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86-408: The Schwarzschild radius is given as r s = 2 G M c 2 , {\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},} where G is the gravitational constant , M is the object mass, and c is the speed of light . In 1916, Karl Schwarzschild obtained the exact solution to Einstein's field equations for the gravitational field outside

172-690: A = J / M c {\displaystyle a=J/Mc} . In this case, "event horizons disappear" means when the solutions are complex for  r ± {\displaystyle r_{\pm }} , or  μ 2 < a 2 {\displaystyle \mu ^{2}<a^{2}} . However, this corresponds to a case where J {\displaystyle J} exceeds G M 2 / c {\displaystyle GM^{2}/c} (or in Planck units , J > M 2 {\displaystyle J>M^{2}} ) ; i.e.

258-403: A stellar black hole . A small mass has an extremely small Schwarzschild radius. A black hole of mass similar to that of Mount Everest would have a Schwarzschild radius much smaller than a nanometre . Its average density at that size would be so high that no known mechanism could form such extremely compact objects. Such black holes might possibly be formed in an early stage of the evolution of

344-406: A (non-rotating) black hole is a spherical region in space that surrounds the singularity at its center; it is not the singularity itself.) With that in mind, the average density of a supermassive black hole can be less than the density of water. The Schwarzschild radius of a body is proportional to its mass and therefore to its volume, assuming that the body has a constant mass-density. In contrast,

430-425: A background field to locate the singularity. A singularity in general relativity, on the other hand, is more complex because spacetime itself becomes ill-defined, and the singularity is no longer part of the regular spacetime manifold. In general relativity, a singularity cannot be defined by "where" or "when". Some theories, such as the theory of loop quantum gravity , suggest that singularities may not exist. This

516-548: A black hole would have a radius of 400 920 754 km (about 2.67 AU ). Classification of black holes by type: A classification of black holes by mass: Gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton 's law of universal gravitation and in Albert Einstein 's theory of general relativity . It

602-454: A certain point (for stars this is the Schwarzschild radius ) would form a black hole, inside which a singularity (covered by an event horizon ) would be formed. The Penrose–Hawking singularity theorems define a singularity to have geodesics that cannot be extended in a smooth manner. The termination of such a geodesic is considered to be the singularity. Modern theory asserts that

688-410: A change of coordinates. An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon . However, spacetime at the event horizon is regular . The regularity becomes evident when changing to another coordinate system (such as

774-445: A complete and precise definition of singularities in the theory of general relativity, the current best theory of gravity, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete . Gravitational singularities are mainly considered in the context of general relativity, where density would become infinite at

860-492: A density of 4.5 g/cm ( ⁠4 + 1 / 2 ⁠ times the density of water), about 20% below the modern value. This immediately led to estimates on the densities and masses of the Sun , Moon and planets , sent by Hutton to Jérôme Lalande for inclusion in his planetary tables. As discussed above, establishing the average density of Earth is equivalent to measuring the gravitational constant, given Earth's mean radius and

946-436: A distance. Motivated by such philosophy of loop quantum gravity, recently it has been shown that such conceptions can be realized through some elementary constructions based on the refinement of the first axiom of geometry, namely, the concept of a point by considering Klein's prescription of accounting for the extension of a small spot that represents or demonstrates a point, which was a programmatic call that he called as

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1032-462: A function of altitude) type. Pendulum experiments still continued to be performed, by Robert von Sterneck (1883, results between 5.0 and 6.3 g/cm ) and Thomas Corwin Mendenhall (1880, 5.77 g/cm ). Cavendish's result was first improved upon by John Henry Poynting (1891), who published a value of 5.49(3) g⋅cm , differing from the modern value by 0.2%, but compatible with

1118-404: A fusion of arithmetic and geometry. Klein's program, according to Born, was actually a mathematical route to consider 'natural uncertainty in all observations' while describing 'a physical situation' by means of 'real numbers'. There is only one type of singularity, each with different physical features that have characteristics relevant to the theories from which they originally emerged, such as

1204-454: A missing piece in the theory, as in the ultraviolet catastrophe , re-normalization , and instability of a hydrogen atom predicted by the Larmor formula . In classical field theories, including special relativity but not general relativity, one can say that a solution has a singularity at a particular point in spacetime where certain physical properties become ill-defined, with spacetime serving as

