How fast is a ringularity^{1} spinning? Spinning black holes are created from spinning stars.

The angular momentum of the 'parent' star must be conserved so spinning black holes are spinning much much faster than their 'parent' star.

Since black holes have generally less mass than their parent star and mass is found only in a tiny amount of space then the ringularity should be spinning incredibly fast.

The ringularity is a ring which contains all the mass of the black hole and it is in the center and it is almost infinitely tiny so it must be spinning almost infinitely fast.

Can it spin faster than c? Is there any way to know if SR can be violated inside such extremely dense objects?

^{1}Ring singularity

$$ K_ ext{min}=2pisqrt{a^2+3(ma^2)^{2/3}} ag{1} $$

where the quantities $m$ and $a$, both of which have units of length, are defined by $$ m=frac{GM}{c^2} hskip2cm a=frac{J}{Mc}. ag{2} $$

The maximum spin is a=m (an extremal Kerr black hole), the borderline case between a black hole and a naked singularity.

More details from Chiral Anomaly here: https://physics.stackexchange.com/a/469282/

## Velocity addition: Physical meaning of non-singularity at c?

In the derivation of the relativistic formula for adding velocities, the Lorentz factor drops out. Mathematically, the formula works for inertial frames with relative velocity c and even gives an answer to Einstein's famous question about what happens if you drive at the speed of light and turn your headlights on.

However, is there a physical meaning to this? Massive light sources can't reach c because the required energy would be infinite. But is there some framework (QFT maybe?) where a photon decays into multiple photons such that one of them can be regarded as "headlights" and another as "light from the headlights"?

## 3 Answers 3

A quick note in the light of some of the comments: I'm interpreting the question to be asking about the escape velocity from a black hole containing a naked singularity rather than the escape velocity from the singularity itself. The escape velocity at the singularity is undefined as GR cannot describe the geometry at that point.

Anyhow, a convenient way to describe the escape velocity from a black hole is to write the metric using the Gullstrand-Painlevé coordinates. In these coordinates spacetime is flowing inwards towards the black hole, and the escape velocity is simply the velocity of the inflow. This is commonly known as the *River Model*, because the analogy is with objects being swept along by a flowing river. For the brave, the details are given in the paper The river model of black holes.

The river model can be used to describe rotating black holes like Gargantua, but the maths involved is scary hard. However there is another class of black hole that can have naked singularities, and that's the charged non-rotating black hole described by the Reissner-Nordström metric. This is a great deal simpler, and the escape velocity can be easily calculated.

So, if you're happy for your naked singularity to be changed and non-rotating, instead of uncharged and rotating, here's how to calculate the escape velocity.

In Gullstrand-Painlevé coordinates coordinates the inflow velocity of a Reissner-Nordström black hole as a function of radial distance $r$ is (see the River Model paper for details):

Since we're only interested in the general behaviour I'll convert this to geometrised units, and normalise the Schwarzschild radius to unity. This simplifies the equation to:

In these units the extremal black hole has a charge of $Q = 0.5$, so $Q lt 0.5$ looks from outside to be ordinary black hole with an event horizon and $Q ge 0.5$ is a naked singularity. If I graph $v^2$ (you'll see why I graph $v^2$ in a moment) against $r$ for a range of different charges the results look like:

The red line is an uncharged black hole, and as we expect the escape velocity goes to $1$ (i.e. $c$) at $r = 1$ (i.e. $r = r_s$).

The green line is for $Q = 0.4$, and the escape velocity goes to $c$ at $r = 0.8r_s$, so the event horizon has contracted a bit. However if you look at the behaviour inside the horizon the escape velocity rises then falls again and returns to $c$ at $r = 0.2r_s$. This is the location of the inner event horizon.

For the extremal black hole, $Q = 0.5$, the escape velocity rises to $c$ at $r = 0.5r_s$ but then falls again. There is a single horizon at $r = 0.5r_s$.

Finally for the naked singularity $Q = 0.6$, the escape velocity never reaches $c$ so there is no horizon.

