Penrose–Hawking singularity theorems

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https://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singularity_theorems 

 

The Penrose–Hawking singularity theorems (after Roger Penrose and Stephen Hawking) are a set of results in general relativity that attempt to answer the question of when gravitation produces singularities. The Penrose singularity theorem is a theorem in semi-Riemannian geometry and its general relativistic interpretation predicts a gravitational singularity in black hole formation. The Hawking singularity theorem is based on the Penrose theorem and it is interpreted as a gravitational singularity in the Big Bang situation. Penrose was awarded the Nobel Prize in Physics in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity", which he shared with Reinhard Genzel and Andrea Ghez.

Singularity

A singularity in solutions of the Einstein field equations is one of three things:

• Spacelike singularities: The singularity lies in the future or past of all events within a certain region. The Big Bang singularity and the typical singularity inside a non-rotating, uncharged Schwarzschild black hole are spacelike.
• Timelike singularities: These are singularities that can be avoided by an observer because they are not necessarily in the future of all events. An observer might be able to move around a timelike singularity. These are less common in known solutions of the Einstein field equations.
• Null singularities: These singularities occur on light-like or null surfaces. An example might be found in certain types of black hole interiors, such as the Cauchy horizon of a charged (Reissner-Nordström) or rotating (Kerr) black hole.

A singularity can be either strong or weak:

• - Weak singularities: A weak singularity is one where the tidal forces (which are responsible for the spaghettification in black holes) are not necessarily infinite. An observer falling into a weak singularity might not be torn apart before reaching the singularity, although the laws of physics would still break down there. The Cauchy horizon inside a charged or rotating black hole might be an example of a weak singularity.
• Strong singularities: A strong singularity is one where tidal forces become infinite. In a strong singularity, any object would be destroyed by infinite tidal forces as it approaches the singularity. The singularity at the center of a Schwarzschild black hole is an example of a strong singularity.

Space-like singularities are a feature of non-rotating uncharged black holes as described by the Schwarzschild metric, while time-like singularities are those that occur in charged or rotating black hole exact solutions. Both of them have the property of geodesic incompleteness, in which either some light-path or some particle-path cannot be extended beyond a certain proper time or affine parameter (affine parameter being the null analog of proper time).

The Penrose theorem guarantees that some sort of geodesic incompleteness occurs inside any black hole whenever matter satisfies reasonable energy conditions. The energy condition required for the black-hole singularity theorem is weak: it says that light rays are always focused together by gravity, never drawn apart, and this holds whenever the energy of matter is non-negative.

Hawking's singularity theorem is for the whole universe, and works backwards in time: it guarantees that the (classical) Big Bang has infinite density. This theorem is more restricted and only holds when matter obeys a stronger energy condition, called the strong energy condition, in which the energy is larger than the pressure. All ordinary matter, with the exception of a vacuum expectation value of a scalar field, obeys this condition. During inflation, the universe violates the dominant energy condition, and it was initially argued (e.g. by Starobinsky) that inflationary cosmologies could avoid the initial big-bang singularity. However, it has since been shown that inflationary cosmologies are still past-incomplete, and thus require physics other than inflation to describe the past boundary of the inflating region of spacetime.

It is still an open question whether (classical) general relativity predicts spacelike singularities in the interior of realistic charged or rotating black holes, or whether these are artefacts of high-symmetry solutions and turn into null or timelike singularities when perturbations are added.

Interpretation and significance

In general relativity, a singularity is a place that objects or light rays can reach in a finite time where the curvature becomes infinite, or spacetime stops being a manifold. Singularities can be found in all the black-hole spacetimes, the Schwarzschild metric, the Reissner–Nordström metric, the Kerr metric and the Kerr–Newman metric, and in all cosmological solutions that do not have a scalar field energy or a cosmological constant.

One cannot predict what might come "out" of a big-bang singularity in our past, or what happens to an observer that falls "in" to a black-hole singularity in the future, so they require a modification of physical law. Before Penrose, it was conceivable that singularities only form in contrived situations. For example, in the collapse of a star to form a black hole, if the star is spinning and thus possesses some angular momentum, maybe the centrifugal force partly counteracts gravity and keeps a singularity from forming. The singularity theorems prove that this cannot happen, and that a singularity will always form once an event horizon forms.

In the collapsing star example, since all matter and energy is a source of gravitational attraction in general relativity, the additional angular momentum only pulls the star together more strongly as it contracts: the part outside the event horizon eventually settles down to a Kerr black hole (see No-hair theorem). The part inside the event horizon necessarily has a singularity somewhere. The proof is somewhat constructive – it shows that the singularity can be found by following light-rays from a surface just inside the horizon. But the proof does not say what type of singularity occurs, spacelike, timelike, null, orbifold, jump discontinuity in the metric. It only guarantees that if one follows the time-like geodesics into the future, it is impossible for the boundary of the region they form to be generated by the null geodesics from the surface. This means that the boundary must either come from nowhere or the whole future ends at some finite extension.

An interesting "philosophical" feature of general relativity is revealed by the singularity theorems. Because general relativity predicts the inevitable occurrence of singularities, the theory is not complete without a specification for what happens to matter that hits the singularity. One can extend general relativity to a unified field theory, such as the Einstein–Maxwell–Dirac system, where no such singularities occur.

Elements of the theorems

In history, there is a deep connection between the curvature of a manifold and its topology. The Bonnet–Myers theorem states that a complete Riemannian manifold that has Ricci curvature everywhere greater than a certain positive constant must be compact. The condition of positive Ricci curvature is most conveniently stated in the following way: for every geodesic there is a nearby initially parallel geodesic that will bend toward it when extended, and the two will intersect at some finite length.

