Black Holes: From Math to Reality

Edison Santos
PPGCosmo, CCE, Universidade Federal do Espírito Santo, 29075-910,
Vitória, ES, Brazil
Correspondence to: edison_cesar@hotmail.com

DOI: https://doi.org/10.70389/PJS.100047

Additional information

• Ethical approval: N/a
• Consent: N/a
• Funding: No industry funding
• Conflicts of interest: N/a
• Author contribution: Edison Santos – Conceptualization, Writing – original draft, review and editing
• Guarantor: Edison Santos
• Provenance and peer-review:
  Commissioned and externally peer-reviewed
• Data availability statement: N/a

Keywords: Black holes, General relativity, History of physics.

Peer-review
Received: 23 August 2024
Revised: 24 December 2024
Accepted: 27 December 2024
Published: 15 January 2025

Introduction

We humans are a curious species by nature. We have looked up to the skies to know when to plant our crops by analyzing the patterns of the constellations, to study the motion of planets and the Sun, or simply for the pure admiration of a starry night. This fascination has been documented since antiquity, in every culture, regardless of the region of the globe. It took centuries to realize that there was much more than what our eyes, or even our basic telescopes, could see. Among the plethora of objects in our universe, such as planets, galaxies, nebulas, cosmic rays, and supernovas, to name a few, there also exists something that essentially tries to hide itself. Black holes (BHs) are some of the most complex and intriguing objects in our universe. They have been depicted in pop culture as methods for traveling vast distances in space: it was portrayed as the main character in Nolan’s film Interstellar or as the title of Muse’s album Black Holes and Revelations. Despite the general difficulty in formally defining a BH, the phrase “an object so heavy that nothing, not even light, can escape its gravitational pull, with a singularity that breaks the laws of physics at its center” is often used in various forms and well establishes the main characteristic of this object: once something goes in, nothing comes out.

Surely, the structure of a BH is not that simple. A general static BH, i.e., one that is not spinning, has three main characteristics: a singularity of infinite density where general relativity (GR) breaks down—a problem we hope to resolve with a full theory of quantum gravity, which is yet to be discovered; an event horizon, which splits the BH into two regions and is the point of no return—once crossed, there is no way to come back, and not even light can escape, rendering the ­inner structure black to any outside observer; a photon sphere just outside the event horizon where light has unstable orbits; and the innermost stable circular orbit (ISCO) region, the last region where stable orbits are possible around it. However, a real astrophysical BH should be spinning and not held static. There are two main differences that come with this: first, the singularity is no longer a point—it gets stretched due to the spinning effect, turning it into a ring; secondly, a new region appears where spacetime itself rotates with the BH, called the ergosphere. In this region, it is impossible to keep any particle at rest; everything must co-rotate with it—even if we try to go against it. Today, it is almost impossible to think that BHs do not exist, although it is easy to forget that it took ­centuries to come to that realization. In Historical Background, we will see how well-known figures in science, such as Laplace, Newton, Einstein, and Hawking, from as early as the late 1700s to the 1900s, contributed to building the theoretical knowledge required to understand these objects.

However, developing the theoretical framework to study BHs is one thing; developing the tools to prove their existence is another. In Observational Milestones, we will uncover how, in the mid-1900s, the scientific community began to be convinced that the only explanation for certain observations was the existence of BHs. However, the definitive proof took a bit longer to be found. It was only with the advent of gravitational-wave astronomy that we finally proved BHs to be a reality, and even more astonishingly, in 2019, we captured the first image of one. BH physics is far from complete; it remains a highly active area of research, both theoretically and observationally. In Future Directions in Black Hole Physics, we will conclude with a discussion of some theoretical challenges still open to debate and the ongoing efforts to enhance the observation of one of the most interesting objects lying in the cosmos.

Historical Background

Early Theoretical Developments

In the eighteenth century, physics was still in its infancy regarding the debate on the nature of light. While Christiaan Huygens argued that light is similar to how we currently understand sound waves, a wave of pressure flowing through a medium called the ether,¹ Newton viewed light as small particles, or corpuscles, bouncing off walls and transmitting through prisms.² This debate was far from settled,^[1] and even further from being fully comprehended. In the meantime, Newton also published his three tomes on “The Mathematical Principles of Natural Philosophy” in 1687, giving birth to what is now known as classical mechanics.³ In the first two books, Newton states his three laws of motion, develops the theory for circular motion, derives Kepler’s and Boyle’s laws, studies the motion of objects with drag forces exerted against them, and lays the basis for hydrostatics and oscillatory motion. In the third book, Newton derives from observation the universal theory of gravitation, stated in the equation

