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Fuzzballs are theorized by some superstring theory scientists to be the true quantum mechanical description of black holes.^[1] The theory resolves two intractable problems that classic black hole theory poses for modern physics:

It dispenses with the gravitational singularity at the heart of the black hole, which is thought to be surrounded by an event horizon, the inside of which is detached from the spacetime of the rest of the universe. Conventional black hole theory holds that a singularity is a zero-dimensional, zero-volume point in which all of a black hole's mass exists at infinite density.^[1]^{[Note 1]} Modern physics breaks down under such extremes because gravity would be so intense that spacetime itself breaks down catastrophically.
It resolves the black hole information paradox wherein conventional black hole theory holds that all the quantum information comprising infalling matter and light is thought to either be extinguished at singularities, or the quantum information is still preserved but since the event horizon and singularity are separated by a large void that is not part of spacetime, the quantum information describing what fell into the singularity cannot reach the event horizon where a certain quantum mechanical process would, in principle, allow it to slowly escape. In either situation, since the event horizon cannot be imprinted with information regarding the composition of what fell into it, this violates a fundamental law of quantum mechanics requiring that quantum information be conserved.^[1]^[2]

Fuzzball theory dispenses with the singularity at the heart of a black hole and preserves infallen quantum information by positing that the entire region within the black hole's event horizon is actually an extended object: a ball of strings, which are advanced as the ultimate building blocks of matter and light. Under string theory, strings are theorized to be bundles of energy vibrating in complex ways in both the three physical dimensions of space as well as in compact directions—extra dimensions interwoven in the quantum foam (also known as spacetime foam).^[1]

Physical characteristics

Samir D. Mathur of Ohio State University, with postdoctoral researcher Oleg Lunin, proposed via two papers in 2002 that black holes are actually sphere-like extended objects with a definite volume; they are not a singularity, which the classic view holds—as shown in the illustration at right—to be a zero-dimensional, zero-volume point into which a black hole's entire mass is concentrated at infinite density, around which, many kilometers away, is an event horizon.^[3]

Both string theory and superstring theory hold that the fundamental constituents of subatomic particles, including the force carriers (e.g. bosons, photons, and gluons), are all composed of strings of energy that take on their identity by vibrating in different modes and/or frequencies. Quite unlike the view of a black hole as a singularity, a small fuzzball can be thought of as an extra-dense neutron star where its neutrons have decomposed, or "melted", liberating the quarks (strings in string theory) composing them. Accordingly, fuzzballs can be regarded as the most extreme form of degenerate matter.

Whereas the event horizon of a classic black hole is thought to be very well defined and distinct, Mathur and Lunin further calculated that the event horizon of a fuzzball would, at an extremely small scale (likely on the order of a few Planck lengths), be very much like a mist: fuzzy, hence the name "fuzzball". They also found that the physical surface of the fuzzball would have a radius equal to that of the event horizon of a classic black hole; for both, the Schwarzschild radius for a non-rotating median-size stellar-mass black hole of 6.8 solar masses (M_☉) is 20 kilometers.

With classical-model black holes, objects passing through the event horizon on their way to the singularity are thought to enter a realm of curved spacetime where the escape velocity exceeds the speed of light—a realm devoid of all structure. Moreover, precisely at the singularity—the heart of a classic black hole—spacetime itself is thought to break down catastrophically since infinite density demands infinite escape velocity; such conditions are problematic with known physics. Under the fuzzball theory, however, the strings comprising matter and photons are believed to fall onto and absorb into the surface of the fuzzball, which is located at the event horizon—the threshold at which the escape velocity has achieved the speed of light.

A fuzzball is a black hole; spacetime, photons, and all else that is not exquisitely close to the surface of a fuzzball are thought to be affected in precisely the same fashion as with the classical model of black holes featuring a singularity at its center. The two theories diverge only at the quantum level; that is, classic black holes and fuzzballs differ only in their internal composiition as well as how they affect virtual particles that form close to their event horizons (see § Information paradox, below). Fuzzball theory is thought by its proponents to be the true quantum description of black holes.

Since the volume of fuzzballs is a function of the Schwarzschild radius (2,954 m per M_☉ for a non-rotating black hole), fuzzballs have a variable density that decreases as the inverse square of their mass (twice the mass is twice the diameter, which is eight times the volume, resulting in one-quarter the density). A typical 6.8 M_☉ fuzzball would have a mean density of 4.0×10¹⁷ kg/m³.^{[Note 2]} A bit of such a fuzzball the size of a drop of water (0.05 mL, 5.0×10⁻⁸ m³) would have a mass of twenty million metric tons, which is the mass of a granite ball 240 meters in diameter.^{[Note 3]}

Though such densities are almost unimaginably extreme, they are, mathematically speaking, infinitely far from infinite density. Although the densities of typical stellar-mass fuzzballs are quite great—about the same as neutron stars.^{[Note 4]}—their densities are many orders of magnitude less than the Planck density (5.155×10⁹⁶ kg/m³), which is equivalent to the mass of the universe packed into the volume of a single atomic nucleus.