1290-712: A non-rotating, spherically symmetric body with mass M {\displaystyle M} (see Schwarzschild metric ). The solution contained terms of the form 1 − r s / r {\displaystyle 1-{r_{\text{s}}}/r} and 1 1 − r s / r {\displaystyle {\frac {1}{1-{r_{\text{s}}}/r}}} , which becomes singular at r = 0 {\displaystyle r=0} and r = r s {\displaystyle r=r_{\text{s}}} respectively. The r s {\displaystyle r_{\text{s}}} has come to be known as

1376-480: A spacetime is considered singular if it is geodesically incomplete , meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the event horizon of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of

1462-484: A standard value for G with a relative standard uncertainty better than 0.1% has therefore remained rather speculative. By 1969, the value recommended by the National Institute of Standards and Technology (NIST) was cited with a relative standard uncertainty of 0.046% (460 ppm), lowered to 0.012% (120 ppm) by 1986. But the continued publication of conflicting measurements led NIST to considerably increase

1548-508: A supermassive black hole. It is thought that supermassive black holes like these do not form immediately from the singular collapse of a cluster of stars. Instead they may begin life as smaller, stellar-sized black holes and grow larger by the accretion of matter, or even of other black holes. The Schwarzschild radius of the supermassive black hole at the Galactic Center of the Milky Way

1634-415: Is inversely related to mass . All known black hole candidates are so large that their temperature is far below that of the cosmic background radiation, which means they will gain energy on net by absorbing this radiation. They cannot begin to lose energy on net until the background temperature falls below their own temperature. This will occur at a cosmological redshift of more than one million, rather than

1720-465: Is a spacetime singularity and cannot be removed. The Schwarzschild radius is nonetheless a physically relevant quantity, as noted above and below. This expression had previously been calculated, using Newtonian mechanics, as the radius of a spherically symmetric body at which the escape velocity was equal to the speed of light. It had been identified in the 18th century by John Michell and Pierre-Simon Laplace . The Schwarzschild radius of an object

1806-553: Is a physical constant that is difficult to measure with high accuracy. This is because the gravitational force is an extremely weak force as compared to other fundamental forces at the laboratory scale. In SI units, the CODATA -recommended value of the gravitational constant is: The relative standard uncertainty is 2.2 × 10 . Due to its use as a defining constant in some systems of natural units , particularly geometrized unit systems such as Planck units and Stoney units ,

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1892-570: Is also known as the universal gravitational constant , the Newtonian constant of gravitation , or the Cavendish gravitational constant , denoted by the capital letter G . In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse square of their distance . In the Einstein field equations , it quantifies

1978-484: Is also true for such classical unified field theories as the Einstein–Maxwell–Dirac equations. The idea can be stated in the form that, due to quantum gravity effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter, or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at

2064-501: Is another form of the Heisenberg uncertainty principle on the Planck scale . (See also Virtual black hole ). The Schwarzschild radius equation can be manipulated to yield an expression that gives the largest possible radius from an input density that doesn't form a black hole. Taking the input density as ρ , For example, the density of water is 1000 kg/m . This means the largest amount of water you can have without forming

2150-453: Is approximately 12 million kilometres. Its mass is about 4.1 million  M ☉ . Stellar black holes have much greater average densities than supermassive black holes. If one accumulates matter at nuclear density (the density of the nucleus of an atom, about 10 kg/m ; neutron stars also reach this density), such an accumulation would fall within its own Schwarzschild radius at about 3  M ☉ and thus would be

2236-405: Is attributed to Henry Cavendish in a 1798 experiment . According to Newton's law of universal gravitation, the magnitude of the attractive force ( F ) between two bodies each with a spherically symmetric density distribution is directly proportional to the product of their masses , m 1 and m 2 , and inversely proportional to the square of the distance, r , directed along

2322-500: Is defined as mass of a black hole divided by the volume of its Schwarzschild sphere. As the Schwarzschild radius is linearly related to mass, while the enclosed volume corresponds to the third power of the radius, small black holes are therefore much more dense than large ones. The volume enclosed in the event horizon of the most massive black holes has an average density lower than main sequence stars. A supermassive black hole (SMBH)