When two nearby parallel geodesics intersect (see conjugate point), the extension of either one is no longer the shortest path between the endpoints. The reason is that two parallel geodesic paths necessarily collide after an extension of equal length, and if one path is followed to the intersection then the other, you are connecting the endpoints by a non-geodesic path of equal length. This means that for a geodesic to be a shortest length path, it must never intersect neighboring parallel geodesics.

Starting with a small sphere and sending out parallel geodesics from the boundary, assuming that the manifold has a Ricci curvature bounded below by a positive constant, none of the geodesics are shortest paths after a while, since they all collide with a neighbor. This means that after a certain amount of extension, all potentially new points have been reached. If all points in a connected manifold are at a finite geodesic distance from a small sphere, the manifold must be compact.

Roger Penrose argued analogously in relativity. If null geodesics, the paths of light rays, are followed into the future, points in the future of the region are generated. If a point is on the boundary of the future of the region, it can only be reached by going at the speed of light, no slower, so null geodesics include the entire boundary of the proper future of a region. When the null geodesics intersect, they are no longer on the boundary of the future, they are in the interior of the future. So, if all the null geodesics collide, there is no boundary to the future.

In relativity, the Ricci curvature, which determines the collision properties of geodesics, is determined by the energy tensor, and its projection on light rays is equal to the null-projection of the energy–momentum tensor and is always non-negative. This implies that the volume of a congruence of parallel null geodesics once it starts decreasing, will reach zero in a finite time. Once the volume is zero, there is a collapse in some direction, so every geodesic intersects some neighbor.

Penrose concluded that whenever there is a sphere where all the outgoing (and ingoing) light rays are initially converging, the boundary of the future of that region will end after a finite extension, because all the null geodesics will converge. This is significant, because the outgoing light rays for any sphere inside the horizon of a black hole solution are all converging, so the boundary of the future of this region is either compact or comes from nowhere. The future of the interior either ends after a finite extension, or has a boundary that is eventually generated by new light rays that cannot be traced back to the original sphere.

Nature of a singularity

The singularity theorems use the notion of geodesic incompleteness as a stand-in for the presence of infinite curvatures. Geodesic incompleteness is the notion that there are geodesics, paths of observers through spacetime, that can only be extended for a finite time as measured by an observer traveling along one. Presumably, at the end of the geodesic the observer has fallen into a singularity or encountered some other pathology at which the laws of general relativity break down.

Assumptions of the theorems

Typically a singularity theorem has three ingredients:

1. An energy condition on the matter,
2. A condition on the global structure of spacetime,
3. Gravity is strong enough (somewhere) to trap a region.

There are various possibilities for each ingredient, and each leads to different singularity theorems.

Tools employed

A key tool used in the formulation and proof of the singularity theorems is the Raychaudhuri equation, which describes the divergence ${\displaystyle \theta }$ of a congruence (family) of geodesics. The divergence of a congruence is defined as the derivative of the log of the determinant of the congruence volume. The Raychaudhuri equation is

${\displaystyle \dot{\theta }=-{\sigma }_{ab}{\sigma }^{ab}-\frac{1}{3}{\theta }^2-{E[\overrightarrow{X}{]}^a}_a}$

where ${\displaystyle {\sigma }_{ab}}$ is the shear tensor of the congruence and ${\displaystyle {E[\overrightarrow{X}{]}^a}_a=R_{mn}{X}^m{X}^n}$ is also known as the Raychaudhuri scalar (see the congruence page for details). The key point is that ${\displaystyle {E[\overrightarrow{X}{]}^a}_a}$ will be non-negative provided that the Einstein field equations hold and

• the null energy condition holds and the geodesic congruence is null, or
• the strong energy condition holds and the geodesic congruence is timelike.

When these hold, the divergence becomes infinite at some finite value of the affine parameter. Thus all geodesics leaving a point will eventually reconverge after a finite time, provided the appropriate energy condition holds, a result also known as the focusing theorem.

This is relevant for singularities thanks to the following argument:

1. Suppose we have a spacetime that is globally hyperbolic, and two points ${\displaystyle p}$ and ${\displaystyle q}$ that can be connected by a timelike or null curve. Then there exists a geodesic of maximal length connecting ${\displaystyle p}$ and ${\displaystyle q}$. Call this geodesic ${\displaystyle \gamma }$.
2. The geodesic ${\displaystyle \gamma }$ can be varied to a longer curve if another geodesic from ${\displaystyle p}$ intersects ${\displaystyle \gamma }$ at another point, called a conjugate point.
3. From the focusing theorem, we know that all geodesics from ${\displaystyle p}$ have conjugate points at finite values of the affine parameter. In particular, this is true for the geodesic of maximal length. But this is a contradiction – one can therefore conclude that the spacetime is geodesically incomplete.

In general relativity, there are several versions of the Penrose–Hawking singularity theorem. Most versions state, roughly, that if there is a trapped null surface and the energy density is nonnegative, then there exist geodesics of finite length that cannot be extended.

These theorems, strictly speaking, prove that there is at least one non-spacelike geodesic that is only finitely extendible into the past but there are cases in which the conditions of these theorems obtain in such a way that all past-directed spacetime paths terminate at a singularity.

Versions

There are many versions; below is the null version:

Assume

1. The null energy condition holds.
2. We have a noncompact connected Cauchy surface.
3. We have a closed trapped null surface ${\displaystyle \mathcal{T}}$.

