# How do we know that black holes are spinning?

How is it possible to know if a black hole is spinning or not?

If a planet is spinning, you can see it clearly but you can't really see a black hole.

Next thing would be that matter interacts with adjacent matter and we could see in which direction the matter surrounding the BH spins (like if you spin a ball on water, the water around would spin too in the same direction) but matter can't interact from inside the event horizon to the outside, so matter right at the event horizon would just be interacting with gravity (like the BH has no friction).

Now gravity. I would think that you could measure the differences in gravity if a large object is not perfectly uniform but I think a BH has the same gravitation pull on all sides.

What am I missing here? How can one even detect or determine by observation that a black hole is spinning, or better yet, measure how fast?

• We believe that BH is spinning (or rotating) to conserve the angular momentum. Also, by No-hair Theorem, a BH is uniquely characterized by mass, charge, and angular momentum. – Kornpob Bhirombhakdi Sep 13 '19 at 15:47
• @KornpobBhirombhakdi The no-hair conjecture isn't a theorem yet. – chrylis -on strike- Sep 14 '19 at 3:03
• @chrylis No, in fact it is multiple theorems. However, the conditions for the theorems allow for several potential "outs" if the universe isn't quite the way we think it is. – mmeent Sep 14 '19 at 10:31
• Kind of related, but rotating objects drag space-time along with them (verified by experiment). nasa.gov/vision/earth/lookingatearth/earth_drag.html – userLTK Sep 14 '19 at 16:40
• @KornpobBhirombhakdi The no-hair theorem tells us that our models of black holes spin. You can't use a mathematical theorem to prove something about a physical object. – David Richerby Sep 15 '19 at 11:53

The gravitational field of spinning matter, or a spinning black hole, causes matter around it to start spinning. This is called "frame dragging" or "gravitomagnetism", the latter name coming from the fact that it's closely analogous to the magnetic effect of moving electric charges. The existence of gravitomagnetism is tied to the finite speed of gravity, so it doesn't exist in Newtonian gravity where that speed is infinite, but it's present in general relativity, and for black holes it's large enough to be detectable.

Also, for purely theoretical reasons we expect that all black holes are spinning because a non-spinning black hole is the same as a spinning black hole with an angular velocity of exactly zero, and there's no reason why a black hole's angular velocity would be exactly zero. On the contrary, because they are so much smaller than the matter that collapses to produce them, even a small, random net angular momentum of the collapsing matter should lead to a rapidly spinning black hole. (The classic analogy for this is an ice skater spinning faster when they pull their arms in.)

• Perhaps worth mentioning that "frame dragging" is an incredibly small effect for say the Earth or our Sun, but, a bigass affect for a black hole. – Fattie Sep 16 '19 at 18:49

The innermost stable circular orbit is different depending on rotation rate. Accretion disks stretch in to the ISCO, so this produces observable changes. From The Spin of Supermassive Black Holes:

For $$a=1$$ (maximal spin in the prograde sense relative to the orbiting particle), we have $$r_{isco}=M$$. This is the same coordinate value as possessed by the event horizon but, in fact, the coordinate system is singular at this location and there exists finite proper distance between the two locations. As a decreases,$$r_{isco}$$ monotonically increases through $$r_{isco}=6M$$ when $$a=0$$ to reach a maximum of $$r=9M$$ when $$a=−1$$ (maximal spin retrograde to the orbiting particle). As we discuss below, the ISCO sets an effective inner edge to the accretion disc (at least for the disc configurations that we shall be considering here). Thus, the spin dependence of the ISCO directly translates into spin-dependent observables; as spin increases and the radius of the ISCO decreases, the disc becomes more efficient at extracting/radiating the gravitational binding energy of the accreting matter, the disc becomes hotter, temporal frequencies associated with the inner disc are increased, and the gravitational redshifts of the disc emission are increased.

Empirically, by looking at the spectra of the accretion disks we can estimate $$a$$.

