Fuzzball (string theory)

Fuzzball theory, which is derived from superstring theory, is advanced by its proponents as a description of black holes that harmonizes quantum mechanics and Albert Einstein's general theory of relativity, which have long been incompatible. Fuzzball theory dispenses with the singularity at the heart of a black hole 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 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 (see Fig. 2, below).[1]

Fuzzball theory addresses 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 space and time—spacetime—of the rest of the universe (see Fig. 1 ). 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 the quantum information describing the light and matter that falls into a classic black hole is thought to either be extinguished at singularities, or is somehow preserved within singularities but the quantum information cannot climb up against the infinite gravitational intensity inside a black hole to reach past the event horizon so it is visible to regular spacetime. Either situation violates a fundamental law of quantum mechanics requiring that quantum information be conserved.[1][2]

Physical properties

Figure 1 Shown over the Hawaiian island of Oahu is a cross-section of an unremarkable 6.8-solar-mass, 40-kilometer-diameter (25 miles) classic black hole (albeit non-spinning and perfectly spherical for simplicity). It comprises a singularity, an event horizon, and a void between them, which is cut off from spacetime. Fuzzball theory posits that black holes are balls of the ultimate form of degenerate matter with a physical surface located precisely at the event horizon.

Structure and composition

Samir D. Mathur of The 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 Fig. 1 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 (see Fig. 2 ). Quite unlike the view of a black hole as a singularity, a small fuzzball can be thought of as an extra-dense neutron star in which its neutrons have undergone a phase transition and decomposed, liberating the quarks (strings in string theory) comprising them. Accordingly, fuzzballs are theorized to be the terminal phase of degenerate matter.

Mathur and Lunin calculated that the physical surface of fuzzballs have radii equal to that of the event horizon of classic black holes; thus, the Schwarzschild radius of a ubiquitous 6.8 solar masses (M_☉) stellar-mass-class black hole—or fuzzball—is 20 kilometers when the effects of spin are excluded. The team also determined 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".

Figure 2 Fuzzball theory posits that in addition to various vibration modes along their lengths, strings have degrees of freedom in additional dimensions (beyond the three dimensions of space) that are "compactified", such as an S¹ mode wrapping helically along strings' lengths, like this toy Slinky.

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.

Figure 3 Fuzzball theory posits that black holes are not voids with infinite-density singularities at their centers but are instead extended objects. A single water drop-size sample from a non-spinning 6.8 M_☉ fuzzball (the size shown in Fig. 1 ) would, on average, have a mass of 20 million metric tons, which is equivalent to this 243-meter-diameter granite ball spanning about four city blocks in Lower Manhattan, New York.

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 composition 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.

Densities

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 precisely equal Schwarzschild radii, which Karl Schwarzschild calculated using an entirely different mathematical technique 87 years earlier.[4]

Since the volume of fuzzballs is a function of the Schwarzschild radius (2953 meters 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. 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. A bit of such a non-spinning fuzzball the size of a drop of water would, on average, have a mass of twenty million metric tons, which is equivalent to that of a granite ball 243 meters in diameter (Fig. 3 ).[Note 2]

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—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.[Note 3]

Figure 4 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 a 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).[5]

Escape velocity

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 the threshold at which the escape velocity equals the speed of light: 299,792,458 meters per second.[Note 4] 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, as shown in Fig. 4 at right, has been curved by gravity to the speed of light in accordance with general relativity.[Note 5]

Gravitational acceleration

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 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.[5] This is general relativity's "accelerating frame of reference" counterpart to special relativity's requirement that infinite energy is required to accelerate an object possessing mass—even a subatomic particle—to precisely the speed of light.

This property of infinite gravitational acceleration (infinite gravitational strength) at event horizons merits further scrutiny because at least as recently as 2023, online popular culture sites such as physics discussion boards, science websites, and even a university physics professor on YouTube writing calculations on a blackboard were promulgating a misunderstanding that objects have non-infinite weights at event horizons. The root cause of some of this misunderstanding was the improper application of Isaac Newton's 336-year-old formula for the law of universal gravitation (upper equation, below) rather than a proper appreciation of the ramifications of Einstein's theory of general relativity and how extreme gravity affects spacetime. Such a mistake is born of a logical non sequitur that while general relativity explains the existence of an event horizon around a black hole, that event horizon somehow remains part of regular un-warped spacetime where Newton's law of universal gravitation applies; it does not. In accordance with general relativity (lower equation, below), event horizons exist because their escape velocity equals the speed of light and gravitational acceleration is infinite, completely cutting them off from spacetime; no further calculations are warranted.

Newton's law of universal gravitation: $F=G{\frac {m_{1}m_{2}}{r^{2}}}$
Schwarzschild radius: $R={\frac {2GM}{c^{2}}}$

Newton's law of universal gravitation yields increasingly inaccurate results as both space and time (spacetime) are increasingly warped by large masses. Even in the mildly gravitationally warped spacetime surrounding Earth, general relativity's gravitational effect on GPS satellites makes their onboard atomic clocks run 45,685 nanoseconds per day (0.01669 second per year) faster when in orbit versus their Earth-centered reference location, 26,562 kilometers below.[6] To make GPS timing signals run at the slower center-of-Earth rate while in orbit, the satellites' reference oscillators receive a "factory offset" before launch, which also compensates for a smaller opposing effect of special relativity due to orbital velocity. At the other extreme, the improper use of Newton's formula to calculate the 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_☉, yields a wildly incorrect (and even survivable) gravitational acceleration of only about 15 times that of Earth's gravity.[5] Regardless of the size of a black hole, from the perspective of an observer outside a black hole's gravitational influence, the escape velocity at event horizons and the surface of fuzzballs equals the speed of light, gravitational strength is infinite, and the flow of time has come to a halt.