Then, we either have null geodesic incompleteness, or closed timelike curves.

Sketch of proof: Proof by contradiction. The boundary of the future of ${\displaystyle \mathcal{T}}$, ${\displaystyle \dot{J}(\mathcal{T})}$ is generated by null geodesic segments originating from ${\displaystyle \mathcal{T}}$ with tangent vectors orthogonal to it. Being a trapped null surface, by the null Raychaudhuri equation, both families of null rays emanating from ${\displaystyle \mathcal{T}}$ will encounter caustics. (A caustic by itself is unproblematic. For instance, the boundary of the future of two spacelike separated points is the union of two future light cones with the interior parts of the intersection removed. Caustics occur where the light cones intersect, but no singularity lies there.) The null geodesics generating ${\displaystyle \dot{J}(\mathcal{T})}$ have to terminate, however, i.e. reach their future endpoints at or before the caustics. Otherwise, we can take two null geodesic segments – changing at the caustic – and then deform them slightly to get a timelike curve connecting a point on the boundary to a point on ${\displaystyle \mathcal{T}}$, a contradiction. But as ${\displaystyle \mathcal{T}}$ is compact, given a continuous affine parameterization of the geodesic generators, there exists a lower bound to the absolute value of the expansion parameter. So, we know caustics will develop for every generator before a uniform bound in the affine parameter has elapsed. As a result, ${\displaystyle \dot{J}(\mathcal{T})}$ has to be compact. Either we have closed timelike curves, or we can construct a congruence by timelike curves, and every single one of them has to intersect the noncompact Cauchy surface exactly once. Consider all such timelike curves passing through ${\displaystyle \dot{J}(\mathcal{T})}$ and look at their image on the Cauchy surface. Being a continuous map, the image also has to be compact. Being a timelike congruence, the timelike curves can't intersect, and so, the map is injective. If the Cauchy surface were noncompact, then the image has a boundary. We're assuming spacetime comes in one connected piece. But ${\displaystyle \dot{J}(\mathcal{T})}$ is compact and boundariless because the boundary of a boundary is empty. A continuous injective map can't create a boundary, giving us our contradiction.

Loopholes: If closed timelike curves exist, then timelike curves don't have to intersect the partial Cauchy surface. If the Cauchy surface were compact, i.e. space is compact, the null geodesic generators of the boundary can intersect everywhere because they can intersect on the other side of space.

Other versions of the theorem involving the weak or strong energy condition also exist.

Modified gravity

In modified gravity, the Einstein field equations do not hold and so these singularities do not necessarily arise. For example, in Infinite Derivative Gravity, it is possible for ${\displaystyle {E[\overrightarrow{X}{]}^a}_a}$ to be negative even if the Null Energy Condition holds.

Roger Penrose

From Wikipedia, the free encyclopedia

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

Sir Roger Penrose OM FRS HonFInstP
Penrose in 2011

Sir Roger Penrose OM FRS HonFInstP (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fellow of Wadham College, Oxford, and an honorary fellow of St John's College, Cambridge, and University College London.

Penrose has contributed to the mathematical physics of general relativity and cosmology. He has received several prizes and awards, including the 1988 Wolf Prize in Physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems, and the 2020 Nobel Prize in Physics "for the discovery that black hole formation is a robust prediction of the general theory of relativity". He is regarded as one of the greatest living physicists, mathematicians and scientists, and is particularly noted for the breadth and depth of his work in both natural and formal sciences.

Early life and education

Born in Colchester, Essex, Roger Penrose is a son of medical doctor Margaret (Leathes) and psychiatrist and geneticist Lionel Penrose. His paternal grandparents were J. Doyle Penrose, an Irish-born artist, and The Hon. Elizabeth Josephine, daughter of Alexander Peckover, 1st Baron Peckover; his maternal grandparents were physiologist John Beresford Leathes and Sonia Marie Natanson, a Russian Jew. His uncle was artist Roland Penrose, whose son with American photographer Lee Miller is Antony Penrose. Penrose is the brother of physicist Oliver Penrose, of geneticist Shirley Hodgson, and of chess Grandmaster Jonathan Penrose. Their stepfather was the mathematician and computer scientist Max Newman.

Penrose spent World War II as a child in Canada where his father worked in London, Ontario. Penrose studied at University College School. He attended and attained a first class degree in mathematics from University College London.

In 1955, while a student, Penrose reintroduced the E. H. Moore generalised matrix inverse, also known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar in 1951. Having started research under the professor of geometry and astronomy, Sir W. V. D. Hodge, Penrose finished his PhD at St John's College, Cambridge, in 1958, with a thesis on tensor methods in algebraic geometry supervised by algebraist and geometer John A. Todd. He devised and popularised the Penrose triangle in the 1950s in collaboration with his father, describing it as "impossibility in its purest form", and exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it. Escher's Waterfall, and Ascending and Descending were in turn inspired by Penrose.

The Penrose triangle

As reviewer Manjit Kumar puts it:

As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up and down. An article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces.

Research and career

Penrose spent the academic year 1956–57 as an assistant lecturer at Bedford College, London and was then a research fellow at St John's College, Cambridge. During that three-year post, he married Joan Isabel Wedge, in 1959. Before the fellowship ended Penrose won a NATO Research Fellowship for 1959–61, first at Princeton and then at Syracuse University. Returning to the University of London, Penrose spent two years, 1961–63, as a researcher at King's College, London, before returning to the United States to spend the year 1963–64 as a visiting associate professor at the University of Texas at Austin. He later held visiting positions at Yeshiva, Princeton, and Cornell during 1966–67 and 1969.

