Schwarzschild metric

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Schwarzschild_metric

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916, and around the same time independently by Johannes Droste, who published his much more complete and modern-looking discussion only four months after Schwarzschild.

According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.

The Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. The boundary is not a physical surface, and if a person fell through the event horizon (before being torn apart by tidal forces), they would not notice any physical surface at that position; it is a mathematical surface which is significant in determining the black hole's properties. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass M, so in principle (according to general relativity theory) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation.

The Schwarzschild metric

The Schwarzschild metric is a spherically symmetric Lorentzian metric (here, with signature convention (-, +, +, +),) defined on (a subset of)

${\displaystyle \mathbb {R} \times \left(E^{3}-O\right)\cong \mathbb {R} \times (0,\infty )\times S^{2}}$

where ${\displaystyle E^{3}}$ is 3 dimensional Euclidean space, and ${\displaystyle S^{2}\subset E^{3}}$ is the two sphere. The rotation group ${\displaystyle SO(3)=SO(E^{3})}$ acts on the ${\displaystyle E^{3}-O}$ or ${\displaystyle S^{2}}$ factor as rotations around the center ${\displaystyle O}$, while leaving the first ${\displaystyle \mathbb {R} }$ factor unchanged. The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body. That is, for a spherical body of radius ${\displaystyle R}$ the solution is valid for ${\displaystyle r>R}$. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at ${\displaystyle r=R}$, such as the interior Schwarzschild metric.

In Schwarzschild coordinates ${\displaystyle (t,r,\theta ,\phi )}$ the Schwarzschild metric (or equivalently, the line element for proper time) has the form

${\displaystyle g=-c^{2}\,{d\tau }^{2}=-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)c^{2}\,dt^{2}+\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}+r^{2}g_{\Omega },}$

where ${\displaystyle g_{\Omega }}$ is the metric on the two sphere, i.e. ${\displaystyle g_{\Omega }=\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right)}$. Furthermore,

• ${\displaystyle d\tau ^{2}}$ is positive for time like curves, and ${\displaystyle \tau }$ is the proper time (time measured by a clock moving along the same world line with the test particle),
• ${\displaystyle c}$ is the speed of light,
• ${\displaystyle t}$ is the time coordinate (measured by a stationary clock located infinitely far from the massive body),
• ${\displaystyle r}$ is the radial coordinate (measured as the circumference, divided by 2π, of a sphere centered around the massive body),
• ${\displaystyle \Omega }$ is a point on the two sphere ${\displaystyle S^{2}}$,
• ${\displaystyle \theta }$ is the colatitude of ${\displaystyle \Omega }$ (angle from north, in units of radians) defined after arbitrarily choosing a z-axis,
• ${\displaystyle \phi }$ is the longitude of ${\displaystyle \Omega }$ (also in radians) around the chosen z-axis, and
• ${\displaystyle r_{s}}$ is the Schwarzschild radius of the massive body, a scale factor which is related to its mass ${\displaystyle M}$ by ${\displaystyle r_{s}={\frac {2GM}{c^{2}}}}$, where ${\displaystyle G}$ is the gravitational constant.

The Schwarzschild metric has a singularity for ${\displaystyle r=0}$ which is an intrinsic curvature singularity. It also seems to have a singularity on the event horizon ${\displaystyle r=r_{s}}$. Depending on the point of view, the metric is therefore defined only on the exterior region ${\displaystyle r>r_{s}}$, only on the interior region ${\displaystyle r$ or their disjoint union. However, the metric is actually non singular across the event horizon as one sees in suitable coordinates (see below). For ${\displaystyle r\gg r_{s}}$, the Schwarzschild metric is asymptotic to the standard Lorentz metric on Minkowski space. For almost all astrophysical objects, the ratio ${\displaystyle {\frac {r_{s}}{r}}}$ is extremely small. For example, the Schwarzschild radius ${\displaystyle r_{s}^{(\mathrm {Earth} )}}$ of the Earth is roughly 8.9 mm, while the Sun, which is 3.3×10⁵ times as massive has a Schwarzschild radius ${\displaystyle r_{s}^{(\mathrm {Sun} )}}$ of approximately 3.0 km. The ratio becomes large only in close proximity to black holes and other ultra-dense objects such as neutron stars.

The radial coordinate turns out to have physical significance as the "proper distance between two events that occur simultaneously relative to the radially moving geodesic clocks, the two events lying on the same radial coordinate line".

The Schwarzschild solution is analogous to a classical Newtonian theory of gravity that corresponds to the gravitational field around a point particle. Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion.

History

The Schwarzschild solution is named in honour of Karl Schwarzschild, who found the exact solution in 1915 and published it in January 1916, a little more than a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I.

Johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler, more direct derivation.

In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of the Einstein field equations. In Schwarzschild's original paper, he put what we now call the event horizon at the origin of his coordinate system. In this paper he also introduced what is now known as the Schwarzschild radial coordinate (r in the equations above), as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius. 

A more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at r = 0 and r = r_s. Although there was general consensus that the singularity at r = 0 was a 'genuine' physical singularity, the nature of the singularity at r = r_s remained unclear.