Gravitational tides

The aforementioned phenomenon of infinite gravitational acceleration at event horizons is distinct from gradients known as gravitational tides. The intense gravitational tides of stellar-mass black holes, intermediate-mass black holes, and smaller supermassive black holes cause 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). For instance, a 10 M_☉ stellar-class fuzzball has a gravitational tide at its surface of 100 billion Earth-gravities per meter, which would stretch an infalling astronaut into a stream of paste well before reaching its surface.[5] Even a relatively small 400,000 M_☉ supermassive fuzzball, which has a gravitational tide of 64 Earth-gravities per meter at its surface, would pull apart the body of a hapless astronaut falling feet-first before reaching its surface due to vertebral tensile forces greater than the weight of one metric ton on Earth.[5]

Information paradox

Figure 5 When charcoal briquettes burn, they react with oxygen from their surroundings to produce gasses, smoke particles, photons, and kinetic heat energy. Burning profoundly scrambles the quantum information comprising this briquettes / surroundings system, making its wave function altogether different. Although perturbations to wave functions are time reversible in a very narrow theoretical sense (by imagining one could acquire perfect knowledge of every quantum perturbation without disturbing them and subtract those changes in reverse order), it is impossible in the full theoretical sense to deduce the original nature of the briquettes / surroundings system. Importantly, although burning irrevocably scrambles the quantum information, none is destroyed, abiding by a law of quantum mechanics requiring it be conserved.

The conservation of quantum information is analogous to taking a digital MPEG movie file saved with lossless compression (a proxy for a perfectly isolated system saturated with random quantum information), and then encrypting it without knowing the cipher key (a proxy for the impossibility of reconstructing an original wave function, which is in accordance with the Born rule requiring that random quantum activity in nature cannot be perfectly known because measurements of it are probabilistic, as well as because the very act of making such measurements affects the outcomes): Though the encrypted file is irreversibly scrambled, the sizes of the two files are the same and so too their information content.

Classical black holes create a problem for physics known as the black hole information paradox; there is no such paradox under fuzzball theory. The paradox was first raised in 1972 by Jacob Bekenstein and later popularized by Stephen Hawking. The information paradox is born of a requirement of quantum mechanics that quantum information must be conserved, which is in conflict with the realization that if black holes have singularities at their centers, quantum information must be extinguished from spacetime. This paradox can be viewed as a contradiction between different parts of a larger theory. Fuzzball theory purports to resolve this tension.

A black hole that fed primarily on the stellar atmosphere (protons, neutrons, and electrons) of a nearby companion star should, if it obeyed the known laws of quantum mechanics, grow to have a quantum composition different from another black hole that fed only on light (photons) from neighboring stars and the cosmic microwave background. 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, no difference in their quantum compositions would theoretically exist. This is because if singularities have zero volume, black holes have no quantum composition. Moreover, even if quantum information was not extinguished in the singularities of classic black holes, it would still be unable to climb against infinite gravitational intensity and reach up to and beyond the event horizon where it could reveal itself in normal spacetime. This situation violates a law of quantum mechanics requiring that quantum information be conserved.

Hawking radiation (undetectable radiation comprising photons and possibly other quanta thought to be emitted from the proximity of black holes) would not circumvent the information paradox for classic black holes with singularities at their centers; it could reveal only the mass, angular momentum, and electric charge of classic black holes. Fuzzball theory however, resolves the paradox by predicting that quantum information regarding what falls onto a fuzzball is not only conserved, but is subtly imprinted in escaping Hawking radiation so it is available to observers in regular spacetime.

Hawking radiation is created whenever massless and truly neutral virtual particle pairs—virtual photons for modern expositions of this topic—form in proximity to, but outside of, an event horizon.[Note 6] One member of a virtual particle pair possesses negative mass-energy (in the absolute $E=mc^{2}$ sense), the other has positive mass-energy, and the average pair's net energy is zero.[7][Note 7] The virtual photon possessing negative energy is captured; it travels down through the event horizon via quantum tunneling whereupon it becomes part of the black hole (robbing it of energy and an equivalent amount of mass). Meanwhile, the pair member with positive energy is ejected, carrying away its share of energy from the black hole as blackbody thermal emissions; this is Hawking radiation wherein the ejected photons are no longer virtual and are real.

It is important to bear in mind that the above description of the origin of Hawking radiation is highly simplified. Even though Hawking's scientific paper, Particle Creation by Black Holes, was directed to theoretical physicists and delved into arcane phenomena like Killing vector fields, Hawking cautioned that his descriptions of the mechanism responsible for black hole thermal emission "are heuristic only and should not be taken too literally." Note that heuristic teaching means "a teaching method where students learn on their own through discovery and problem-solving in lieu of pure instruction," however, in theoretical physics the verb heuristic can connote "treated in a simpler manner than it really is" whereas the compound noun heuristic approach tends to mean "a simpler or more intuitive way to examine or explain a phenomenon."[8] Nonetheless, Hawking's advisement to his peers to not take his explanations too literally bears witness to the complexities underlying Hawking radiation. His advisement also underscores his remarkable achievement of producing a mathematical formula that relates photon emissions from black holes of any given mass to a blackbody temperature. Within that formula, Hawking linked thermodynamics to a variety of disparate disciplines in physics: quantum mechanics, relativity, Newtonian mechanics, and gravitation, as shown below.