In 1964, while a reader at Birkbeck College, London, (and having had his attention drawn from pure mathematics to astrophysics by the cosmologist Dennis Sciama, then at Cambridge) in the words of Kip Thorne of Caltech, "Roger Penrose revolutionised the mathematical tools that we use to analyse the properties of spacetime". Until then, work on the curved geometry of general relativity had been confined to configurations with sufficiently high symmetry for Einstein's equations to be solvable explicitly, and there was doubt about whether such cases were typical. One approach to this issue was by the use of perturbation theory, as developed under the leadership of John Archibald Wheeler at Princeton. The other, and more radically innovative, approach initiated by Penrose was to overlook the detailed geometrical structure of spacetime and instead concentrate attention just on the topology of the space, or at most its conformal structure, since it is the latter – as determined by the lay of the lightcones – that determines the trajectories of lightlike geodesics, and hence their causal relationships. The importance of Penrose's epoch-making paper "Gravitational Collapse and Space-Time Singularities" was not its only result, summarised roughly as that if an object such as a dying star implodes beyond a certain point, then nothing can prevent the gravitational field getting so strong as to form some kind of singularity. It also showed a way to obtain similarly general conclusions in other contexts, notably that of the cosmological Big Bang, which he dealt with in collaboration with Dennis Sciama's most famous student, Stephen Hawking.

Predicted view from outside the event horizon of a black hole lit by a thin accretion disc

It was in the local context of gravitational collapse that the contribution of Penrose was most decisive, starting with his 1969 cosmic censorship conjecture, to the effect that any ensuing singularities would be confined within a well-behaved event horizon surrounding a hidden space-time region for which Wheeler coined the term black hole, leaving a visible exterior region with strong but finite curvature, from which some of the gravitational energy may be extractable by what is known as the Penrose process, while accretion of surrounding matter may release further energy that can account for astrophysical phenomena such as quasars.

Following up his "weak cosmic censorship hypothesis", Penrose went on, in 1979, to formulate a stronger version called the "strong censorship hypothesis". Together with the Belinski–Khalatnikov–Lifshitz conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. Also from 1979, dates Penrose's influential Weyl curvature hypothesis on the initial conditions of the observable part of the universe and the origin of the second law of thermodynamics. Penrose and James Terrell independently realised that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation. This effect has come to be called the Terrell rotation or Penrose–Terrell rotation.

A Penrose tiling

In 1967, Penrose invented the twistor theory, which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature (2,2).

Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, and are the first tilings to exhibit fivefold rotational symmetry. In 1984, such patterns were observed in the arrangement of atoms in quasicrystals. Another noteworthy contribution is his 1971 invention of spin networks, which later came to form the geometry of spacetime in loop quantum gravity. He was influential in popularizing what are commonly known as Penrose diagrams (causal diagrams).

In 1983, Penrose was invited to teach at Rice University in Houston, by the then provost Bill Gordon. He worked there from 1983 to 1987. His doctoral students have included, among others, Andrew Hodges, Lane Hughston, Richard Jozsa, Claude LeBrun, John McNamara, Tristan Needham, Tim Poston, Asghar Qadir, and Richard S. Ward.

In 2004, Penrose released The Road to Reality: A Complete Guide to the Laws of the Universe, a 1,099-page comprehensive guide to the Laws of Physics that includes an explanation of his own theory. The Penrose Interpretation predicts the relationship between quantum mechanics and general relativity, and proposes that a quantum state remains in superposition until the difference of space-time curvature attains a significant level.

Penrose is the Francis and Helen Pentz Distinguished Visiting Professor of Physics and Mathematics at Pennsylvania State University.

An earlier universe

WMAP image of the (extremely tiny) anisotropies in the cosmic background radiation

In 2010, Penrose reported possible evidence, based on concentric circles found in Wilkinson Microwave Anisotropy Probe data of the cosmic microwave background sky, of an earlier universe existing before the Big Bang of our own present universe. He mentions this evidence in the epilogue of his 2010 book Cycles of Time, a book in which he presents his reasons, to do with Einstein's field equations, the Weyl curvature C, and the Weyl curvature hypothesis (WCH), that the transition at the Big Bang could have been smooth enough for a previous universe to survive it. He made several conjectures about C and the WCH, some of which were subsequently proved by others, and he also popularized his conformal cyclic cosmology (CCC) theory. In this theory, Penrose postulates that at the end of the universe all matter is eventually contained within black holes, which subsequently evaporate via Hawking radiation. At this point, everything contained within the universe consists of photons, which "experience" neither time nor space. There is essentially no difference between an infinitely large universe consisting only of photons and an infinitely small universe consisting only of photons. Therefore, a singularity for a Big Bang and an infinitely expanded universe are equivalent.

In simple terms, Penrose believes that the singularity in Einstein's field equation at the Big Bang is only an apparent singularity, similar to the well-known apparent singularity at the event horizon of a black hole. The latter singularity can be removed by a change of coordinate system, and Penrose proposes a different change of coordinate system that will remove the singularity at the big bang. One implication of this is that the major events at the Big Bang can be understood without unifying general relativity and quantum mechanics, and therefore we are not necessarily constrained by the Wheeler–DeWitt equation, which disrupts time. Alternatively, one can use the Einstein–Maxwell–Dirac equations.

