Gravitational Singularities

• Type of physical science: Relativity
• Field of study: General relativity

Various solutions to the field equations of Albert Einstein’s general theory of relativity describe an infinite distortion of space and time, which is known as a black hole. The heart of a black hole is a singularity—a region where matter has collapsed into a point (or ring) of infinite density and zero volume, compressing finite mass into an infinitesimal region.

Overview

To discuss the concept of a gravitational singularity, it is first necessary to gain some understanding of Albert Einstein’s general theory of relativity. Unlike Sir Isaac Newton’s universal law of gravitation, which holds that gravity is an attractive force, general relativity shows that a gravitational field is a consequence of the shape of the universe. A large mass, such as the sun, distorts the fabric of space-time around it.

Consider four people tightly holding a sheet by each of its corners. If a heavy ball is then placed in the center of the sheet, the sheet is stretched or distorted. A smaller ball placed on an edge of the sheet would then move according to the curvature of the sheet. This example illustrates the concept of warped space-time. The smaller ball is moving toward the larger ball, not because of any force acting between the two balls, but because of the shape of the sheet.

When it was first proposed in 1916, the general theory of relativity was thought to be beyond comprehension. In fact, it was often stated that only a few scientists fully understood its complexities. To more fully appreciate the problem, consider that the theory contains sixteen nonlinear, partial differential equations. These equations involve ten independent components of the space-time metric. In his original paper, Einstein had worked out only an approximate solution to the equations.

The first exact solution was accomplished by the German astronomer Karl Schwarzschild (1873-1916). In a 1916 paper, Schwarzschild investigated the consequences of infinite gravity around a spherical, nonrotating body. Schwarzschild’s solution describes a bizarre object: a gravitational singularity, or what is commonly called a black hole. Schwarzschild concluded that, if a star were to collapse within a certain radius, then the escape velocity at its surface would exceed the speed of light. In other words, light would not be able to escape from this body and, in effect, it would disappear from the universe. The radius that Schwarzschild identified is the event horizon or, as it is often called, the Schwarzschild radius. The event horizon or Schwarzschild radius is a boundary in the geometry of space-time beyond which events cannot be detected. Anything inside of an event horizon is totally disconnected from the universe.

The size of this radius depends on the mass that formed the singularity and is equal to 2.95 kilometers (1.83 miles) multiplied by the mass of the black hole in solar mass units. For example, a 10-solar-mass black hole has a radius of 29.5 kilometers (18.3 miles).

The effects on matter encountering an event horizon can be predicted. Only matter on the outside moving inward can cross the event horizon; matter on the inside is forever trapped.

Consider two spacecraft, A and B, in the vicinity of an event horizon. As ship A approaches the event horizon, B will observe that A’s clocks are running more slowly. The crew of spaceship A will, however, find that everything is normal. As A approaches the event horizon, B observes A’s signals becoming increasingly redshifted and delayed near the boundary. The instant that A crosses the event horizon, it will appear to A’s crew members that they are about to strike the collapsed star. This image, however, is only the fossil remains of the light that was emitted as the star collapsed through the event horizon, probably millions of years earlier. The image quickly vanishes as A is pulled inward at increasing velocities toward the singularity. As the spacecraft nears the singularity, immense tidal forces will distort its shape, stretching it along its axis of motion. As the ship is stretched, its volume is being reduced. Eventually, the ship and its crew would be crushed out of existence. The crew of spaceship B, however, would observe A suspended at the end of the event horizon forever.

After the 1916 Schwarzschild solution, other physicists proposed solutions to the equations of general relativity. Shortly after the publication of Schwarzschild’s paper, the German physicist Heinrich Reissner and the Finnish physicist Gunnar Nordström solved the equations for a body with both mass and electrical charge. The electrically charged black hole would have two event horizons that separate it from the rest of the universe.

One event horizon would be caused by the mass of the black hole; the other would be attributable to its charge. Most scientists believe that the chances of such an object as a Reissner-Nordström black hole actually existing are highly unlikely. Because such a massive object with a charge would project a huge electric field, particles of the opposite charge would quickly be attracted to it. The overall charge on the black hole would soon be neutralized.

In the early 1960s, Roy Kerr solved Einstein’s equations of general relativity for rotating black holes, predicting a singularity shaped like a ring rather than a single point. This conclusion differs significantly from the Schwarzschild and Reissner-Nordström solutions, where the singularity is a point. Kerr’s solution mathematically allows for the theoretical possibility of passing through the center of the ring singularity without encountering infinite tidal forces. However, modern theoretical physics considers such traversals physically unrealistic due to immense tidal forces, instabilities, and quantum effects near the singularity. Twenty-first-century models suggest that any interpretation involving negative space-time, reversed time, or repulsive gravity is speculative and likely a mathematical artifact rather than a physically achievable reality.