Fuzzballs become less dense as their mass increases due to fractional tension. When matter or energy (strings) fall onto a fuzzball, more strings are not simply added to the fuzzball; strings fuse together, and in doing so, all the quantum information of the in‑falling strings becomes part of larger, more complex strings. Due to fractional tension, string tension exponentially decreases as they become more complex with more modes of vibration, relaxing to considerable lengths. The string theory formulas of Mathur and Lunin produce fuzzball surface radii that are precisely equal Schwarzschild radii, which Karl Schwarzschild calculated using an entirely different mathematical technique 87 years earlier.

Einstein's 1915 theory of general relativity established how gravity affects spacetime, as illustrated in these three panes depicting a type of Minkowski spacetime diagram. Far away from a black hole, particles and photons can move in any direction, represented by the curvy arrows. The limiting rays at ±45° represent photons traveling directly leftwards and rightwards at the speed of light as time moves upwards at the speed of light.

Close to an event horizon, photon paths not heading directly at the black hole are sheared to one extent or another to the right, and photons escaping the black hole lose energy and become redshifted as they climb against gravity. Since "straight" is "the path taken by photons in a vacuum," mass that distorts photon paths distorts spacetime itself.

At an event horizon—depicted here as inside the void surrounding a singularity—all photons have lost all energy (are infinitely redshifted) and none can escape. Moreover, no amount of force can lift away a particle possessing mass. Fuzzball theory holds that matter and photons collide with a physical surface precisely at the event horizon.

Due to the mass-density inverse-square rule, fuzzballs need not all have unimaginable densities. Supermassive black holes, which are found at the center of virtually all galaxies, can have modest densities. For instance, Sagittarius A*, the black hole at the center of our Milky Way galaxy, is 4.3 million M_☉. Fuzzball theory predicts that a non-spinning supermassive black hole with the same mass as Sagittarius A* has mean density "only" 51 times that of gold. Moreover, at 3.9 billion M_☉ (a rather large super-massive black hole), a non-spinning fuzzball would have a radius of 77 astronomical units—about the same size as the termination shock of the Solar System's heliosphere—and a mean density equal to that of the Earth's atmosphere at sea level (1.2 kg/m³).^[4]

Irrespective of a fuzzball's mass, resultant mean density, or even its spin (which affects the Schwarzschild radius; see also Ergosphere and Rotating black hole), its physical surface is located exactly at the event horizon, which is threshold at which the escape velocity equals the speed of light: 299,792,458 meters per second.^{[Note 5]} Escape velocity, as its name suggests, is the velocity a smaller body must achieve to escape from a much more massive one; at 11,186 m/s, Earth's escape velocity is only 3.7 thousandths of one percent that of event horizons. Thus, event horizons—those either surrounding singularities or the surface of fuzzballs—lie at the point where spacetime has been curved by gravity to the local speed of light in accordance with Albert Einstein's general theory of relativity.^{[Note 6]} Fuzzball theory holds that infalling photons and particles of matter impact the physical surface of a fuzzball at the event horizon, whereupon their individual strings are liberated to dissolve into and contribute to the fuzzball's quantum makeup.

Note that escape velocity, which has the unit of measure m/s, is distinct from gravitational strength, which is a different property known as acceleration and has m/s² as its unit of measure. Though the escape velocity at an event horizon is a finite value (the speed of light), the gravitational strength at event horizons (and the surface of theorized fuzzballs) is infinite, which imbues particles of matter possessing any mass whatsoever with infinite weight. Thus, an imaginary uncrushable rocket with its center of mass located at an event horizon would require infinite thrust to merely hover.^[4]

Gravitational strength at event horizons cannot be properly calculated using Newton's 338-year-old law of universal gravitation. For instance, Newton's formula for gravitational strength at the event horizon of the largest known supermassive black hole, Phoenix A (see List of most massive black holes), which is estimated to be 100 billion M_☉, incorrectly yields a rather uncomfortable but survivable gravitational acceleration of only about 15 times that of Earth's gravity. However, when properly calculated in accordance to Einstein's general theory of relativity to account for the way mass warps spacetime, gravitational strength is infinite at the event horizon surrounding singularities and at the surface of fuzzballs.

The aforementioned phenomenon of infinite gravitational acceleration at event horizons is distinct from a stretching effect on objects known as spaghettification, lethal amounts of which can occur hundreds of kilometers above the surface of stellar-mass fuzzballs (or above the event horizon surrounding a singularity). Spaghettification is a consequence of intense gravitational field gradients known as tidal forces.

Information paradox

Classical black holes create a problem for physics known as the black hole information paradox, an issue first raised in 1972 by Jacob Bekenstein and later popularized by Stephen Hawking. The information paradox is born out of the realization that all the quantum nature (information) of the matter and light that falls into a classic black hole is thought to entirely vanish from existence into the zero-volume singularity at its heart. For instance, a black hole that is feeding on the stellar atmosphere (protons, neutrons, and electrons) from a nearby companion star should, if it obeyed the known laws of quantum mechanics, technically grow to be increasingly different in composition from one that is feeding on light (photons) from neighboring stars. Yet, the implications of classic black hole theory are inescapable: other than the fact that the two classic black holes would become increasingly massive due to the infalling matter and light, they would undergo zero change in their relative composition because their singularities have no composition. Bekenstein noted that this theorized outcome violated the quantum mechanical law of reversibility, which essentially holds that quantum information must not be lost in any process. This field of study is today known as black hole thermodynamics.