However something rather odd happens at small $r$ for all the charged black holes. $v^2$ falls to zero then goes negative. A negative value of $v^2$ means the escape velocity is imaginary. This is generally interpreted as meaning that the Reissner-Nordström metric ceases to be physically meaningful at smaller values of $r$.

So, subject to worries about the behaviour at small $r$, it's straightforward to calculate the escape velocity and it doesn't do anything particularly weird. In principle the same calculation can be done for a rotating black hole, but as I said out the outset it's a big jump in difficulty so I shall leave it to Kip Thorne.

## Contents

After a classical education at a Jesuit secondary school, the Collège du Sacré-Coeur, in Charleroi, Lemaître began studying civil engineering at the Catholic University of Louvain at the age of 17. In 1914, he interrupted his studies to serve as an artillery officer in the Belgian army for the duration of World War I. At the end of hostilities, he received the Belgian War Cross with palms. [12]

After the war, he studied physics and mathematics, and began to prepare for the diocesan priesthood, not for the Jesuits. [13] He obtained his doctorate in 1920 with a thesis entitled *l'Approximation des fonctions de plusieurs variables réelles* (*Approximation of functions of several real variables*), written under the direction of Charles de la Vallée-Poussin. [14] He was ordained a priest on 22 September 1923 by Cardinal Désiré-Joseph Mercier. [15] [16]

In 1923, he became a research associate in astronomy at Cambridge UK, spending a year at St Edmund's House (now St Edmund's College, University of Cambridge). He worked with Arthur Eddington, who introduced him to modern cosmology, stellar astronomy, and numerical analysis. He spent the next year at Harvard College Observatory in Cambridge, Massachusetts, with Harlow Shapley, who had just gained renown for his work on nebulae, and at the Massachusetts Institute of Technology (MIT), where he registered for the doctoral program in sciences.

On his return to Belgium in 1925, he became a part-time lecturer at the Catholic University of Louvain and began the report that was published in 1927 in the *Annales de la Société Scientifique de Bruxelles* (*Annals of the Scientific Society of Brussels*) under the title "Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extragalactiques" ("A homogeneous Universe of constant mass and growing radius accounting for the radial velocity of extragalactic nebulae"), that was later to bring him international fame . [2] In this report, he presented the new idea that the universe is expanding, which he derived from General Relativity. This later became known as Hubble's law, even though Lemaître was the first to provide an observational estimate of the Hubble constant. [17] The initial state he proposed was taken to be Einstein's own model of a finitely sized static universe. The paper had little impact because the journal in which it was published was not widely read by astronomers outside Belgium. Arthur Eddington reportedly helped translate the article into English in 1931, but the part of it pertaining to the estimation of the "Hubble constant" was not included in the translation for reasons that remained unknown for a long time. [18] [7] This issue was clarified in 2011 by Mario Livio: Lemaître omitted those paragraphs himself when translating the paper for the Royal Astronomical Society, in favour of reports of newer work on the subject, since by that time Hubble's calculations had already improved on Lemaître's earlier ones. [4]

At this time, Einstein, while not taking exception to the mathematics of Lemaître's theory, refused to accept that the universe was expanding Lemaître recalled his commenting "*Vos calculs sont corrects, mais votre physique est abominable*" [19] ("Your calculations are correct, but your physics is atrocious"). In the same year, Lemaître returned to MIT to present his doctoral thesis on *The gravitational field in a fluid sphere of uniform invariant density according to the theory of relativity*. [20] Upon obtaining his PhD, he was named ordinary professor at the Catholic University of Louvain.

In 1931, Arthur Eddington published in the *Monthly Notices of the Royal Astronomical Society* a long commentary on Lemaître's 1927 article, which Eddington described as a "brilliant solution" to the outstanding problems of cosmology. [21] The original paper was published in an abbreviated English translation later on in 1931, along with a sequel by Lemaître responding to Eddington's comments. [22] Lemaître was then invited to London to participate in a meeting of the British Association on the relation between the physical universe and spirituality. There he proposed that the universe expanded from an initial point, which he called the "Primeval Atom". He developed this idea in a report published in *Nature*. [11] Lemaître's theory appeared for the first time in an article for the general reader on science and technology subjects in the December 1932 issue of Popular Science. [23] Lemaître's theory became better known as the "Big Bang theory," a picturesque term playfully coined during a 1949 BBC radio broadcast by the astronomer Fred Hoyle, [24] [25] who was a proponent of the steady state universe and remained so until his death in 2001.

