User talk:Greg L/Fuzzballs (string theory)

Key assumptions used in article

• Schwarzschild radius, per $R_{s}={\frac {2GM}{c^{2}}}$, = 2953.25 ±0.07 meters per solar mass (M_☉) (updated 14 August, 2023)

where:

M = mass in kilograms and where solar mass = 1.988435(27)×10³⁰ kg (from Measurement of Newton’s Constant Using a Torsion Balance with Angular Acceleration Feedback, Jens H. Gundlach and Stephen M. Merkowitz, PHYSICAL REVIEW LETTERS, VOLUME 85, NUMBER 14, (hereinafter referred to as the "University of Washington" group), wherein their value for (M_☉) (for use as M) was derived from their measurement of Big G).

G = 6.674215(92)×10⁻¹¹ m³ kg⁻¹ s⁻² for the purposes here only of calculating Schwarzschild radius since the above value for solar mass was derived from this Big G value from the same group (from Measurement of Newton’s Constant Using a Torsion Balance with Angular Acceleration Feedback, Jens H. Gundlach and Stephen M. Merkowitz, PHYSICAL REVIEW LETTERS, VOLUME 85, NUMBER 14. When one measures Big G, one also measures the masses of the Sun and Earth, and the relationship is proportional since the uncertainties in Earth's orbital radius and the length of a year are far smaller than Big G's).

c = 299,792,458 m/s

Note that the latest CODATA value for G, 6.67430(15)×10⁻¹¹ m³ kg⁻¹ s⁻², is slightly different from the U-of-W value. Since measurements of G establishes the mass of the Sun by solving Kepler's third law of planetary motion, as follows:

$$M_{\odot }={\frac {(\mathrm {au} )^{3}\times \pi ^{2}4}{(\mathrm {yr} )^{2}\times G}}$$

…the CODATA value requires slightly different masses for both the Sun and Earth. However, due to the way solar masses (M_☉) and G are used in the formula for calculating the Schwarzschild radius (top formula), an increase in Big G (G) in papers under circumstances where G is used to calculate M_☉ (like the U-of-W paper) means a proportional decrease in solar mass—the two (G and M_☉) are reciprocals—and the net result is no effect at all on the Schwarzschild radius because G and M (M_☉) are multiplication factors on the same line.

Using the U-of-W value's for M_☉ and G, where both come from the same paper at the same time, has the effect of adopting the U-of-W values for the number of SI atomic seconds assumed for a sidereal year as well as the number of meters in an astronomical unit (Earth-Sun distance) that the authors elected to use while writing their journal paper; those two variables (the au and the length of a year) influence the equality between G to 1 M_☉. These values used for the length of a year and the au are central to establishing the value of M_☉ because other terms in the M_☉ formula—like pi and “4”—are either locked down by definition or can be accurately expressed to an arbitrarily excessive precision.

Note that the Earth's orbit is an ellipse and travels faster when closer to the sun than when further away. Per Kepler's second law of planetary motion, the orbits of planets sweep out equal areas in equal time. The Schwarzschild formula will return a radius for any arbitrary mass used for M. However, since the number of kilograms in M_☉ is determined by the measurement of G, the magnitude assigned to M_☉ (in kilograms) is equivalent to asking “What centripetal force is there between the Earth and Sun?” And this is the same as asking “What combination of solar mass and strength of gravity, where mass could be bigger if the strength of gravity is weaker, accounts for the centripetal force between the Earth and Sun?” And that question is effectively asks “What combination of Sun-Earth distance and length of sidereal year causes the Earth to sweep out 2.228×10¹⁵ m²/s in its orbit?” Ergo, the Schwarzschild radius produces any radius for any arbitrary inputed value of M, but if M is M_☉, which is the product of of Kepler’s third law, the value of R_s, in meters, is an abstracted value describing “the Earth’s motion around the sun” that is dependent upon two values: the au and length of sidereal year.

Critical point: According to the authors' October 2000 paper, they relied upon the Astronomical Almanac for 2001 (U.S. Government Publishing Office, Washington, D.C., 1999), p. k6 for those two astronomical values.

The U-of-W authors’ decision to use the Astronomical Almanac for 2001 was a professional move and a beyond-vast improvement over the ad hoc way amateur laypersons on the Internet use when calculating terms like the length of a year based on the number of days that are supposedly in a year (the length of a day, as measured in SI seconds, varies and has longterm drift). Moreover, many laypersons when attempting to calculate the mass of the sun from G use the length of a tropical year (how long it takes the sun to appear in the same place in the sky), which is doubly wrong. Note that the sidereal year and the au in epoch 2001 was before the 2012 redefinition of the au as being precisely fixed at 149,597,870,700 meters. Accordingly, as regards the values that Astronomical Almanac for 2001 used for the length of year and the au, which came from the U.S. Naval Observatory, it was crucial that the length of the sidereal year and the au be in harmony and were calculated at the same time using the same set of assumptions and methodology since the two measures are intimately linked.