2408-403: Is diffeomorphism invariant, is infinite. While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity", in a rotating black hole, also known as a Kerr black hole , the singularity occurs on a ring (a circular line), known as a " ring singularity ". Such a singularity may also theoretically become a wormhole . More generally,

2494-436: Is equivalent to G ≈ 8 × 10  m ⋅kg ⋅s . The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton's death, by Henry Cavendish. He determined a value for G implicitly, using a torsion balance invented by the geologist Rev. John Michell (1753). He used a horizontal torsion beam with lead balls whose inertia (in relation to

2580-941: Is exact only within the approximation of the Earth's orbit around the Sun as a two-body problem in Newtonian mechanics, the measured quantities contain corrections from the perturbations from other bodies in the solar system and from general relativity. From 1964 until 2012, however, it was used as the definition of the astronomical unit and thus held by definition: 1   A U = ( G M 4 π 2 y r 2 ) 1 3 ≈ 1.495979 × 10 11   m . {\displaystyle 1\ \mathrm {AU} =\left({\frac {GM}{4\pi ^{2}}}\mathrm {yr} ^{2}\right)^{\frac {1}{3}}\approx 1.495979\times 10^{11}\ \mathrm {m} .} Since 2012,

2666-478: Is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere the coordinate system is used. An example of such a conical singularity is a cosmic string and a Schwarzschild black hole . Solutions to the equations of general relativity or another theory of gravity (such as supergravity ) often result in encountering points where

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2752-449: Is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between general relativity and quantum mechanics ; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity . Trying to find

2838-464: Is proportional to its mass. Accordingly, the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi), whereas Earth 's is approximately 9 mm (0.35 in) and the Moon 's is approximately 0.1 mm (0.0039 in). The simplest way of deriving the Schwarzschild radius comes from the equality of the modulus of a spherical solid mass' rest energy with its gravitational energy: So,

2924-449: Is quite difficult to measure because gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies. Measurements with pendulums were made by Francesco Carlini (1821, 4.39 g/cm ), Edward Sabine (1827, 4.77 g/cm ), Carlo Ignazio Giulio (1841, 4.95 g/cm ) and George Biddell Airy (1854, 6.6 g/cm ). Cavendish's experiment

3010-427: Is referred to as the cosmic censorship hypothesis . However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make

3096-630: Is the cosmological constant , g μν is the metric tensor , T μν is the stress–energy tensor , and κ is the Einstein gravitational constant , a constant originally introduced by Einstein that is directly related to the Newtonian constant of gravitation: κ = 8 π G c 4 ≈ 2.076647 ( 46 ) × 10 − 43 N − 1 . {\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.076647(46)\times 10^{-43}\mathrm {\,N^{-1}} .} The gravitational constant

3182-511: Is the author of the article "Gravitation" in the Encyclopædia Britannica Eleventh Edition (1911). Here, he cites a value of G = 6.66 × 10  m ⋅kg ⋅s with a relative uncertainty of 0.2%. Paul R. Heyl (1930) published the value of 6.670(5) × 10  m ⋅kg ⋅s (relative uncertainty 0.1%), improved to 6.673(3) × 10  m ⋅kg ⋅s (relative uncertainty 0.045% = 450 ppm) in 1942. However, Heyl used

3268-412: Is the largest type of black hole, though there are few official criteria on how such an object is considered so, on the order of hundreds of thousands to billions of solar masses. (Supermassive black holes up to 21 billion (2.1 × 10)  M ☉ have been detected, such as NGC 4889 .) Unlike stellar mass black holes , supermassive black holes have comparatively low average densities. (Note that

3354-555: Is the local gravitational field of Earth (also referred to as free-fall acceleration). Where M ⊕ {\displaystyle M_{\oplus }} is the mass of the Earth and r ⊕ {\displaystyle r_{\oplus }} is the radius of the Earth , the two quantities are related by: g = G M ⊕ r ⊕ 2 . {\displaystyle g=G{\frac {M_{\oplus }}{r_{\oplus }^{2}}}.} The gravitational constant appears in

