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Black Holes: Some Recent Observations

lpetrich

lpetrich

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Not directly, of course, but from their gravity.

The Gaia satellite recently detected evidence of three black holes, though very indirectly. It is an astrometric satellite, one designed to make precision measurements of the angles between stars. It does so repeatedly, and from repeated observations of each star, one can find
  • Parallax from the Earth's orbit
  • Angular velocity (proper motion)
  • Being pulled on by another celestial body
Gaia has observed a large number of binary stars - Gaia Data Release 3 - Astrometric binary star processing | Astronomy & Astrophysics (A&A) - and also an exoplanet - HIP 66074 | NASA Exoplanet Archive - though Gaia's data may contain evidence of many more - [2403.08226] Astrometric detection of exoplanets

Among Gaia's discoveries are these binary stars with black holes in them (distance, BH mass, major axis, period, eccentricity, companion mass, companion radius, companion luminocity, companion surface temperature):
  •  Gaia BH1 - 1,560 ly, 478 pc - 9.62 Ms - 1.40 AU - 185.59 d, 0.508 y - 0.451 - 0.93 Ms - 0.99 Rs - 1.06 Ls - 5,850 K (Sunlike)
  •  Gaia BH2 - 1,276.7 ly, 4.96 pc - 8.94 Ms - 4.96 AU - 1276.7 d, 3.495 y - 0.5176 - 1.07 Ms - 7.77 Rs - 24.6 Ls - 4,604 K (red giant)
  •  Gaia BH3 - 1,930 ly, 591 pc - 32.70 Ms - 16.17 AU - 4253.1 d, 11.644 y - 0.7291 - 0.76 Ms - 4.936 Rs - 16.3 Ls - 5,212 K (red giant)
Neither of the first two black holes have any observable X-ray emissions: [2311.05685] No X-Rays or Radio from the Nearest Black Holes and Implications for Future Searches Other observed ones do have such emissions: Observed Black Hole Masses | stellarcollapse.org Like  Cygnus X-1 a 21-solar-mass black hole which orbits a blue giant star called HDE 226868. These emissions come from material from the companion star falling onto the BH.
 
The Event Horizon Telescope is not any one radio telescope, but several radio telescopes whose signals are combined to create the effect of a single Earth-sized telescope. That makes it possible for the EHT to resolve the event horizons of the black holes that it has observed so far, the two with the largest angular size of event horizon: the central black holes of our Galaxy (Sgr A*) and of M87 (M87*).

These most recent observations were done by looking for radio-wave polarization, and this polarization is a probe of the magnetic fields at the emitting electrons. These fields make the electrons orbit their field lines, making those electrons' emissions polarized.

March 24, 2021:
Astronomers Image Magnetic Fields at the Edge of M87’s Black Hole | Event Horizon Telescope
Harvard, Smithsonian Astronomers Help Capture First Image of Black Hole’s Magnetic Fields | Center for Astrophysics | Harvard & Smithsonian
and
March 27, 2024:
Astronomers Unveil Strong Magnetic Fields Spiraling at the Edge of Milky Way’s Central Black Hole | Event Horizon Telescope
Astronomers Unveil Strong Magnetic Fields Spiraling at the Edge of Milky Way’s Central Black Hole | Center for Astrophysics | Harvard & Smithsonian

A remarkable result is that the accretion disks of Sgr A* and M87* are very similar in structure, despite M87* being 1,500 times more massive than Sgr A*.
 
General-relativity effects have been observed in the orbit of one of the closest stars to  Sagittarius A* (Sgr A*) --  S2 (star) -- a blue giant

This star's orbit:
Period = 16.0518 years (astronomers use Julian years of 365.25)
Major axis = 995.68 AU (observed angular) 0.12540"
Mean orbital velocity = 1847.55 km/s = 0.0061628 c
Eccentricity = 0.88466
Distance range = 114.84 to 1876.52 AU
Velocity range = 457.06 km/s to 7468.3 km/s = 0.0015246 to 0.024912 c (1/40 c)


Black-hole masses: Gaia BH3: 32.70 Msun, Sgr A*: 4.297*106 Msun, M87*: 6.5*109 Msun
Black-hole (Schwarzschild) radii: Earth: 8.87 mm, Sun: 2.953 km, Gaia BH3: 98.6 km, Sgr A*: 0.0848 AU, M87*: 130 AU


From observations of S2's radial velocity and angular separation:
A Geometric Determination of the Distance to the Galactic Center - IOPscience - 2003 October 24

Distance: 7,940 parsecs = 25,900 light years.

Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole | Astronomy & Astrophysics (A&A) - 26 July 2018
Relativistic redshift of the star S0-2 orbiting the Galactic Center supermassive black hole | Science - 25 Jul 2019

Both teams found agreement with general relativity to within 20%, or at least special relativity with the equivalence principle (gravity ~ inertia).


Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole | Astronomy & Astrophysics (A&A) - 16 April 2020

They also find agreement to within 20%. But unlike redshift, precession is cumulative, so with more observations, one can get stronger results.

This precession was first observed in the planet Mercury's orbit, which precesses about 43"/cy more than what one finds from the gravitational pulls of the other planets on that planet. It is also observed in binary pulsars.
 
First gravitational-wave detection of a mass-gap object merging with a neutron star - Northwestern Now - "The ‘mass gap’ might be ‘less empty than previously thought,’ researcher says"
The LIGO-Virgo-KAGRA collaboration detected the signal from GW230529 in May 2023, shortly after the start of its fourth observing run. By analyzing the signal, astrophysicists determined it came from the merger of two compact objects: One with a mass between 1.2 to 2.0 times the mass of our sun and the other with a mass between 2.5 to 4.5 times the mass of our sun. The researchers say the less massive object likely is a neutron star and the more massive object is potentially a black hole. But scientists are confident the more massive object is within the mass gap.
Click to expand...
[2210.00425] On the Neutron Star/Black Hole Mass Gap and Black Hole Searches - the mass gap is between the most massive known neutron stars at around 2 Msun and the least massive known black holes at around 5 Msun (rather approximate numbers).

 Tolman–Oppenheimer–Volkoff limit also mentions that mass gap. Computing that limit from stellar-structure theory is very difficult, because nucleons, protons and neutrons, interact very strongly with each other, unlike for a similar limit for white dwarfs, the  Chandrasekhar limit where electrons and nuclei are close to non-interacting.

There is now a sizable  List of gravitational wave observations They are mostly very distant objects, in the billions of light-years away, and most parent objects observed so far are black holes, with the rest being neutron stars and mass-gap objects.
 
For inspiral, most of the observations are of the last half-second, and after the merger, the resulting black hole "rings" for a short time more.

With more than one observation of some event, one can find the direction of the event's source. With two, it is in a ring in the sky, with three, it is two points, and with four or more, one has more than enough data, and one can see if one has a good fit.

 Gravitational-wave astronomy and  Gravitational-wave observatory -- these are two long tubes with mirrors at their ends. A laser beam is split in two and sent down each tube, then combined when it returns. This combination will make interference, and this interference is watched for evidence of one tube getting stretched or squeezed relative to the other.

The G-wave observatories that have detect G-waves are

As with position, more observations of an event can give us a better idea of the polarizations of G-waves. In GR, there are two possible polarizations, much like the polarizations of EM waves. In the EM case, looking along the direction of motion:

Linear: a combination of
  • horizontal: left, then right
  • vertical: up, then down
Circular: combination of these modes in sequence
  • up, left, down, right
  • up, right, down, left
These are all transverse, with no longitudinal component, a component along the direction of motion.

In GR, G-waves make stretching, squeezing, and shearing:
  • horizontal stretch vertical squeeze, then horizontal squeeze vertical stretch
  • diagonal 1 stretch diagonal 2 squeeze, then diagonal 1 squeeze diagonal 2 stretch
Circular polarization is combinations of these modes in sequence analogous to EM circular polarization.