Hawking's formula for calculating the blackbody temperature of black hole thermal emission.

The amount of Hawking radiation emitted by black holes, or their luminosity, is inversely proportional to the square of their mass. Such calculations assume that Hawking radiation comprises only photons; that assumption is used throughout this and related articles on Wikipedia. That equation is as follows: [5]

L={\frac {\delta E}{\delta t}}={\frac {1}{M^{2}}}{\frac {\hbar c^{6}}{15360\pi G^{2}}}

The term $L$ (luminosity) represents power in watts (an exceedingly small portion of a watt for Hawking radiation), which can be converted to other measures such as mass loss rates. Details on the formula's other terms are beyond the scope of this article and are covered at Bekenstein–Hawking formula. The formula's name honors Jacob Bekenstein (1947–2015), who laid down important foundations to black hole theory that predated Hawking's contributions by several years.

In a purely theoretical sense, the fuzzball theory advanced by Mathur and Lunin satisfies the requirement that quantum information be conserved because it holds, in part, that the quantum information of the strings that fall onto a fuzzball is preserved as those strings dissolve into and contribute to the fuzzball's quantum makeup. The theory further holds that a fuzzball's quantum information is not only expressed at its surface, but tunnels up through the tunneling fuzziness of the event horizon where it can be imprinted on Hawking radiation, which very slowly carries that information into regular spacetime in the form of delicate correlations in 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 explained all aspects of the universe.[9][10] 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][11]

Testability of the theory

As no direct experimental evidence supports string theory or fuzzball theory, both are products purely of calculations and theoretical research.[12] However, theories must be experimentally testable if there is to be a possibility of ascertaining their validity.[13] 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 cannot be substantiated by observing its predicted subtle effects on Hawking radiation because the radiation itself is, for all practical purposes, undetectable.[14] However, Fuzzball theory may be testable through gravitational-wave astronomy.[15]

Figure 6 This is a false-color view of the supermassive black hole M87* (pronounced "Em eighty-seven star") at the center of the galaxy Messier 87. This image was captured using 230 GHz (1.3 mm) microwaves. At around 6.5×10⁹ M_☉, M87* emits invisible Hawking radiation with a peak-emission wavelength 43 times larger than the orbital diameter of Neptune.

Fuzzball theory resolves a long-standing conflict between general relativity and quantum mechanics by holding that quantum information is preserved in fuzzballs and that Hawking radiation originating within the Planck-scale quantum foam just above a fuzzball's surface is subtly encoded with that information. As a practical matter however, Hawking radiation is virtually impossible to detect because black holes emit it at astronomically low power levels and the individual photons comprising Hawking radiation have extraordinarily little energy.[14] This underlies why theoretically perfectly quiescent black holes (ones in a universe containing no matter or other types of electromagnetic radiation to absorb) evaporate so slowly as they lose energy (and equivalent amounts of mass) via Hawking radiation; even a modest 4.9 M_☉ black hole would require 10⁵⁹ times the current age of the Universe to vanish. Moreover, a top-of-the-list 106 billion M_☉ supermassive black hole would require ten-million-trillion-trillion times longer still to evaporate: 10⁹⁰ times the age of the Universe.[5]

Hawking showed that the energy of photons released by Hawking radiation is inversely proportional to the mass of a black hole, consequently, the smallest black holes emit the most energetic photons that are the least difficult to detect. However, the radiation emitted by even a minimum-size, 2.7 M_☉ black hole (or fuzzball) comprises extremely low-energy photons that are equivalent to those emitted by a black body with a temperature of around 23 billionths of one kelvin above absolute zero. More challenging still, such a black hole has a radiated power—for the entire black hole—of 1.2×10⁻²⁹ watt (12 billion-billion-billionths of one milliwatt).[5] Such an infinitesimal transmitted power 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.

Critically though, when signals are this weak, the challenge is no longer one of classic radio astronomy technological issues like gain and signal-to-noise ratio; Hawking radiation comprises individual photon quanta, so such a weak signal means a 2.7 M_☉ black hole is emitting at most only ten photons per second.[Note 8] Even if such a black hole was only 100 lightyears away, the odds of just one of its Hawking radiation photons landing anywhere on Earth—let alone being captured by an antenna—while a human is watching are astronomically improbable.[Note 9] Importantly, the above values are for the smallest possible stellar-mass black holes; far more difficult yet to detect is the Hawking radiation emitted by supermassive black holes at the center of galaxies. For instance, M87* (Fig. 6), which is an unremarkable supermassive black hole, emits Hawking radiation at a near-nonexistent radiant power of at most 13 photons per century, and does so with a wavelength so great that a receiving antenna possessing even a modest degree of absorption efficiency would be larger than the Solar System.[5]

Fuzzball theory may though, be readily testable though gravitational-wave astronomy. 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 gravity-wave observatories are capable of discerning fuzzball-theory-supporting evidence in the signals from merging binary black holes (and the 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 that would also be dominated by "echos" from earlier ring oscillations.[15] 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.[16]