Lemaître's proposal met with skepticism from his fellow scientists. Eddington found Lemaître's notion unpleasant. [26] Einstein thought it unjustifiable from a physical point of view, although he encouraged Lemaître to look into the possibility of models of non-isotropic expansion, so it is clear he was not altogether dismissive of the concept. Einstein also appreciated Lemaître's argument that Einstein's model of a static universe could not be sustained into the infinite past.

With Manuel Sandoval Vallarta, Lemaître discovered that the intensity of cosmic rays varied with latitude because these charged particles are interacting with the Earth's magnetic field. [27] In their calculations, Lemaître and Vallarta made use of MIT's differential analyzer computer developed by Vannevar Bush. They also worked on a theory of primary cosmic radiation and applied it to their investigations of the sun's magnetic field and the effects of the galaxy's rotation.

Lemaître and Einstein met on four occasions: in 1927 in Brussels, at the time of a Solvay Conference in 1932 in Belgium, at the time of a cycle of conferences in Brussels in California in January 1933 [28] and in 1935 at Princeton. In 1933 at the California Institute of Technology, after Lemaître detailed his theory, Einstein stood up, applauded, and is supposed to have said, "This is the most beautiful and satisfactory explanation of creation to which I have ever listened." [29] However, there is disagreement over the reporting of this quote in the newspapers of the time, and it may be that Einstein was not referring to the theory as a whole, but only to Lemaître's proposal that cosmic rays may be the leftover artifacts of the initial "explosion".

In 1933, when he resumed his theory of the expanding universe and published a more detailed version in the *Annals of the Scientific Society of Brussels*, Lemaître achieved his greatest public recognition. [30] Newspapers around the world called him a famous Belgian scientist and described him as the leader of the new cosmological physics. Also in 1933, Lemaître served as a visiting professor at The Catholic University of America. [31]

On July 27, 1935, he was named an honorary canon of the Malines cathedral by Cardinal Josef Van Roey. [32]

He was elected a member of the Pontifical Academy of Sciences in 1936, and took an active role there, serving as its president from March 1960 until his death. [33]

In 1941, he was elected a member of the Royal Academy of Sciences and Arts of Belgium. [34] In 1946, he published his book on *L'Hypothèse de l'Atome Primitif* (*The Primeval Atom Hypothesis*). It was translated into Spanish in the same year and into English in 1950. [* citation needed *]

By 1951, Pope Pius XII declared that Lemaître's theory provided a scientific validation for Catholicism. [35] However, Lemaître resented the Pope's proclamation, stating that the theory was neutral and there was neither a connection nor a contradiction between his religion and his theory. [36] [37] [16] Lemaître and Daniel O'Connell, the Pope's scientific advisor, persuaded the Pope not to mention Creationism publicly, and to stop making proclamations about cosmology. [38] Lemaître was a devout Catholic, but opposed mixing science with religion, [38] although he held that the two fields were not in conflict. [39]

During the 1950s, he gradually gave up part of his teaching workload, ending it completely when he took emeritus status in 1964. In 1962, strongly opposed to the expulsion of French speakers from the Catholic University of Louvain, he created the ACAPSUL movement together with Gérard Garitte to fight against the split. [40]

During the Second Vatican Council of 1962–65 he was asked by Pope John XXIII to serve on the 4th session of the Pontifical Commission on Birth Control. [41] However, since his health made it impossible for him to travel to Rome – he suffered a heart attack in December 1964 – Lemaître demurred, expressing surprise that he was chosen. He told a Dominican colleague, Père Henri de Riedmatten, that he thought it was dangerous for a mathematician to venture outside of his area of expertise. [42] He was also named Domestic prelate (*Monsignor*) in 1960 by Pope John XXIII. [34]

At the end of his life, he was increasingly devoted to problems of numerical calculation. He was a remarkable algebraicist and arithmetical calculator. Since 1930, he had used the most powerful calculating machines of the time, the Mercedes-Euklid. In 1958, he was introduced to the University's Burroughs E 101, its first electronic computer. Lemaître maintained a strong interest in the development of computers and, even more, in the problems of language and computer programming.