3440-873: The Compton wavelength ( 2 π ℏ / M c {\displaystyle 2\pi \hbar /Mc} ) corresponding to a given mass are similar when the mass is around one Planck mass ( M = ℏ c / G {\textstyle M={\sqrt {\hbar c/G}}} ), when both are of the same order as the Planck length ( ℏ G / c 3 {\textstyle {\sqrt {\hbar G/c^{3}}}} ). Thus, r s r ∼ ℓ P 2 {\displaystyle r_{s}r\sim \ell _{P}^{2}} or Δ r s Δ r ≥ ℓ P 2 {\displaystyle \Delta r_{s}\Delta r\geq \ell _{P}^{2}} , which

3526-417: The Einstein field equations of general relativity , G μ ν + Λ g μ ν = κ T μ ν , {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,} where G μν is the Einstein tensor (not the gravitational constant despite the use of G ), Λ

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3612-629: The Gaussian gravitational constant was historically in widespread use, k = 0.017 202 098 95 radians per day , expressing the mean angular velocity of the Sun–Earth system. The use of this constant, and the implied definition of the astronomical unit discussed above, has been deprecated by the IAU since 2012. The existence of the constant is implied in Newton's law of universal gravitation as published in

3698-718: The Kruskal coordinates ), where the metric is perfectly smooth . On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest a singularity exists. The existence of the singularity can be verified by noting that the Kretschmann scalar , being the square of the Riemann tensor i.e. R μ ν ρ σ R μ ν ρ σ {\displaystyle R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }} , which

3784-396: The Schwarzschild radius . The physical significance of these singularities was debated for decades. It was found that the one at r = r s {\displaystyle r=r_{\text{s}}} is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that was used; while the one at r = 0 {\displaystyle r=0}

3870-522: The cgs system. Richarz and Krigar-Menzel (1898) attempted a repetition of the Cavendish experiment using 100,000 kg of lead for the attracting mass. The precision of their result of 6.683(11) × 10  m ⋅kg ⋅s was, however, of the same order of magnitude as the other results at the time. Arthur Stanley Mackenzie in The Laws of Gravitation (1899) reviews the work done in the 19th century. Poynting

3956-458: The earliest moments of the Big Bang , but in general, quantum mechanics does not permit particles to inhabit a space smaller than their Compton wavelengths . Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for

4042-430: The mean gravitational acceleration at Earth's surface, by setting G = g R ⊕ 2 M ⊕ = 3 g 4 π R ⊕ ρ ⊕ . {\displaystyle G=g{\frac {R_{\oplus }^{2}}{M_{\oplus }}}={\frac {3g}{4\pi R_{\oplus }\rho _{\oplus }}}.} Based on this, Hutton's 1778 result

4128-457: The metric blows up to infinity. However, many of these points are completely regular , and the infinities are merely a result of using an inappropriate coordinate system at this point . To test whether there is a singularity at a certain point, one must check whether at this point diffeomorphism invariant quantities (i.e. scalars ) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by

4214-496: The universe contains a causal singularity at the start of time ( t =0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite spacetime curvature. Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon , making naked singularities impossible. This

4300-502: The 1680s (although its notation as G dates to the 1890s), but is not calculated in his Philosophiæ Naturalis Principia Mathematica where it postulates the inverse-square law of gravitation. In the Principia , Newton considered the possibility of measuring gravity's strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable. Nevertheless, he had

4386-465: The 2006 CODATA value. An improved cold atom measurement by Rosi et al. was published in 2014 of G = 6.671 91 (99) × 10  m ⋅kg ⋅s . Although much closer to the accepted value (suggesting that the Fixler et al. measurement was erroneous), this result was 325 ppm below the recommended 2014 CODATA value, with non-overlapping standard uncertainty intervals. As of 2018, efforts to re-evaluate

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4472-501: The 2010 value, and one order of magnitude below the 1969 recommendation. The following table shows the NIST recommended values published since 1969: In the January 2007 issue of Science , Fixler et al. described a measurement of the gravitational constant by a new technique, atom interferometry , reporting a value of G = 6.693(34) × 10  m ⋅kg ⋅s , 0.28% (2800 ppm) higher than