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 predict—one minute into the future—the location of a buoy floating in storm-tossed seas to a precision of one millimeter. If singularities exist, and if they have non-zero diameters with a density equal to the Planck density (5.155×10⁹⁶ kg/m³), then even a minimal 2.7 M_☉ singularity would have a definite non-zero volume with a diameter of 7.8×10¹² Planck lengths (1.26×10⁻²² m), which may seem large but is still far smaller than an electron and is even a thousand times smaller than the minimum dimension that can be probed with a world-class 10 TeV particle accelerator (10⁻¹⁹ m). Alternatively, if a minimal 2.7 M_☉ singularity has a quantum-limited size of one Planck volume, then it has a "fuzzy" density averaging 1.27×10¹³⁵ kg/m³, which far exceeds the Planck density. Lastly, if singularities have truly infinite density, they necessarily have literally zero volume; which is to say, not even one Planck length in diameter. This inability to describe the exact nature of singularities speaks to the dilemma of physics theories wherever their mathematical formulas have a zero being used as a divisor and the known laws of physics have been declared to have "broken down"; it is often an indicator that a theory is incomplete.
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: 12.46 billion metric tons per teaspoon
- 6.8 M_☉ fuzzball: 1.965 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 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 = mc², 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), "If a body gives off the energy L in the form of radiation, its mass diminishes by L/V²." Note Einstein's early use of L instead of $E$ as the symbol for energy and V instead of $c$ as the symbol for the velocity of light, which could be expressed entirely symbolically as m = L/V² and L = mV². 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 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 in German, 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...
In his 1975 paper, Hawking wrote that the radiation emitted by black holes, now called "Hawking radiation," would comprise only of massless virtual particles. He specified the composition as photons, gravitons, neutrinos, and other "massless fermions." His paper was before the discovery that neutrinos actually have mass.
The meanings of "negative mass-energy” or simply "negative energy" (in the absolute $E=mc^{2}$ sense) in discussions of virtual photons at black holes, differs somewhat from what "negative energy" normally means for virtual photons in the lab (in regular spacetime). Virtual photons are oscillations in the background electromagnetic field that prevent an otherwise pure vacuum from containing (possessing) zero energy. Virtual photons are characterized by their wavelength (frequency, or "color"), momentum (which, unlike real photons, is exceedingly variable), and polarization (spin-angular momentum).
Unfortunately, terms like "negative energy," "antiparticle," and "antimatter" can add confusion to a topic that has long fallen victim to popular misunderstanding. Moreover, the issue of whether photons are best described as "waves" or "particles" can needlessly belabor a simple and accessible exposition on Hawking radiation if not formally addressed. Accordingly, a short treatise on the broad subject is required to establish context for how Hawking radiation can be viewed as "arising from virtual photons possessing negative energy tunneling through an event horizon."
The "wave–particle duality" adds complexity to a topic that is already challenging to understand. This duality is commonly encountered when photons are referred to as "quantized wave packets propagating in the electromagnetic field," rather than simply "particles." Referring to photons as "particles" more accurately and conveniently describes the nature of photons after they hit a light detector in a double-slit experiment; it is needlessly ponderous to say "the photon's wave function then collapses to a point." Especially in a treatise on Hawking radiation, where Hawking himself wrote, "there will be pairs of particles, one with negative energy and one with positive energy," it is very useful here to use particle-based vernacular. Nonetheless, the wave nature of virtual photons must be addressed to properly cover Hawking radiation.
Though one member of a pair of virtual photons can possesses negative energy, this is neither the product of charge conjugation (the reversing of electric charge as permitted by "C-symmetry"), nor is it antimatter because, by definition, antimatter is "matter (which has mass) possessing an electric charge opposite that of ordinary matter." Though photons are considered to be their own antiparticle (which is a broad family that confusingly includes antimatter like antiprotons), photons are more specifically a truly neutral particle / antiparticle. Furthermore, real photons must always possess energy equal to the speed of light times their momentum vector and must have zero rest mass.
In laboratories, virtual photons possess different kinds of momentum and interact with matter and its accompanying electromagnetic fields in different ways. Virtual photons exist everywhere and their effects are observable as the Lamb shift as they interact with the electromagnetic field of electrons surrounding atoms. This activity also underlies zero-point energy, which jostles matter to such an extent it prevents helium at near-absolute zero from freezing at room pressure. Virtual photons are also responsible for the Casimir effect, which squeezes two closely spaced plates together. Virtual photons can also be polarized; this is to say, they have the quantum property of spin-angular momentum, which can couple to the angular momentum of charged particles.
In simple terms, all three of the above effects: the Lamb shift, the Casimir force, and the inability of helium to freeze at room pressure due to zero-point energy, arise from the collective activity of virtual photons. More precisely—and ponderously—these three effects are the result of oscillations in the quantum electrodynamic field, resulting in a non-zero QED vacuum (or simply vacuum energy). The QED vacuum is the lowest energy state of the all-pervasive electromagnetic field permeating the Universe; real photons are traveling excitations in this electromagnetic field. Note that the quantum electrodynamic field and the electromagnetic field are essentially the same thing except that the former is the quantum-based view of electromagnetism that accounts for a non-zero vacuum energy. Note also that in discussions of Hawking radiation, the term "zero-point energy" is interchangeable in practice with "vacuum energy" but the former is broader and encompasses other zero-point fields, including the quantum chromodynamic vacuum (QCD), which governs interactions at the quark level.
This is a wave-based view showing the momentum disturbance of a single oscillation in the quantum electrodynamic (QED) field, also known as "a pair of virtual photons" in a particle-based view. These are the source of the photons comprising Hawking radiation.
Here, the 3D QED field that permeates the Universe is projected onto a 2D plane and the vertical axis represents the vector momentum, $p$ in an absolute (relativistic) sense. The upper and lower bulges represent virtual photons possessing positive and negative relativistic energy, respectively. This oscillation has no spin-angular momentum (polarization), which would appear as helical twists. The diameter of the oscillation is its wavelength, ${\lambda }$ (lambda), which cannot factor into the energy of oscillations possessing zero net momentum.
A single oscillation in the QED field (the quantum version of the electromagnetic field) comprises momentum components that are on average—but by no means always—equal and opposite. Consequently, the average net relativistic mass-energy of these oscillations is zero. QED oscillations last one full cycle of their wavelength, so one with a wavelength of 600 nanometers (orange for real photons) exists for only about 2 femtoseconds.
Zero-point energy comprises oscillations in all types of quantum fields and is the subject of ongoing research in theoretical physics. In part, zero-point energy arises from the Heisenberg uncertainty principle's effect on QED vacuum energy—which allows for non-symmetrical virtual photon momenta (asymmetric bulges). Thus, some oscillations in the QED field possess non-zero net momentum and non-zero net relativistic mass-energy before quickly vanishing. While momentum asymmetries contribute to vacuum energy, they are not required to produce Hawking radiation.
The relativistic mass-energy of real photons (their absolute $E=mc^{2}$ energy that, as Hawking wrote, is "relative to infinity") is proportional to their momentum vector times the speed of light per $E = p c$ , where…
$E$ is energy,