In 1921 Paul Painlevé and in 1922 Allvar Gullstrand independently produced a metric, a spherically symmetric solution of Einstein's equations, which we now know is coordinate transformation of the Schwarzschild metric, Gullstrand–Painlevé coordinates, in which there was no singularity at r = r_s. They, however, did not recognize that their solutions were just coordinate transforms, and in fact used their solution to argue that Einstein's theory was wrong. In 1924 Arthur Eddington produced the first coordinate transformation (Eddington–Finkelstein coordinates) that showed that the singularity at r = r_s was a coordinate artifact, although he also seems to have been unaware of the significance of this discovery. Later, in 1932, Georges Lemaître gave a different coordinate transformation (Lemaître coordinates) to the same effect and was the first to recognize that this implied that the singularity at r = r_s was not physical. In 1939 Howard Robertson showed that a free falling observer descending in the Schwarzschild metric would cross the r = r_s singularity in a finite amount of proper time even though this would take an infinite amount of time in terms of coordinate time t.

In 1950, John Synge produced a paper that showed the maximal analytic extension of the Schwarzschild metric, again showing that the singularity at r = r_s was a coordinate artifact and that it represented two horizons. A similar result was later rediscovered by George Szekeres, and independently Martin Kruskal. The new coordinates nowadays known as Kruskal-Szekeres coordinates were much simpler than Synge's but both provided a single set of coordinates that covered the entire spacetime. However, perhaps due to the obscurity of the journals in which the papers of Lemaître and Synge were published their conclusions went unnoticed, with many of the major players in the field including Einstein believing that singularity at the Schwarzschild radius was physical.

Real progress was made in the 1960s when the more exact tools of differential geometry entered the field of general relativity, allowing more exact definitions of what it means for a Lorentzian manifold to be singular. This led to definitive identification of the r = r_s singularity in the Schwarzschild metric as an event horizon (a hypersurface in spacetime that can be crossed in only one direction).

Singularities and black holes

The Schwarzschild solution appears to have singularities at r = 0 and r = r_s; some of the metric components "blow up" (entail division by zero or division by infinity) at these radii. Since the Schwarzschild metric is expected to be valid only for those radii larger than the radius R of the gravitating body, there is no problem as long as R > r_s. For ordinary stars and planets this is always the case. For example, the radius of the Sun is approximately 700000 km, while its Schwarzschild radius is only 3 km.

The singularity at r = r_s divides the Schwarzschild coordinates in two disconnected patches. The exterior Schwarzschild solution with r > r_s is the one that is related to the gravitational fields of stars and planets. The interior Schwarzschild solution with 0 ≤ r < r_s, which contains the singularity at r = 0, is completely separated from the outer patch by the singularity at r = r_s. The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions. The singularity at r = r_s is an illusion however; it is an instance of what is called a coordinate singularity. As the name implies, the singularity arises from a bad choice of coordinates or coordinate conditions. When changing to a different coordinate system (for example Lemaitre coordinates, Eddington–Finkelstein coordinates, Kruskal–Szekeres coordinates, Novikov coordinates, or Gullstrand–Painlevé coordinates) the metric becomes regular at r = r_s and can extend the external patch to values of r smaller than r_s. Using a different coordinate transformation one can then relate the extended external patch to the inner patch.

The case r = 0 is different, however. If one asks that the solution be valid for all r one runs into a true physical singularity, or gravitational singularity, at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant, which is given by

${\displaystyle R^{\alpha \beta \gamma \delta }R_{\alpha \beta \gamma \delta }={\frac {12r_{\mathrm {s} }^{2}}{r^{6}}}={\frac {48G^{2}M^{2}}{c^{4}r^{6}}}\,.}$

At r = 0 the curvature becomes infinite, indicating the presence of a singularity. At this point the metric, and spacetime itself, is no longer well-defined. For a long time it was thought that such a solution was non-physical. However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case. 

The Schwarzschild solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations, although it has some rather bizarre properties. For r < r_s the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike. A curve at constant r is no longer a possible worldline of a particle or observer, not even if a force is exerted to try to keep it there; this occurs because spacetime has been curved so much that the direction of cause and effect (the particle's future light cone) points into the singularity. The surface r = r_s demarcates what is called the event horizon of the black hole. It represents the point past which light can no longer escape the gravitational field. Any physical object whose radius R becomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole.

Alternative coordinates

The Schwarzschild solution can be expressed in a range of different choices of coordinates besides the Schwarzschild coordinates used above. Different choices tend to highlight different features of the solution.

Flamm's paraboloid

A plot of Flamm's paraboloid. It should not be confused with the unrelated concept of a gravity well.

 

The spatial curvature of the Schwarzschild solution for r > r_s can be visualized as the graphic shows. Consider a constant time equatorial slice through the Schwarzschild solution (θ = ​^π⁄₂, t = constant) and let the position of a particle moving in this plane be described with the remaining Schwarzschild coordinates (r, φ). Imagine now that there is an additional Euclidean dimension w, which has no physical reality (it is not part of spacetime). Then replace the (r, φ) plane with a surface dimpled in the w direction according to the equation (Flamm's paraboloid)

${\displaystyle w=2{\sqrt {r_{\mathrm {s} }\left(r-r_{\mathrm {s} }\right)}}.}$

This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above,

${\displaystyle dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}=-c^{2}\,d\tau ^{2}={\frac {dr^{2}}{1-{\frac {r_{\mathrm {s} }}{r}}}}+r^{2}\,d\varphi ^{2}}$

Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric. It should not, however, be confused with a gravity well. No ordinary (massive or massless) particle can have a worldline lying on the paraboloid, since all distances on it are spacelike (this is a cross-section at one moment of time, so any particle moving on it would have an infinite velocity). Even a tachyon would not move along the path that one might naively expect from a "rubber sheet" analogy: in particular, if the dimple is drawn pointing upward rather than downward, the tachyon's path still curves toward the central mass, not away. See the gravity well article for more information. 