This set of modes is called transverse traceless (TT). Alternatives to GR often have more possible modes, like horizontal and vertical together stretch then together squeeze, transverse-longitudinal diagonal stretch then squeeze, and all-longitudinal stretch then squeeze.

With more than two observations of G-waves, one can look for departures from TT, though for a full test, one will need at least six observations.
 
I'll now consider the prospects of observing black hole Sgr A* from star S2.  Main sequence - for definiteness, I will make it a B0V (main-sequence) start, with mass 18 Msun, radius 7.4 Rsun, luminosity 20,000 Lsun, and temperature 30,000 K.

One would have to be in orbit around S2, but what would be a good distance? From  Wien's displacement law the peak emission of S2 is at 100 nm, compared to the Sun at 500 nm. That's well within the ultraviolet range, at the short end of UV-C, and its visible light is mostly bluish.

To get where its flux of light equals the average of the Sun on the Earth, one will need to be 140 AU away. The star will be about 1.6 arcminutes across, barely resolvable without a telescope. Its light will be dangerous; one can easily get a bad case of sunburn (starburn?) from it.

One will orbit with a period of 400 years, and since that is much longer than S2's period around Sgr A*, one's orbit will not be stable.

Even so, Sgr A*'s angular diameter will not be very great. Using its Schwarzschild diameter, it has minimum 0.3', mean 0.6', maximum 5' (arcminutes). Multiplying by 3*sqrt(3)/2 to get the minimum observable angular distance of anything beyond it, I find minimum 0.8', mean 1.5', maximum 13' (arcminutes).

The BH's gravitational-lens effect will be much larger, and a star directly behind the BH will look like a ring around it, an  Einstein ring Its radius is the  Einstein radius and it has minimum 0.54d, mean 0.75d, and maximum 2.20d (degrees of arc) -- about 4 times the angular size of the Sun and the Moon. So one will see its lensing.
 
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I will now turn to what the Solar System would be like with Gaia BH3 in its orbit around its companion. For definiteness, I will make its mass 16 Msun, its distance 16 AU, and its eccentricity 0.451. It will have period 11.1 years.

It would make the orbit unstable of anything between Mars and the Oort cloud. The inner Solar System would remain but the outer Solar System would not exist. In its main-sequence days, it would have been a blue giant, and it would have lasted only a few million years. But that would have been enough to keep the inner Solar System from forming in anything like its actual form, with temperatures there of some 1,000 - 2,000 K.

Leaving that aside, what would the BH look like?

Its Schwarzschild radius would be 47 km, and its minimum-passing-light diameter 3*sqrt(3) this or 246 km. Its size would be minimum 15, mean 21, maximum 39 milliarcseconds.

But its gravitational-lens effect would be much larger, with Einstein radius minimum 0.57', mean 0.68', maximum 0.92' (arcminutes), at the limit of human visual acuity. But that effect should be easily visible in a small telescope.
 
I will now consider gravitational lensing.

First, the Schwarzschild or black-hole radius for mass M, gravitational constant G, and speed of light c:

\( \displaystyle{ R_{bh} = \frac{2GM}{c^2} } \)

For initial angle asrc, final angle aobs, and impact parameter b

\( \displaystyle{ a_{obs} = a_{src} + \frac{2R_{bh}}{b} } \)

The impact parameter is \( \displaystyle{ b = D a_{obs} } \)

where D is the reduced distance: D(src-mass) * D(mass-obs) / D(src-obs) very close to D(mass-obj) if D(src-mass) >> D(mass-obj). That gives us

\( \displaystyle{ a_{obs} = a_{src} + \frac{2R_{bh}}{D a_{obs}} = a_{src} + \frac{(a_{ring})^2}{a_{obs}} } \)

where the Einstein-ring radius is

\( \displaystyle{ a_{ring} = \sqrt{ \frac{ 2R_{bh} }{ D } } } \)

Note that aring >> Rbh / D unless D is not much greater than Rbh. So one will see a ring around the black-holey effects except if the source is close to the BH, like an accretion disk.

If an object is behind the mass, with asrc = 0, then aobs = aring and one sees a ring around the mass.
 