A few years after the publication of the Kerr solution, Ezra Newman and his colleagues expanded that solution to include charged, rotating bodies. Shortly after this development, still another solution predicted the possibility of a singularity being exposed, with no surrounding event horizon. In this “super extreme Kerr object” (SEKO), the event horizon does not exist.

A Kerr black hole consists of two horizons, an outer event horizon and an inner Cauchy horizon.

The gravitational attraction is the dominating force as long as the event horizon for mass is larger than that for angular momentum. As the angular momentum becomes larger, its event horizon expands toward the event horizon for mass. If the angular momentum event horizon continues to expand, then it will eventually meet and fuse with the event horizon for mass. According to theory, any further expansion in angular momentum will cause the fused event horizons to vanish and, subsequently, will expose the ring singularity. Physicists refer to such an object as a “naked singularity.” In principle, a naked singularity could be visited by astronauts without the danger of becoming entangled in the infinite space-time warp that is caused by an event horizon. While the Kerr solution allows for the theoretical existence of naked singularities, most modern physicists believe such objects are unlikely to form naturally, due to what Roger Penrose termed the Cosmic Censorship Conjecture, which suggests nature prevents singularities from existing without event horizons.

According to the general theory of relativity, infinite forces exist at a singularity. The theory also indicates that termination at the singularity is the fate of all particles that become trapped within the event horizon. Further models developed by physicists, however, indicate the possibility of a somewhat different fate for those particles.

In the 1930s, Einstein and Anders Rosen developed mathematical models of the gravitational fields around static black holes. They found that, as the singularity is approached, the field becomes increasingly strong and the distortion in space-time becomes more pronounced. The tool that they used in their investigation is called an embedding diagram or a gravity well diagram. Embedding diagrams were first used to explain the general theory of relativity in its early days. A large mass produces a distortion in space-time, and an embedding diagram is simply a drawing of this phenomenon. The more massive is the body, the deeper is the well or depression in the embedding diagram. As the depression becomes deeper, the energy needed to escape or climb out of the well becomes greater as well. A singularity should produce a gravity well of infinite depth, according to theory, but the equations of Einstein and Rosen produced a model in which the embedding diagram suddenly opens up again. Exactly what is connected to the other end of this so-called Einstein-Rosen bridge is not known. It may connect to another remote part of the universe or, perhaps, another universe altogether.

In the 1960s, British theoretical physicists Stephen W. Hawking and Roger Penrose developed theorems showing that gravitational collapse leads to singularities under general relativity. Other universes, however, are beyond the domain of experimental science. Physics recognizes that black holes emit radiation, known as Hawking radiation, due to quantum effects near the event horizon, leading to the theoretical possibility that black holes can gradually evaporate over immense time scales.

Applications

According to gravitational theory, singularities are formed when massive stars go through their evolutionary stages and eventually die out. During the lifetime of a star, two major forces work in opposition to each other. Within the core of a star, the process of nuclear fusion occurs, and a vast quantity of heat flows from the core toward the outer layers of the star.

Because of this outward flow of heat, the tendency would be for the star to expand. This result would occur if it were not for the crushing force of gravity. During the main portion of a star’s lifetime, these two forces are balanced and the star is in equilibrium.

When a sufficiently massive star exhausts its supply of nuclear fuel in its core, nuclear burning begins in the layers surrounding the core. This burning causes an expansion of the outer layers of the star against the force of gravity. In the case of a massive star, the core may have been entirely converted into iron and at this stage, electron degeneracy pressure supports the core; that is, further compression is resisted by quantum effects. New iron, which is being made in the shell around the core, now begins to fall inward upon the core. This tremendous buildup in mass causes core temperatures to skyrocket, perhaps exceeding 5 billion degrees Celsius (9 billion degrees Fahrenheit). At this temperature, energetic photons break up the iron atoms into elementary particles. Protons and electrons are combined in a process called neutronization to form a core of pure neutrons. The core’s mass is so great that not even the neutron’s degenerate pressure can stop the collapse. The collapse continues until the matter that made up the core is crushed down to a point, the singularity.

For each gravitational body in the universe, there exists a Schwarzschild radius. If the body in question collapses within that radius, according to theory, a singularity must exist.

Consider the sun as an example. The velocity that it takes to escape from the sun’s surface is about 625 kilometers (388.35 miles) per second. If the sun were collapsing, then the escape velocity would increase as the sun became smaller. At about one-tenth of the sun’s current radius, the escape velocity would be nearly 2,000 kilometers (1,243 miles) per second. As the collapse continued, light leaving the surface of the sun would be distorted in its path and would be strongly redshifted (the spectrum would be displaced toward longer wavelengths). When the sun’s radius became 2.95 kilometers, the escape velocity from its surface would exactly equal the speed of light; the Schwarzschild radius will have been reached. If the sun were to continue its collapse, then a singularity would be formed.