Flamm's paraboloid may be derived as follows. The Euclidean metric in the cylindrical coordinates (r, φ, w) is written

${\displaystyle ds^{2}=dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}\,.}$

Letting the surface be described by the function w = w(r), the Euclidean metric can be written as

${\displaystyle ds^{2}=\left(1+\left({\frac {dw}{dr}}\right)^{2}\right)\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}$

Comparing this with the Schwarzschild metric in the equatorial plane (θ = π/2) at a fixed time (t = constant, dt = 0)

${\displaystyle ds^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}$

yields an integral expression for w(r):

${\displaystyle w(r)=\int {\frac {dr}{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}}=2r_{\mathrm {s} }{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}+{\mbox{constant}}}$

whose solution is Flamm's paraboloid.

Orbital motion

Comparison between the orbit of a test particle in Newtonian (left) and Schwarzschild (right) spacetime; note the Apsidal precession on the right.

A particle orbiting in the Schwarzschild metric can have a stable circular orbit with r > 3r_s. Circular orbits with r between 1.5r_s and 3r_s are unstable, and no circular orbits exist for r < 1.5r_s. The circular orbit of minimum radius 1.5r_s corresponds to an orbital velocity approaching the speed of light. It is possible for a particle to have a constant value of r between r_s and 1.5r_s, but only if some force acts to keep it there.

Noncircular orbits, such as Mercury's, dwell longer at small radii than would be expected classically. This can be seen as a less extreme version of the more dramatic case in which a particle passes through the event horizon and dwells inside it forever. Intermediate between the case of Mercury and the case of an object falling past the event horizon, there are exotic possibilities such as knife-edge orbits, in which the satellite can be made to execute an arbitrarily large number of nearly circular orbits, after which it flies back outward.

Symmetries

The group of isometries of the Schwarzschild metric is the subgroup of the ten-dimensional Poincaré group which takes the time axis (trajectory of the star) to itself. It omits the spatial translations (three dimensions) and boosts (three dimensions). It retains the time translations (one dimension) and rotations (three dimensions). Thus it has four dimensions. Like the Poincaré group, it has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time reversed and spatially inverted.

Curvatures

The Ricci curvature scalar and the Ricci curvature tensor are both zero. Non-zero components of the Riemann curvature tensor are

${\displaystyle R^{t}{}_{rrt}=2R^{\theta }{}_{r\theta r}=2R^{\phi }{}_{r\phi r}={\frac {r_{s}}{r^{2}(r_{s}-r)}},}$

${\displaystyle 2R^{t}{}_{\theta \theta t}=2R^{r}{}_{\theta \theta r}=R^{\phi }{}_{\theta \phi \theta }={\frac {r_{s}}{r}},}$

${\displaystyle 2R^{t}{}_{\phi \phi t}=2R^{r}{}_{\phi \phi r}=-R^{\theta }{}_{\phi \phi \theta }={\frac {r_{s}\sin ^{2}(\theta )}{r}},}$

${\displaystyle R^{r}{}_{trt}=-2R^{\theta }{}_{t\theta t}=-2R^{\phi }{}_{t\phi t}=c^{2}{\frac {r_{s}(r_{s}-r)}{r^{4}}}}$

Components which are obtainable by the symmetries of the Riemann tensor are not displayed.

To understand the physical meaning of these quantities, it is useful to express the curvature tensor in an orthonormal basis. In an orthonormal basis of an observer the non-zero components in geometric units are

${\displaystyle R^{\hat {r}}{}_{{\hat {t}}{\hat {r}}{\hat {t}}}=-R^{\hat {\theta }}{}_{{\hat {\phi }}{\hat {\theta }}{\hat {\phi }}}=-{\frac {r_{s}}{r^{3}}},}$

${\displaystyle R^{\hat {\theta }}{}_{{\hat {t}}{\hat {\theta }}{\hat {t}}}=R^{\hat {\phi }}{}_{{\hat {t}}{\hat {\phi }}{\hat {t}}}=-R^{\hat {r}}{}_{{\hat {\theta }}{\hat {r}}{\hat {\theta }}}=-R^{\hat {r}}{}_{{\hat {\phi }}{\hat {r}}{\hat {\phi }}}={\frac {r_{s}}{2r^{3}}}.}$

Again, components which are obtainable by the symmetries of the Riemann tensor are not displayed. These results are invariant to any Lorentz boost, thus the components do not change for non-static observers. The geodesic deviation equation shows that the tidal acceleration between two observers separated by ${\displaystyle \xi ^{\hat {j}}}$ is ${\displaystyle D^{2}\xi ^{\hat {j}}/D\tau ^{2}=-R^{\hat {j}}{}_{{\hat {t}}{\hat {k}}{\hat {t}}}\xi ^{\hat {k}}}$, so a body of length ${\displaystyle L}$ is stretched in the radial direction by an apparent acceleration ${\displaystyle (r_{s}/r^{3})c^{2}L}$ and squeezed in the perpendicular directions by ${\displaystyle -(r_{s}/(2r^{3}))c^{2}L}$.