Even if quantum information was not extinguished in the singularity of a classic black hole and it somehow still existed, quantum data would be unable to climb up against infinite gravitational intensity to reach the surface of its event horizon and escape. Hawking radiation (so-far undetected particles and photons thought to be emitted from the proximity of black holes) would not circumvent the information paradox; it could reveal only the mass, angular momentum, and electric charge of classic black holes. Hawking radiation is thought to be created when virtual particles—particle / antiparticle pairs of all sorts plus photons, which are their own antiparticle—form very close to the event horizon and one member of a pair spirals in while the other escapes, carrying away the energy of the black hole but no information about it.

The fuzzball theory advanced by Mathur and Lunin satisfies the law of reversibility because the quantum nature of all the strings that fall into a fuzzball is preserved as new strings contribute to the fuzzball's makeup; no quantum information is squashed out of existence. Moreover, this aspect of the theory is testable since its central tenet holds that a fuzzball's quantum data do not stay trapped at its center but reach up to its fuzzy surface and that Hawking radiation carries away this information, which is encoded in the delicate correlations between the outgoing quanta.^[1]

Fuzzball theory's proposed solution to the black hole information paradox resolves a significant incompatibility between quantum mechanics and general relativity. While Einstein made important contributions to quantum mechanics, he had objections to it. Throughout the remainder of his career, Einstein searched in vain for a unifying theory—a Theory of Everything, so to speak, that linked together all aspects of the universe.^[5]^[6] To this day, there is no widely accepted theory of quantum gravity—a quantum description of gravity—that is in harmony with general relativity, however, both fuzzball theory and other forms of string theory such as M-theory have been advanced as candidates.^[1]^[7]

Testability of the theory

As there is no direct experimental evidence supporting string theory or fuzzball theory, both are currently products purely of calculations and theoretical research.^[8] However, theories must be experimentally testable if there is to be a possibility of ascertaining their validity.^[9] To be in full accordance with the scientific method—and one day be widely accepted as true, as are Einstein's theories of special and general relativity—theories regarding the natural world must make predictions that are consistently affirmed through observations of nature.

Fuzzball theory predicts that quantum information is preserved in fuzzballs and that Hawking radiation originating within the Planck-scale quantum foam just above a fuzzball's surface would theoretically be encoded with that information. As a practical matter, Hawking radiation is virtually impossible to detect because the individual photons comprising Hawking radiation have extraordinarily little energy. Moreover, black holes radiate at astronomically low power levels. This underlies why black holes require so long to evaporate: 10⁶⁷ times the age of the universe for a 4.9 M_☉ black hole.^[4] Hawking showed that the energy released by Hawking radiation is inversely proportional to the mass of a black hole; ergo, the smallest black holes emit the most energetic Hawking radiation that is least difficult to detect. However, the radiation emitted from even a minimum-size, 2.7 M_☉ fuzzball does so at wavelengths equivalent to a black body temperature of around 23 billionths of one kelvin above absolute zero, and with a radiated power—for the entire black hole—of 1.2×10⁻²⁹ watt (12 billion-billion-billionths of one milliwatt).^[4] Such a minuscule signal would be vastly overwhelmed by even the 2.73 K blackbody temperature of the cosmic microwave background coming from all directions of space.^[10]

Ever since the first direct detection of gravity waves, a 2015 event known as GW150914, which was a merger between a binary pair of stellar-mass black holes, the gravity-wave signals detected by the LIGO and Virgo gravitational-wave observatories have so far matched the predictions of general relativity for classical black holes with singularities at their centers. However, an Italian team of scientists that ran computer simulations suggested in 2021 that existing observatories are capable of discerning fuzzball-theory-supporting evidence in the gravity-wave signals from merging binary black holes (and resultant effects on ringdowns) by virtue of the nontrivial unique attributes of fuzzballs, which are extended objects with a physical structure. The team's simulations predicted slower-than-expected decay rates for certain vibration modes and that those modes would be dominated by "echos" from earlier ring oscillations.^[11] Moreover, a separate Italian team a year earlier posited that future gravity-wave detectors, such as the proposed Laser Interferometer Space Antenna (LISA), which is intended to have the ability to observe high-mass binary mergers at frequencies far below the limits of current observatories, would improve the ability to confirm aspects of fuzzball theory by orders of magnitude.^[12]