He died on 20 June 1966, shortly after having learned of the discovery of cosmic microwave background radiation, which provided further evidence for his proposal about the birth of the universe. [43]

Lemaître was a pioneer in applying Albert Einstein's theory of general relativity to cosmology. In a 1927 article, which preceded Edwin Hubble's landmark article by two years, Lemaître derived what became known as Hubble's law and proposed it as a generic phenomenon in relativistic cosmology. Lemaître was also the first to estimate the numerical value of the Hubble constant.

Einstein was skeptical of this paper. When Lemaître approached Einstein at the 1927 Solvay Conference, the latter pointed out that Alexander Friedmann had proposed a similar solution to Einstein's equations in 1922, implying that the radius of the universe increased over time. (Einstein had also criticized Friedmann's calculations, but withdrew his comments.) In 1931, his *annus mirabilis*, [44] Lemaître published an article in *Nature* setting out his theory of the "primeval atom." [11]

Friedmann was handicapped by living and working in the USSR, and died in 1925, soon after inventing the Friedmann–Lemaître–Robertson–Walker metric. Because Lemaître spent his entire career in Europe, his scientific work is not as well known in the United States as that of Hubble or Einstein, both well known in the U.S. by virtue of residing there. Nevertheless, Lemaître's theory changed the course of cosmology. This was because Lemaître:

- Was well acquainted with the work of astronomers, and designed his theory to have testable implications and to be in accord with observations of the time, in particular to explain the observed redshift of galaxies and the linear relation between distances and velocities
- Proposed his theory at an opportune time, since Edwin Hubble would soon publish his velocity–distance relation that strongly supported an expanding universe and, consequently, Lemaître's Big Bang theory
- Had studied under Arthur Eddington, who made sure that Lemaître got a hearing in the scientific community.

Both Friedmann and Lemaître proposed relativistic cosmologies featuring an expanding universe. However, Lemaître was the first to propose that the expansion explains the redshift of galaxies. He further concluded that an initial "creation-like" event must have occurred. In the 1980s, Alan Guth and Andrei Linde modified this theory by adding to it a period of inflation.

Einstein at first dismissed Friedmann, and then (privately) Lemaître, out of hand, saying that not all mathematics lead to correct theories. After Hubble's discovery was published, Einstein quickly and publicly endorsed Lemaître's theory, helping both the theory and its proposer get fast recognition. [45]

Lemaître was also an early adopter of computers for cosmological calculations. He introduced the first computer to his university (a Burroughs E 101) in 1958 and was one of the inventors of the Fast Fourier transform algorithm. [46]

In 1931, Lemaître was the first scientist to propose the expansion of the universe was actually accelerating which was confirmed observationally in the 1990s through observations of very distant Type IA supernova with the Hubble Space Telescope which led to the 2011 Nobel Prize in Physics. [47] [48] [49]

In 1933, Lemaître found an important inhomogeneous solution of Einstein's field equations describing a spherical dust cloud, the Lemaître–Tolman metric.

In 1948 Lemaître published a polished mathematical essay "Quaternions et espace elliptique" which clarified an obscure space. [50] William Kingdon Clifford had cryptically described elliptic space in 1873 at a time when versors were too common to mention. [* ambiguous *] Lemaître developed the theory of quaternions from first principles so that his essay can stand on its own, but he recalled the Erlangen program [* further explanation needed *] in geometry while developing the metric geometry of elliptic space. [* citation needed *]

Lemaître was the first theoretical cosmologist ever nominated in 1954 for the Nobel Prize in Physics for his prediction of the expanding universe. Remarkably, he was also nominated for the 1956 Nobel Prize in Chemistry for his primeval atom theory.