Quran desecration

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quran_desecration

The term "Quran desecration" is defined as insulting the Quran—which Muslims believe to be the literal word of God, in its original Arabic form—by defiling or defacing copies. Intentionally insulting the Quran is regarded by Muslims as blasphemous.

Most traditional schools of Islamic law require wudu (ritual handwashing) before a Muslim may touch the Quran. Muslims must always treat the printed book with reverence, which may even extend to excerpts of text.

Disposal of worn copies is also of concern to Muslims. Because the Quran contains no specifics on how to dispose of a worn or defective text, different and conflicting methods of disposal have been adopted in different regions by different sects. According to Islamic historian Michael Cook the Quran should be wrapped in cloth and buried on holy ground where it is unlikely to be trampled on or "safely" placed where it is unlikely to come into contact with impurity. According to Arab News, Muslims are forbidden to recycle, pulp, or shred worn-out copies of the text; instead, burning or burying the worn-out copies in a respectful manner is required.

Respect for the written text of the Quran is an important element of religious faith in Islam. Intentionally desecrating a copy of the Quran is punishable by imprisonment in some countries (up to life imprisonment in Pakistan, according to Article 295-B of the Penal Code) and could lead to death in Afghanistan, Saudi Arabia, Somalia and Pakistan.

Notable instances

2005 - Guantanamo

In mid-2005, allegations of deliberate desecration of the Quran in front of Muslim prisoners at the United States military Guantanamo Bay detention camp, Cuba fueled widespread controversy and led to ensuing Muslim riots. A US military investigation confirmed four instances of Quran desecration by US personnel (two of which were described as "unintentional"), and fifteen instances of desecration by Muslim prisoners. According to CBC News, "The statement did not provide any explanation about why the detainees might have abused their own Holy books." In May 2005, a report in Newsweek, claiming that it was U.S. interrogators who desecrated the Quran at the Guantanamo Bay base, further sparking Muslim unrest.

2007 - Nigeria

In 2007, Nigerian Christian teacher Christianah Oluwatoyin Oluwasesin was stabbed to death after allegations that she had desecrated a Quran.

2010/2011 - United States

In 2010, Christian pastor Terry Jones of the Dove World Outreach Center, a church in Gainesville, Florida, provoked international condemnation after announcing plans to burn a Quran on the anniversary of the Islamic terrorist September 11 attacks on the United States. He later cancelled the plans; however, on March 20, 2011, he oversaw the burning of a Quran. In response, Muslims in Afghanistan rioted and 12 people were killed.

In the 2011 Louis Theroux documentary America's Most Hated Family in Crisis, Megan Phelps of the Westboro Baptist Church explained in an interview that they deliberately and publicly burned a copy of the Quran.

2012 - Bangladesh

On September 29, an Islamic mob estimated at 25,000 vandalized and torched Buddhist temples, shrines, and houses, along with Hindu temples as incited by an alleged Facebook Buddhist posting of an image depicting the desecration of a Quran. The violence started in Ramu Upazila in Cox's Bazar District and later spread to other areas of Bangladesh.

2012/2015 - Afghanistan

In February 2012, protests broke out in various parts of Afghanistan over the improper disposal of Qurans at the US military Bagram Air Base. Protesters shouted "Death to America" and burned US flags. At least 30 people were killed and hundreds injured. Also,6 U.S. soldiers were killed after members of the Afghan National Security Forces turned their weapons on them and the Afghan protesters.

On March 19, 2015, Farkhunda Malikzada, a 27-year-old Afghan woman, was publicly beaten and slain by a mob of hundreds of people in Kabul. Farkhunda had previously been arguing with a mullah named Zainuddin, in front of a mosque where she worked as a religious teacher, about his practice of selling charms at the Shah-Do Shamshira Mosque, the Shrine of the King of Two Swords, a religious shrine in Kabul. During this argument, Zainuddin reportedly falsely accused her of burning the Quran. Police investigations revealed that she had not burned anything. A number of prominent public officials turned to Facebook immediately after the death to endorse the murder. After it was revealed that she did not burn the Quran, the public reaction in Afghanistan turned to shock and anger. Her murder led to 49 arrests; three adult men received twenty-year prison sentences, eight other adult males received sixteen year sentences, a minor received a ten-year sentence, and eleven police officers received one-year prison terms for failing to protect Farkhunda. Her murder and the subsequent protests served to draw attention to women's rights in Afghanistan.

Others

Saudi Arabia destroys Qurans of pilgrims that fall short of state standards. The preferred method is by burning, to avoid soiling the pages.