• Does this apply only to super-massive black holes (the topic of the quoted source) or does it work for smaller black holes, as long as they have acretion disks? – uhoh Sep 14 '19 at 0:14
• you might also be interested in adding an answer to What defines the plane of an accretion disk around a black hole? – uhoh Sep 14 '19 at 0:17
• @uhoh - Yes, this applies to all black holes with accretion disks. I think supermassive black holes are better studied since the larger scale makes discerning the signature of the disk end easier to see than for a stellar mass black hole. – Anders Sandberg Sep 14 '19 at 15:21

The gravitational field of a black hole depends on both its mass and its spin. This has a number of observable consequences:

• As mentioned in Anders Sandberg's answer, there is a smallest possible circular orbit around a black hole (the ISCO), whose radius depends on the spin of the black hole. So, if you see matter orbitting a black hole in an accretion disk, the inner edge will give a lower bound on the spin.
• When two black holes merge, the resulting object settles down by oscillating and emitting gravitational waves with a charactaristic frequency and decay rate determined by the mass and spin of the final black hole. For loud mergers (such as GW150914) this so-called ringdown can be measured, giving a direct measure of the mass and spin of the formed black hole.
• Before such a merger, the spins of the individual black holes will affect how the inspiral evolves, which imprints on the gravitational waveform observed. By comparing the observed waveform with theoretically expected templates for different spins, one can (try to) measure the spins of the merging black holes. (Thusfar most observed (published) mergers could be consistent with both BHs being non-spinning.)
• The spin of a black hole also affects how it deflects light. Consequently, the pictures of a black hole's shadow such as taken by the event horizon telescope can be used to determine the spin of the black hole (if we happen to view it under the right angle).
• +1 this is an excellent answer, thanks! I don't quite understand the third item, so I've asked a new question: What is a black hole's "shadow" and the best angle to view it to measure the spin? – uhoh Sep 14 '19 at 23:14
• Also, the event horizon bulges at the "equator", just as the Earth's radius is slightly larger at its equator. – Acccumulation Sep 16 '19 at 17:58
• @Acccumulation That is a coordinate dependent statement. For example, it is not true in the commonly used Boyer-Lindquist coordinates. As such, I doubt it can be translated to an observable effect. – mmeent Sep 17 '19 at 6:50

As mentioned in Rory's comment, an object in space must at some point in time acquire spin. Any object has gravity, and with a rotational rate of zero it would have no spin, as soon as it contacts another object spin is imparted on it.

While it is true, but unlikely, that it could be struck by another object that exactly cancelled out its spin it's only a matter of time before yet another object comes along - therefore objects in space are far more likely to spin than not.

See for example the SXS Collaboration video: "Inspiral and merger of binary black hole GW151226":

Angular momentum is the rotational equivalent of linear momentum and a conserved quantity — the total angular momentum of a closed system remains constant. The greater the density the faster the spin of the object, to conserve its angular momentum.

For anyone seeking additional information I'll include these references:

• "Inferring black hole spins and probing accretion/ejection flows in AGNs with the Athena X-ray Integral Field Unit" (Jun 6 2019), by Didier Barret (IRAP) and Massimo Cappi (INAF-OAS):

"Context. Active Galactic Nuclei (AGN) display complex X-ray spectra which exhibit a variety of emission and absorption features, that are commonly interpreted as a combination of i) a relativistically smeared reflection component, resulting from the irradiation of an accretion disk by a compact hard X-ray source, ii) one or several warm/ionized absorption components produced by AGN-driven outflows crossing our line of sight, and iii) a non relativistic reflection component produced by more distant material. Disentangling these components via detailed model fitting can thus be used to constrain the black hole spin, the geometry and characteristics of the accretion flow, as well as of the outflows and surroundings of the black hole.
Aims. We investigate how a high throughput high resolution X-ray spectrometer, such as the Athena X-ray Integral Field Unit (X-IFU) can be used to this aim, using the state of the art reflection model relxill in a lamp post geometrical configuration.
Methods. We simulate a representative sample of AGN spectra, including all necessary model complexities, as well as a range of model parameters going from standard to more extreme values, and considered X-ray fluxes that are representative of known AGN and Quasars (QSOs) populations. We also present a method to estimate the systematic errors related to the uncertainties in the calibration of the X-IFU.
Results. In a conservative setting, in which the reflection component is computed self consistently by the relxill model from the preset geometry and no iron over abundance, the mean errors on the spin and height of the irradiating source are < 0.05 and ∼ 0.2 R$$_g$$ (in units of gravitational radius). Similarly the absorber parameters (column density, ionization parameter, covering factor and velocity) are measured to an accuracy typically less than ∼ 5% over their allowed range of variations. Extending the simulations to include blue shifted ultra fast outflows, we show that X-IFU could measure their velocity with statistical errors < 1%, even for high redshift objects (e.g. at redshifts ∼ 2.5).
Conclusions. The simulations presented here demonstrate the potential of the X-IFU to understand how black holes are powered and how they shape their host galaxies. The accuracy to recover the physical model parameters encoded in their X-ray emission is reached thanks to the unique capability of X-IFU to separate and constrain, narrow and broad, emission and absorption components.".