(1) That is, the force of gravity F is exerted between two masses, m₁ and m₂, and is inversely proportional to their distance squared, 1/r2. Newton not only provided a satisfactory explanation of how gravity works, satisfying the observations made by Kepler and Galileo, but also unified the heavens with Earth: the laws that govern the planets, comets, and stars are the same as those that explain how apples fall from trees and how pendulums oscillate. In his first definition of the first book, Newton develops the idea of mass and density:³

“The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. Thus air of double density, in double space, is quadruple in quantity; (…) The same thing is to be understood of snow and fine dust of powders that are condensed by compression or liquefaction; and of all bodies that are by any causes whatever differently condensed. (…) It is this quantity that I mean hereafter everywhere under the name of body or mass.”

That is, for Newton, everything has mass. Everybody, particle, or corpuscle, independent of its state or behavior, possesses this innate characteristic of having mass, a measure of inertia. Amid the debate about the nature of light and the (fairly) recent formulation of Newton’s theory of gravity, the geologist and physicist John Michell wrote a private letter to his colleague Henry Cavendish in 1784:⁴

“If the semi-diameter of a sphere of the same density as the Sun were to exceed that of the Sun in the proportion of 500 to 1, a body falling from an infinite height towards it would have acquired at its surface a greater velocity than that of light, and consequently, assuming light to be attracted by the same force in proportion to its vis inertiae as other bodies, all light emitted from such a body would be made to return towards it by its own proper gravity. This assumes that gravity influences light in the same way as massive objects.”

In this letter, the idea of a body sufficiently compact to trap light within it was first conveyed. Michell coined the term dark star for this theoretical object. Independently of Michell, about a decade later in 1796, Pierre-Simon Laplace evaluated the minimum radius needed for a body of mass M to entrap light of mass m.⁵ This can be evaluated by calculating how much initial E₀ kinetic energy is needed to be fully converted into E_f gravitational potential energy as

(2) Solving the previous equation for the velocity v_s, we obtain

(3) which is known as the escape velocity. More precisely, this is the minimum velocity needed for a body to escape the gravitational field of a body of mass M. If we instead solve for r, swapping vs for c in equation 2, we get

(4) This equation specifies the minimum radius for a body of mass M to contain any body traveling at c, the speed of light.[2] Despite the progress made on this concept, ­Newton’s interpretation of light became less popular over the ­decades, and the massless wave interpretation of light gained strength. Since Newtonian gravity relies on the masses of two different bodies, the idea of dark stars was forgotten for the next few decades.

Mid-Twentieth-Century Breakthroughs

Nineteenth century witnessed tremendous progress in fundamental physics. James Clerk Maxwell released his two-volume treatise on electromagnetism, summarizing the recent developments on the subject while adding his own ideas.^6,7 Heinrich Hertz published his paper showing that electromagnetic waves travel at a finite speed, c.⁸ Max Planck gave birth to quantum mechanics by showing that energy must be quantized at a fundamental level.[3]^,9 In 1905, during Albert Einstein’s annus mirabilis, he presented his four renowned papers.^10–13 In one of these papers, Einstein formulated his special theory of relativity. His theory was “special” in the sense that it was restricted and yet to be generalized to accommodate gravity. It took Einstein ten more years to refine Newton’s theory of gravity, culminating in GR, stated in Einstein’s field equations (EFE):

(5) setting the foundation for our current understanding of gravity. His theory was no longer fixed to a ­particular reference frame. We now have a general theory of relativity.¹⁴ The left-hand side of the EFE in (5), known as ­Einstein’s tensor, contains information about the ­geometry of spacetime, i.e., how it is curved. The right-hand side, the stress-energy (or energy-momentum) tensor, encloses the information about the mass/energy[4] that curves the given spacetime. As John Wheeler succinctly stated:

“Spacetime tells matter how to move; matter tells spacetime how to curve.”

Despite the deceptive simplicity of equation (5), it encapsulates a set of ten non-linear, coupled partial differential equations. To put it mildly, it seemed impossible to find any analytical solution that would unveil the physics behind Einstein’s theory, and who knew how long it would take for anyone to finally solve Einstein’s equations. One year. Actually, less than a year. In January of 1916, Karl Schwarzschild, amid the First World War, found the first exact solution to the EFE.¹⁵ He assumed the simplest possible case to accomplish this achievement: a solution in a vacuum, T_µν = 0, with spherical symmetry, and a body that is static, not evolving in time and thus not depending on the time coordinate either. This led to what is now known as Schwarzschild’s metric:

(6) The term dΩ is the solid angle defined as dΩ = dθ2 + sin2(θ)dΦ, and r_s is what is commonly known as the Schwarzschild’s radius given by

(7) Schwarzschild’s solution presented two main issues, at r_s = 0 and r = r_s. The former causes the dt2 term to diverge, while the latter does the same with dr2. At the time, it was agreed that r = 0 was a legitimate singularity, but the meaning of the singularity at r = r_s was unclear. Investigating the geodesics, i.e., the motion of any light-ray or massive particle approaching r = r_s, they never seem to reach it, let alone cross it.