It is believed that only massive stars can undergo such a process as a gravitational collapse to the Schwarzschild radius and beyond. It is the high mass of the core that prevents electron or neutron degenerate pressure from stopping the collapse and forming a white dwarf or a neutron star.

Advances in technology have taken the world of singularities beyond the exercise in mathematics stage. Scientists have gathered data which, perhaps, may prove the existence of these bizarre objects. Advancements in computational relativity have allowed physicists to accurately simulate black hole mergers and accretion phenomena, closely matching gravitational wave observations and expanding the scientific understanding of black holes.

Although black holes cannot be optically observed, they may be detected by the gravitational influence that they have on a neighboring star. For example, gases falling into a black hole might be detected, or physicists may discover strong gravitational perturbations in the movement of a star in a binary system with a black hole. Another effect of black holes may be “gravitational lensing,” in which a massive body such as a black hole distorts the light heading toward Earth from a distant galaxy. As the light bends around either side of the black hole and is recorded on Earth, it may appear that there are two galaxies instead of one. Direct observational evidence for black holes significantly advanced in the 2010s and 2020s, notably through gravitational wave detections by the LIGO and Virgo observatories and images from the Event Horizon Telescope (EHT), confirming predictions of Einstein’s theory, and transforming black holes from theoretical concepts into observable astrophysical objects. In 2019, the EHT collaboration released the first direct image of a black hole centered in the elliptical galaxy M87. This was followed in 2022 by images of Sagittarius A, the black hole at the center of the Milky Way galaxy, with improved techniques enabling more detailed imaging of black hole environments and measurements of accretion flows and magnetic fields.

An additional method of black-hole detection may be in the location of X-ray emitting objects. One model for X-ray emission suggests that, as the black hole attracts gas and dust into its gravitational field, the matter spreads out and forms a thin accretion disk. Because the belts of matter that are nearest to the black hole are orbiting faster, friction between the belts causes the generation of tremendous amounts of heat. As the heat builds up, radiation in X-ray frequencies is released. Astronomers have speculated that many black holes may be identified from the dozens of known X-ray sources in the universe.

Context

The idea of a gravitational singularity was first suggested in 1796 by the French astronomer and mathematician Pierre-Simon Laplace. At that time, light was thought to consist of tiny little particles or corpuscles. Laplace considered the possibility that a massive body might have enough gravitational attraction to keep these particles from escaping. He speculated on the possibility that space may contain an infinite number of massive gravitational bodies. Since there was no way to substantiate his hypothesis, however, it was soon discarded.

The idea was revived again after the Schwarzschild solution was published. Although he had described a static black hole, Schwarzschild had no idea whether such an object could actually exist. His paper was purely theoretical mathematics.

In 1939, Einstein published a paper in which he attempted to demonstrate that it was impossible for matter to be so highly compressed that it could collapse within its Schwarzschild radius. At that time, it was generally believed that, when a star decayed, it would lose most of its mass during the supernova stage. It was assumed that the small remnant of the core would decay into a white dwarf.

Later that same year, physicist J. Robert Oppenheimer of the University of California, Berkeley and his colleagues proposed that, if a star exceeded the sun’s mass by a small amount, it would be massive enough to collapse through its Schwarzschild radius. Oppenheimer discussed various types of condensed states of matter and proposed laws that would apply to matter in the ultradense state. He found that, with a massive hypothetical body, there was no opposing force that could stop the collapse into a singularity.

Interest in the possibility of black holes in space was rekindled in the early 1960s with the discovery of quasi-stellar radio sources (Quasars). The mass of these objects was believed to be far in excess of the mass that Oppenheimer had predicted as being capable of infinite collapse.

Calculations completed by John Wheeler at Princeton University regarding spherical, static bodies indicate that such a body, if massive enough, will collapse indefinitely. Eventually, an event horizon would form around such an object, effectively shielding it from observation by any other part of the universe.

Principal terms

BLACK HOLE: the remains of the core of a massive star after its nuclear fuel is exhausted and the star has gone through the supernova stage

EVENT HORIZON: the Schwarzschild radius, a boundary within which events cannot be observed by the outside universe

GENERAL THEORY OF RELATIVITY: Albert Einstein’s theory of gravity, proposed in 1916, which states that the curvature of space-time causes gravitational fields

SCHWARZSCHILD RADIUS: the distance from the surface of a star at which the velocity needed to escape the star’s gravity is equal to the speed of light

SINGULARITY: a point in space-time at which matter is crushed to a state of zero volume and infinite density

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