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There are two solutions, one for the light coming nearly directly and one for the light making a detour around the mass:

\( \displaystyle{ a_{obs} = \frac12 \left( a_{src} + \sqrt{(a_{src})^2 + (2 a_{ring})^2} \right) } \)
and
\( \displaystyle{ a_{obs} = \frac12 \left( a_{src} - \sqrt{(a_{src})^2 + (2 a_{ring})^2} \right) = \frac{ 2(a_{ring})^2 }{ a_{src} + \sqrt{(a_{src})^2 + (2 a_{ring})^2} } } \)

with long-distance limits

\( \displaystyle{ a_{obs} \to a_{src} + \frac{(a_{ring})^2}{a_{src}} ,\ a_{obs} \to \frac{(a_{ring})^2}{a_{src}} } \)

The relative luminosity of each image is

\( \displaystyle{ L = \frac{ (a_{obs})^2 }{ a_{src} \sqrt{(a_{src})^2 + (2 a_{ring})^2} } } \)

with asymptotic values

\( L \to 1 \) and \( \displaystyle{ L \to \left( \frac{a_{ring}}{a_{src}} \right)^4 } \)

Which adds to the difficulty of seeing black-holey effects in light deflection. Images close to the black-hole radius will be very faint.
 
In Newtonian mechanics, if an object's escape velocity is greater than c, then if light is a particle, then it will not escape. That was pointed out by astronomer John Michell in a letter in 1784.

But if light is a wave, that is very doubtful, and by the late 19th cy., it was firmly established that light is a wave. Or was it? Half a century later, it was firmly established that light also has particlelike properties, without negating any of the earlier observations of wavelike properties: wave-particle duality. But that's another subject.

Maxwell's equations feature a fixed value of c, something contrary to Newtonian mechanics. That was a major quandary in physics, given the great success of both Newtonian mechanics and Maxwellian electrodynamics. Even worse, there were no observable environmental effects on c -- it was fixed to within experiments' precision. It was only resolved by Albert Einstein's special relativity, a modification of Newtonian mechanics that makes c a sort of cosmic speed limit. When one gets close to c, it is very hard to go much faster, and the closer, the harder. It also makes time relative the way that space is relative, though the effect is very tiny for ordinary speeds.

Einstein noticed that Newtonian gravity does not fit in very well with SR, so he tried to construct a theory of gravity that does. He first noticed that to high experimental precision, gravity is independent of composition, something called the equivalence principle, and he concluded from that that space-time must be curved. His teacher Hermann Minkowski showed that SR was equivalent to time being like another space dimension, but contributing in the opposite direction to Pythagoras's theorem. He first dismissed that as "superfluous learnedness", but he then decided that he had to work with time in that way.

To express curvature, he used a generalization of Pythagoras's theorem for coordinate differences, using what's called the "metric tensor" of coefficients that can be arbitrary functions. For n dimensions, it has (1/2)*n*(n+1) components, so for space-time, it has 10 components. The mathematics of deriving curvature from these coefficients is rather horribly complicated, and I won't go into it here, but it is math that he had to learn.

After some experimenting, he published in 1915 his general theory of relativity. Gravity is due to space-time being curved, and the amount of that curvature is controlled by the mass-energy-momentum content of this space-time.
 
The wave-particle duality.

When propagating in space we model ilight as a wave with electrc and magnetic fields.

When interacting with matter we model it as a particle with a wavelength.

AE demonstrated the quantization of light in the Photoelectric Effect. It is what made him famous.

AE predicted light would be affected by gravity.

If I remember right it was first demonstrated by light from a star behind the sun being observed before there was a visual line of sight.
 
Einstein reportedly despaired of ever being able to solve his complicated nonlinear equations as anything more than a departure from flat space-time. But the next year, Karl Schwarzschild succeeded in doing exactly that.