Consciousness

Penrose at a conference

Penrose has written books on the connection between fundamental physics and human (or animal) consciousness. In The Emperor's New Mind (1989), he argues that known laws of physics are inadequate to explain the phenomenon of consciousness. Penrose proposes the characteristics this new physics may have and specifies the requirements for a bridge between classical and quantum mechanics (what he calls correct quantum gravity). Penrose uses a variant of Turing's halting theorem to demonstrate that a system can be deterministic without being algorithmic. (For example, imagine a system with only two states, ON and OFF. If the system's state is ON when a given Turing machine halts and OFF when the Turing machine does not halt, then the system's state is completely determined by the machine; nevertheless, there is no algorithmic way to determine whether the Turing machine stops.)

Penrose believes that such deterministic yet non-algorithmic processes may come into play in the quantum mechanical wave function reduction, and may be harnessed by the brain. He argues that computers today are unable to have intelligence because they are algorithmically deterministic systems. He argues against the viewpoint that the rational processes of the mind are completely algorithmic and can thus be duplicated by a sufficiently complex computer. This contrasts with supporters of strong artificial intelligence, who contend that thought can be simulated algorithmically. He bases this on claims that consciousness transcends formal logic because factors such as the insolubility of the halting problem and Gödel's incompleteness theorem prevent an algorithmically based system of logic from reproducing such traits of human intelligence as mathematical insight. These claims were originally espoused by the philosopher John Lucas of Merton College, Oxford.

The Penrose–Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence has been criticised by mathematicians, computer scientists and philosophers. Many experts in these fields assert that Penrose's argument fails, though different authors may choose different aspects of the argument to attack. Marvin Minsky, a leading proponent of artificial intelligence, was particularly critical, stating that Penrose "tries to show, in chapter after chapter, that human thought cannot be based on any known scientific principle." Minsky's position is exactly the opposite – he believed that humans are, in fact, machines, whose functioning, although complex, is fully explainable by current physics. Minsky maintained that "one can carry that quest [for scientific explanation] too far by only seeking new basic principles instead of attacking the real detail. This is what I see in Penrose's quest for a new basic principle of physics that will account for consciousness."

Penrose responded to criticism of The Emperor's New Mind with his follow-up 1994 book Shadows of the Mind, and in 1997 with The Large, the Small and the Human Mind. In those works, he also combined his observations with those of anesthesiologist Stuart Hameroff.

Penrose and Hameroff have argued that consciousness is the result of quantum gravity effects in microtubules, which they dubbed Orch-OR (orchestrated objective reduction). Max Tegmark, in a paper in Physical Review E, calculated that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 10,000,000,000. The reception of the paper is summed up by this statement in Tegmark's support: "Physicists outside the fray, such as IBM's John A. Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior'". Tegmark's paper has been widely cited by critics of the Penrose–Hameroff position.

In their reply to Tegmark's paper, also published in Physical Review E, the physicists Scott Hagan, Jack Tuszyński and Hameroff claimed that Tegmark did not address the Orch-OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch-OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, but still well short of the 25 ms required if the quantum processing in the theory was to be linked to the 40 Hz gamma synchrony, as Orch-OR suggested. To bridge this gap, the group made a series of proposals. They supposed that the interiors of neurons could alternate between liquid and gel states. In the gel state, it was further hypothesized that the water electrical dipoles are oriented in the same direction, along the outer edge of the microtubule tubulin subunits. Hameroff et al. proposed that this ordered water could screen any quantum coherence within the tubulin of the microtubules from the environment of the rest of the brain. Each tubulin also has a tail extending out from the microtubules, which is negatively charged, and therefore attracts positively charged ions. It is suggested that this could provide further screening. Further to this, there was a suggestion that the microtubules could be pumped into a coherent state by biochemical energy.

Penrose in the University of Santiago de Compostela to receive the Fonseca Prize

Finally, he suggested that the configuration of the microtubule lattice might be suitable for quantum error correction, a means of holding together quantum coherence in the face of environmental interaction.

Hameroff, in a lecture in part of a Google Tech talks series exploring quantum biology, gave an overview of current research in the area, and responded to subsequent criticisms of the Orch-OR model. In addition to this, a 2011 paper by Roger Penrose and Stuart Hameroff published in the Journal of Cosmology gives an updated model of their Orch-OR theory, in light of criticisms, and discusses the place of consciousness within the universe.

Phillip Tetlow, although himself supportive of Penrose's views, acknowledges that Penrose's ideas about the human thought process are at present a minority view in scientific circles, citing Minsky's criticisms and quoting science journalist Charles Seife's description of Penrose as "one of a handful of scientists" who believe that the nature of consciousness suggests a quantum process.

In January 2014, Hameroff and Penrose ventured that a discovery of quantum vibrations in microtubules by Anirban Bandyopadhyay of the National Institute for Materials Science in Japan supports the hypothesis of Orch-OR theory. A reviewed and updated version of the theory was published along with critical commentary and debate in the March 2014 issue of Physics of Life Reviews.

