Schwarzschild Solves the Equations of General Relativity Summary

  • Last updated on November 10, 2022

Karl Schwarzschild’s exact solution to the equations of Einstein’s theory of general relativity described a gravitational black hole.

Summary of Event

In 1915, Albert Einstein published his theory of general relativity. This revolutionary new concept of curved space-time Space-time[Space time] stood in opposition to the universal law of gravitation, which had been developed by Sir Isaac Newton in 1665. According to Newton, gravity is an attractive force acting between all particles of matter in the universe. Einstein believed that gravity is a consequence of the shape of space-time. Local space-time, according to the theory of general relativity, is distorted by the presence of a large mass such as a star or a planet. Objects traveling in close proximity to a massive body would therefore travel in a curved path—hence the appearance of a gravitational field. Gravitation theories General relativity Relativity;general Black holes Physics;Schwarzschild solution Schwarzschild solution Astrophysics;black holes [kw]Schwarzschild Solves the Equations of General Relativity (1916) [kw]Equations of General Relativity, Schwarzschild Solves the (1916) [kw]General Relativity, Schwarzschild Solves the Equations of (1916) [kw]Relativity, Schwarzschild Solves the Equations of General (1916) General relativity Relativity;general Black holes Physics;Schwarzschild solution Schwarzschild solution Astrophysics;black holes [g]Germany;1916: Schwarzschild Solves the Equations of General Relativity[03940] [c]Science and technology;1916: Schwarzschild Solves the Equations of General Relativity[03940] [c]Physics;1916: Schwarzschild Solves the Equations of General Relativity[03940] [c]Astronomy;1916: Schwarzschild Solves the Equations of General Relativity[03940] [c]Mathematics;1916: Schwarzschild Solves the Equations of General Relativity[03940] Schwarzschild, Karl Einstein, Albert Newton, Sir Isaac

When the theory of general relativity was first proposed, its mathematics were thought to be beyond comprehension. In fact, it was frequently stated that only about twelve or so scientists in the world completely understood the theory. To appreciate the complexity, consider that the theory contains sixteen separate equations, each of which is a nonlinear partial differential equation for sixteen separate unknown functions. In addition, problems arise even after the equations are solved because interpretation of the solutions is extremely difficult. Even today, many of the theory’s implications remain to be understood.

The first person to find an exact solution to the equations of the theory of general relativity was the German physicist Karl Schwarzschild. Prior to Schwarzschild’s work, the only solutions to the equations had been approximations. In 1916, when Schwarzschild was working on his solution, Germany was at war. Because of his patriotism, the forty-year-old Schwarzschild insisted on serving in the German armed forces. Various campaigns took him to Belgium, France, and finally to Russia. While serving in Russia he contracted pemphigus, a fatal autoimmune disease. Although he became too ill to continue in the military service, he continued to work on the equations.

Shortly after Schwarzschild returned to Germany, he completed his paper titled “Über das Gravitationsfeld eines Mass enpunktes nach der Einstein Theorie” "Über das Gravitationsfeld eines Mass enpunktes nach der Einstein Theorie" (Schwarzschild)[Über das Gravitationsfeld eines Mass enpunktes nach der Einstein Theorie] (on the field of gravity of a point mass in the theory of Einstein) and sent a copy to Einstein. Within a few months following completion of his work, Schwarzschild died. His paper was published in the 1916 edition of Sitzungsberichte der Preussichen Akademie der Wissenschaften zu Berlin (the journal of the Royal Prussian Academy of Sciences).

Schwarzschild had sought to investigate what would happen if gravity around a spherical body became infinitely powerful. He was also concerned with finding the least complex explanation. The result is known as the Schwarzschild solution. This solution describes a bizarre object: a black hole. The French astronomer Pierre-Simon Laplace Laplace, Pierre-Simon first considered the possibility of an object so dense that light itself could not escape from its surface. Unfortunately, at the time (1796) there was no way to test his ideas, and they were soon discarded. When Schwarzschild derived his solution to Einstein’s equations, the scientific community did not readily recognize that, like Laplace, he was describing a model for a black hole. Difficulties in interpreting the Schwarzschild solution cast some doubt on its validity until the 1960’s, when its true significance was recognized.

The formation of a black hole is believed to be the final stage in the decay of a massive star. During its life, the star produces energy in its core through the process of nuclear fusion. Nuclear fusion The resulting outward flow of heat and energy enables the various layers of the star to balance the inward force of gravity. When nuclear fuel is exhausted in the core, the star begins its decay stage. The exact sequence of events depends entirely on the mass of the star.