$p$ is the magnitude of the momentum vector, and

$c$ is the speed of light.
Individual virtual photons are different from real ones; they may carry any momentum, or relativistic mass-energy, permitted by the Heisenberg uncertainty principle. Thus, any given pair of virtual photons may possess opposite and unequal momenta. However, across a large population of virtual photon pairs, their net momentum averages to zero and so too does their rest mass-energy and relativistic energy.
With regard to measurements in the lab of virtual photon momentum, the labels "positive energy" and "negative energy" are relative classifications established by the direction of their momentum vector, p (and accompanying energy) in relation to an external electromagnetic field (from one or more nearby charged particles). This convention comes from the behavior of real photons, which possess positive energy with respect to electrons; this underlies spectral lines where the electrons surrounding atoms transition from a lower-energy atomic orbital to a higher-energy one after absorbing photons. When an individual virtual photon is exchanged between two particles with like charges (followed soon after by its partner), it is considered to have positive energy when its momentum adds energy to the electromagnetic force between them and they more vigorously repel each other. Note that a virtual photon with a positive-energy momentum direction when it is exchanged between two electrons would be classified as possessing a negative-energy momentum if the exchange was between positrons.
In the context of Hawking radiation however, the labels "positive energy" and "negative energy" for virtual photons are in an absolute sense, or "relative to infinity," as Hawking wrote. Virtual photons with negative mass-energies in an absolute sense are generally considered as not physically real. This is because a virtual photon possessing positive momentum and positive mass-energy behaves just like easy-to-study real photons whereas a virtual photon possessing negative momentum and negative mass-energy cannot be isolated in the lab and its distinctive property studied. This inability to isolate a virtual photon possessing negative energy is to be expected since in the wave-based view, it is actually an integral part of an individual oscillation in the quantum electrodynamic field (electromagnetic field) permeating the Universe (see image at right) that humans cannot bifurcate.
Black holes have the unique ability to do what cannot be done in the lab: separate virtual photon pairs. This is in part due to the extremely small radius of curvature of spacetime near their event horizons. From the perspective of an outside observer viewing wave-based phenomena, black holes can shear, stretch, and bifurcate the components of a QED oscillation possessing opposite momenta. Hawking radiation arises when the portion of the oscillation possessing negative momentum and negative relativistic mass-energy (the lower bulge of the QED oscillation at right) tunnels through to a black hole. This liberates the positive-energy half of the QED oscillation as a real photon, which becomes exceedingly gravitationally redshifted as it climbs up the extreme gravity well surrounding the black hole and escapes to infinity. The peak-emission wavelength of fully redshifted photons is about ten times the diameter of the event horizon surrounding a non-spinning black hole regardless of its mass; a 6.77 M_☉ non-spinning black hole with an event horizon diameter of 40.0 kilometers emits Hawking radiation where the most common photons have a wavelength of 403 kilometers.
Finally, it is important to bear in mind that gravity and spacetime are so agitated near a black hole that even the mathematics describing Hawking radiation can be viewed in different ways. For instance, instead of negative-energy particles tunneling through the horizon in forward-directed time, they can be thought of, as Hawking wrote in his paper, "as positive-energy particles crossing the horizon on past-directed world-lines." As mentioned earlier, Hawking cautioned that, "It should be emphasized that the mechanism responsible for the thermal emission and area decrease are heuristic only and should not be taken too literally." Even the basic premiss that Hawking radiation is the product of a stationary person observing a highly accelerating region of spacetime has its general relativity inverse known as the Unruh effect, which predicts that thermal radiation surrounds an accelerating observer. A helpful YouTube video, "Hawking radiation", by ScienceClic English provides a very visual and detailed explanation of virtual particles, Hawking radiation, and how there are different ways of looking at these phenomena.
A 2.7 M_☉ black hole (or fuzzball) emits peak-emission photons that have been very gravitationally redshifted to due to having climbed out of an extreme gravity well and possess an energy of only 7.719×10⁻¹² electron-volt, which is 1.234×10⁻³⁰ joule per photon. With such a black hole radiating at a power of 1.235×10⁻²⁹ watt, it would be emitting 10 quanta packets (photons) per second. This rate assumes though, that the radiated energy comprises solely of photons. Hawking predicted (a PDF of his paper is here) that black holes would radiate not only photons, but gravitons, neutrinos, and other "massless fermions" as well. However, Hawking submitted his paper in 1975, long before the 2001 discovery that neutrinos changed "flavor," so he incorrectly believed neutrinos (a fermion) to be massless. Consequently, gravitons are undetectable hypothetical entities, neutrinos possess a small but as-yet indeterminant mass and are very nearly undetectable, and the only other known massless fermion is the Weyl fermion, which is an emergent quasiparticle first detected inside synthetic crystals in 2015. Accordingly, 10 photons per second is the upper bound for the emission rate and is the only quanta of radiated energy that can be readily detected.
For a 2.7 M_☉ black hole (or fuzzball) that is emitting 10 photons per second and is 100 lightyears away (closer than the nearest known black hole), by the time the photons traveled those 100 lightyears, they will have disbursed over an imaginary sphere with an area of 1.125×10³⁷ square meters, or 8.82×10²² Earth-silhouettes. Even if people continually watched for 279,000 years (roughly as long as Homo sapiens have existed) the odds of just one of those photons impacting somewhere on Earth are one in a billion.