Up to this point, Schwarzschild’s solution was considered nothing more than a mathematical solution to the EFE, with no real application or object to be described by it. However, Oppenheimer and Snyder showed in 1939 that if a sufficiently heavy star collapses after exhausting any source of nuclear energy in its core, the star will emit its mass as radiation, and what remains after the gravitational collapse process can be described by Schwarzschild’s solution.16 Although ­Oppenheimer and Snyder’s work was not fully appreciated initially, it was the first time that the mathematical solution of Schwarzschild was given a sense of reality, representing what remains after a star dies. In 1958, Finkelstein realized that by using a different coordinate system than Schwarzschild’s, anything that fell past r_s could not escape back, creating a one-way membrane.¹⁷ Thus, r_s was an apparent singularity caused by a poor choice of coordinates. This region was recognized as an event horizon, a term coined by ­Rindler in 1956.[5]^,18

The Golden Age of General Relativity

Around the middle of the twentieth century, GR became a mature theory, well-accepted in mainstream physics and widely studied for its implications. There was a pivotal 10-year period between the 1960s and 1970s during which previous ideas underwent significant development and became central topics of debate in astrophysics and GR. Until that time, several other analytical solutions to Einstein’s equations had been obtained. However, none of them described a rotating solution, despite it being well-known that everything in the universe exhibits rotational motion. But it was with Roy Kerr’s solution in 1963, where he generalized the static Schwarzschild metric to a rotating body.¹⁹ In 1965, Roger Penrose proved that singularities are formed in a gravitational collapse under general conditions.²⁰ Then in 1966, Stephen Hawking applied Penrose’s work to cosmological models, proving that our universe also contains a singularity (what is now known as the Big Bang).²¹ In 1970, this time together, they summarized their previous work, presenting what has since been recognized as the Hawking-Penrose Singularity Theorems,[6] proving that GR is inherently plagued by singularities, and they are a recurrent feature of the theory itself.²²

It was finally in 1967, during a lecture at the American Association for the Advancement of Science, that John Wheeler coined the term black hole, replacing the more cumbersome “gravitationally completely collapsed object.” Whatever falls inside the event horizon is lost within it, and no information about the nature of the matter can be retrieved. For example, we cannot determine whether a BH with 10 solar masses is composed of matter or antimatter. Israel demonstrated this in 1967,²³ and the following year, he showed that the same applies to any electric charge.²⁴ Carter then generalized the result to include angular momentum. Thus, a BH is fully described by only three parameters: its mass M, electric charge Q, and spin a, with no other distinguishing features; as Wheeler famously said, “a BH has no hair.”[7] This principle is now known as the no-hair theorem. In 1975 and 1976, Stephen Hawking showed that by examining quantum effects in curved spacetime—specifically how particles are created and annihilated near the event horizon of a BH—an outside observer would see particles being emitted from the BH itself. He concluded that this particle emission would cause the BH to lose mass, leading to its gradual evaporation through what is now known as Hawking radiation.^25,26 In just over a decade, we finally found a suitable name for the “gravitationally completely collapsed object”; we discovered that BHs are described by only three parameters; they must contain a singularity; Kerr’s metric was crucial in our understanding of real, astrophysical BHs; and we realized that BHs are not completely “black” due to Hawking radiation. With the theory now well-developed and well-established, the next step was to seek out these objects.

Observational Milestones

Cygnus XR-1

In a survey conducted in 1965 during two rocket flights, eight new sources of cosmic X-rays were detected.²⁷ Among these objects, the second strongest source was emitted by Cygnus XR-1 (Cyg XR-1), located in the constellation of the same name. Curiously, this immense source of X-rays, shown in Figure 1, lacked any corresponding radio or optical sources at its position. In other words, we could not see whatever was emitting it.

At the time, the notion of Cyg XR-1 being a BH was considered implausible:

“We suggest that multiple-star models of the Cyg XR-1 system must be considered further before it can be concluded that Cyg XR-1 is a black hole.”