Publications

His popular publications include:

• The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics (1989)
• Shadows of the Mind: A Search for the Missing Science of Consciousness (1994)
• The Road to Reality: A Complete Guide to the Laws of the Universe (2004)
• Cycles of Time: An Extraordinary New View of the Universe (2010)
• Fashion, Faith, and Fantasy in the New Physics of the Universe (2016)

His co-authored publications include:

• The Nature of Space and Time (with Stephen Hawking) (1996)
• The Large, the Small and the Human Mind (with Abner Shimony, Nancy Cartwright, and Stephen Hawking) (1997)
• White Mars: The Mind Set Free (with Brian Aldiss) (1999)

His academic books include:

• Techniques of Differential Topology in Relativity (1972, ISBN 0-89871-005-7)
• Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (with Wolfgang Rindler, 1987) ISBN 0-521-33707-0 (paperback)
• Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (with Wolfgang Rindler, 1988) (reprint), ISBN 0-521-34786-6 (paperback)

His forewords to other books include:

• Foreword to “The Map and the Territory: Exploring the foundations of science, thought and reality” by Shyam Wuppuluri and Francisco Antonio Doria. Published by Springer in "The Frontiers Collection", 2018.
• Foreword to Beating the Odds: The Life and Times of E. A. Milne, written by Meg Weston Smith. Published by World Scientific Publishing Co in June 2013.
• Foreword to "A Computable Universe" by Hector Zenil. Published by World Scientific Publishing Co in December 2012.
• Foreword to Quantum Aspects of Life by Derek Abbott, Paul C. W. Davies, and Arun K. Pati. Published by Imperial College Press in 2008.
• Foreword to Fearful Symmetry by Anthony Zee's. Published by Princeton University Press in 2007.

Awards and honours

Penrose during a lecture

Penrose has been awarded many prizes for his contributions to science. In 1971, he was awarded the Dannie Heineman Prize for Astrophysics. He was elected a Fellow of the Royal Society (FRS) in 1972. In 1975, Stephen Hawking and Penrose were jointly awarded the Eddington Medal of the Royal Astronomical Society. In 1985, he was awarded the Royal Society Royal Medal. Along with Stephen Hawking, he was awarded the prestigious Wolf Foundation Prize for Physics in 1988.

In 1989, Penrose was awarded the Dirac Medal and Prize of the British Institute of Physics. He was also made an Honorary Fellow of the Institute of Physics (HonFInstP).

In 1990, Penrose was awarded the Albert Einstein Medal for outstanding work related to the work of Albert Einstein by the Albert Einstein Society. In 1991, he was awarded the Naylor Prize of the London Mathematical Society. From 1992 to 1995, he served as President of the International Society on General Relativity and Gravitation. In 1994, Penrose was knighted for services to science. In the same year, he was also awarded an Honorary Degree (Doctor of Science) by the University of Bath, and became a member of Polish Academy of Sciences. In 1998, he was elected Foreign Associate of the United States National Academy of Sciences. In 2000, he was appointed a Member of the Order of Merit (OM).

In 2004, he was awarded the De Morgan Medal for his wide and original contributions to mathematical physics. To quote the citation from the London Mathematical Society:

His deep work on General Relativity has been a major factor in our understanding of black holes. His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.

In 2005, Penrose was awarded an honorary doctorate by Warsaw University and Katholieke Universiteit Leuven (Belgium), and in 2006 by the University of York. In 2006, he also won the Dirac Medal given by the University of New South Wales. In 2008, Penrose was awarded the Copley Medal. He is also a Distinguished Supporter of Humanists UK and one of the patrons of the Oxford University Scientific Society.

He was elected to the American Philosophical Society in 2011. The same year, he was also awarded the Fonseca Prize by the University of Santiago de Compostela.

In 2012, Penrose was awarded the Richard R. Ernst Medal by ETH Zürich for his contributions to science and strengthening the connection between science and society. In 2015 Penrose was awarded an honorary doctorate by CINVESTAV-IPN (Mexico).

In 2017, he was awarded the Commandino Medal at the Urbino University for his contributions to the history of science.

In 2020, Penrose was awarded one half of the Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity, a half-share also going to Reinhard Genzel and Andrea Ghez for the discovery of a supermassive compact object at the centre of our galaxy.

Personal life

Penrose's first marriage was to American Joan Isabel Penrose (née Wedge), whom he married in 1959. They had three sons. Penrose is now married to Vanessa Thomas, director of Academic Development at Cokethorpe School and former head of mathematics at Abingdon School. They had one son.

Religious views

During an interview with BBC Radio 4 on 25 September 2010, Penrose stated, "I'm not a believer myself. I don't believe in established religions of any kind." He regards himself as an agnostic. In the 1991 film A Brief History of Time, he also said, "I think I would say that the universe has a purpose, it's not somehow just there by chance … some people, I think, take the view that the universe is just there and it runs along—it's a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it."

Penrose is a patron of Humanists UK.

Religious interpretations of the Big Bang theory

From Wikipedia, the free encyclopedia

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

 

Since the emergence of the Big Bang theory as the dominant physical cosmological paradigm, there have been a variety of reactions by religious groups regarding its implications for religious cosmologies. Some accept the scientific evidence at face value, some seek to harmonize the Big Bang with their religious tenets, and some reject or ignore the evidence for the Big Bang theory.

Background

The Big Bang itself is a scientific theory, and as such, stands or falls by its agreement with observations. However, as a theory which addresses the nature of the universe since its earliest discernible existence, the Big Bang carries possible theological implications regarding the concept of creation out of nothing. Many atheist philosophers have argued against the idea of the Universe having a beginning – the universe might simply have existed for all eternity, but with the emerging evidence of the Big Bang theory, both theists and physicists have viewed it as capable of being explained by theism; a popular philosophical argument for the existence of God known as the Kalam cosmological argument rests in the concepts of the Big Bang. In the 1920s and 1930s, almost every major cosmologist preferred an eternal steady state universe, and several complained that the beginning of time implied by the Big Bang imported religious concepts into physics; this objection was later repeated by supporters of the steady-state theory, who rejected the implication that the universe had a beginning.