In March 2013, the al Qaeda English-language magazine Inspire published a poster stating "Wanted dead or alive for crimes against Islam" with a prominent image of Terry Jones, known for public Quran burning events. Iran's news agency, IRIB, reported on April 8, 2013, that Terry Jones planned another Quran burning event on September 11, 2013. On April 11, IRIB published statements from an Iranian MP who said the West must stop the event and warned that "the blasphemous move will spark an uncontrollable wave of outrage among over 1.6 billion people across the globe who follow Islam." In Pakistan, protesters set the American flag and effigy of the US pastor Terry Jones on fire, condemning the 9/11 plan, according to an April 14, 2013 article in The Nation.

In October 2013, a Turkish woman was arrested on suspicion of blasphemy and inciting religious hatred after allegedly stepping on Quran and then posting the picture on Twitter.

Proposals to recycle old Qurans in Pakistan have met with opposition.

On July 31, 2016, a couple of days after the Normandy church attack, several copies of the Quran at the multi-faith room of Mater Dei Hospital in Malta were desecrated when slices of pork were laid inside the book. The perpetrators also left a photo of Jacques Hamel, the Catholic priest murdered during the attack, with the caption "Victim of Islam".

Cygnus X-1

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Cygnus_X-1

 

Cygnus X-1/HDE 226868

The location of Cygnus X-1 (circled) to the left of Eta Cygni in the constellation Cygnus based on known coordinates
Observation data Epoch J2000      Equinox J2000
Constellation	Cygnus
Right ascension	19h 58m 21.67595s
Declination	+35° 12′ 05.7783″
Apparent magnitude (V)	8.95
Characteristics
Spectral type	O9.7Iab
U−B color index	−0.30
B−V color index	+0.81
Variable type	Ellipsoidal variable
Astrometry
Radial velocity (Rv)	−13 km/s
Proper motion (μ)	RA: −3.37 mas/yr Dec.: −7.15 mas/yr
Parallax (π)	0.539 ± 0.033 mas
Distance	6,100 ± 400 ly (1,900 ± 100 pc)
Absolute magnitude (MV)	−6.5±0.2
Details
Mass	14–16 M☉
Radius	20–22 R☉
Luminosity	3–4×105 L☉
Surface gravity (log g)	3.31±0.07 cgs
Temperature	31000 K
Rotation	every 5.6 days
Age	5 Myr
Other designations
AG (or AGK2)+35 1910, BD+34 3815, HD (or HDE) 226868, HIP 98298, SAO 69181, V1357 Cyg.
Database references
SIMBAD	data

Cygnus X-1 (abbreviated Cyg X-1) is a galactic X-ray source in the constellation Cygnus, and the first such source widely accepted to be a black hole. It was discovered in 1964 during a rocket flight and is one of the strongest X-ray sources seen from Earth, producing a peak X-ray flux density of 2.3×10⁻²³ Wm⁻² Hz⁻¹ (2.3×10³ Jansky). It remains among the most studied astronomical objects in its class. The compact object is now estimated to have a mass about 14.8 times the mass of the Sun and has been shown to be too small to be any known kind of normal star, or other likely object besides a black hole. If so, the radius of its event horizon has 300 km "as upper bound to the linear dimension of the source region" of occasional X-ray bursts lasting only for about 1 ms.

Cygnus X-1 belongs to a high-mass X-ray binary system, located about 6,070 light-years from the Sun, that includes a blue supergiant variable star designated HDE 226868 which it orbits at about 0.2 AU, or 20% of the distance from the Earth to the Sun. A stellar wind from the star provides material for an accretion disk around the X-ray source. Matter in the inner disk is heated to millions of degrees, generating the observed X-rays. A pair of jets, arranged perpendicularly to the disk, are carrying part of the energy of the infalling material away into interstellar space.

This system may belong to a stellar association called Cygnus OB3, which would mean that Cygnus X-1 is about five million years old and formed from a progenitor star that had more than 40 solar masses. The majority of the star's mass was shed, most likely as a stellar wind. If this star had then exploded as a supernova, the resulting force would most likely have ejected the remnant from the system. Hence the star may have instead collapsed directly into a black hole.

Cygnus X-1 was the subject of a friendly scientific wager between physicists Stephen Hawking and Kip Thorne in 1974, with Hawking betting that it was not a black hole. He conceded the bet in 1990 after observational data had strengthened the case that there was indeed a black hole in the system. This hypothesis lacks direct empirical evidence but has generally been accepted from indirect evidence.

Discovery and observation

Observation of X-ray emissions allows astronomers to study celestial phenomena involving gas with temperatures in the millions of degrees. However, because X-ray emissions are blocked by the Earth's atmosphere, observation of celestial X-ray sources is not possible without lifting instruments to altitudes where the X-rays can penetrate. Cygnus X-1 was discovered using X-ray instruments that were carried aloft by a sounding rocket launched from White Sands Missile Range in New Mexico. As part of an ongoing effort to map these sources, a survey was conducted in 1964 using two Aerobee suborbital rockets. The rockets carried Geiger counters to measure X-ray emission in wavelength range 1–15 Å across an 8.4° section of the sky. These instruments swept across the sky as the rockets rotated, producing a map of closely spaced scans.

As a result of these surveys, eight new sources of cosmic X-rays were discovered, including Cyg XR-1 (later Cyg X-1) in the constellation Cygnus. The celestial coordinates of this source were estimated as right ascension 19^h53^m and declination 34.6°. It was not associated with any especially prominent radio or optical source at that position.