Note also that the U-of-W paper's uncertainty in the magnitude of 1 M_☉ equals the uncertainty in R_s; for both values it is a relative standard uncertainty of 1.4×10⁻⁵. The U-of-W's stated uncertainty in the magnitude of G has no effect on the uncertainty of R_s due, as mentioned above, to the way G and M are reciprocals of each other in all contexts where a measurement of G is used to establish the magnitude of M_☉. Since there is a linear and proportional equality between the magnitude of R_s and 1 M_☉, the uncertainty in R_s (a relative uncertainty of 1.4×10⁻⁵) therefore equals ±0.04 meters; ergo, R_s = 2953.25 ±0.04 m/M_☉.

However, the U-of-W measurement of G, while precise, is not the only high-precision measurement out there; other measurements of G have been done by other groups and they vary from the U-of-W's. The measurement of G is a modern mess in comparison to that of other fundamental constants of nature, which are known with precisions that are either many more orders of magnitude more precise or are locked down by definition. While the CODATA value of G was heavily influenced by the U-of-W paper, it incorporates measurements from other groups around the world and the resultant uncertainty in the CODATA value is a complex evaluation of competing uncertainties, some of which don’t overlap; it is a weighted assessment of all measurements of G. The CODATA value of 6.67430(15)×10⁻¹¹ m³ kg⁻¹ s⁻² is a relative standard uncertainty of 2.2×10⁻⁵. Thus, the proper value and uncertainty for R_s is 2953.25 ±0.07 m/M_☉. This article rounds the value to 2953 m/M_☉ so as to avoid excess precision.

Summary: Due to the simple and elegant (and sometimes reciprocal and self-nulling) nature of the formulas for calculating the mass of the sun (M_☉) and the Schwarzschild radius (R_s), the magnitude of R_s is strongly influenced by the values of two key astronomical values taken from the Astronomical Almanac for 2001 (U.S. Government Publishing Office, Washington, D.C., 1999), p. k6; and the uncertainty, or tolerance, of R_s comes from the CODATA value for G. Both sources are the most authoritative ones for the values of the length of a year and au (to calculate M_☉), and G to plug into the formulas for calculating R_s.

• Planck density: $\rho _{P}={\frac {m_{P}}{l_{P}^{3}}}$ = 5.15489(36)×10⁹⁶ kg/m³

where:

m_P is the Planck mass (CODATA value) with a relative standard uncertainty of 5.0×10⁻⁵

l_P is the Planck length (CODATA value) with a relative standard uncertainty of 5.0×10⁻⁵