Hinduism

The view from the Hindu Puranas is that of an eternal universe cosmology, in which time has no absolute beginning, but rather is infinite and cyclic, rather than a universe which originated from a Big Bang. However, the Encyclopædia of Hinduism, referencing Katha Upanishad 2:20, states that the Big Bang theory reminds humanity that everything came from the Brahman which is "subtler than the atom, greater than the greatest." It consists of several "Big Bangs" and "Big Crunches" following each other in a cyclical manner.

The Nasadiya Sukta, the Hymn of Creation in the Rigveda (10:129), mentions the world beginning from nothing through the power of heat. This can be seen as corresponding to the Big Bang theory.

THEN was not non-existent nor existent: there was no realm of air, no sky beyond it. What covered in, and where? and what gave shelter? Was water there, unfathomed depth of water?

— Rig Veda X.129.1

Death was not then, nor was there aught immortal: no sign was there, the day's and night's divider. That One Thing, breathless, breathed by its own nature: apart from it was nothing whatsoever

— Rig Veda X.129.2

Several prominent modern scientists have remarked that Hinduism (and also Buddhism and Jainism by extension as all three faiths share most of these philosophies) is the only religion (or civilization) in all of recorded history, that has timescales and theories in astronomy (cosmology), that appear to correspond to those of modern scientific cosmology, e.g. Carl Sagan, Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Robert Oppenheimer, George Sudarshan, Fritjof Capra etc. Sir Roger Penrose is among the present-day physicists that believe in a cyclical model for the Universe, wherein there are alternating cycles consisting of Big Bangs and Big Crunches, and he describes this model to be "a bit more like Hindu philosophy" as compared to the Abrahamic faiths.

Christianity

Further information: Christianity and science

The Big Bang theory was partly developed by a Catholic priest, Georges Lemaître, who believed that there was neither a connection nor a conflict between his religion and his science. At the November 22, 1951, opening meeting of the Pontifical Academy of Sciences, Pope Pius XII declared that the Big Bang theory does not conflict with the Catholic concept of creation. Some Conservative Protestant Christian denominations have also welcomed the Big Bang theory as supporting a historical interpretation of the doctrine of creation; however, adherents of Young Earth creationism, who advocate a very literal interpretation of the Book of Genesis, tend to reject the theory.

Baháʼí Faith

Further information: Baháʼí Faith and science

Bahá’u’lláh, the founder of the Baháʼí Faith, has taught that the universe has "neither beginning nor ending". In the Tablet of Wisdom ("Lawh-i-Hikmat", written 1873–1874). Bahá'u'lláh states: "That which hath been in existence had existed before, but not in the form thou seest today. The world of existence came into being through the heat generated from the interaction between the active force and that which is its recipient. These two are the same, yet they are different." The terminology used here refers to ancient Greek and Islamic philosophy (al-Kindi, Avicenna, Fakhr al-Din al-Razi and Shaykh Ahmad). In an early text, Bahá’u’lláh describes the successive creation of the four natures heat and cold (the active force), dryness and moisture (the recipients), and the four elements fire, air, water and earth. About the phrase "That which hath been in existence had existed before, but not in the form thou seest today," 'Abdu'l-Bahá has stated that it means that the universe is evolving.^[34] He also states that "the substance and primary matter of contingent beings is the ethereal power, which is invisible and only known through its effects... Ethereal matter is itself both the active force and the recipient... it is the sign of the Primal Will in the phenomenal world... The ethereal matter is, therefore, the cause, since light, heat, and electricity appear from it. It is also the effect, for as vibrations take place in it, they become visible...".

Jean-Marc Lepain, Robin Mihrshahi, Dale E. Lehman and Julio Savi suggest a possible relation of this statement with the Big Bang theory.

Islam

Further information: Islamic attitudes towards science

Writing for the Kyoto Bulletin of Islamic Area Studies, Haslin Hasan and Ab. Hafiz Mat Tuah wrote that modern scientific ideas on cosmology are creating new ideas on how to interpret the Quran's cosmogonical terms. In particular, some modern-day Muslim groups have advocated for interpreting the term al-sama, traditionally believed to be a reference to both the sky and the seven heavens, as instead referring to the universe as a whole.

Mirza Tahir Ahmad, head of the Ahmadiyya community, asserted in his book Revelation, Rationality, Knowledge & Truth that the Big Bang theory was foretold in the Quran. He referenced the verse 30 of the Sūrat al-Anbiyāʼ, which says that the heavens and the earth were a joined entity:

Have those who disbelieved not considered that the heavens and the earth were a joined entity, and We separated them and made from water every living thing? Then will they not believe?

— Quran 21:30

This view that the Qu'ran references the initial singularity of the Big Bang is also accepted by many Muslim scholars outside of the Ahmadiyya community such as Muhammad Tahir-ul-Qadri, who is a Sufi scholar, and Muhammad Asad, who was a nondenominational Muslim scholar. Further, some scholars such as Faheem Ashraf of the Islamic Research Foundation International, Inc. and Sheikh Omar Suleiman of the Yaqeen Institute for Islamic Research argue that the scientific theory of an expanding universe is described in Sūrat adh-Dhāriyāt:

And the heaven We constructed with strength, and indeed, We are [its] expander.