This was stated in reference28 nearly a decade after the detection of Cyg XR-1 in 1974. They attempted to explain the unusual X-ray emission using two different theoretical models of a triple star system:[8] the hierarchical triple system model and the dynamical interaction model. In the former, two stars form a close binary system, with a third star orbiting farther away; the X-ray emissions arise due to interactions between the binary and the tertiary components, such as mass transfer or gravitational perturbations. The latter model, however, envisions the three stars in a more involved and complex dynamical interaction; one of the stars may act as an intermediary, transferring mass ­between the other two or causing angular momentum exchanges that lead to X-ray emissions. These two models aim to mimic the main observational features, such as the mass function and the variability in the X-ray source intensity, as the system evolves dynamically over time. However, in the same year, Shipman published a straightforward paper titled “The Implausible History of Triple Star Models for Cygnus XR-1: Evidence for a Black Hole.”³⁰ Still, in that same year, Stephen Hawking and Kip Thorne made a bet on whether Cyg XR-1 was a BH or not, with Hawking betting against the idea but hoping to lose. Cyg XR-1 became one of the most researched astronomical objects, and even today, we do not have direct evidence of it being a BH. Despite this, it is widely accepted within the scientific community that Cyg XR-1 is a 21.1 solar mass BH³¹ in a binary system with a blue supergiant.

…and Hawking conceded the bet to Thorne back in 1990.

Sagittarius A*

In the early days of radio astronomy, in 1933, Karl ­Jansky discovered a source of high-frequency electromagnetic waves coming from a fixed position in the constellation Sagittarius,³² which would later be ­identified as our galactic center in 1954.³³ Within this region, in 1974 Balick and Brown observed a substructure of “very bright radio waves” emanating from a compact object that could have been a quasar.³⁴ A decade later, in 1982 to be precise, this compact object emitting radio waves was named ­Sagittarius A* (Sgr A*).³⁵ Although it was evident since the 1980s that Sgr A* was a BH, it was only in 1994 that Reinhard Genzel[9] demonstrated that it should be a massive BH with a mass on the order of millions of solar masses.³⁶ However, after ten years of high-resolution astrometric imaging, in 2002, a team led by Genzel ruled out any other possible explanation for Sgr A*.³⁷ Even though the only plausible explanation for the observations could be a BH, definitive proof was still elusive.

Gravitational-Wave Astronomy

In 1992, two sites were chosen in the United States—Hanford and Livingston—to build an upscaled version of the Michelson-Morley experiment. The goal was very clear: to find the ripples of spacetime in the form of gravitational waves. Both observatories were completed in 1999, and the Laser Interferometer Gravitational-Wave Observatory (LIGO) was successfully established. In parallel, in 1997, the construction of the Franco- Italian Virgo interferometer in Cascina, Italy, began, being inaugurated in 2003. The design is essentially the same as LIGO’s; however, while LIGO’s interferometers were each 4 km long, Virgo’s were 3 km. LIGO began operations in 2002, and in 2007, it joined forces with Virgo, creating the LIGO-Virgo Collaboration (LVC) with three observatories working as a single machine. This initial run ended in 2010; even though it placed important limits on gravitational-wave sources, no detection of any kind was found. It would take five more years of significant improvements to LIGO’s observatories to finally have another chance at detecting these waves[10]. On September 12, 2015, the first science observation run (O1) by the LVC began. Two days later, on September 14, a new window on the universe was opened. The signal of a binary system of two BHs, one with 36 and the other with 29 solar masses forming a 62 solar mass BH remnant was detected.[11]^,38 In their own words:

“This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.”

The LVC collaboration was joined by the Japanese Kamioka Gravitational Wave Detector (KAGRA) in 2021, and they are currently running the fourth observation run (O4). Together, they have found 90 mergers of binary BH systems, 2 neutron star-BH systems, and 2 binary neutron star coalescences (Figure 2).³⁹

The Image of M87*

Despite a BH being essentially black, it should be found in an environment clustered with other stars, gases, and accretion matter revolving around it. Even though the BH itself could never be directly seen, the region outside its event horizon interacting with the medium around it must be visible. In 2009, the Event Horizon Telescope (EHT) project began. It is composed of several radio telescopes spread around the world, working together as a single observatory. To put it simply: an effective telescope the size of Earth. In the beginning, the EHT collaboration had four main goals: i) test GR in the strong field regime; ii) check the existence of an event horizon; iii) assess the plausibility of estimating the BH spin by observing orbits near the event horizon; iv) attempt to explain how BHs accrete matter and create powerful jets. And they were not shy about what it could accomplish:⁴⁰

“direct imaging of black holes can be achieved within the next decade.”