Notes

^ The smallest linear dimension in physics that has any meaning in the measurement of spacetime is the Planck length, which is 1.616255(18)×10⁻³⁵ m (CODATA value). Below the Planck length, the effects of quantum foam dominate and it is meaningless to conjecture about length at a finer scale—much like how meaningless it would be to measure ocean tides at a precision of one millimeter in storm-tossed seas. A singularity is thought to have a diameter that does not amount to even one Planck length; which is to say, zero.
^ This is a mean bulk density; as with neutron stars, the sun, and its planets, a fuzzball's density varies from the surface where it is less dense, to its center where it is most dense.
^ Smaller fuzzballs would be denser yet. The smallest black hole yet discovered, XTE J1650-500, is 3.8 ±0.5 M_☉. Theoretical physicists believe that the transition point separating neutron stars and black holes is 1.7 to 2.7 M_☉ (Goddard Space Flight Center: NASA Scientists Identify Smallest Known Black Hole). A very small, 2.7 M_☉ fuzzball would be over six times as dense as a median-size fuzzball of 6.8 M_☉, with a mean density of 2.53×10¹⁸ kg/m³. A bit of such a fuzzball the size of a drop of water would have a mass of 126 million metric tons, which is the mass of a granite ball 449 meters in diameter.
^
Neutron stars have a mean density thought to be in the range of 3.7–5.9×10¹⁷ kg/m³, which is equal to median-size fuzzballs ranging from 7.1 to 5.6 M_☉. However, the smallest fuzzballs are denser than neutron stars; a small, 2.7 M_☉ fuzzball would be four to seven times denser than a neutron star. On a "teaspoon" (≈4.929 mL) basis, which is a common measure for conveying density in the popular press to a general-interest readership, comparative mean densities are as follows:
- 2.7 M_☉ fuzzball: 12.45 billion metric tons per teaspoon
- 6.8 M_☉ fuzzball: 1.963 billion metric tons per teaspoon
- Neutron star: 1.8–2.9 billion metric tons per teaspoon.
^ Event horizons are located where the escape velocity equals the speed of light in vacuo for all observers stationary with respect to a black hole and outside its gravitational sphere of influence.
^
The warpage of space by mass is described in Einstein's second theory of relativity, later known as "general relativity," which includes the effects of accelerating frames of reference and gravity (another type of acceleration)—not his first theory of relativity (later known as "special relativity"). The theoretical physicist John A. Wheeler, who was largely responsible for reviving interest in general relativity in the United States after World War II, wrote the following oft-cited summarization of general relativity: “Matter tells spacetime how to curve, and curved spacetime tells matter how to move.”
A Minkowski spacetime diagram illustrating special relativity

How these two theories ("special" and "general") were related, described the laws of nature, and eventually got their names (which describe their scope, or meaning) was an evolving, multi-year process as Einstein endeavored to incorporate the effects of gravity into a unified theory that correctly predicted observations for all observers in all frames of reference and enabled Karl Schwarzschild to precisely calculate the radius of event horizons.
Having authored or coauthored nearly 500 scientific journal papers (an average of one paper every six weeks) and 16 books over his 54-year-long career, Einstein was a prolific writer (see List of scientific publications by Albert Einstein). In his 1905 paper, Zur Elektrodynamik bewegter Körper, published in a German scientific journal and later re-published in English as On the Electrodynamics of Moving Bodies (and what would later be known as "special relativity"), Einstein—as illustrated in the animation at right—established the following:
1. The laws of physics are identical in all non-accelerating frames of reference, and
2. The speed of light in a vacuum is the same for all observers irrespective of the relative motion between the light source and observer.
Note that Einstein’s famous formula regarding mass–energy equivalence, $E = mc 2$ , as Einstein began writing the equation in the 1920s and which entered popular culture at the start of the post-World War II Atomic Age, was neither part of his paper on special relativity nor general relativity; it was from a separate 1905 journal paper, Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (Does the Inertia of a Body Depend upon its Energy-Content?). In that paper, Einstein originally expressed the equivalency partly in prose by writing (when translated to English) " $mass$ $diminishes$ $by$ $L / V 2$ ", which used $L$ instead of $E$ as the symbol for energy and $V$ instead of $c$ as the symbol for the velocity of light, and which could be expressed entirely symbolically as $m = L / V 2$ and $L = mV 2$ .
Einstein's 1914 paper, Die Formale Grundlage der allgemeinen Relativitätstheorie (known as The Foundation of the Generalised Theory of Relativity) was the first to mention the term "General Theory" and refer to his previous theory as "Special Relativity theory." From the preamble of the paper:

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as "the theory of Relativity;" this latter theory I differentiate from the former "Special Relativity theory," and suppose it to be known.

In 1916, Einstein expanded upon general relativity and tied it together with special relativity in the German-language journal paper, Die Grundlage der allgemeinen Relativitätstheorie, (Relativity: The Special and the General Theory), which comprised 54 pages in the German-language physics journal, Annalen der Physik (Annals of Physics), Volume 354, Issue 7. A 2.4 MB downloadable and searchable German-language PDF is available here at Wiley Online Library.
Later, Einstein in collaboration with the British physicist Robert W. Lawson who translated Einstein's works, further expanded upon his 1916 journal paper and consolidated his theories into an English-language hard-cover book given the same title as the paper. Two versions—with different forewords by Lawson on the dust jackets—were published in 1920: 1) In the U.S., as a 182-page (168 numbered body pages) book titled Relativity: The Special and the General Theory, by Henry Holt and Company, New York; and 2) In England with a 138-page printing titled Relativity: The Special and the General Theory. A Popular Exposition, by Methuen & Co., Ltd, London.
In the book, Einstein explained the basis for referring to his first theory (On the Electrodynamics of Moving Bodies) as "special relativity"; it was valid only for a particular, or special, subset of reference frames (non-accelerating ones). What Einstein had been striving for was a unified theory applicable to all observers, regardless if they were in an inertial or accelerating frame of reference. Such a unified theory would, in Einstein's view, have the virtue of being compliant with an all-encompassing universal law of nature. The German adjective "allgemeinen," (in Die Grundlage der allgemeinen Relativitätstheorie, or Relativity: The Special and the General Theory) translates to "general" but has a subtly different meaning than in English technical writing, where it commonly connotes "broad but not necessarily specific." The word "allgemeinen" is a declension of the root adjective "allgemein" (a close pronunciation for English-only speakers is I’ll-guh-mine, where the syllable I’ll is pronounced like the contraction for "I will"), which has multiple context-sensitive connotations, one of which—especially in technical matters—means "universal." The following is from his 1920 book, Relativity: The Special and the General Theory:

