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The reason for the belief that the solution could be unphysical is due to the fact that the high degree of symmetry could indeed conduct to the appearance of singularities, as it already did in Newtonian gravity. For example, a spherically symmetric cloud of dust in Newtonian gravity will collapse at the origin and form a singularity due to the matter achieving infinity density at that point. However, it the cloud was rotating in the slighest bit, the centrifugal effect would keep the singularity from forming. Hence, it was not clear whether the singularity on the Schwarzschild solution was a mere accident due to symmetry or an actual prediction of General Relativity.

There are more black hole solutions within General Relativity. Usually, one considers the stationary solutions (i.e., those that possess a time translation symmetry). If one assumes all matter on spacetime is comprised of electromagnetic fields, the no-hair theorems imply that the only possible solutions are the reissner-nordstrom-metric, the kerr-metric, and the kerr-newman-metric (the latter having the remaining ones as special cases).

Black holes can be formed by means of gravitational collapse of a sufficiently massive star, when it can continue interacting gravitationally with its surroundings. For example, gravitational-wave-detectors, such as LIGO, Virgo, and KAGRA, have observed a number of black hole and neutron star mergers. Notice then that, even though black holes do not emit light, they can be studied experimentally by means of their gravitational influence.

While this approach involved gravitational-waves, there are also other possibilities. Analyzing the orbits of stars at the center of the Milky Way led to the discovery of the presence of a "supermassive compact object at the center of our galaxy", in the words chosen by the Nobel Prize when justifying half of the 2020 Nobel Prize in Physics being awarded to Andrea Ghez and Reinhard Genzel for this discovery.

The reason for the belief that the solution could be unphysical is because the high degree of symmetry could indeed conduct the appearance of singularities, as it already did in Newtonian gravity. For example, a spherically symmetric cloud of dust in Newtonian gravity will collapse at the origin and form a singularity due to the matter achieving infinity density at that point. However, if the cloud was rotating in the slightest bit, the centrifugal effect would keep the singularity from forming. Hence, it was not clear whether the singularity on the Schwarzschild solution was a mere accident due to symmetry or an actual prediction of General Relativity.

There are more black hole solutions within General Relativity. Usually, one considers the stationary solutions (i.e., those that possess a time translation symmetry). If one assumes all matter on spacetime is comprised of electromagnetic fields, the no-hair theorems implies that the only possible solutions are the reissner-nordstrom-metric, the kerr-metric, and the kerr-newman-metric (the latter having the remaining ones as special cases).

Black holes can be formed by means of the gravitational collapse of a sufficiently massive star when it can continue interacting gravitationally with its surroundings. For example, gravitational-wave-detectors, such as LIGO, Virgo, and KAGRA, have observed a number of black hole and neutron star mergers. Notice then that even though black holes do not emit light, they can be studied experimentally by means of their gravitational influence.

While this approach involved gravitational-waves, there are also other possibilities. Analyzing the orbits of stars at the center of the Milky Way led to the discovery of the presence of a "supermassive compact object at the center of our galaxy." In the words, chosen by the Nobel Prize when justifying half of the 2020 Nobel Prize in Physics being awarded to Andrea Ghez and Reinhard Genzel for this discovery.

A black hole is a geometrically defined region of spacetime exhibiting such strong gravitational effects that nothing—including particles and electromagnetic radiation such as light—can escape from inside it.[1] The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole.[2][3] The boundary of the region from which no escape is possible is called the event horizon. Although crossing the event horizon has enormous effect on the fate of the object crossing it, it appears to have no locally detectable features. In many ways a black hole acts like an ideal black body, as it reflects no light.[4][5] Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it essentially impossible to observe.

Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Black holes were long considered a mathematical curiosity; it was during the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality.

Black holes of stellar mass are expected to form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings. By absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses (M☉) may form. There is general consensus that supermassive black holes exist in the centers of most galaxies.

Despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Matter falling onto a black hole can form an accretion disk heated by friction, forming some of the brightest objects in the universe. If there are other stars orbiting a black hole, their orbit can be used to determine its mass and location. Such observations can be used to exclude possible alternatives (such as neutron stars). In this way, astronomers have identified numerous stellar black hole candidates in binary systems, and established that the radio source known as Sagittarius A*, at the core of our own Milky Way galaxy, contains a super massive black hole of about 4.3 million solar masses.

Brief Summary

Intuitively, black holes are compact regions of spacetime from which nothing, not even light, can escape. They were originally conceived within newtonian-gravity by considering a body so massive that the escape velocity at its surface would be larger than the speed of light (and hence they would be seen as black), but their study became much more profound within the framework of general-relativity.

