wall of infinite energy—a curvature singularity at the Cauchy horizon. However, because the spacetime region inside the horizon contains closed timelike curves, it follows periodic boundary conditions, a situation akin to the Casimir effect. This violates the AWEC. If the interior spacetime does violate AWEC, the horizon becomes stable, and the frequency-boosting effects that would otherwise increase energy density near the horizon are canceled by the spacetime acting as a divergent lens. If confirmed empirically, this would contradict the strong cosmic censorship conjecture. In 2018, it was shown that the spacetime behind the Cauchy horizon of a charged, rotating black hole does exist, but it is not smooth, meaning the strong cosmic censorship conjecture is false.
**Cauchy horizon singularity**
As energy density rises near the Cauchy horizon and infalling matter causes spacetime backreactions, a weak, null curvature singularity forms at the horizon. Approaching the horizon, the gravitational field at the singularity grows strong, and the internal mass function diverges to infinity, making tidal forces on an observer increase without bound. Yet the total tidal deformation on the observer stays finite. Along the Cauchy horizon, the radial Schwarzschild coordinate decreases steadily until it reaches zero, at which point the singularity becomes spacelike.
**In popular media**
In the 2020 film *Palm Springs*, the character Sarah mentions the Cauchy horizon while devising a plan to escape a time loop.
- Field
- Physics
- Known for
- Boundary of validity of Cauchy problems; internal horizon of Reissner–Nordström black holes
- Related concept
- Augustin-Louis Cauchy
Lore & Background
Under the averaged weak energy condition (AWEC), Cauchy horizons are inherently unstable and severely susceptible to time-dependent perturbations. The smallest perturbation to the horizon would cause a contraction of proper time and an increase of energy density that would grow exponentially for an observer approaching the horizon. Such an observer would see the entire future history of the universe pass by as they approached the horizon until they suddenly hit a wall of infinite energy: a curvature singularity at the Cauchy horizon. However, since the region of spacetime inside the Cauchy horizon has closed timelike curves, it is subject to periodic boundary conditions, an example of the Casimir effect. This violates the average weak energy condition. If the spacetime inside the Cauchy horizon violates AWEC, then the horizon becomes stable and frequency boosting effects causing the increase in energy density near the horizon would be canceled out by the tendency of the spacetime to act as a divergent lens. Were this conjecture shown to be empirically true, it would provide a counter-example to the strong cosmic censorship conjecture. In 2018, it was shown that the spacetime behind the Cauchy horizon of a charged, rotating black hole exists, but is not smooth, so the strong cosmic censorship conjecture is false.
Reader's Guide
Due to the increase in energy density near the Cauchy horizon and spacetime backreactions by infalling matter, a weak, null curvature singularity forms at the horizon. As the Cauchy horizon is approached, the gravitational field at the singularity becomes strong and the internal mass function diverges to infinity, causing tidal forces on an observer to increase without bound. However, the total tidal deformation on the observer remains finite. Along the Cauchy horizon, the radial Schwarzschild coordinate r decreases monotonically until it reaches r = 0, at which point the singularity becomes spacelike. The concept appears in popular media: in the 2020 film Palm Springs, the character Sarah mentions the Cauchy horizon as she formulates a plan to escape a time loop; in the pilot episode of the 2021 Amazon original series Solos, the character Leah solves time travel with 'the Cauchy horizon', which is central to the episode.
Did You Know?
- The simplest example of a Cauchy horizon is the internal horizon of a Reissner–Nordström black hole.
- Under the averaged weak energy condition, Cauchy horizons are inherently unstable and susceptible to time-dependent perturbations.
- The Cauchy horizon is named after Augustin-Louis Cauchy.
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