Seeing a need for longer duration studies, in 1963 Riccardo Giacconi and Herb Gursky proposed the first orbital satellite to study X-ray sources. NASA launched their Uhuru Satellite in 1970, which led to the discovery of 300 new X-ray sources. Extended Uhuru observations of Cygnus X-1 showed fluctuations in the X-ray intensity that occurs several times a second. This rapid variation meant that the energy generation must take place over a relatively small region of roughly 10⁵ km, as the speed of light restricts communication between more distant regions. For a size comparison, the diameter of the Sun is about 1.4×10⁶ km.

In April–May 1971, Luc Braes and George K. Miley from Leiden Observatory, and independently Robert M. Hjellming and Campbell Wade at the National Radio Astronomy Observatory, detected radio emission from Cygnus X-1, and their accurate radio position pinpointed the X-ray source to the star AGK2 +35 1910 = HDE 226868. On the celestial sphere, this star lies about half a degree from the 4th-magnitude star Eta Cygni. It is a supergiant star that is, by itself, incapable of emitting the observed quantities of X-rays. Hence, the star must have a companion that could heat gas to the millions of degrees needed to produce the radiation source for Cygnus X-1. 

Louise Webster and Paul Murdin, at the Royal Greenwich Observatory, and Charles Thomas Bolton, working independently at the University of Toronto's David Dunlap Observatory, announced the discovery of a massive hidden companion to HDE 226868 in 1971. Measurements of the Doppler shift of the star's spectrum demonstrated the companion's presence and allowed its mass to be estimated from the orbital parameters. Based on the high predicted mass of the object, they surmised that it may be a black hole as the largest possible neutron star cannot exceed three times the mass of the Sun.

With further observations strengthening the evidence, by the end of 1973 the astronomical community generally conceded that Cygnus X-1 was most likely a black hole. More precise measurements of Cygnus X-1 demonstrated variability down to a single millisecond. This interval is consistent with turbulence in a disk of accreted matter surrounding a black hole—the accretion disk. X-ray bursts that last for about a third of a second match the expected time frame of matter falling toward a black hole.

This X-ray image of Cygnus X-1 was taken by a balloon-borne telescope, the High-Energy Replicated Optics (HERO) project. NASA image.

 

Cygnus X-1 has since been studied extensively using observations by orbiting and ground-based instruments. The similarities between the emissions of X-ray binaries such as HDE 226868/Cygnus X-1 and active galactic nuclei suggests a common mechanism of energy generation involving a black hole, an orbiting accretion disk and associated jets. For this reason, Cygnus X-1 is identified among a class of objects called microquasars; an analog of the quasars, or quasi-stellar radio sources, now known to be distant active galactic nuclei. Scientific studies of binary systems such as HDE 226868/Cygnus X-1 may lead to further insights into the mechanics of active galaxies.

Binary system

The compact object and blue supergiant star form a binary system in which they orbit around their center of mass every 5.599829 days. From the perspective of the Earth, the compact object never goes behind the other star; in other words, the system does not eclipse. However, the inclination of the orbital plane to the line of sight from the Earth remains uncertain, with predictions ranging from 27–65°. A 2007 study estimated the inclination is 48.0±6.8°, which would mean that the semi-major axis is about 0.2 AU, or 20% of the distance from the Earth to the Sun. The orbital eccentricity is thought to be only 0.0018±0.002; a nearly circular orbit. Earth's distance to this system is about 1,860 ± 120 parsecs (6,070 ± 390 light-years).

The HDE 226868/Cygnus X-1 system shares a common motion through space with an association of massive stars named Cygnus OB3, which is located at roughly 2,000 parsecs from the Sun. This implies that HDE 226868, Cygnus X-1 and this OB association may have formed at the same time and location. If so, then the age of the system is about 5±1.5 Ma. The motion of HDE 226868 with respect to Cygnus OB3 is 9±3 km/s; a typical value for random motion within a stellar association. HDE 226868 is about 60 parsecs from the center of the association, and could have reached that separation in about 7±2 Ma—which roughly agrees with estimated age of the association.

With a galactic latitude of 4 degrees and galactic longitude 71 degrees, this system lies inward along the same Orion Spur in which the Sun is located within the Milky Way, near where the spur approaches the Sagittarius Arm. Cygnus X-1 has been described as belonging to the Sagittarius Arm, though the structure of the Milky Way is not well established. 

Compact object

From various techniques, the mass of the compact object appears to be greater than the maximum mass for a neutron star. Stellar evolutionary models suggest a mass of 20±5 solar masses, while other techniques resulted in 10 solar masses. Measuring periodicities in the X-ray emission near the object has yielded a more precise value of 14.8±1 solar masses. In all cases, the object is most likely a black hole—a region of space with a gravitational field that is strong enough to prevent the escape of electromagnetic radiation from the interior. The boundary of this region is called the event horizon and has an effective radius called the Schwarzschild radius, which is about 44 km for Cygnus X-1. Anything (including matter and photons) that passes through this boundary is unable to escape.