Toward the end of its life, a massive star goes through the supernova Supernovas stage, where much of its outer material is blasted off into space. The core—no longer producing energy to counteract the crushing force of gravity—begins to collapse. If the star’s mass is 1.4 or fewer solar masses, the final decay product is a white dwarf. Dwarf stars White dwarf stars At this stage, the pressure exerted by the electrons of the atoms making up the core is sufficient to stop the collapse. This hot carbon mass will eventually cool to become a black dwarf. Black dwarf stars If the mass of the decayed star is between 1.4 and 3.1 times the mass of the Sun, gravity causes a much more extensive collapse. At this point, the pressure is so great that electrons cannot support the core. Electrons and protons combine to form neutrons in a process called neutronization. The result is a neutron star. Neutron stars If the remnant is greater than 3.1 solar masses, neutrons are not able to counteract the force of gravity, and the star continues to collapse. As this collapse proceeds, the surface gravity for the collapsed star becomes greater and greater. As a result, the velocity needed to escape this gravitational body increases. When the escape velocity reaches the velocity of light, further collapse results in the formation of a black hole.

A massive star may end its life as a black hole. During its main sequence (left), radiation emits outward. As the core burns (center), the star begins to collapse in on itself. Finally (right), the increasing mass at the core is so great that gravity is extremely strong, preventing any radiation (including light) from escaping.





This distance, at which the escape velocity is equal to the velocity of light, is the distance calculated by Schwarzschild in his solution to Einstein’s equations; it is known as the Schwarzschild radius Schwarzschild radius or the event horizon. Event horizon Beyond this point, there is no way of determining events. It is an area that is totally disconnected from our space and time. When a star collapses within its Schwarzschild radius, it becomes a black hole. The term “black hole” comes from the fact that no visible light or any other form of electromagnetic radiation can escape from such an object. It therefore cannot be detected optically. In theory, any object could become a black hole if it were compressed enough. If Earth were shrunk to slightly less than 1 centimeter (0.39 inch) in radius, it would become a black hole. If the Sun were to be compressed to a radius of less than 3 kilometers (1.86 miles), it would become a black hole.

The diameter of the Schwarzschild radius of a black hole depends on the mass of the decayed stellar core. For example, a decayed core with a mass five times greater than the Sun would have an event horizon with a radius of 30 kilometers (18.64 miles). A stellar remnant of twenty solar masses would have an event horizon with a 60-kilometer (37.28-mile) radius. Within this boundary, however, the remains of the star continue to collapse to a point of infinite pressure, infinite density, and infinite curvature of space-time. This point is known as the Schwarzschild singularity. Schwarzschild singularity

The Schwarzschild solution describes a black hole that has neither electrical charge nor rotation, but only mass. Consider a body falling toward such an object. As the body passed through the Schwarzschild radius, it would feel nothing unusual. For observers watching from some distance away, however, the situation would appear quite different. Observers would notice the effects on time predicted by the theory of general relativity. As the body approached the event horizon, its onboard clock would appear to slow. When the body reached the event horizon, it and its clock would both appear to stop, frozen in time. The body would enter the event horizon and plunge—at nearly the speed of light—toward the singularity. At this point, what would happen is not possible to predict because the laws of physics are no longer valid. An interesting solution to this problem was presented in the 1930’s: The Einstein-Rosen bridge interpretation described space-time warped into a tunnel or a bridge to another universe or to some other part of the universe. As of the early twenty-first century, however, the concept of infinite space-time warps can be the topic of little more than interesting speculation for cosmologists.


In 1916, when Schwarzschild published his exact solution to the equations of general relativity, the work was thought to be an exercise in mathematics without any actual application. Schwarzschild believed that his solution would have no more application in the natural sciences than the approximate solution that Einstein had worked out previously. The true significance of Schwarzschild’s work was not recognized until more study had been done on stellar structure and evolution. An important step was taken in 1931 when the astronomer Subrahmanyan Chandrasekhar Chandrasekhar, Subrahmanyan completed calculations that described the interior of a white dwarf star. At that time, he did not consider the fate of very massive stars, but English astronomer Arthur Stanley Eddington Eddington, Arthur Stanley proposed that massive stars in their death stages continue to radiate energy as they become smaller and smaller. At some point, they reach equilibrium.