The validity of the principle of relativity was assumed only for these reference-bodies, but not for others (e.g. those possessing motion of a different kind). In this sense we speak of the special principle of relativity, or special theory of relativity.
....
Or, in brief: General laws of nature are co-variant with respect to Lorentz transformations.
This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. If a general law of nature were to be found...

References

^ ^a ^b ^c ^d ^e ^f The Fuzzball Fix for a Black Hole Paradox, Jennifer Ouellette, Quanta Magazine, (June 23, 2015)
^ The fuzzball paradigm for black holes: FAQ, Samir D. Mathur, (January 22, 2009) (395 KB)
^ AdS/CFT duality and the black hole information paradox, SD Mathur and Oleg Lunin, Nuclear Physics B, 623, (2002), pp. 342–394 (arxiv); and Statistical interpretation of Bekenstein entropy for systems with a stretched horizon, SD Mathur and Oleg Lunin, Physical Review Letters, 88 (2002) (arxiv).
^ ^a ^b ^c ^d Vttoth.com: Hawking radiation calculator
^ Abraham Pais (September 23, 1982). Subtle is the Lord : The Science and the Life of Albert Einstein: The Science and the Life of Albert Einstein. Oxford University Press. ISBN 978-0-19-152402-8.
^ Steven Weinberg (April 20, 2011). Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature. Knopf Doubleday Publishing Group. ISBN 978-0-307-78786-6.
^ Overbye, Dennis (January 24, 2023). "Where is Physics Headed (and How Soon Do We Get There)? - Two leading scientists discuss the future of their field - Comment". The New York Times. Archived from the original on January 25, 2023. Retrieved January 28, 2023.
^ Why String Theory?, Joseph Conlon, CRC Press, (2016) ISBN-13: 978-1482242478
^ Philosophy of science for scientists, Lars-Göran Johansson, Springer–Cham, (2016), doi: 10.1007/978-3-319-26551-3
^ What is Hawking radiation?, Dr. Alastair Gunn, BBC Science Focus, (April 16, 2022)
^ A Way to Experimentally Test String Theory’s “Fuzzball” Prediction, APS Journals, (September 16, 2021)
^ Phenomenological Imprints of the String-Theory ‘Fuzzball’ Scenario, University of Rome–La Sapienza, (November 24, 2020)

External links

Are Black Holes Fuzzballs? — Space Today Online
The Fuzzball Fix for a Black Hole Paradox, June 23, 2015 — Quanta Magazine
Information paradox solved? If so, Black Holes are "Fuzzballs" — Ohio State University
The fuzzball paradigm for black holes: FAQ — Samir D. Mathur
ArXiv.org link: Unwinding of strings thrown into a fuzzball — Stefano Giusto and Samir D. Mathur
Astronomers take virtual plunge into black hole (84 MB) (10 MB version), a 40-second animation produced by JILA — a joint venture of the University of Colorado at Boulder and the NIST
Video lecture series at CERN (four parts approximately an hour each): “The black hole information problem and the fuzzball proposal,” Part 1, Part 2, Part 3, Part 4

[2] The smallest linear dimension in physics that has any meaning in the measurement of spacetime is the Planck length, which is 1.616255(18)×10⁻³⁵ m (CODATA value). Below the Planck length, the effects of quantum foam dominate and it is meaningless to conjecture about length at a finer scale—much like how meaningless it would be to measure ocean tides at a precision of one millimeter in storm-tossed seas. A singularity is thought to have a diameter that does not amount to even one Planck length; which is to say, zero.

[5] This is a mean bulk density; as with neutron stars, the sun, and its planets, a fuzzball's density varies from the surface where it is less dense, to its center where it is most dense.

[6] Smaller fuzzballs would be denser yet. The smallest black hole yet discovered, XTE J1650-500, is 3.8 ±0.5 M_☉. Theoretical physicists believe that the transition point separating neutron stars and black holes is 1.7 to 2.7 M_☉ (Goddard Space Flight Center: NASA Scientists Identify Smallest Known Black Hole). A very small, 2.7 M_☉ fuzzball would be over six times as dense as a median-size fuzzball of 6.8 M_☉, with a mean density of 2.53×10¹⁸ kg/m³. A bit of such a fuzzball the size of a drop of water would have a mass of 126 million metric tons, which is the mass of a granite ball 449 meters in diameter.