• "Observing Black Holes Spin" (Mar 27 2019), by Christopher S. Reynolds:

"... black holes are nature’s simplest objects, defined solely by their electrical charge (which is neutralized to zero in realistic astrophysical settings), mass, and angular momentum.

...

In this Review, I will survey the current state and future promise of black hole spin measurements. For much of the past 20 years, quantitative measures of spin have been the domain of X-ray astronomy, and these techniques continue to be refined as the quality of the data improves. With the recent advent of gravitational wave astronomy, we now have a completely new and complementary window on spinning black holes. Furthermore, we stand on the threshold of anothernmajor breakthrough, the direct imaging of the shadow of the event horizon by global mm-band Very Long Baseline Interferometry, aka, the Event Horizon Telescope (EHT). We are truly entering a goldenbage for the study of black hole physics and black hole spin.

...

While the original Penrose process may be hard to realise in nature, Roger Blandford and Roman Znajek showed that magnetic fields can similarly extract rotational energy from the ergosphere. Magnetic spin-extraction is the.leading theoretical model for the driving of relativistic jets from black hole systems.
To be more quantitative, we consider a black hole with mass $$M$$ and angular momentum $$J$$. We can define the unitless “spin parameter” by $$a = cJ/GM^2$$ where $$c$$ is the speed of light and $$G$$ is Newton’s constant of Gravitation. The Kerr solution tells us that the structure of the spacetime around a spinning black hole depends only on $$M$$ and $$a$$. As well as greatly simplifying any GR treatments of black hole astrophysics, this provides a route to observational explorations of gravity theories beyond GR — once the mass and spin of an astrophysical black hole has been measured, we can in principle search for deviations of the inferred gravitational field (including any gravitational radiation) from the predictions of GR.
If one were to spin a planet or a star too quickly, it would fly apart as the centrifugal forces overwhelm the gravity that binds the object together. There is an equivalent situation for a black hole. The Kerr solution shows that, if $$|a| > 1$$, there is no longer an event horizon. GR would then predict a naked spacetime singularity, an outcome that is abhorrant to physical law and the notion of predictability and thus forbidden by the Cosmic Censorship Hypothesis. Of course, it is of great interest to physicists to test whether nature respects this Kerr limit.".

Page 3:

Figure 1: Location of some special orbits in the equatorial plane of a Kerr black hole as a function of spin parameter. Shown here is the innermost stable circular orbit (red line), photon circular orbit (blue line), static limit (dashed white line), and event horizon (bounding the grey shade). Positive/negative spin parameter corresponds to spin that is prograde/retrograde, respectively, relative to the orbiting matter (or photons). The vertical dashed red line separates the prograde and retrograde cases. Circular orbits are stable outside of the innermost stable orbit but become unstable inside of this radius (region denoted by light red shading). Circular orbits do not exist interior to the photon circular orbit (region denoted by solid red shading). For concreteness, a 10 solar mass black hole is assumed. Radii for other masses can be obtained using linear proportionality.

One way of thinking of the gravitational field outside a black hole is that it is a kind of fossil, or frozen impression. It reflects the gravity of the matter that formed/fell into the black hole at the moment when it became "locked away" inside the event horizon, and so unable to affect anything outside, including the gravitational field.

If the matter at that stage had net angular momentum, the gravitational field outside the black hole is different. Mathematically, it is described by the Kerr solution to Einstein's equations, instead of the Schwarzschild solution. This difference can be observed in a number of ways, for instance in the behaviour of light or matter close to the black hole.

• Ya, but how do we know that black holes are spinning? Only the last half of the last sentence starts to answer this, but "in the behaviour of light or matter" doesn't really say anything. – uhoh Sep 13 '19 at 14:33