4558-550: The AU is defined as 1.495 978 707 × 10  m exactly, and the equation can no longer be taken as holding precisely. The quantity GM —the product of the gravitational constant and the mass of a given astronomical body such as the Sun or Earth—is known as the standard gravitational parameter (also denoted μ ). The standard gravitational parameter GM appears as above in Newton's law of universal gravitation, as well as in formulas for

4644-574: The Kerr metric to  Boyer–Lindquist coordinates , it can be shown  that the coordinate (which is not the radius) of the event horizon is, r ± = μ ± ( μ 2 − a 2 ) 1 / 2 {\displaystyle r_{\pm }=\mu \pm \left(\mu ^{2}-a^{2}\right)^{1/2}} , where  μ = G M / c 2 {\displaystyle \mu =GM/c^{2}} , and 

4730-505: The Schwarzschild radius reads as Any object whose radius is smaller than its Schwarzschild radius is called a black hole . The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body (a rotating black hole operates slightly differently). Neither light nor particles can escape through this surface from the region inside, hence the name "black hole". Black holes can be classified based on their Schwarzschild radius, or equivalently, by their density, where density

4816-468: The average density of Earth and the Earth's mass . His result, ρ 🜨 = 5.448(33) g⋅cm , corresponds to value of G = 6.74(4) × 10  m ⋅kg ⋅s . It is surprisingly accurate, about 1% above the modern value (comparable to the claimed relative standard uncertainty of 0.6%). The accuracy of the measured value of G has increased only modestly since the original Cavendish experiment. G

4902-463: The average density of a planet and the period of a satellite orbiting just above its surface. For elliptical orbits, applying Kepler's 3rd law , expressed in units characteristic of Earth's orbit : where distance is measured in terms of the semi-major axis of Earth's orbit (the astronomical unit , AU), time in years , and mass in the total mass of the orbiting system ( M = M ☉ + M E + M ☾ ). The above equation

4988-457: The center of a black hole without corrections from quantum mechanics , and within astrophysics and cosmology as the earliest state of the universe during the Big Bang . Physicists have not reached a consensus about what actually happens at the extreme densities predicted by singularities (including at the start of the Big Bang). General relativity predicts that any object collapsing beyond

5074-403: The charge exceeds what is normally viewed as the upper limit of its physically possible values. Also, actual astrophysical black holes are not expected to possess any appreciable charge. A black hole possessing the lowest M {\displaystyle M} value consistent with its J {\displaystyle J} and Q {\displaystyle Q} values and

5160-513: The conflicting results of measurements are underway, coordinated by NIST, notably a repetition of the experiments reported by Quinn et al. (2013). In August 2018, a Chinese research group announced new measurements based on torsion balances, 6.674 184 (78) × 10  m ⋅kg ⋅s and 6.674 484 (78) × 10  m ⋅kg ⋅s based on two different methods. These are claimed as the most accurate measurements ever made, with standard uncertainties cited as low as 12 ppm. The difference of 2.7 σ between

5246-534: The deflection of light caused by gravitational lensing , in Kepler's laws of planetary motion , and in the formula for escape velocity . This quantity gives a convenient simplification of various gravity-related formulas. The product GM is known much more accurately than either factor is. Calculations in celestial mechanics can also be carried out using the units of solar masses, mean solar days and astronomical units rather than standard SI units. For this purpose,

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5332-433: The different shapes of the singularities, conical and curved . They have also been hypothesized to occur without event horizons, structures that delineate one spacetime section from another in which events cannot affect past the horizon; these are called naked. A conical singularity occurs when there is a point where the limit of some diffeomorphism invariant quantity does not exist or is infinite, in which case spacetime

5418-404: The experiment had at least proved that the Earth could not be a hollow shell , as some thinkers of the day, including Edmond Halley , had suggested. The Schiehallion experiment , proposed in 1772 and completed in 1776, was the first successful measurement of the mean density of the Earth, and thus indirectly of the gravitational constant. The result reported by Charles Hutton (1778) suggested

5504-417: The gravitational constant is: G ≈ 4.3009 × 10 − 3   p c ⋅ ( k m / s ) 2 M ⊙ − 1 . {\displaystyle G\approx 4.3009\times 10^{-3}\ {\mathrm {pc{\cdot }(km/s)^{2}} \,M_{\odot }}^{-1}.} For situations where tides are important,

