Axisymmetry
Short answer: Modelers often assume the gas around a black hole looks the same at every angle around the spin axis. Quantities then depend only on distance and polar angle, and a 3D simulation becomes a much cheaper 2D one.
The shortcut has known limits. Cowling's anti-dynamo theorem (1933) says an axisymmetric magnetized flow can't regenerate its magnetic field, so 2D simulations lose the turbulence that drives accretion unless an artificial 'sub-grid' dynamo is added. And because infalling gas generally doesn't share the black hole's spin direction, many disks are expected to be tilted. Tilted disks break the symmetry and must be simulated in full 3D. The classic prediction that a thin disk's inner part aligns with the spin (Bardeen–Petterson, 1975) didn't appear in some GRMHD simulations of moderately thick disks.
What should you know about Axisymmetry?
Quick facts. Each one stands alone and has a stable ID you can cite.
2D axisymmetric simulations can't sustain a black hole disk's magnetic field on their own, because of Cowling's anti-dynamo theorem.
Tilted accretion disks break axisymmetry and have to be simulated in full 3D.
Video transcript: Axisymmetry: Why Black Hole Models Assume Symmetry
DID YOU KNOW? An axisymmetric distribution of matter around black holes refers to a configuration where physical properties such as density, pressure, and velocity are symmetric about a central axis, typically aligned with the black hole's spin axis. This assumption simplifies the complex geometry of accretion disks and relativistic jets, allowing for models that only vary with radial distance and polar angle, not azimuth. NOW YOU KNOW!
Burned-in captions, read frame by frame (see transcript.md) · 22.3s Reel ·
The write-up
Why black hole models assume symmetry. Treating the disk as identical all the way around its axis turns a 3D problem into 2D, at a cost.
- Exhibit A
Why assume it
Axisymmetric, two-dimensional simulations are much cheaper to run than full 3D ones, so researchers can explore a wider range of black hole accretion systems.
- Exhibit B
The dynamo problem
Cowling's anti-dynamo theorem (1933) says axisymmetric ideal-MHD systems can't regenerate their magnetic fields. In 2D simulations the field decays and accretion dies out, unless a sub-grid dynamo is added to stand in for the missing 3D turbulence.
- Exhibit C
Near a spinning black hole
Numerical tests in the Kerr geometry of a rotating black hole also found that an axisymmetric field couldn't be sustained, suggesting Cowling's theorem still holds there.
- Exhibit D
Tilted disks
Infalling gas doesn't 'know' which way the black hole spins, so many disks are expected to be tilted. Tilted disks break the symmetries simulations rely on and must be modeled in full 3D.
- Exhibit E
Alignment isn't guaranteed
Theory predicts that the inner part of a thin tilted disk aligns with the black hole's spin (the Bardeen–Petterson effect, 1975). But GRMHD simulations of moderately thick disks didn't find this alignment.
Questions people ask about Axisymmetry
In an axisymmetric model, quantities don't vary with which coordinate?
They vary only with radius and polar angle.
Source: Write-up · Exhibit A (Why assume it) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421)
Which theorem says axisymmetric MHD can't regenerate magnetic fields?
Cowling stated it in 1933.
Source: Write-up · Exhibit B (The dynamo problem) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421); GRMHD simulations of accreting tori: resolution study (arXiv:2403.16236)
What do 2D simulations add to work around it?
It stands in for the missing 3D turbulence.
Source: Write-up · Exhibit B (The dynamo problem) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421); GRMHD simulations of accreting tori: resolution study (arXiv:2403.16236)
Why must tilted disks be simulated in 3D?
No axial or reflection symmetry is left to exploit.
Source: Write-up · Exhibit D (Tilted disks) · Tilted accretion disks (review, arXiv:2404.10052); Bardeen–Petterson alignment, jets and magnetic truncation in GRMHD simulations of tilted thin accretion discs (Liska et al., MNRAS 2019)
The predicted alignment of a thin disk's inner part with the spin is named after whom?
It was predicted in 1975.
Source: Write-up · Exhibit E (Alignment isn't guaranteed) · Bardeen–Petterson alignment, jets and magnetic truncation in GRMHD simulations of tilted thin accretion discs (Liska et al., MNRAS 2019)
Test Your Knowledge
All quizzes →5 questions, all from this case file. Answers are revealed instantly.
In an axisymmetric model, quantities don't vary with which coordinate?
- Radius
- Polar angle
- Azimuth
- Time
Show the answer
Answer: C. Azimuth They vary only with radius and polar angle. Source: Write-up · Exhibit A (Why assume it) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421)
Which theorem says axisymmetric MHD can't regenerate magnetic fields?
- Noether's theorem
- Cowling's anti-dynamo theorem
- Birkhoff's theorem
- No-hair theorem
Show the answer
Answer: B. Cowling's anti-dynamo theorem Cowling stated it in 1933. Source: Write-up · Exhibit B (The dynamo problem) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421); GRMHD simulations of accreting tori: resolution study (arXiv:2403.16236)
What do 2D simulations add to work around it?
- A sub-grid dynamo
- Dark matter
- A second black hole
- Extra dimensions
Show the answer
Answer: A. A sub-grid dynamo It stands in for the missing 3D turbulence. Source: Write-up · Exhibit B (The dynamo problem) · Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo (Sądowski et al., arXiv:1407.4421); GRMHD simulations of accreting tori: resolution study (arXiv:2403.16236)
Why must tilted disks be simulated in 3D?
- They're hotter
- They break axisymmetry
- They have no magnetic field
- They're too small
Show the answer
Answer: B. They break axisymmetry No axial or reflection symmetry is left to exploit. Source: Write-up · Exhibit D (Tilted disks) · Tilted accretion disks (review, arXiv:2404.10052); Bardeen–Petterson alignment, jets and magnetic truncation in GRMHD simulations of tilted thin accretion discs (Liska et al., MNRAS 2019)
The predicted alignment of a thin disk's inner part with the spin is named after whom?
- Kerr and Newman
- Bardeen and Petterson
- Blandford and Znajek
- Lense and Thirring
Show the answer
Answer: B. Bardeen and Petterson It was predicted in 1975. Source: Write-up · Exhibit E (Alignment isn't guaranteed) · Bardeen–Petterson alignment, jets and magnetic truncation in GRMHD simulations of tilted thin accretion discs (Liska et al., MNRAS 2019)
Cite this case file
Easterling, Marcus. "Axisymmetry." The Detail Department (DetailEnjoyer), case file DD-054, last updated May 21, 2025. https://detailenjoyer.com/case/axisymmetry/
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