References

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).
Information Paradox Solved? If So, Black Holes Are "Fuzzballs", The Ohio State University, (February 29, 2004)
Vttoth.com: Hawking radiation calculator
Effects of the Theory of Relativity in the GPS, Mario Haustein, Chemnitz University of Technology, equation 19 p. 9, (February 25, 2009)
Peskin, Michael E. (2018-05-04). An Introduction To Quantum Field Theory. doi:10.1201/9780429503559. ISBN 978-0-429-97210-2.
Examples: An Introduction to QED & QCD, (PDF) F. Hautmann, Dept. of Theoretical Physics, University of Oxford (September 2010); Quantum Electro and Chromodynamics treated by Thompson’s heuristic approach, (PDF) Cláudio Nassif, The Brazilian Center for Research in Physics (October 24, 2018)
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 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" — The Ohio State University
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

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[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 predict—one minute into the future—the location of a buoy floating in storm-tossed seas to a precision of one millimeter. If singularities exist, and if they have non-zero diameters with a density equal to the Planck density (5.155×10⁹⁶ kg/m³), then even a minimal 2.7 M_☉ singularity would have a definite non-zero volume with a diameter of 7.8×10¹² Planck lengths (1.26×10⁻²² m), which may seem large but is still far smaller than an electron and is even a thousand times smaller than the minimum dimension that can be probed with a world-class 10 TeV particle accelerator (10⁻¹⁹ m). Alternatively, if a minimal 2.7 M_☉ singularity has a quantum-limited size of one Planck volume, then it has a "fuzzy" density averaging 1.27×10¹³⁵ kg/m³, which far exceeds the Planck density. Lastly, if singularities have truly infinite density, they necessarily have literally zero volume; which is to say, not even one Planck length in diameter. This inability to describe the exact nature of singularities speaks to the dilemma of physics theories wherever their mathematical formulas have a zero being used as a divisor and the known laws of physics have been declared to have "broken down"; it is often an indicator that a theory is incomplete.

[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: 12.46 billion metric tons per teaspoon

6.8 M_☉ fuzzball: 1.965 billion metric tons per teaspoon

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

[4] 2.7 M_☉ fuzzball: 12.46 billion metric tons per teaspoon

[5] 6.8 M_☉ fuzzball: 1.965 billion metric tons per teaspoon

[6] 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 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), " $If$ $a$ $body$ $gives$ $off$ $the$ $energy$ $L$ $in$ $the$ $form$ $of$ $radiation,$ $its$ $mass$ $diminishes$ $by$ $L / V 2$ ." Note Einstein's early use of $L$ instead of $E$ as the symbol for energy and $V$ instead of $c$ as the symbol for the velocity of light, 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 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 in German, 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...

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

[10] 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] In his 1975 paper, Hawking wrote that the radiation emitted by black holes, now called "Hawking radiation," would comprise only of massless virtual particles. He specified the composition as photons, gravitons, neutrinos, and other "massless fermions." His paper was before the discovery that neutrinos actually have mass.