Black Holes in General Relativity

The first black hole solution of the Einstein Field Equations, the main equations of General Relativity, was the Schwarzschild solution, which describes a spherically symmetric, vacuum spacetime. By comparing the solution to the predictions of Newtonian gravity, one finds out that it is characterized by a parameter $M$ which can be identified as the mass of a spherical body at the center of the spherical system of coordinates. Spacetime is then described by the metric $$\textrm{d}s^2 = - \left(1 - \frac{2M}{r}\right)\textrm{d}t^2 + \left(1 - \frac{2M}{r}\right)^{-1}\textrm{d}r^2 + r^2 \textrm{d}\theta^2 + r^2 \sin^2\theta \textrm{d}\phi^2,$$ where geometrical units, with $G = c = 1$, are employed.

One can then notice the presence of two disturbing features of this metric-tensor: it is ill-defined at $r = 2M$ and at $r = 0$. While the former was eventually seen to be a coordinate singularity, just like the issues one has at $\theta = 0$ and $\theta = \pi$ due to the use of spherical coordinates, the latter was deemed problematic and possibly unphysical.

The reason for the belief that the solution could be unphysical is due to the fact that the high degree of symmetry could indeed conduct to the appearance of singularities, as it already did in Newtonian gravity. For example, a spherically symmetric cloud of dust in Newtonian gravity will collapse at the origin and form a singularity due to the matter achieving infinity density at that point. However, it the cloud was rotating in the slighest bit, the centrifugal effect would keep the singularity from forming. Hence, it was not clear whether the singularity on the Schwarzschild solution was a mere accident due to symmetry or an actual prediction of General Relativity.

While the Schwarzschild solution was proposed in the 1910s, the matter would only be settled in the 1960s, with the work of R. Penrose and S. Hawking concerning the singularity theorems, which employed global techniques to show that the formation of singularities is indeed a robust prediction of General Relativity and not a mere consequence of symmetry considerations. Penrose would later be awarded half of the 2020 Nobel Prize in Physics due to his work on this.

There are more black hole solutions within General Relativity. Usually, one considers the stationary solutions (i.e., those that possess a time translation symmetry). If one assumes all matter on spacetime is comprised of electromagnetic fields, the no-hair theorems imply that the only possible solutions are the reissner-nordstrom-metric, the kerr-metric, and the kerr-newman-metric (the latter having the remaining ones as special cases).

Astrophysical Considerations

Black holes can be formed by means of gravitational collapse of a sufficiently massive star, when it can continue interacting gravitationally with its surroundings. For example, gravitational-wave-detectors, such as LIGO, Virgo, and KAGRA, have observed a number of black hole and neutron star mergers. Notice then that, even though black holes do not emit light, they can be studied experimentally by means of their gravitational influence.

While this approach involved gravitational-waves, there are also other possibilities. Analyzing the orbits of stars at the center of the Milky Way led to the discovery of the presence of a "supermassive compact object at the center of our galaxy", in the words chosen by the Nobel Prize when justifying half of the 2020 Nobel Prize in Physics being awarded to Andrea Ghez and Reinhard Genzel for this discovery.

Furthermore, the presence of matter surrounding the black hole and the gravitational-lensing effects due to the black hole's massive gravity also provide a way of performing experimental observations. By exploiting this, the [Event Horizon Telescope] (https://en.wikipedia.org/wiki/Event_Horizon_Telescope) was able to take the first picture of a black hole.

A black hole is a geometrically defined region of spacetime exhibiting such strong gravitational effects that nothing—including particles and electromagnetic radiation such as light—can escape from inside it.[1] The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole.[2][3] The boundary of the region from which no escape is possible is called the event horizon. Although crossing the event horizon has enormous effect on the fate of the object crossing it, it appears to have no locally detectable features. In many ways a black hole acts like an ideal black body, as it reflects no light.[4][5] Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it essentially impossible to observe.

Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Black holes were long considered a mathematical curiosity; it was during the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality.

Black holes of stellar mass are expected to form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings. By absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses (M☉) may form. There is general consensus that supermassive black holes exist in the centers of most galaxies.

Despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Matter falling onto a black hole can form an accretion disk heated by friction, forming some of the brightest objects in the universe. If there are other stars orbiting a black hole, their orbit can be used to determine its mass and location. Such observations can be used to exclude possible alternatives (such as neutron stars). In this way, astronomers have identified numerous stellar black hole candidates in binary systems, and established that the radio source known as Sagittarius A*, at the core of our own Milky Way galaxy, contains a super massive black hole of about 4.3 million solar masses.