In 1939, the American physicist J. Robert Oppenheimer Oppenheimer, J. Robert and his student Hartland S. Snyder Snyder, Hartland S. showed that a star that possesses enough mass will collapse indefinitely. In their paper, they actually expanded on the work of Schwarzschild and proposed that black holes may exist in the universe. It is now fully recognized that the Schwarzschild solution describes a black hole. The type of black hole described by the Schwarzschild equations is static. A static black hole does not rotate and has no electrical charge. The only property possessed by such a body is mass.

Based on the work of Schwarzschild, Oppenheimer, Chandrasekhar, and others, scientists have formulated a model for the formation of a black hole. Although there is some debate on the lower limit of solar masses necessary for black hole formation, a value of about 3.1 is generally acceptable. This means that if the decayed star has a mass of 3.1 solar masses or greater, it will become a black hole. As the stellar remnant begins to collapse, gravity compresses the star’s matter into a smaller and smaller volume. The curvature of space-time around the body becomes more pronounced, and, as a result, beams of light no longer leave the surface in a straight path. As the gravitational collapse continues, more light beams are drawn back to the surface. When the star collapses within the Schwarzschild radius, light can no longer escape; the object has become a black hole. The object described by Schwarzschild consists of only one event horizon, or Schwarzschild radius. Within this boundary lies the singularity, the point that constitutes the remains of the collapsed star.

Variations on the work of Schwarzschild have produced such theoretical objects as rotating black holes, black holes with electrical charge, and those with both charge and rotation. At this time, there is no actual proof that these objects exist. Astrophysicists, however, have identified several objects in the universe that may be black holes. Further research will determine whether the Schwarzschild solution describes a real subject or whether it was only an exercise in theoretical mathematics. General relativity Relativity;general Black holes Physics;Schwarzschild solution Schwarzschild solution Astrophysics;black holes

Further Reading
  • citation-type="booksimple"

    xlink:type="simple">Asimov, Isaac. The Collapsing Universe. 1977. Reprint. New York: Pocket Books, 1986. A very readable volume describing such topics as the forces of nature, planets and planetary formation, and the stages of stellar evolution. Suitable for lay readers.
  • citation-type="booksimple"

    xlink:type="simple">Greenstein, George. Frozen Star. New York: Freundlich Books, 1983. Discusses pulsars, black holes, and stellar evolution. Recommended for readers with some general physics and astronomy background.
  • citation-type="booksimple"

    xlink:type="simple">Kaufmann, William J., III. Black Holes and Warped Spacetime. New York: W. H. Freeman, 1979. Well-illustrated volume deals with stellar evolution and the warped space-time of general relativity. Presents the study of the structure and properties of black holes. Suitable for the average reader.
  • citation-type="booksimple"

    xlink:type="simple">Krane, Kenneth S. Modern Physics. 2d ed. New York: John Wiley & Sons, 1995. A highly technical treatment of selected topics in modern physics, including special and general relativity, quantum mechanics, and nuclear physics. Intended as a textbook for advanced undergraduate courses in modern physics.
  • citation-type="booksimple"

    xlink:type="simple">Miller, Arthur I. Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes. Boston: Houghton Mifflin, 2005. Provides background on the history of the idea of black holes and describes the debate between Chandrasekhar and Eddington concerning the nature of black holes as well as the implications of that debate for twentieth century science.
  • citation-type="booksimple"

    xlink:type="simple">Shipman, Harry L. Black Holes, Quasars, and the Universe. 2d ed. Boston: Houghton Mifflin, 1980. Covers such topics as stellar evolution, galaxies, active galaxies, cosmology, and astrophysics. Designed for readers with some background in elementary physics and astronomy.
  • citation-type="booksimple"

    xlink:type="simple">Sullivan, Walter. Black Holes: The Edge of Space, the End of Time. Garden City, N.Y.: Doubleday, 1979. Describes the evolution of the concept of black holes from early theory to modern cosmology. Well illustrated. Suitable for the general reader.
  • citation-type="booksimple"

    xlink:type="simple">Susskind, Leonard, and James Lindesay. An Introduction to Black Holes, Information, and the String Theory Revolution: The Holographic Universe. Hackensack, N.J.: World Scientific, 2004. Explains concepts that physicists of the early twenty-first century have developed in relation to black holes and thinking about space, time, matter, and information.

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Oppenheimer Calculates the Nature of Black Holes

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