[7] Neutron stars have a mean density thought to be in the range of 3.7–5.9×10¹⁷ kg/m³, which is equal to median-size fuzzballs ranging from 7.1 to 5.6 M_☉. However, the smallest fuzzballs are denser than neutron stars; a small, 2.7 M_☉ fuzzball would be four to seven times denser than a neutron star. On a "teaspoon" (≈4.929 mL) basis, which is a common measure for conveying density in the popular press to a general-interest readership, comparative mean densities are as follows:
2.7 M_☉ fuzzball: 12.45 billion metric tons per teaspoon

6.8 M_☉ fuzzball: 1.963 billion metric tons per teaspoon

Neutron star: 1.8–2.9 billion metric tons per teaspoon.

[5] 2.7 M_☉ fuzzball: 12.45 billion metric tons per teaspoon

[6] 6.8 M_☉ fuzzball: 1.963 billion metric tons per teaspoon

[7] Neutron star: 1.8–2.9 billion metric tons per teaspoon.

[9] Event horizons are located where the escape velocity equals the speed of light in vacuo for all observers stationary with respect to a black hole and outside its gravitational sphere of influence.

[10] The warpage of space by mass is described in Einstein's second theory of relativity, later known as "general relativity," which includes the effects of accelerating frames of reference and gravity (another type of acceleration)—not his first theory of relativity (later known as "special relativity"). The theoretical physicist John A. Wheeler, who was largely responsible for reviving interest in general relativity in the United States after World War II, wrote the following oft-cited summarization of general relativity: “Matter tells spacetime how to curve, and curved spacetime tells matter how to move.”
A Minkowski spacetime diagram illustrating special relativity

How these two theories ("special" and "general") were related, described the laws of nature, and eventually got their names (which describe their scope, or meaning) was an evolving, multi-year process as Einstein endeavored to incorporate the effects of gravity into a unified theory that correctly predicted observations for all observers in all frames of reference and enabled Karl Schwarzschild to precisely calculate the radius of event horizons.
Having authored or coauthored nearly 500 scientific journal papers (an average of one paper every six weeks) and 16 books over his 54-year-long career, Einstein was a prolific writer (see List of scientific publications by Albert Einstein). In his 1905 paper, Zur Elektrodynamik bewegter Körper, published in a German scientific journal and later re-published in English as On the Electrodynamics of Moving Bodies (and what would later be known as "special relativity"), Einstein—as illustrated in the animation at right—established the following:

The laws of physics are identical in all non-accelerating frames of reference, and

The speed of light in a vacuum is the same for all observers irrespective of the relative motion between the light source and observer.

Note that Einstein’s famous formula regarding mass–energy equivalence, $E = mc 2$ , as Einstein began writing the equation in the 1920s and which entered popular culture at the start of the post-World War II Atomic Age, was neither part of his paper on special relativity nor general relativity; it was from a separate 1905 journal paper, Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (Does the Inertia of a Body Depend upon its Energy-Content?). In that paper, Einstein originally expressed the equivalency partly in prose by writing (when translated to English) " $mass$ $diminishes$ $by$ $L / V 2$ ", which used $L$ instead of $E$ as the symbol for energy and $V$ instead of $c$ as the symbol for the velocity of light, and which could be expressed entirely symbolically as $m = L / V 2$ and $L = mV 2$ .
Einstein's 1914 paper, Die Formale Grundlage der allgemeinen Relativitätstheorie (known as The Foundation of the Generalised Theory of Relativity) was the first to mention the term "General Theory" and refer to his previous theory as "Special Relativity theory." From the preamble of the paper:

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as "the theory of Relativity;" this latter theory I differentiate from the former "Special Relativity theory," and suppose it to be known.

In 1916, Einstein expanded upon general relativity and tied it together with special relativity in the German-language journal paper, Die Grundlage der allgemeinen Relativitätstheorie, (Relativity: The Special and the General Theory), which comprised 54 pages in the German-language physics journal, Annalen der Physik (Annals of Physics), Volume 354, Issue 7. A 2.4 MB downloadable and searchable German-language PDF is available here at Wiley Online Library.
Later, Einstein in collaboration with the British physicist Robert W. Lawson who translated Einstein's works, further expanded upon his 1916 journal paper and consolidated his theories into an English-language hard-cover book given the same title as the paper. Two versions—with different forewords by Lawson on the dust jackets—were published in 1920: 1) In the U.S., as a 182-page (168 numbered body pages) book titled Relativity: The Special and the General Theory, by Henry Holt and Company, New York; and 2) In England with a 138-page printing titled Relativity: The Special and the General Theory. A Popular Exposition, by Methuen & Co., Ltd, London.
In the book, Einstein explained the basis for referring to his first theory (On the Electrodynamics of Moving Bodies) as "special relativity"; it was valid only for a particular, or special, subset of reference frames (non-accelerating ones). What Einstein had been striving for was a unified theory applicable to all observers, regardless if they were in an inertial or accelerating frame of reference. Such a unified theory would, in Einstein's view, have the virtue of being compliant with an all-encompassing universal law of nature. The German adjective "allgemeinen," (in Die Grundlage der allgemeinen Relativitätstheorie, or Relativity: The Special and the General Theory) translates to "general" but has a subtly different meaning than in English technical writing, where it commonly connotes "broad but not necessarily specific." The word "allgemeinen" is a declension of the root adjective "allgemein" (a close pronunciation for English-only speakers is I’ll-guh-mine, where the syllable I’ll is pronounced like the contraction for "I will"), which has multiple context-sensitive connotations, one of which—especially in technical matters—means "universal." The following is from his 1920 book, Relativity: The Special and the General Theory:

The validity of the principle of relativity was assumed only for these reference-bodies, but not for others (e.g. those possessing motion of a different kind). In this sense we speak of the special principle of relativity, or special theory of relativity.
....
Or, in brief: General laws of nature are co-variant with respect to Lorentz transformations.
This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. If a general law of nature were to be found...

[10] The laws of physics are identical in all non-accelerating frames of reference, and

[11] The speed of light in a vacuum is the same for all observers irrespective of the relative motion between the light source and observer.

[12] The laws of physics are identical in all non-accelerating frames of reference, and

[13] The speed of light in a vacuum is the same for all observers irrespective of the relative motion between the light source and observer.

[Ouellette-1] ^ ^a ^b ^c ^d ^e ^f The Fuzzball Fix for a Black Hole Paradox, Jennifer Ouellette, Quanta Magazine, (June 23, 2015)

[3] The fuzzball paradigm for black holes: FAQ, Samir D. Mathur, (January 22, 2009) (395 KB)

[4] AdS/CFT duality and the black hole information paradox, SD Mathur and Oleg Lunin, Nuclear Physics B, 623, (2002), pp. 342–394 (arxiv); and Statistical interpretation of Bekenstein entropy for systems with a stretched horizon, SD Mathur and Oleg Lunin, Physical Review Letters, 88 (2002) (arxiv).

[Vttoth-8] Vttoth.com: Hawking radiation calculator

[Pais1982-11] Abraham Pais (September 23, 1982). Subtle is the Lord : The Science and the Life of Albert Einstein: The Science and the Life of Albert Einstein. Oxford University Press. ISBN 978-0-19-152402-8.

[Weinberg2011-12] Steven Weinberg (April 20, 2011). Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature. Knopf Doubleday Publishing Group. ISBN 978-0-307-78786-6.

[NYT-20230124-13] Overbye, Dennis (January 24, 2023). "Where is Physics Headed (and How Soon Do We Get There)? - Two leading scientists discuss the future of their field - Comment". The New York Times. Archived from the original on January 25, 2023. Retrieved January 28, 2023.

[14] Why String Theory?, Joseph Conlon, CRC Press, (2016) ISBN-13: 978-1482242478

[15] Philosophy of science for scientists, Lars-Göran Johansson, Springer–Cham, (2016), doi: 10.1007/978-3-319-26551-3

[16] What is Hawking radiation?, Dr. Alastair Gunn, BBC Science Focus, (April 16, 2022)

[17] A Way to Experimentally Test String Theory’s “Fuzzball” Prediction, APS Journals, (September 16, 2021)

[18] Phenomenological Imprints of the String-Theory ‘Fuzzball’ Scenario, University of Rome–La Sapienza, (November 24, 2020)