In one of the largest and most massive galaxies in our local universe, in the constellation of Virgo, a rather strong source of radio-frequency energy was ­detected in 1949 from M87.⁴¹ At the time, it was still unknown what this object was; however, it was acknowledged since 1918 that it emitted a strong, straight jet from its nucleus.⁴² Later, in 1978, it was calculated that the central mass concentration of M87 should be of the order of 5 billion solar masses, which would be entirely consistent with a BH.⁴³ It was then, in 2017, that the EHT collaboration gathered the first set of data^44–49 to ultimately release the first processed image of a supermassive BH in the center of M87[12] in 2019, Figure 1. This BH contains a mass of 6.5 billion solar masses and an event horizon 120 times the distance from Earth to the Sun^50–55 in 2019 (Figure 3).

From there, the EHT collaboration continued to reveal images of BHs around the universe. In 2021, they released the reconstructed image of Centaurus A’s jet emission;⁵⁶ in 2022, the image of the supermassive BH in the center of our galaxy, Sagittarius A*;^57–62 and sharper images of M87* in 2023.⁶³

Future Directions in Black Hole Physics

We could say that BH physics is experiencing its second golden age, this time on the observational side. However, many questions remain unresolved in the theoretical domain as well. GR is a very well-tested theory that continues to provide us with a deeper understanding of the universe as a whole. Despite this, it is clear that GR is not our final theory of gravity. The singularity within BHs reveals one of the limitations of GR. Singularities cannot exist in reality; it seems curious, to say the least, to wonder how such a region would be created due to the gravitational collapse of a star, for example. Explaining what a singularity would look like is another struggle in itself. Singularities are in the extreme-field regime, where it is impossible to ignore gravity, but the region is also small enough for quantum mechanics to come into play. The unfortunate news is that despite several different attempts to resolve this issue,⁶⁴ we still do not know how to merge these two theories. We could say that this is the holy grail of current theoretical physics, and we hope that a future “quantum-gravity theory” may solve the singularity problem in some manner.

Information cannot be lost. This is not only a plausible idea but a foundational consequence of quantum mechanics. However, whatever enters a BH is lost forever; we cannot retrieve any information from whatever went inside. Recall that a BH does not distinguish between matter and antimatter, for example, and Hawking radiation emits thermal energy only, which also does not carry any other information about the BH itself. This is what Hawking called the information paradox.⁶⁵ Many different proposals for the resolution of this paradox have been suggested over the years.⁶⁶ Hawking returned to the issue 40 years later, in 2015, proposing his own resolution: although information can theoretically be recovered, it would be so scrambled that for all practical purposes it would be lost.⁶⁷ Despite this, the information paradox remains an open debate that challenges our understanding of gravity and quantum mechanics
simultaneously.

On the observational side, we have just discovered a new way to investigate the universe using gravitational waves. The KAGRA observatory is still operating at around 10% of its maximum efficiency,⁶⁸ LIGO-India (INDIGO) is set to be commissioned in 2030,⁶⁹ and LIGO is expected to upgrade its system by a factor of two within this decade.⁷⁰ The Laser Interferometer Space Antenna (LISA), the first space-based interferometer, will be far more sensitive than any ground-based gravitational-wave observatory. Led by the European Space Agency (ESA) in collaboration with NASA, LISA will consist of three spacecraft in a triangular formation, separated by 2.5 million kilometers, using laser beams to detect minute changes in distance caused by passing GWs. Operating in space and free from terrestrial noise, LISA will observe sources of GWs inaccessible to ground-based detectors, including those from BHs of various masses, such as intermediate-mass and supermassive ones, shedding light on their formation, evolution, and mergers. The mission will also detect GWs emitted shortly after the Big Bang, offering new insights into the early universe’s structure and dynamics. The LISA mission is scheduled to launch in 2035.⁷¹

A new generation of radio telescopes, the Square Kilometre Array (SKA), currently under construction in South Africa and Australia, will have an effective area of 1 km². It is set to achieve its first light in 2028. BHs are far from being mere objects of imagination or just pop culture phenomena. They are not only a central part of contemporary physics but also one of the biggest puzzles that allow us to further understand our universe. We have progressed from not understanding what light even was, to literally observing the light emitted from a BH’s accretion disk; from struggling to interpret preposterous observations, to measuring almost 100 BHs using gravitational waves; from mathematically being unable to cross the event horizon, to pondering how to resolve the singularities within them. It has taken us about 300 years to reach our current understanding, and it seems almost impossible to predict what discoveries the next few decades will bring as we continue to satiate our curiosity about the universe.

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Santos E. Black Holes: From Math to Reality. Premier Journal of Science 2024;1:100047.

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