Chat history
You said:
is the event horizon of a black hole always spherical, even if the singularity spins really fast?
ChatGPT said:
Excellent question — and it touches on some really interesting aspects of general relativity and rotating black holes.
Short answer:
No, the event horizon of a spinning (rotating) black hole is not perfectly spherical. It’s
oblate — slightly flattened at the poles and bulging at the equator — due to the black hole’s rotation.
Here’s why:
1. Non-spinning black holes: perfectly spherical
A non-rotating black hole is described by the
Schwarzschild metric, which is spherically symmetric.
→ Its event horizon is a sphere with radius
rs=2GMc2.rs=c22GM.
2. Spinning black holes: the Kerr metric
When a black hole spins, it’s described by the
Kerr metric. Rotation drags spacetime around with it (a phenomenon called
frame dragging). This rotation changes the shape of the event horizon.
The radius of the horizon depends on both the mass MM and the spin parameter aa (angular momentum per unit mass):
r+=GMc2+(GMc2)2−a2.r+=c2GM+(c2GM)2−a2.
3. Shape and flattening
Although this formula gives you the
radial coordinate of the horizon, the
actual spatial geometry of the horizon is not a perfect sphere. It’s
oblate, meaning:
- Equatorial circumference > Polar circumference.
- The faster the spin (larger aa), the more the flattening.
In the extreme case — a
maximally rotating Kerr black hole (a=GM/c2a=GM/c2) — the event horizon is as “squashed” as it can get. It’s still a smooth surface with spherical topology (homeomorphic to a sphere), but it’s not geometrically spherical.
4. So, topology vs. geometry
- Topology: Always spherical (no holes, still a 2D surface enclosing a 3D region).
- Geometry: Not a perfect sphere when spinning; oblate due to rotation.
In summary:
| Property | Non-spinning (Schwarzschild) | Spinning (Kerr) |
|---|
| Horizon shape | Perfect sphere | Oblate (flattened at poles) |
| Topology | Spherical | Spherical |
| Cause | Static spacetime | Frame dragging & rotation |
Would you like a diagram comparing a spherical and Kerr horizon? I can generate a clear visual showing the flattening effect of spin.
You said:
in gravitational lensing, is the effect of rotation visible?
ChatGPT said:
Fantastic follow-up — and you’re asking one of the subtler and more fascinating questions in modern astrophysics.
Yes — in principle, the rotation (spin) of a black hole does affect gravitational lensing, but the effect is
subtle and often
very hard to detect observationally.
Let’s unpack this:
1. Lensing by a non-spinning black hole
For a non-rotating (Schwarzschild) black hole:
- The lensing pattern is perfectly symmetric around the center.
- Light rays bend only due to the mass; there’s no preferred direction.
- The result is the familiar Einstein ring (if alignment is perfect) or symmetric multiple images.
2. Enter the spin: Kerr lensing
A rotating black hole (Kerr black hole)
drags spacetime around itself — this is
frame dragging or the
Lense–Thirring effect.
That means light traveling
with the rotation (prograde) and
against the rotation (retrograde) experiences
different amounts of curvature and time delay.
Observable consequences:
- The photon trajectories are asymmetric.
- The “shadow” or silhouette of the black hole is slightly displaced and distorted.
- The Einstein ring becomes asymmetric — brighter or thicker on one side.
- The polarization of light can also be affected by the black hole’s spin.
3. How big is the effect?
It depends on the spin parameter aa and how close the light passes to the black hole.
- For light rays far from the event horizon:
→ Spin effects are tiny (negligible for most gravitational lensing by galaxies or clusters).
- For light rays skimming near the horizon:
→ Spin effects are significant — can change image positions, magnifications, and time delays measurably.
This is the regime of
strong gravitational lensing, such as what we expect near
Sgr A* or
M87*, the supermassive black holes imaged by the
Event Horizon Telescope (EHT).
4. What the EHT actually sees
In the EHT images of M87* (2019) and Sgr A* (2022):
- The ring looks almost circular, but brightness is asymmetric — brighter on one side.
- That brightness asymmetry is consistent with relativistic beaming and spin: the side of the accretion flow moving toward us appears brighter.
So while the
mass dominates the lensing, the
spin subtly warps and skews the photon trajectories, producing the
characteristic “D-shaped” shadow of a Kerr black hole.
5. Summary table
| Feature | Non-spinning (Schwarzschild) | Spinning (Kerr) |
|---|
| Symmetry | Perfectly circular | Slightly asymmetric |
| Frame dragging | None | Yes — affects light paths |
| Brightness | Uniform (theoretically) | Asymmetric (beamed side brighter) |
| Shadow shape | Circular | “D-shaped” (flattened on one side) |
| Detectability | Easy | Very subtle — needs horizon-scale imaging (EHT) |
In short:
Rotation does affect gravitational lensing, but you only see it near the black hole itself.
For normal astrophysical lenses (galaxies, clusters), the effect is too small to notice — but for strong-field lensing, it’s key.
Would you like me to show a
side-by-side visualization of lensing patterns for a Schwarzschild vs. Kerr black hole (including the asymmetric “D-shaped” shadow)?
You said:
can you use that effect to calculate the rotational speed and/or mass of the black hole?
ChatGPT said:
Thought for 33s
Short answer:
yes for mass (already done); maybe for spin (hard, but getting better).
How lensing lets you measure mass
Near-horizon lensing sets a characteristic ring size. In GR, the apparent “shadow/ring” diameter scales almost linearly with GM/c2GM/c2. Measuring that angular diameter θθ gives M∝θDM∝θD (with DD the distance). That’s how the EHT got:
- M87*: ring ∼42 μ∼42 μas → M≈6.5×109 M⊙M≈6.5×109M⊙ (using D≈16.
≈16.8 Mpc). arXiv
- Sgr A*: ring 51.8±2.3 μ51.8±2.3 μas → M≈4.0−0.6+1.1×106 M⊙M≈4.0−0.6+1.1×106M⊙ (consistent with stellar-orbit results). arXiv+1
What about spin?
Rotation skews the lensing: it displaces/warps the shadow, changes photon-ring structure, and affects time delays. In principle, those features encode
spin a∗a∗ and
viewing angle:
- Asymmetry & displacement of the shadow/ring (Kerr lensing) carry spin information. arXiv
- “Photon-ring subrings” imprint universal interferometric signatures that, with long enough baselines, can measure mass and spin precisely. This is a key target for next-gen EHT. PMC
- Polarimetry (the pattern of polarized light around the ring) links to spin through magnetosphere geometry; early studies show promise for constraining a∗a∗. arXiv
Reality check (today)
- With current EHT data, mass has been robustly inferred from ring size (above).
- Spin is still weakly constrained because measurements depend on inclination, plasma models, variability, and (for Sgr A*) interstellar scattering. Multiple reviews/analyses note that EHT so far has not yielded strong spin constraints—the direction of rotation for Sgr A* is even ambiguous. NSF Public Access+2A&A+2
Takeaways
- Mass from lensing: yes — already demonstrated with M87* and Sgr A*.
- Spin from lensing: in principle yes, using shadow asymmetry, photon-ring structure, timing, and polarization — but present constraints are model-dependent and not yet tight. Next-gen arrays (more stations, higher frequencies, longer baselines, better polarimetry) should make spin measurements much more definitive. PMC
If you’d like, I can sketch how a hypothetical set of ring measurements (diameter, displacement, brightness asymmetry) would translate into posteriors on MM, a∗a∗, and inclination, and show where the main degeneracies come from.
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