On 17 March 1934, Lemaître received the Francqui Prize, the highest Belgian scientific distinction, from King Leopold III. [34] His proposers were Albert Einstein, Charles de la Vallée-Poussin and Alexandre de Hemptinne. The members of the international jury were Eddington, Langevin, Théophile de Donder and Marcel Dehalu. The same year he received the Mendel Medal of the Villanova University. [51]

In 1936, Lemaître received the Prix Jules Janssen, the highest award of the Société astronomique de France, the French astronomical society. [52]

Another distinction that the Belgian government reserves for exceptional scientists was allotted to him in 1950: the decennial prize for applied sciences for the period 1933–1942. [34]

In 1953, he was given the inaugural Eddington Medal awarded by the Royal Astronomical Society. [53] [54]

In 2005, Lemaître was voted to the 61st place of *De Grootste Belg* ("The Greatest Belgian"), a Flemish television program on the VRT. In the same year he was voted to the 78th place by the audience of the *Les plus grands Belges* ("The Greatest Belgians"), a television show of the RTBF.

On 17 July 2018, Google Doodle celebrated Georges Lemaître's 124th birthday. [55]

On 26 October 2018, an electronic vote among all members of the International Astronomical Union voted 78% to recommend changing the name of the Hubble law to the Hubble–Lemaître law. [6] [56]

## Tip 3: Use the Right Programming Solution

Singularities are not easy to spot in your robot code. It’s very normal not to notice them until you download your program to the robot and it behaves strangely. By then, however, you have already taken the robot out of production to reprogram it.

But, it doesn’t have to be like this. You can detect singularities very easily by using the right programming solution in the first place.

RoboDK has automatic singularity detection. It will not let you program the robot to move through a singularity. Instead, it will give you a helpful warning which tells you that the move would have been a problem.

For example, I programmed the example welding scenario into RoboDK and it told me: *“Movement is not possible. Joint 5 crosses 0 degrees. This is a singularity and is not allowed for a linear move.”*

## Singularity

The longer a star lives the denser it becomes and the denser it becomes the more distortion it makes. However this distortion is pointing towards the singularity. So the longer a star lives the more it distorts spacetime towards singularity.

**Singularity is a location in the future of stars.** An observer far away from a black hole sees the events near a back hole in slow motion. If he shines a beam of light into this black hole he will have to wait forever but still this beam of light will never reach the singularity.

The singularity is a location in the future of stars where gravity goes so mad that space and time become indistinguishable. From general relativity we know that this is a location where the structure of spacetime becomes singular (hence the name singularity). However singular (Ahad in Arabic أَحَدٌ ) is one of God's 99 names. In the Quran God swears by the locations of stars which turned out to carry His own name:

##### Quran 56:75-77

I swear by the locations of stars, it is a great swear if you knew, it is a noble Quran.

٧٥ فَلَا أُقْسِمُ بِمَوَاقِعِ النُّجُومِ

٧٦ وَإِنَّهُ لَقَسَمٌ لَوْ تَعْلَمُونَ عَظِيمٌ

٧٧ إِنَّهُ لَقُرْآنٌ كَرِيمٌ

Here God swears not by the stars themselves but rather by their locations (mawakeh in Arabic). Today we know that stars distort spacetime in the direction of the singularity, which turned out to carry God's own name: "Ahad أَحَدٌ ".

**How could an illiterate man who lived 1400 years ago have known about the singularity?**

## Ask Ethan: What Happens When A Black Hole's Singularity Evaporates?

The event horizon of a black hole is a spherical or spheroidal region from which nothing, not even . [+] light, can escape. Although conventional radiation is emitted from outside the event horizon, it is unclear where, when, or how the entropy/information encoded on the surface behaves in a merger scenario. But we do believe there is a singularity at the black hole's core.

NASA Dana Berry, SkyWorks Digital, Inc.

