LOG#199. M-sigma.

Black holes are coming! No, winter is coming! NOOOO! Winter is here, and black holes, as shown by LIGO, are not coming. They are already here, as they were already supposed to exist due to X-ray astronomy. Moreover, LIGO-VIRGO, LISA, ET, KAGRA, LIGO-India, Event Horizon, Ska (clustering radiotelescopes),…are coming and are detecting BH. The darkness is here…Prepare yourself for a BH diet! And no, it is NOT about M-theory today…


Firstly, let me review some stuff already mentioned in this blog. Astrophysical black holes are believed to come in the following mass-sizes:

  1. Stellar mass black holes (SBH). 3M_\odot\leq M\leq 100M_\odot. The first detection of gravitational waves came from medium size BH species in this window (circa 2016 and 2017!).
  2. Intermediate mass black holes (IMBH). 100M_\odot\leq M\leq 10^6M_\odot. A mysterious population not yet discovered (not with precision!). The title of this post is related to the biggest hint pointing out their existence. The so-called M-sigma (M-\sigma) relation. NO, it has nothing to do with Vector Sigma.
  3. Supermassive black holes (SMBH). 10^6M_\odot\leq M\leq 10^{10}M_\odot. There is no clear evidence of more massive BH, and this window includes quasars, blazars, AGN (active galactic nuclei), and other monsters.

Other species by size:

  1. Mini-BH and micro-BH.
  2. Extremal BH.
  3. Elementary particle BH (electron BH is a concrete example).
  4. Virtual BH (Hawking’s idea!).
  5. Planckian sized BH.

In space-based gravitational wave detectors, you will hear about EMRIs (Extreme Mass Ratio Inspirals), and IMRIs (Intermediate Mass Ration Inspirals), that can be the result of compact objects (mainly black hole species from very different sizes but also weird compact stuff!). More binary or even multiple BH inspirals could be found in the future!

From the viewpoint of general relativity and generalizations, BH are soliton-like solutions to the Einstein-Field-Equations (EFE). Remarkable (and simple) solutions include:

  1. Schwarzschild BH (and Tangherlini’s solution in high dimensional spacetimes).
  2. Taub-NUT.
  3. Reissner-Nördstrom BH (charged BH).
  4. Kerr BH.
  5. Kerr-Newman BH.
  6. dS and AdS solutions.
  7. Type D solutions. Sometimes, they are also named PlebanskiDemianski black holes.
  8. Myers-Perry rotating BH in higher dimensions.
  9. Black rings, black saturns, black p-branes and black folds in higher dimensions.
  10. The pp-wave.

A map I show from time to time everywhere

Binary black hole mergers are becoming more and more popular since LIGO’s GW discovery. The phases of coalescence are dubbed generally as inspiral (GW emission, quite newtonian, postnewtonian-like), merger, plunge and ringdown.

Beyond BH, you do know compact objects like white dwarfs (WD) and neutron stars (NS). The Chandrasekhar’s limit of WD is about 1.4 solar masses. The Tolmann-Oppenheimer-Volkoff (and Landau) limit for NS is 2-3 solar masses (generally 2 solar masses but you can take 3 for safety!). No compact object between 3-5 solar masses is known. However, that is not a limit for the imagination of scientists. Exotic wonderful compact objects (even those suggested by anti-black hole fans; oh, yes! BH followers have enemies!) do exist:

  1. Quark stars.
  2. Preon stars.
  3. Strange stars.
  4. Electroweak stars.
  5. Boson stars.
  6. Axion stars.
  7. Gravastars.
  8. Planck stars (I love this one!).
  9. Dark energy stars.
  10. Dark stars (different from the original dark star name for black holes!).
  11. Quasi-stars.
  12. Q-stars.
  13. Wormholes.
  14. MECOs.
  15. Exotic (not erotic!) stars.

Any BH has “features” (sometimes called hair in some properties!). For instance:

  1. BH thermodynamics (this topic is even more general than BH theirselves!).
  2. Event horizons (apparent horizons?)
  3. Photon sphere.
  4. Ergosphere.
  5. Space-time singularity (naked? Do they exist naked singularities? Ring or higher order singularities?).
  6. Schwarzschild radius.
  7. Quasi-periodic oscillations (QPO).
  8. BZ processes (Blandford-Znajek).
  9. Spaghettification.
  10. Bondi accretion.
  11. Immirzi parameter.
  12. Kugelblitz.
  13. Fuzzball proposal.
  14. White hole-BH duality?
  15. Membrane paradigm.
  16. Area theorem.
  17. Penrose processes.

BH theoretical issues (not exhaustive list) and controversial themes:

  1. No hair theorem.
  2. BH information paradox.
  3. Cosmic censorship conjecture.
  4. Alternative BH models?
  5. Holographic principle.
  6. Firewalls.
  7. Complexity.
  8. Entropy and degrees of freedom of BH.
  9. ER=EPR (or GR=EPR).
  10. Final parsec problem (LISA target?).
  11. Entanglement and BH states.
  12. Supertraslations and BMS algebras.

Finaly, what is M-\sigma? It is quite simple: it is an empirical correlation between the stellar velocity dispersion \sigma of a galaxy bulge and the mass M of the SMBH of its center! Putting it into mathematics

    \[\boxed{\dfrac{M}{10^{8}M_\odot}\equiv M_8\approx 3.1\left(\dfrac{\sigma}{200km\cdot s^{-1}}\right)^4}\]

Sometimes it is written as well as

    \[\boxed{\dfrac{M}{10^{8}M_\odot}\equiv M_8\approx 1.9\left(\dfrac{\sigma}{200km\cdot s^{-1}}\right)^{5.1}}\]

The main thing with this relation is…What if you scale it DOWN to lower masses? Then, M-sigma (M-\sigma) implies that IMBH should exist! They could be hidden in stellar clusters or the DM halo and we can not see them because they are “dormant”. Indeed, this relation is so amazing, that you can use  to estimate BH masses in galaxies that are too distant for direct mass measurements to be made, and to assay the overall BH content of the Universe. GW astronomy has the challenge to identify and calculate the number of sources (mainly BH!) that pervade the Universe in the dark gravitational sector, specially the sources that are BH and that do not emit any other form of radiation beyond gravitational waves!

Indeed, I have a question for you, related to my previous post:

Has BH mass any upper and/or lower limit? What can quantum gravity say about it? What physics can explain M_\bullet\leq 10^{10-11}M_\odot and/or M_\bullet\geq M_0? What is the origin (if any, as it is a conjecture) of BH maximons and minimons?

May the M-\sigma be with you! Always!

P. S.: The Universe is a gas of galaxies and BH! The Multiverse/Polyverse is a gas of Universes! What is “in-between” the Universes? A void or a full network of wormholes (BH?), tying the universes together?

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