5590-693: The gravitational radius in the form r s = 2 ( G / c 3 ) M c {\displaystyle r_{s}=2\,(G/c^{3})Mc} , (see also virtual black hole ). Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows: t r t = 1 − r s r {\displaystyle {\frac {t_{r}}{t}}={\sqrt {1-{\frac {r_{\mathrm {s} }}{r}}}}} where: The Schwarzschild radius ( 2 G M / c 2 {\displaystyle 2GM/c^{2}} ) and

5676-482: The initial state of the universe , at the beginning of the Big Bang, was a singularity. In this case, the universe did not collapse into a black hole, because currently-known calculations and density limits for gravitational collapse are usually based upon objects of relatively constant size, such as stars, and do not necessarily apply in the same way to rapidly expanding space such as the Big Bang. Neither general relativity nor quantum mechanics can currently describe

5762-601: The limits noted above; i.e., one just at the point of losing its event horizon, is termed extremal . Before Stephen Hawking came up with the concept of Hawking radiation , the question of black holes having entropy had been avoided. However, this concept demonstrates that black holes radiate energy, which conserves entropy and solves the incompatibility problems with the second law of thermodynamics . Entropy, however, implies heat and therefore temperature. The loss of energy also implies that black holes do not last forever, but rather evaporate or decay slowly. Black hole temperature

5848-414: The line connecting their centres of mass : F = G m 1 m 2 r 2 . {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}.} The constant of proportionality , G , in this non-relativistic formulation is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g" ( g ), which

5934-555: The metric is regular for all positive values of  r {\displaystyle r} , or in other words, the singularity has no event horizon. However, this corresponds to a case where Q / 4 π ϵ 0 {\displaystyle Q/{\sqrt {4\pi \epsilon _{0}}}} exceeds M G {\displaystyle M{\sqrt {G}}} (or in Planck units, Q > M {\displaystyle Q>M} ) ; i.e.

6020-402: The modern value within the cited relative standard uncertainty of 0.55%. In addition to Poynting, measurements were made by C. V. Boys (1895) and Carl Braun (1897), with compatible results suggesting G = 6.66(1) × 10  m ⋅kg ⋅s . The modern notation involving the constant G was introduced by Boys in 1894 and becomes standard by the end of the 1890s, with values usually cited in

6106-450: The opportunity to estimate the order of magnitude of the constant when he surmised that "the mean density of the earth might be five or six times as great as the density of water", which is equivalent to a gravitational constant of the order: A measurement was attempted in 1738 by Pierre Bouguer and Charles Marie de La Condamine in their " Peruvian expedition ". Bouguer downplayed the significance of their results in 1740, suggesting that

6192-402: The period P of an object in circular orbit around a spherical object obeys G M = 3 π V P 2 , {\displaystyle GM={\frac {3\pi V}{P^{2}}},} where V is the volume inside the radius of the orbit, and M is the total mass of the two objects. It follows that This way of expressing G shows the relationship between

6278-488: The physical radius of the body is proportional to the cube root of its volume. Therefore, as the body accumulates matter at a given fixed density (in this example, 997 kg/m , the density of water), its Schwarzschild radius will increase more quickly than its physical radius. When a body of this density has grown to around 136 million solar masses (1.36 × 10  M ☉ ), its physical radius would be overtaken by its Schwarzschild radius, and thus it would form

6364-457: The relation between the geometry of spacetime and the energy–momentum tensor (also referred to as the stress–energy tensor ). The measured value of the constant is known with some certainty to four significant digits. In SI units , its value is approximately 6.6743 × 10  N⋅m /kg . The modern notation of Newton's law involving G was introduced in the 1890s by C. V. Boys . The first implicit measurement with an accuracy within about 1%

6450-458: The relevant length scales are solar radii rather than parsecs. In these units, the gravitational constant is: G ≈ 1.90809 × 10 5   ( k m / s ) 2 R ⊙ M ⊙ − 1 . {\displaystyle G\approx 1.90809\times 10^{5}\mathrm {\ (km/s)^{2}} \,R_{\odot }M_{\odot }^{-1}.} In orbital mechanics ,