It's hard to imagine, given the full diversity of forms that matter takes in this Universe, that for millions of years, there were only neutral atoms of hydrogen and helium gas. It's perhaps equally hard to imagine that someday, quadrillions of years from now, all the stars will have gone dark. Only the remnants of our now-vibrant Universe will be left, including some of the most spectacular objects of all: black holes. But even they won't last forever. David Weber wants to know how that happens for this week's Ask Ethan, inquiring:

What happens when a black hole has lost enough energy due to hawking radiation that its energy density no longer supports a singularity with an event horizon? Put another way, what happens when a black hole ceases to be a black hole due to hawking radiation?

In order to answer this question, it's important to understand what a black hole actually is.

The anatomy of a very massive star throughout its life, culminating in a Type II Supernova when the . [+] core runs out of nuclear fuel.

Black holes generally form during the collapse of a massive star's core, where the spent nuclear fuel ceases to fuse into heavier elements. As fusion slows and ceases, the core experiences a severe drop in radiation pressure, which was the only thing holding the star up against gravitational collapse. While the outer layers often experience a runaway fusion reaction, blowing the progenitor star apart in a supernova, the core first collapses into a single atomic nucleus — a neutron star — but if the mass is too great, the neutrons themselves compress and collapse to such a dense state that a black hole forms. (A black hole can also form if a neutron star accretes enough mass from a companion star, crossing the threshold necessary to become a black hole.)

When a neutron star accretes enough matter, it can collapse to a black hole. When a black hole . [+] accretes matter, it grows an accretion disk and will increase its mass as matter gets funneled into the event horizon.

NASA/ESA Hubble Space Telescope collaboration

From a gravitational point of view, all it takes to become a black hole is to gather enough mass in a small enough volume of space that light cannot escape from within a certain region. Every mass, including planet Earth, has an escape velocity: the speed you'd need to achieve to completely escape from the gravitational pull at a given distance (e.g., the distance from Earth's center to its surface) from its center-of-mass. But if there's enough mass so that the speed you'd need to achieve at a certain distance from the center of mass is the speed of light or greater, then nothing can escape from it, since nothing can exceed the speed of light.

The mass of a black hole is the sole determining factor of the radius of the event horizon, for a . [+] non-rotating, isolated black hole.

That distance from the center of mass where the escape velocity equals the speed of light — let's call it **R** — defines the size of the black hole's event horizon. But the fact that there's matter inside under these conditions has another consequence that's less-well appreciated: this matter *must* collapse down to a singularity. You might think there could be a state of matter that's stable and has a finite volume within the event horizon, but that's not physically possible.

In order to exert an outward force, an interior particle would have to send a force-carrying particle away from the center-of-mass and closer to the event horizon. But that force-carrying particle is also limited by the speed of light, and no matter where you are inside the event horizon, all light-like curves wind up at the center. The situation is even worse for slower, massive particles. Once you form a black hole with an event horizon, all the matter inside gets crunched into a singularity.

The exterior spacetime to a Schwarzschild black hole, known as Flamm's Paraboloid, is easily . [+] calculable. But inside an event horizons, all geodesics lead to the central singularity.

Wikimedia Commons user AllenMcC

And since nothing can escape, you might think a black hole would remain a black hole forever. If it weren't for quantum physics, this would be exactly what happens. But in quantum physics, there's a non-zero amount of energy inherent to space itself: the quantum vacuum. In curved space, the quantum vacuum takes on slightly different properties than in flat space, and there are no regions where the curvature is greater than near the singularity of a black hole. Combining these two laws of nature — quantum physics and the General Relativistic spacetime around a black hole — gives us the phenomenon of Hawking radiation.

A visualization of QCD illustrates how particle/antiparticle pairs pop out of the quantum vacuum for . [+] very small amounts of time as a consequence of Heisenberg uncertainty.

Performing the quantum field theory calculation in curved space yields a surprising solution: that thermal, blackbody radiation is emitted in the space surrounding a black hole's event horizon. And the smaller the event horizon is, the greater the curvature of space near the event horizon is, and thus the greater the rate of Hawking radiation. If our Sun were a black hole, the temperature of the Hawking radiation would be about 62 nanokelvin if you took the black hole at the center of our galaxy, 4,000,000 times as massive, the temperature would be about 15 femtokelvin, or just 0.000025% the temperature of the less massive one.