Evidence of just such an event horizon may have been detected in 1992 using ultraviolet (UV) observations with the High Speed Photometer on the Hubble Space Telescope. As self-luminous clumps of matter spiral into a black hole, their radiation will be emitted in a series of pulses that are subject to gravitational redshift as the material approaches the horizon. That is, the wavelengths of the radiation will steadily increase, as predicted by general relativity. Matter hitting a solid, compact object would emit a final burst of energy, whereas material passing through an event horizon would not. Two such "dying pulse trains" were observed, which is consistent with the existence of a black hole.

Chandra X-ray Observatory image of Cygnus X-1

 

The spin of the compact object is not yet well determined. Past analysis of data from the space-based Chandra X-ray Observatory suggested that Cygnus X-1 was not rotating to any significant degree. However, evidence announced in 2011 suggests it is rotating extremely rapidly, approximately 790 times per second.

Formation

The largest star in the Cygnus OB3 association has a mass 40 times that of the Sun. As more massive stars evolve more rapidly, this implies that the progenitor star for Cygnus X-1 had more than 40 solar masses. Given the current estimated mass of the black hole, the progenitor star must have lost over 30 solar masses of material. Part of this mass may have been lost to HDE 226868, while the remainder was most likely expelled by a strong stellar wind. The helium enrichment of HDE 226868's outer atmosphere may be evidence for this mass transfer. Possibly the progenitor may have evolved into a Wolf–Rayet star, which ejects a substantial proportion of its atmosphere using just such a powerful stellar wind.

If the progenitor star had exploded as a supernova, then observations of similar objects show that the remnant would most likely have been ejected from the system at a relatively high velocity. As the object remained in orbit, this indicates that the progenitor may have collapsed directly into a black hole without exploding (or at most produced only a relatively modest explosion).

Accretion disk

A Chandra X-ray spectrum of Cygnus X-1 showing a characteristic peak near 6.4 keV due to ionized iron in the accretion disk, but the peak is gravitationally red-shifted, broadened by the Doppler effect, and skewed toward lower energies

 

The compact object is thought to be orbited by a thin, flat disk of accreting matter known as an accretion disk. This disk is intensely heated by friction between ionized gas in faster-moving inner orbits and that in slower outer ones. It is divided into a hot inner region with a relatively high level of ionization—forming a plasma—and a cooler, less ionized outer region that extends to an estimated 500 times the Schwarzschild radius, or about 15,000 km.

Though highly and erratically variable, Cygnus X-1 is typically the brightest persistent source of hard X-rays—those with energies from about 30 up to several hundred keV—in the sky. The X-rays are produced as lower-energy photons in the thin inner accretion disk, then given more energy through Compton scattering with very high-temperature electrons in a geometrically thicker, but nearly transparent corona enveloping it, as well as by some further reflection from the surface of the thin disk.^[55] An alternative possibility is that the X-rays may be Compton scattered by the base of a jet instead of a disk corona.

The X-ray emission from Cygnus X-1 can vary in a somewhat repetitive pattern called quasi-periodic oscillations (QPO). The mass of the compact object appears to determine the distance at which the surrounding plasma begins to emit these QPOs, with the emission radius decreasing as the mass decreases. This technique has been used to estimate the mass of Cygnus X-1, providing a cross-check with other mass derivations.

Pulsations with a stable period, similar to those resulting from the spin of a neutron star, have never been seen from Cygnus X-1. The pulsations from neutron stars are caused by the neutron star's magnetic field; however, the no-hair theorem guarantees that black holes do not have magnetic poles. For example, the X-ray binary V 0332+53 was thought to be a possible black hole until pulsations were found. Cygnus X-1 has also never displayed X-ray bursts similar to those seen from neutron stars. Cygnus X-1 unpredictably changes between two X-ray states, although the X-rays may vary continuously between those states as well. In the most common state, the X-rays are "hard", which means that more of the X-rays have high energy. In the less common state, the X-rays are "soft", with more of the X-rays having lower energy. The soft state also shows greater variability. The hard state is believed to originate in a corona surrounding the inner part of the more opaque accretion disk. The soft state occurs when the disk draws closer to the compact object (possibly as close as 150 km), accompanied by cooling or ejection of the corona. When a new corona is generated, Cygnus X-1 transitions back to the hard state.

 

The spectral transition of Cygnus X-1 can be explained using a two component advective flow solution, as proposed by Chakrabarti and Titarchuk. A hard state is generated by the inverse Comptonisation of seed photons from the Keplarian disk and likewise synchrotron photons produced by the hot electrons in the Centrifugal Pressure-supported Boundary Layer (CENBOL).

 

The X-ray flux from Cygnus X-1 varies periodically every 5.6 d, especially during superior conjunction when the orbiting objects are most closely aligned with the Earth and the compact source is the more distant. This indicates that the emissions are being partially blocked by circumstellar matter, which may be the stellar wind from the star HDE 226868. There is a roughly 300 d periodicity in the emission that could be caused by the precession of the accretion disk.

Jets

As accreted matter falls toward the compact object, it loses gravitational potential energy. Part of this released energy is dissipated by jets of particles, aligned perpendicular to the accretion disk, that flow outward with relativistic velocities. (That is, the particles are moving at a significant fraction of the speed of light.) This pair of jets provide a means for an accretion disk to shed excess energy and angular momentum. They may be created by magnetic fields within the gas that surrounds the compact object.