6536-420: The simulation were removed. However, it is hypothesized that light entering a singularity would similarly have its geodesics terminated, thus making the naked singularity look like a black hole. Disappearing event horizons exist in the  Kerr metric , which is a spinning black hole in a vacuum, if the  angular momentum  ( J {\displaystyle J} ) is high enough. Transforming

6622-985: The spin exceeds what is normally viewed as the upper limit of its physically possible values. Similarly, disappearing event horizons can also be seen with the  Reissner–Nordström  geometry of a charged black hole if the charge ( Q {\displaystyle Q} ) is high enough. In this metric, it can be shown  that the singularities occur at r ± = μ ± ( μ 2 − q 2 ) 1 / 2 {\displaystyle r_{\pm }=\mu \pm \left(\mu ^{2}-q^{2}\right)^{1/2}} , where  μ = G M / c 2 {\displaystyle \mu =GM/c^{2}} , and  q 2 = G Q 2 / ( 4 π ϵ 0 c 4 ) {\displaystyle q^{2}=GQ^{2}/\left(4\pi \epsilon _{0}c^{4}\right)} . Of

6708-428: The standard uncertainty in the 1998 recommended value, by a factor of 12, to a standard uncertainty of 0.15%, larger than the one given by Heyl (1930). The uncertainty was again lowered in 2002 and 2006, but once again raised, by a more conservative 20%, in 2010, matching the relative standard uncertainty of 120 ppm published in 1986. For the 2014 update, CODATA reduced the uncertainty to 46 ppm, less than half

6794-416: The statistical spread as his standard deviation, and he admitted himself that measurements using the same material yielded very similar results while measurements using different materials yielded vastly different results. He spent the next 12 years after his 1930 paper to do more precise measurements, hoping that the composition-dependent effect would go away, but it did not, as he noted in his final paper from

6880-420: The three possible cases for the relative values of  μ {\displaystyle \mu } and  q {\displaystyle q} , the case where  μ 2 < q 2 {\displaystyle \mu ^{2}<q^{2}}  causes both  r ± {\displaystyle r_{\pm }} to be complex. This means

6966-415: The torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. In spite of the experimental design being due to Michell, the experiment is now known as the Cavendish experiment for its first successful execution by Cavendish. Cavendish's stated aim was the "weighing of Earth", that is, determining

7052-415: The two results suggests there could be sources of error unaccounted for. Analysis of observations of 580 type Ia supernovae shows that the gravitational constant has varied by less than one part in ten billion per year over the last nine billion years. Spacetime singularity A gravitational singularity , spacetime singularity or simply singularity is a theoretical condition in which gravity

7138-419: The universe, just after the Big Bang , when densities of matter were extremely high. Therefore, these hypothetical miniature black holes are called primordial black holes . When moving to the Planck scale ℓ P = ( G / c 3 ) ℏ {\displaystyle \ell _{P}={\sqrt {(G/c^{3})\,\hbar }}} ≈ 10 m , it is convenient to write

7224-543: The value of the gravitational constant will generally have a numeric value of 1 or a value close to it when expressed in terms of those units. Due to the significant uncertainty in the measured value of G in terms of other known fundamental constants, a similar level of uncertainty will show up in the value of many quantities when expressed in such a unit system. In astrophysics , it is convenient to measure distances in parsecs (pc), velocities in kilometres per second (km/s) and masses in solar units M ⊙ . In these units,

7310-437: The year 1942. Published values of G derived from high-precision measurements since the 1950s have remained compatible with Heyl (1930), but within the relative uncertainty of about 0.1% (or 1000 ppm) have varied rather broadly, and it is not entirely clear if the uncertainty has been reduced at all since the 1942 measurement. Some measurements published in the 1980s to 2000s were, in fact, mutually exclusive. Establishing

7396-423: Was first repeated by Ferdinand Reich (1838, 1842, 1853), who found a value of 5.5832(149) g⋅cm , which is actually worse than Cavendish's result, differing from the modern value by 1.5%. Cornu and Baille (1873), found 5.56 g⋅cm . Cavendish's experiment proved to result in more reliable measurements than pendulum experiments of the "Schiehallion" (deflection) type or "Peruvian" (period as

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