An X-ray / Infrared composite image of the black hole at the center of our galaxy: Sagittarius A*. . [+] It has a mass of about four million Suns, and is found surrounded by hot, X-ray emitting gas. However, it also emits (undetectable) Hawking radiation, at much, much lower temperatures.

X-ray: NASA/UMass/D.Wang et al., IR: NASA/STScI

This means the smallest black holes decay the fastest, and the largest ones live the longest. Doing the math, a solar mass black hole would live for about 10^67 years before evaporating, but the black hole at the center of our galaxy would live for 10^20 times as long before decaying. The crazy thing about it all is that right up until the final fraction-of-a-second, the black hole still has an event horizon. Once you form a singularity, you remain a singularity — and you retain an event horizon — right up until the moment your mass goes to zero.

Hawking radiation is what inevitably results from the predictions of quantum physics in the curved . [+] spacetime surrounding a black hole's event horizon.

That final second of a black hole's life, however, will result in a very specific and very large release of energy. When the mass drops down to 228 metric tonnes, that's the signal that exactly one second remains. The event horizon size at the time will be 340 yoctometers, or 3.4 × 10^-22 meters: the size of one wavelength of a photon with an energy greater than any particle the LHC has ever produced. But in that final second, a total of 2.05 × 10^22 Joules of energy, the equivalent of five million megatons of TNT, will be released. It's as though a million nuclear fusion bombs went off all at once in a tiny region of space that's the final stage of black hole evaporation.

As a black hole shrinks in mass and radius, the Hawking radiation emanating from it becomes greater . [+] and greater in temperature and power.

What's left? Just outgoing radiation. Whereas previously, there was a singularity in space where mass, and possibly charge and angular momentum existed in an infinitesimally small volume, now there is none. Space has been restored to its previously non-singular state, after what must have seemed like an eternity: enough time for the Universe to have done all it's done to date trillions upon trillions of times over. There will be no other stars or sources of light left when this occurs for the first time in our Universe there will be no one to witness this spectacular explosion. But there's no "threshold" where this occurs. Rather, the black hole needs to evaporate completely. When it does, to the best of our knowledge, there will be nothing left behind at all but outgoing radiation.

Against a seemingly eternal backdrop of everlasting darkness, a single flash of light will emerge: . [+] the evaporation of the final black hole in the Universe.

In other words, if you were to watch the last black hole in our Universe evaporate, you would see an empty void of space, that displayed no light or signs of activity for perhaps 10^100 years or more. All of a sudden, a tremendous outrush of radiation of a very particular spectrum and magnitude would appear, leaving a single point in space at 300,000 km/s. For the last time in our observable Universe, an event would have occurred to bathe the Universe in radiation. The last black hole evaporation of all would, in a poetic way, be the final time that the Universe would ever say, "Let there be light!"

## Rupture velocity of plane strain shear cracks

Propagation of plane strain shear cracks is calculated numerically by using finite difference equations with second-order accuracy. The rupture model, in which stress drops gradually as slip increases, combines two different rupture criteria: (1) slip begins at a finite stress level (2) finite energy is absorbed per unit area as the crack advances. Solutions for this model are nonsingular. In some cases there may be a transition from rupture velocity less than Rayleigh velocity to rupture velocity greater than shear wave velocity. The locus of this transition is surveyed in the parameter space of fracture energy, upper yield stress, and crack length. A solution for this model can be represented as a convolution of a singular solution having abrupt stress drop with a ‘rupture distribution function.’ The convolution eliminates the singularity and spreads out the rupture front in space-time. If the solution for abrupt stress drop has an inverse square root singularity at the crack tip, as it does for sub-Rayleigh rupture velocity, then the rupture velocity of the convolved solution is independent of the rupture distribution function and depends only on the fracture energy and crack length. On the other hand, a crack with abrupt stress drop propagating faster than the shear wave velocity has a lower-order singularity. A supershear rupture front must necessarily be spread out in space-time if a finite fracture energy is absorbed as stress drops.

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