The Cygnus X-1 jets are inefficient radiators and so release only a small proportion of their energy in the electromagnetic spectrum. That is, they appear "dark". The estimated angle of the jets to the line of sight is 30° and they may be precessing. One of the jets is colliding with a relatively dense part of the interstellar medium (ISM), forming an energized ring that can be detected by its radio emission. This collision appears to be forming a nebula that has been observed in the optical wavelengths. To produce this nebula, the jet must have an estimated average power of 4–14×10³⁶ erg/s, or (9±5)×10²⁹ W. This is more than 1,000 times the power emitted by the Sun. There is no corresponding ring in the opposite direction because that jet is facing a lower density region of the ISM.

In 2006, Cygnus X-1 became the first stellar mass black hole found to display evidence of gamma ray emission in the very high energy band, above 100 GeV. The signal was observed at the same time as a flare of hard X-rays, suggesting a link between the events. The X-ray flare may have been produced at the base of the jet while the gamma rays could have been generated where the jet interacts with the stellar wind of HDE 226868.

HDE 226868

An artist's impression of the HDE 226868–Cygnus X-1 binary system. ESA/Hubble illustration.

 

HDE 226868 is a supergiant star with a spectral class of O9.7 Iab, which is on the borderline between class O and class B stars. It has an estimated surface temperature of 31,000 K and mass approximately 20–40 times the mass of the Sun. Based on a stellar evolutionary model, at the estimated distance of 2,000 parsecs this star may have a radius equal to about 15–17 times the solar radius and is approximately 300,000–400,000 times the luminosity of the Sun. For comparison, the compact object is estimated to be orbiting HDE 226868 at a distance of about 40 solar radii, or twice the radius of this star.

The surface of HDE 226868 is being tidally distorted by the gravity of the massive companion, forming a tear-drop shape that is further distorted by rotation. This causes the optical brightness of the star to vary by 0.06 magnitudes during each 5.6-day binary orbit, with the minimum magnitude occurring when the system is aligned with the line of sight. The "ellipsoidal" pattern of light variation results from the limb darkening and gravity darkening of the star's surface.

When the spectrum of HDE 226868 is compared to the similar star Epsilon Orionis, the former shows an overabundance of helium and an underabundance of carbon in its atmosphere. The ultraviolet and hydrogen alpha spectral lines of HDE 226868 show profiles similar to the star P Cygni, which indicates that the star is surrounded by a gaseous envelope that is being accelerated away from the star at speeds of about 1,500 km/s.

Like other stars of its spectral type, HDE 226868 is thought to be shedding mass in a stellar wind at an estimated rate of 2.5×10⁻⁶ solar masses per year. This is the equivalent of losing a mass equal to the Sun's every 400,000 years. The gravitational influence of the compact object appears to be reshaping this stellar wind, producing a focused wind geometry rather than a spherically symmetrical wind.^[72] X-rays from the region surrounding the compact object heat and ionize this stellar wind. As the object moves through different regions of the stellar wind during its 5.6-day orbit, the UV lines, the radio emission, and the X-rays themselves all vary.

The Roche lobe of HDE 226868 defines the region of space around the star where orbiting material remains gravitationally bound. Material that passes beyond this lobe may fall toward the orbiting companion. This Roche lobe is believed to be close to the surface of HDE 226868 but not overflowing, so the material at the stellar surface is not being stripped away by its companion. However, a significant proportion of the stellar wind emitted by the star is being drawn onto the compact object's accretion disk after passing beyond this lobe.

The gas and dust between the Sun and HDE 226868 results in a reduction in the apparent magnitude of the star as well as a reddening of the hue—red light can more effectively penetrate the dust in the interstellar medium. The estimated value of the interstellar extinction (A_V) is 3.3 magnitudes. Without the intervening matter, HDE 226868 would be a fifth-magnitude star and, thus, visible to the unaided eye.

Stephen Hawking and Kip Thorne

Cygnus X-1 was the subject of a bet between physicists Stephen Hawking and Kip Thorne, in which Hawking bet against the existence of black holes in the region. Hawking later described this as an "insurance policy" of sorts. In his book A Brief History of Time he wrote:

This was a form of insurance policy for me. I have done a lot of work on black holes, and it would all be wasted if it turned out that black holes do not exist. But in that case, I would have the consolation of winning my bet, which would win me four years of the magazine Private Eye. If black holes do exist, Kip will get one year of Penthouse. When we made the bet in 1975, we were 80% certain that Cygnus X-1 was a black hole. By now [1988], I would say that we are about 95% certain, but the bet has yet to be settled.

According to the updated tenth-anniversary edition of A Brief History of Time, Hawking has conceded the bet due to subsequent observational data in favor of black holes. In his own book, Black Holes and Time Warps, Thorne reports that Hawking conceded the bet by breaking into Thorne's office while he was in Russia, finding the framed bet, and signing it. (While Hawking referred to the bet as taking place in 1975, the written bet itself (in Thorne's handwriting, with his and Hawking's signatures) bears additional witness signatures under a legend stating "Witnessed this tenth day of December 1974". This date was confirmed by Kip Thorne on the January 10, 2018 episode of Nova on PBS.)