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 ==Testability of the theory==
+<!--
-As there is no direct experimental evidence supporting string theory or fuzzball theory, both are currently products purely of calculations and theoretical research.<ref>''Why String Theory?'', Joseph Conlon, CRC Press, (2016) ISBN-13: 978-1482242478</ref> However, theories must be [[Testability|experimentally testable]] if there is to be a possibility of ascertaining their validity.<ref>''Philosophy of science for scientists'', Lars-Göran Johansson, Springer–Cham, (2016), doi: [https://link.springer.com/book/10.1007/978-3-319-26551-3 10.1007/978-3-319-26551-3]</ref> To be in full accordance with the [[scientific method]]—and one day be widely accepted as true, as are Einstein's theories of special and general relativity—theories regarding the natural world  must make predictions that are consistently affirmed through observations of nature.[[File:Cosmic Microwave Background (CMB).jpeg|400px|thumb|right|In addition to enumerable other electromagnetic-emitting sources in the universe, even the all-pervasive cosmic microwave background, shown here, vastly overwhelms Hawking radiation.]]
+EDITORS NOTE FOR Wikipedians (I link, therefore I am): Please don’t over-link this section (and the rest of the article). Links should well-anticipate what a given readership would be interested in learning more about. Readers who are still engaged at this late point in this article, which regards an advanced scientific subject, will invariably be familiar with basic concepts like "power," "watts," "milliwatts," and "wavelength." Similarly, please don’t replace spelled-out units of measure like "watt," with unit symbols (W)—particularly on first occurrences. Many wikipedians feel that using nothing but unit symbols makes articles somehow appear professional, but Wikipedia is a *general interest* encyclopedia, not a paper in a scientific journal. Prose that is “sciency” for the sake of simply looking so or that calls attention to itself by following cultural fads (…“an astronaut falling into a black hole would notice that she"…) begs to be rephrased.
+END OF NOTE
+-->As there is no direct experimental evidence supporting string theory or fuzzball theory, both are currently products purely of calculations and theoretical research.<ref>''Why String Theory?'', Joseph Conlon, CRC Press, (2016) ISBN-13: 978-1482242478</ref> However, theories must be [[Testability|experimentally testable]] if there is to be a possibility of ascertaining their validity.<ref>''Philosophy of science for scientists'', Lars-Göran Johansson, Springer–Cham, (2016), doi: [https://link.springer.com/book/10.1007/978-3-319-26551-3 10.1007/978-3-319-26551-3]</ref> To be in full accordance with the [[scientific method]]—and one day be widely accepted as true, as are Einstein's theories of special and general relativity—theories regarding the natural world  must make predictions that are consistently affirmed through observations of nature.[[File:Cosmic Microwave Background (CMB).jpeg|400px|thumb|right|In addition to enumerable other electromagnetic-emitting sources in the universe, even the all-pervasive cosmic microwave background, shown here, vastly overwhelms Hawking radiation.]]
 Fuzzball theory predicts that quantum information is preserved in fuzzballs and that Hawking radiation originating within the Planck-scale quantum foam just above a fuzzball's surface would theoretically be encoded with that information. As a practical matter, Hawking radiation is virtually impossible to detect because the individual photons comprising Hawking radiation have extraordinarily little energy. Moreover, black holes radiate at astronomically low power levels. This underlies why black holes require so long to evaporate: {{val|e=67}} times the age of the universe for a {{Solar mass|4.9}} black hole.<ref name="Vttoth"/> Hawking showed that the energy released by Hawking radiation is inversely proportional to the mass of a black hole; ergo, the smallest black holes emit the most energetic Hawking radiation that is least difficult to detect. However, the radiation emitted from even a minimum-size, {{Solar mass|2.7}} fuzzball does so at wavelengths equivalent to a [[black body]] temperature of around 23 billionths of one [[kelvin]] above [[absolute zero]], and with a radiated power—for the entire black hole—of {{val|1.2|e=-29|u=watt}} ({{nowrap|12 billion-}}billion-billionths of one milliwatt).<ref name="Vttoth"/> Such a minuscule signal would be vastly overwhelmed by even the 2.73&nbsp;K blackbody temperature of the [[cosmic microwave background]] coming from all directions of space.<ref>[https://www.sciencefocus.com/space/what-is-hawking-radiation ''What is Hawking radiation?''], Dr. Alastair Gunn, BBC Science Focus, (April 16, 2022)</ref>

v t e Black holes
Outline
Types	BTZ black hole Schwarzschild Rotating Charged Virtual Kugelblitz Supermassive Primordial Direct collapse Rogue Malament–Hogarth spacetime
Size	Micro Extremal Electron Stellar Microquasar Intermediate-mass Supermassive Active galactic nucleus Quasar LQG Blazar OVV
Formation	Stellar evolution Gravitational collapse Neutron star Related links Tolman–Oppenheimer–Volkoff limit White dwarf Related links Supernova Micronova Hypernova Related links Gamma-ray burst Binary black hole Quark star Supermassive star Quasi-star Supermassive dark star X-ray binary
Properties	Astrophysical jet Gravitational singularity Ring singularity Theorems Event horizon Photon sphere Innermost stable circular orbit Ergosphere Penrose process Blandford–Znajek process Accretion disk Hawking radiation Gravitational lens Microlens Bondi accretion M–sigma relation Quasi-periodic oscillation Thermodynamics Bekenstein bound Bousso's holographic bound Immirzi parameter Schwarzschild radius Spaghettification
Issues	Black hole complementarity Information paradox Cosmic censorship ER = EPR Final parsec problem Firewall (physics) Holographic principle No-hair theorem
Metrics	Schwarzschild (Derivation) Kerr Reissner–Nordström Kerr–Newman Hayward
Alternatives	Nonsingular black hole models Black star Dark star Dark-energy star Gravastar Magnetospheric eternally collapsing object Planck star Q star Fuzzball Geon
Analogs	Optical black hole Sonic black hole
Lists	Black holes Most massive Nearest Quasars Microquasars
Related	Outline of black holes Black Hole Initiative Black hole starship Black holes in fiction Big Bang Big Bounce Compact star Exotic star Quark star Preon star Gravitational waves Gamma-ray burst progenitors Gravity well Hypercompact stellar system Membrane paradigm Naked singularity Population III star Supermassive star Quasi-star Supermassive dark star Rossi X-ray Timing Explorer Superluminal motion Timeline of black hole physics White hole Wormhole Tidal disruption event Planet Nine
Notable	Cygnus X-1 XTE J1650-500 XTE J1118+480 A0620-00 Sagittarius A* Centaurus A Phoenix Cluster PKS 1302-102 OJ 287 SDSS J0849+1114 TON 618 MS 0735.6+7421 NeVe 1 Hercules A 3C 273 Q0906+6930 Markarian 501 ULAS J1342+0928 PSO J030947.49+271757.31 AT2018hyz Swift J1644+57 1ES 1927+654
Category Commons