[14] The meanings of "negative mass-energy” or simply "negative energy" (in the absolute $E=mc^{2}$ sense) in discussions of virtual photons at black holes, differs somewhat from what "negative energy" normally means for virtual photons in the lab (in regular spacetime). Virtual photons are oscillations in the background electromagnetic field that prevent an otherwise pure vacuum from containing (possessing) zero energy. Virtual photons are characterized by their wavelength (frequency, or "color"), momentum (which, unlike real photons, is exceedingly variable), and polarization (spin-angular momentum).
Unfortunately, terms like "negative energy," "antiparticle," and "antimatter" can add confusion to a topic that has long fallen victim to popular misunderstanding. Moreover, the issue of whether photons are best described as "waves" or "particles" can needlessly belabor a simple and accessible exposition on Hawking radiation if not formally addressed. Accordingly, a short treatise on the broad subject is required to establish context for how Hawking radiation can be viewed as "arising from virtual photons possessing negative energy tunneling through an event horizon."
The "wave–particle duality" adds complexity to a topic that is already challenging to understand. This duality is commonly encountered when photons are referred to as "quantized wave packets propagating in the electromagnetic field," rather than simply "particles." Referring to photons as "particles" more accurately and conveniently describes the nature of photons after they hit a light detector in a double-slit experiment; it is needlessly ponderous to say "the photon's wave function then collapses to a point." Especially in a treatise on Hawking radiation, where Hawking himself wrote, "there will be pairs of particles, one with negative energy and one with positive energy," it is very useful here to use particle-based vernacular. Nonetheless, the wave nature of virtual photons must be addressed to properly cover Hawking radiation.
Though one member of a pair of virtual photons can possesses negative energy, this is neither the product of charge conjugation (the reversing of electric charge as permitted by "C-symmetry"), nor is it antimatter because, by definition, antimatter is "matter (which has mass) possessing an electric charge opposite that of ordinary matter." Though photons are considered to be their own antiparticle (which is a broad family that confusingly includes antimatter like antiprotons), photons are more specifically a truly neutral particle / antiparticle. Furthermore, real photons must always possess energy equal to the speed of light times their momentum vector and must have zero rest mass.
In laboratories, virtual photons possess different kinds of momentum and interact with matter and its accompanying electromagnetic fields in different ways. Virtual photons exist everywhere and their effects are observable as the Lamb shift as they interact with the electromagnetic field of electrons surrounding atoms. This activity also underlies zero-point energy, which jostles matter to such an extent it prevents helium at near-absolute zero from freezing at room pressure. Virtual photons are also responsible for the Casimir effect, which squeezes two closely spaced plates together. Virtual photons can also be polarized; this is to say, they have the quantum property of spin-angular momentum, which can couple to the angular momentum of charged particles.
In simple terms, all three of the above effects: the Lamb shift, the Casimir force, and the inability of helium to freeze at room pressure due to zero-point energy, arise from the collective activity of virtual photons. More precisely—and ponderously—these three effects are the result of oscillations in the quantum electrodynamic field, resulting in a non-zero QED vacuum (or simply vacuum energy). The QED vacuum is the lowest energy state of the all-pervasive electromagnetic field permeating the Universe; real photons are traveling excitations in this electromagnetic field. Note that the quantum electrodynamic field and the electromagnetic field are essentially the same thing except that the former is the quantum-based view of electromagnetism that accounts for a non-zero vacuum energy. Note also that in discussions of Hawking radiation, the term "zero-point energy" is interchangeable in practice with "vacuum energy" but the former is broader and encompasses other zero-point fields, including the quantum chromodynamic vacuum (QCD), which governs interactions at the quark level.
This is a wave-based view showing the momentum disturbance of a single oscillation in the quantum electrodynamic (QED) field, also known as "a pair of virtual photons" in a particle-based view. These are the source of the photons comprising Hawking radiation.
Here, the 3D QED field that permeates the Universe is projected onto a 2D plane and the vertical axis represents the vector momentum, $p$ in an absolute (relativistic) sense. The upper and lower bulges represent virtual photons possessing positive and negative relativistic energy, respectively. This oscillation has no spin-angular momentum (polarization), which would appear as helical twists. The diameter of the oscillation is its wavelength, ${\lambda }$ (lambda), which cannot factor into the energy of oscillations possessing zero net momentum.
A single oscillation in the QED field (the quantum version of the electromagnetic field) comprises momentum components that are on average—but by no means always—equal and opposite. Consequently, the average net relativistic mass-energy of these oscillations is zero. QED oscillations last one full cycle of their wavelength, so one with a wavelength of 600 nanometers (orange for real photons) exists for only about 2 femtoseconds.
Zero-point energy comprises oscillations in all types of quantum fields and is the subject of ongoing research in theoretical physics. In part, zero-point energy arises from the Heisenberg uncertainty principle's effect on QED vacuum energy—which allows for non-symmetrical virtual photon momenta (asymmetric bulges). Thus, some oscillations in the QED field possess non-zero net momentum and non-zero net relativistic mass-energy before quickly vanishing. While momentum asymmetries contribute to vacuum energy, they are not required to produce Hawking radiation.
The relativistic mass-energy of real photons (their absolute $E=mc^{2}$ energy that, as Hawking wrote, is "relative to infinity") is proportional to their momentum vector times the speed of light per $E = p c$ , where…
$E$ is energy,