Here's how he did that. He knew that it's easy to find a spherically symmetric solution in Newtonian gravity, so he decided to try that in GR, setting two of the coordinates to spherical-coordinate angles. He found that the 10 metric-tensor parameters reduced to 4 distinct ones, making solution much easier. He then imposed time independence, making one of the remaining coordinates a time coordinate, making one parameter drop out, leaving 3. He then recognized that he fixed all but the radial coordinate, so he fixed that one also, as 1/(2*pi)*circumference, so he got only 2. He could then solve the resulting GR equations in the empty-space case, giving his solution. A little over two decades later, Richard C. Tolman, J. Robert Oppenheimer, and George Volkoff discovered a version of this solution with matter in it.

The Schwarzschild solution has a remarkable feature, a point of no return, a distance from the center where any smaller distance one will be unable to escape from, an event horizon. Its radius is the "Schwarzschild radius" or black-hole radius, and it is the distance where the Newtonian escape velocity equals c.

The Earth's black-hole radius is 8.87 mm, and the Sun's is 2.95 kilometers.

The black-hole radius is linear in mass: 2*G*M/c2 -- for gravitational constant G and mass M.

A solution for a rotating black hole, the Kerr solution, was not discovered until 1963, and it is much more complicated. Its only symmetries are in time and azimuthal angle, angle around the spin axis, leaving the radial and polar-angle coordinates. That means 6 metric-tensor parameters, and fixing those two coordinates gives 4.


I note in passing that using symmetry also greatly simplifies finding cosmology solutions. One imposes constant space curvature, whether zero or nonzero, and one finds 2 parameters. Fixing the time coordinate gives only one to solve for.

Extending this symmetry to time gives not only flat space-time, but also the De Sitter and Anti De Sitter cosmological solutions. Those ones are for (pressure) = - (density) * c2 where the minus sign is a feature and not a bug, and where the density and pressure are constant. The De Sitter ones are for positive density and the Anti De Sitter ones for negative density. The De Sitter solutions include an exponentially-expanding one, a solution in the steady-state model and also in the inflationary phase of the Universe's expansion.
 
lpetrich said:
In Newtonian mechanics, if an object's escape velocity is greater than c, then if light is a particle, then it will not escape.
Click to expand...
Sure, but as Hawking pointed out, if a particle-antiparticle pair forms just outside the Schwarzschild limit, and only one of the pair is subsequently captured by crossing the event horizon, this has the same effect as the un-captured member of the pair having escaped from inside the event horizon.

So despite their inescapability, black holes should (if not supplied with matter from other sources), steadily lose mass to the outside universe, until they evaporate completely.
 
From a bio AE had help from mathematicians and peer reviews, he did notwork in isolation on relativity.

All they had were slide rules, mechanical calculators, along with paper and pencil/

A bit of computational trivia.

en.wikipedia.org

Curta - Wikipedia

en.wikipedia.org en.wikipedia.org

Solving both linear and non linear arbitrary systems of equations became practical with the computer.

In electronics the general solver is SPICE. Electronics has a lot of non linear systems. Simulating/calculating a chip would be impossible without a solver.
 
Before getting into tests of general relativity, I will mention  Tests of special relativity - a variety of tests of features of SR. For instance, recent versions of the  Michelson–Morley experiment experiment have found upper limits on anisotropy of c of around 10-17 for instance. The original one, in 1887, found an upper limit of around 10-9, about 1/5 of what one expects from the Earth's motion around the Sun. Also,  Experimental testing of time dilation includes successful demonstration of the  Twin paradox with the "twins" being muons in storage rings vs. stationary muons, and atomic clocks sent around the world vs. stay-at-home ones.

Now to  Tests of general relativity

Equivalence principle - acceleration by gravity is independent of composition. Three kinds:

Weak EP - independent of nongravitational composition. Every experiment I could find anything on has been done with chemical elements, meaning that they mostly test for differences that come from (proton + electron) vs (neutron) and nuclear binding energy. The champion upper limit to date is 10-15

Einstein EP - WEP with local Lorentz invariance (special relativity) and local positional invariance (plenty of tests for both space and time). EEP implies that gravity is caused by space-time curvature, with objects moving on "geodesics", straight lines generalized to curved spaces.