• With regard to how 6.8 solar masses is a “median-size” stellar-mass black hole, this is based on this list of black hole candidates. Excluding one useless entry with a mass range of 3–100 solar masses, the median value of all the rest of the candidates with known masses is 7.2 solar masses with a 3.1 standard deviation. I chose 6.8 solar masses (0.13 of a standard deviation off the median value) because its Schwarzschild radius produced a nice round value of 20 km and its density was also a nice round number.
• According to this article by Goddard Space Flight Center (NASA), the minimum mass for a black hole lies in the range of 1.7–2.7 solar masses and the smallest black hole discovered so far is 3.8 solar masses.
• Note: The neutron star density range of 3.7–5.9×10¹⁷ kg/m³, comes from Wikipedia’s article on the subject.
• The “70-story tall building” is the Four Seasons Hotel Miami, which is independently listed as “789 feet (240 meters)” tall.
• The Empire State Building is 1,453 ft–89⁄16 in (443.09 meters) tall (according to Wikipedia here in the article) and independently at Empire State Building Official Internet Site (although they erred with the metric conversion by 0.1 meter).
• With an area of 1430.4 square kilometers, Kauai has an equivalent radius of 21.338 km.
• Drop of water: Long-standing rule of thumb in science is 0.05 mL (20 drops per milliliter) [1], [2]
• Mean density of granite = 2.667 g/mL (used to be 2.75 g/mL) (Citation: Geological Society of America, Daly et al., 1966)
• Density of gold = 19.32 g/mL. Though gold’s density is often stated as 19.3 g/mL (Web Elements and Wikipedia), the customary, high-precision value is 10.18 troy ounces per cubic inch, which works out to 19.32 g/mL (Google search, World Gold Council, and hypertextbook)
• One astronomical unit = 1.49597870691(6)×10¹¹ m
• Our solar system’s termination shock radius was measured at 83.7 and 94 AU by Voyager 2 and 1 respectively [3]
• Planck density = 5.1×10⁹⁶ kg/m³
• Mass of Sagittarius A* is 4.31 ±0.06 million solar masses: Monitoring stellar orbits around the Massive Black Hole in the Galactic Center by S. Gillessen et al.
• With regard to 3.9 billion solar masses being “near the upper bounds for supermassive black holes”, the biggest known supermassive black hole is 18 billion solar masses. Cite: NewScientist.com, Biggest black hole in the cosmos discovered, which tells of a paper by Mauri Valtonen of Tuorla Observatory in Finland. I chose 3.9 billion solar masses because its density is equal to that of a fluid with which everyone is familiar: air.
• Range of mass of the universe = 10⁵³–10⁶⁰ kg [4] with 10⁵³–10⁵⁵ being common values.
• Rounded, high-end mass universe at 10⁵⁵ kg at the Planck density of 5.1×10⁹⁶ kg/m³ = volume of 1.96×10⁻⁴² m³ or radius of 7.76 fm, which is about the size of uranium nucleus.
• Rounded, low-end mass of the universe at 10⁵³ kg at the Planck density = volume of 1.96×10⁻⁴⁴ m³ or radius of 1.67 fm, which is the size of helium‑4 nucleus.
• Precision estimate: Assuming Guth’s Inflationary theory is correct and Ω is precisely 1, then the critical density of the universe is 1.8791×10⁻²⁹ g/cm³ (1.8791×10⁻²⁶ kg/m³) [5], (8.988×10¹⁶ J/m³) and assuming the universe has a radius of 46.5 billion light years, then the mass of the universe is 6.7×10⁵⁴ kg, which is close to two other commonly quoted masses for the universe (2×10⁵⁴ kg and 2.5×10⁵⁴ kg) (google search). At the Planck density of 5.1×10⁹⁶ kg/m³ = volume of 1.314×10⁻⁴² m³ = radius of 6.79 fm, which is about the size of a gold nucleus
• A total mass of 2×10⁵⁴ kg (a commonly cited mass) at the Planck density is a volume of 3.92×10⁻⁴³ m³ = radius of 4.54 fm, which is about 84% the diameter of an iron nucleus and 129% the size of an oxygen‑16 nucleus.
• Uranium nucleus radius = 7.4×10⁻¹⁵ m (7.4 fm) (rochester.edu)
• Gold nucleus radius is 6.5×10⁻¹⁵ m (6.5 fm) [6]
• Iron nucleus radius = 5.4×10⁻¹⁵ m (5.4 fm) [7].
• Oxygen-16 nucleus radius = 3.53×10⁻¹⁵ m (3.53 fm) [8]
• Helium-4 nucleus = 1.7×10⁻¹⁵ m (1.7 fm) [9]
• The most likely and plausible mass range for the universe (1×10⁵³ to 1×10⁵⁵ kg), when compacted to the Planck density results in radii equal to that of atomic nuclei ranging from helium‑4 to uranium (1.67–7.76 fm).
• (Added Aug. 14, 2023) With regard to …with a radiated power—for the entire black hole—of 1.2×10⁻²⁹ watt (12 billion-billion-billionths of one milliwatt). Such a minuscule signal is to one watt as ¹⁄₃₀₀₀th of a drop of water (about one-quarter the volume of a common grain of table salt) is to all the Earth's oceans, per NOAA, the volume of earth’s oceans is 1,335,000,000 cubic kilometers (1.335×10²⁴ milliliters). Moreover, 20 drops are assumed per milliliter (Excel Medical, confirmed via personal experiences decades ago). Ergo, (2.670×10²⁵ drops) in earth’s oceans × (1.2×10⁻²⁹ watt) = 3.204×10⁻⁴ of a drop = ¹⁄₃₁₂₁th of a drop, rounded to ¹⁄₃₀₀₀ so as to not exceed a precision of one part in twenty (drops per ml). The size of a grain of common table salt is taken to be a cube measuring 400 micrometers per side (my own value for common Morton-brand table salt using a video-measuring stereo microscope at work while taking photos of solder paste… and also after finding a wild range of values for salt on the Internet), thus, 0.2503 of a salt grain, rounded to "one-quarter." Greg L (talk)

Excess-precision values:

• 6.8 solar masses is Schwarzschild radius of 20.087 kilometers and is 3.98364×10¹⁷ kg/m³ and 0.05 mL (“one drop”) at that density has a mass of 19.918 million metric tons, which is equivalent to a sphere 242.517 meters in diameter at a density of 2.667 g/mL, and a teaspoon of such a fuzzball would have a mass of 1.9635 billion metric tons.
• Assuming a conservative lower limit of 2.7 solar masses for a black hole (maximum upper limit for a neutron star), a black hole can have a density of up to 2.5268×10¹⁸ kg/m³, and 0.05 mL (“one drop”) at that density has a mass of 126.340 million metric tons, which is equivalent to a sphere 448.924 meters in diameter at a density of 2.667 g/mL, and a teaspoon of such a fuzzball would have a mass of 12.454 billion metric tons.
• Cygnus X-1, 8.7‑solar-mass black hole, Schwarzschild diameter = 51.400 kilometers.
• Cross-sectional area of 51.400-kilometer-diameter is 2074.96 km² whereas Walker County, Texas is listed as 2,075 km² (Wikipedia and Census results).
• The neutron star density range of 3.7–5.9×10¹⁷ kg/m³, (1.8237–2.9081 billion metric tons per teaspoon) equates to fuzzballs with masses ranging from 7.056 to 5.588 solar masses
• 4.31 million solar masses = 9.91615×10⁵ kg/m³ = 51.326 times that of gold.
• 3.9 billion solar masses = 77.0104 astronomical units and 1.211068 kg/m³, which was contrasted to atmospheric density = 1.1968 kg/m³, which was taken at 13.94 °C (global mean temperature), 194 meters (median altitude of human habitation), and 9 °C dewpoint, (which is 1.16% molar concentration of water). — Preceding unsigned comment added by Greg L (talk • contribs) 21:21, 19 August 2023 (UTC)

A public thank-you to a valued behind-the-scenes contributors to this article

I want to thank Dr. Mathur, the Ph.D. who advanced the theory of fuzzballs; he assisted me in writing this exposition. I exchanged many emails with Dr. Mathur while working on this and spoke with him over the phone on occasion. Dr. Mathur even offered the assistance of one of his graduate students to add a math section to the article (which I declined as I feel those abstruse math sections are wholly unnecessary for a general-interest encyclopedia).

I also want to give a well-deserved public shout-out to Viktor T. Toth, who created the Hawking radiation calculator cited frequently in the article. Mr. Toth provided invaluable feedback while I struggled with this article.

It's worth noting that I independently calculated (which is to say, double-checked) many of the values cited to his online calculator until it was clear that the calculator A) uses correct formulas, and B) of course, isn’t capable of occasional calculation mistakes like I am.

During our many email exchanges, it’s clear that Mr. Toth not only truly and deeply understands the direct and tangential theories underlying the formulas of black holes and Hawking radiation, but also has an uncanny ability to use plain-speak when communicating arcane scientific issues. In particular, Mr. Toth utterly shamed me into diverging from a near-exclusive particle-based treatment of Hawking radiation towards a wave-based treatment. Without connecting the dots between waves and particles, one would be doing a disservice to our readership.

Thank you, Viktor.

Greg L (talk) 01:46, 4 October 2023 (UTC)

An interesting picture of everything in the Universe that is or ever was

I didn’t want to lose this scientific paper and a fascinating graph in the paper that shows every size and mass of object in the universe that ever existed. It is beyond the scope of this article, but many people who are interested enough in this article to visit this talk page will find this interesting.

• GRAPH OF EVERYTHING: Masses, sizes, and relative densities of objects in our Universe. I find it interesting that according to fuzzball theory, all portions of a fuzzball lie precisely on the diagonal “black hole” line along the top of the graph. This is unlike a classic black hole where the void between the event horizon and the singularity occupies the forbidden zone at top and where the singularity is way beyond the top of the graph.

• JOURNAL ARTICLE: That image comes from the journal paper, “All objects and some questions” at the American Journal of Physics.

• NEWS STORIES ABOUT THE ARTICLE: Two simplified synopses of the journal paper are available at “A new view of all objects in the universe” at Phys.org, and at “Astrophysicists Put All Objects in Universe into One Pedagogical Plot” at Sci.News.

In the graph, the white “instanton” dot at the intersection of “quantum uncertainty” and “forbidden by gravity” (general relativity) is the Planck mass (2.176×10⁻⁵ g, or the mass of a cubic salt crystal measuring 0.216 mm on a side) compressed into a volume of one Planck length cubed, at the Planck density. And, as the paper says, the temperature of the Hawking radiation emitted from the instanton is the Planck temperature. At the upper right-hand corner of the graph, the big black dot is the Universe, which fits on the “black hole” line.

I hope others find this helpful in connecting the dots between general relativity and quantum physics. Greg L (talk) 03:07, 19 October 2023 (UTC)