$p$ is the magnitude of the momentum vector, and

$c$ is the speed of light.
Individual virtual photons are different from real ones; they may carry any momentum, or relativistic mass-energy, permitted by the Heisenberg uncertainty principle. Thus, any given pair of virtual photons may possess opposite and unequal momenta. However, across a large population of virtual photon pairs, their net momentum averages to zero and so too does their rest mass-energy and relativistic energy.
With regard to measurements in the lab of virtual photon momentum, the labels "positive energy" and "negative energy" are relative classifications established by the direction of their momentum vector, p (and accompanying energy) in relation to an external electromagnetic field (from one or more nearby charged particles). This convention comes from the behavior of real photons, which possess positive energy with respect to electrons; this underlies spectral lines where the electrons surrounding atoms transition from a lower-energy atomic orbital to a higher-energy one after absorbing photons. When an individual virtual photon is exchanged between two particles with like charges (followed soon after by its partner), it is considered to have positive energy when its momentum adds energy to the electromagnetic force between them and they more vigorously repel each other. Note that a virtual photon with a positive-energy momentum direction when it is exchanged between two electrons would be classified as possessing a negative-energy momentum if the exchange was between positrons.
In the context of Hawking radiation however, the labels "positive energy" and "negative energy" for virtual photons are in an absolute sense, or "relative to infinity," as Hawking wrote. Virtual photons with negative mass-energies in an absolute sense are generally considered as not physically real. This is because a virtual photon possessing positive momentum and positive mass-energy behaves just like easy-to-study real photons whereas a virtual photon possessing negative momentum and negative mass-energy cannot be isolated in the lab and its distinctive property studied. This inability to isolate a virtual photon possessing negative energy is to be expected since in the wave-based view, it is actually an integral part of an individual oscillation in the quantum electrodynamic field (electromagnetic field) permeating the Universe (see image at right) that humans cannot bifurcate.
Black holes have the unique ability to do what cannot be done in the lab: separate virtual photon pairs. This is in part due to the extremely small radius of curvature of spacetime near their event horizons. From the perspective of an outside observer viewing wave-based phenomena, black holes can shear, stretch, and bifurcate the components of a QED oscillation possessing opposite momenta. Hawking radiation arises when the portion of the oscillation possessing negative momentum and negative relativistic mass-energy (the lower bulge of the QED oscillation at right) tunnels through to a black hole. This liberates the positive-energy half of the QED oscillation as a real photon, which becomes exceedingly gravitationally redshifted as it climbs up the extreme gravity well surrounding the black hole and escapes to infinity. The peak-emission wavelength of fully redshifted photons is about ten times the diameter of the event horizon surrounding a non-spinning black hole regardless of its mass; a 6.77 M_☉ non-spinning black hole with an event horizon diameter of 40.0 kilometers emits Hawking radiation where the most common photons have a wavelength of 403 kilometers.
Finally, it is important to bear in mind that gravity and spacetime are so agitated near a black hole that even the mathematics describing Hawking radiation can be viewed in different ways. For instance, instead of negative-energy particles tunneling through the horizon in forward-directed time, they can be thought of, as Hawking wrote in his paper, "as positive-energy particles crossing the horizon on past-directed world-lines." As mentioned earlier, Hawking cautioned that, "It should be emphasized that the mechanism responsible for the thermal emission and area decrease are heuristic only and should not be taken too literally." Even the basic premiss that Hawking radiation is the product of a stationary person observing a highly accelerating region of spacetime has its general relativity inverse known as the Unruh effect, which predicts that thermal radiation surrounds an accelerating observer. A helpful YouTube video, "Hawking radiation", by ScienceClic English provides a very visual and detailed explanation of virtual particles, Hawking radiation, and how there are different ways of looking at these phenomena.

[23] A 2.7 M_☉ black hole (or fuzzball) emits peak-emission photons that have been very gravitationally redshifted to due to having climbed out of an extreme gravity well and possess an energy of only 7.719×10⁻¹² electron-volt, which is 1.234×10⁻³⁰ joule per photon. With such a black hole radiating at a power of 1.235×10⁻²⁹ watt, it would be emitting 10 quanta packets (photons) per second. This rate assumes though, that the radiated energy comprises solely of photons. Hawking predicted (a PDF of his paper is here) that black holes would radiate not only photons, but gravitons, neutrinos, and other "massless fermions" as well. However, Hawking submitted his paper in 1975, long before the 2001 discovery that neutrinos changed "flavor," so he incorrectly believed neutrinos (a fermion) to be massless. Consequently, gravitons are undetectable hypothetical entities, neutrinos possess a small but as-yet indeterminant mass and are very nearly undetectable, and the only other known massless fermion is the Weyl fermion, which is an emergent quasiparticle first detected inside synthetic crystals in 2015. Accordingly, 10 photons per second is the upper bound for the emission rate and is the only quanta of radiated energy that can be readily detected.

[24] For a 2.7 M_☉ black hole (or fuzzball) that is emitting 10 photons per second and is 100 lightyears away (closer than the nearest known black hole), by the time the photons traveled those 100 lightyears, they will have disbursed over an imaginary sphere with an area of 1.125×10³⁷ square meters, or 8.82×10²² Earth-silhouettes. Even if people continually watched for 279,000 years (roughly as long as Homo sapiens have existed) the odds of just one of those photons impacting somewhere on Earth are one in a billion.

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

[16] 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] 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).

[5] Information Paradox Solved? If So, Black Holes Are "Fuzzballs", The Ohio State University, (February 29, 2004)

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

[11] Effects of the Theory of Relativity in the GPS, Mario Haustein, Chemnitz University of Technology, equation 19 p. 9, (February 25, 2009)

[13] Peskin, Michael E. (2018-05-04). An Introduction To Quantum Field Theory. doi:10.1201/9780429503559. ISBN 978-0-429-97210-2.

[15] Examples: An Introduction to QED & QCD, (PDF) F. Hautmann, Dept. of Theoretical Physics, University of Oxford (September 2010); Quantum Electro and Chromodynamics treated by Thompson’s heuristic approach, (PDF) Cláudio Nassif, The Brazilian Center for Research in Physics (October 24, 2018)

[Pais1982-16] 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.

[Weinberg2011-17] 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-20230124-18] 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.

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

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

[Gunn-21] What is Hawking radiation?, Dr. Alastair Gunn, BBC Science Focus, (April 16, 2022)

[Italian2021-22] "A Way to Experimentally Test String Theory's 'Fuzzball' Prediction", APS Journals, (September 16, 2021)

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

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 Radio-Quiet Radio-Loud
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
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 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 SDSS J150243.09+111557.3 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 P172+18
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