Strong EP - EEP with gravitational binding energy.

Gravitational redshift and time dilation - an outcome of the equivalence princple - tested with a variety of experiments: Earth-based, Earth orbit, the Sun, white dwarfs, galaxy clusters, and S2, a star near our Galaxy's central black hole Sgr A*.

GPS and Relativity - at their altitude, GPS satellites run fast by about 38 microseconds per day. This is:

45 mcsec/d from the Earth's surface having more gravitational time dilation than the satellites
-
7 mcsec/d from orbital-motion time dilation

So these satellites are run slow to avoid timing discrepancies that would make them unusable.
 
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Deflection and delay of light by gravity

Deflection of light was calculated using Newtonian mechanics by Johann Georg von Soldner in 1801, and the GR result is twice the Newton-Soldner value. Alternatives to GR has a parameter, g (gamma), that is (space curvature) / (time curvature) that GR predicts to be 1.

Deflection = (Newton-Soldner) * (1 + g) = (GR) * (1 + g) / 2

It was first measured in a solar eclipse in 1919, and it agreed wiith GR, though with big error bars, about 30%. Einstein said if the numbers turned out incorrect, "Then I would feel sorry for the dear Lord. The theory is correct anyway."

Since then, the tests have become much improved with the best visible-light tests being with the Hipparcos astrometry telescope - 10-3 and VLBI observations of quasars - 104

Gravitational deflection produces gravitational lensing, an effect that has been observed in several galaxies.

A similar effect is the Shapiro or gravitational time delay, here also (1 + g) * its Newtonian value or (1 + g)/2 its GR value. With the Cassini spacecraft it's been measured to 10-5.

The Confrontation between General Relativity and Experiment | Living Reviews in Relativity - from 2014

Extra precession

Planet Mercury is known to have some extra orbit precession, in excess of what one calculates from the other planets pulling on it. GR and Mercury's Orbital Precession

In arcseconds per century: total: 574, other planets: 531, discrepancy: 43

Alternatives to GR have another parameter, b (beta), one that multiplies (time-curvature nonlinearity). GR predicts it to be 1.

(Precession) = (GR) * (2 + 2*g - b)/3

Parameter b agrees with GR to about 10-4.

It's not just Mercury that has this precession, though the other planets have much less.
Mercury 43"/cy, Venus 8.6"/cy, Earth 3.8"/cy, Mars 1.35"/cy, Jupiter 0.06"/cy, Saturn 0.01"/cy

This precession has also been observed in the star S2 near central black hole Sgr A*.
 
Gravity of gravitational binding energy

General relativity predicts that it will have the same gravity as every other kind of mass/energy, but many GR alternatives predict some discrepancy.

This can be tested using the "Nordtvedt effect", perturbation of an orbit from these different amounts of gravity. This has been searched for by observing the Moon by bouncing laser beams off of retroreflectors left there by some Apollo astronauts, and then calculating other gravitational effects, mostly from the Sun. Once that is done, one finds that the observed departure from GR is less than about 10-3.

In terms of other alternate-gravity parameters, this effect has a multiplier of
4*b - g - 3 - (10/3)*xi - a1 + (2/3)*a2 - (2/3)*z1 - (1/3)*z2
Of these, in GR, xi = a1 = a2 = z1 = z2 = 0.

Spin precession

This has two types, which may be called spin-orbit and spin-spin.

Spin-orbit or geodetic precession has two components: a SR effect called Thomas precession and a curved-space-time effect, and they add up to be
(GR) * (2 + g) / 3

Spin-spin, Lense-Thirring precession, frame dragging, or gravitomagnetic precession is a curved-space-time effect from the gravitational field of an object's rotation. That also causes orbit precession, though in most cases, that is smaller than the precession caused by the object's equatorial bulge.
(GR) * (1 + g + a1/4) / 2

The Gravity Probe B spacecraft, operated over 2004 - 2005, contained four super precise gyroscopes for looking for these precession effects. It confirmed geodetic precession to within 0.2%, and Lense-Thirring to within 20%.
 
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