— Quran 51:47

Symmetry breaking

From Wikipedia, the free encyclopedia

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

 A ball is initially located at the top of the central hill (C). This position is an unstable equilibrium: a very small perturbation will cause it to fall to one of the two stable wells left (L) or right (R). Even if the hill is symmetric and there is no reason for the ball to fall on either side, the observed final state is not symmetric.

For other uses, see Symmetry breaking (disambiguation).

In physics, symmetry breaking is a phenomenon where a disordered but symmetric state collapses into an ordered, but less symmetric state. This collapse is often one of many possible bifurcations that a particle can take as it approaches a lower energy state. Due to the many possibilities, an observer may assume the result of the collapse to be arbitrary. This phenomenon is fundamental to quantum field theory (QFT), and further, contemporary understandings of physics. Specifically, it plays a central role in the Glashow–Weinberg–Salam model which forms part of the Standard model modelling the electroweak sector.

A (black) particle is always driven to lowest energy. In the proposed ${\displaystyle {\mathbb{Z}}_2}$-Symmetric system, it has two possible (purple) states. When it spontaneously breaks symmetry, it collapses into one of the two states. This phenomenon is known as spontaneous symmetry breaking.

A 3D representation of a particle in a symmetric system (a Higgs Mechanism) before assuming a lower energy state

In an infinite system (Minkowski spacetime) symmetry breaking occurs, however in a finite system (that is, any real super-condensed system), the system is less predictable, but in many cases quantum tunneling occurs. Symmetry breaking and tunneling relate through the collapse of a particle into non-symmetric state as it seeks a lower energy.

Symmetry breaking can be distinguished into two types, explicit and spontaneous. They are characterized by whether the equations of motion fail to be invariant, or the ground state fails to be invariant.

Non-technical description

This section describes spontaneous symmetry breaking. In layman's terms, this is the idea that for a physical system, the lowest energy configuration (the vacuum state) is not the most symmetric configuration of the system. Roughly speaking there are three types of symmetry that can be broken: discrete, continuous and gauge, ordered in increasing technicality.

An example of a system with discrete symmetry is given by the figure with the red graph: consider a particle moving on this graph, subject to gravity. A similar graph could be given by the function ${\displaystyle f(x)=(x^2-a^2{)}^2}$. This system is symmetric under reflection in the y-axis. There are three possible stationary states for the particle: the top of the hill at ${\displaystyle x=0}$, or the bottom, at ${\displaystyle x=\pm a}$. When the particle is at the top, the configuration respects the reflection symmetry: the particle stays in the same place when reflected. However, the lowest energy configurations are those at ${\displaystyle x=\pm a}$. When the particle is in either of these configurations, it is no longer fixed under reflection in the y-axis: reflection swaps the two vacuum states.

An example with continuous symmetry is given by a 3d analogue of the previous example, from rotating the graph around an axis through the top of the hill, or equivalently given by the graph ${\displaystyle f(x,y)=(x^2+y^2-a^2{)}^2}$. This is essentially the graph of the Mexican hat potential. This has a continuous symmetry given by rotation about the axis through the top of the hill (as well as a discrete symmetry by reflection through any radial plane). Again, if the particle is at the top of the hill it is fixed under rotations, but it has higher gravitational energy at the top. At the bottom, it is no longer invariant under rotations but minimizes its gravitational potential energy. Furthermore rotations move the particle from one energy minimizing configuration to another. There is a novelty here not seen in the previous example: from any of the vacuum states it is possible to access any other vacuum state with only a small amount of energy, by moving around the trough at the bottom of the hill, whereas in the previous example, to access the other vacuum, the particle would have to cross the hill, requiring a large amount of energy.

Gauge symmetry breaking is the most subtle, but has important physical consequences. Roughly speaking, for the purposes of this section a gauge symmetry is an assignment of systems with continuous symmetry to every point in spacetime. Gauge symmetry forbids mass generation for gauge fields, yet massive gauge fields (W and Z bosons) have been observed. Spontaneous symmetry breaking was developed to resolve this inconsistency. The idea is that in an early stage of the universe it was in a high energy state, analogous to the particle being at the top of the hill, and so had full gauge symmetry and all the gauge fields were massless. As it cooled, it settled into a choice of vacuum, thus spontaneously breaking the symmetry, thus removing the gauge symmetry and allowing mass generation of those gauge fields. A full explanation is highly technical: see electroweak interaction.

Spontaneous symmetry breaking

Main article: Spontaneous symmetry breaking

In spontaneous symmetry breaking (SSB), the equations of motion of the system are invariant, but any vacuum state (lowest energy state) is not.

For an example with two-fold symmetry, if there is some atom which has two vacuum states, occupying either one of these states breaks the two-fold symmetry. This act of selecting one of the states as the system reaches a lower energy is SSB. When this happens, the atom is no longer ${\displaystyle {\mathbb{Z}}_2}$ symmetric (reflectively symmetric) and has collapsed into a lower energy state.

Such a symmetry breaking is parametrized by an order parameter. A special case of this type of symmetry breaking is dynamical symmetry breaking.

In the Lagrangian setting of Quantum field theory (QFT), the Lagrangian ${\displaystyle L}$ is a functional of quantum fields which is invariant under the action of a symmetry group ${\displaystyle G}$. However, the vacuum expectation value formed when the particle collapses to a lower energy may not be invariant under ${\displaystyle G}$. In this instance, it will partially break the symmetry of ${\displaystyle G}$, into a subgroup ${\displaystyle H}$. This is spontaneous symmetry breaking.

Within the context of gauge symmetry however, SSB is the phenomenon by which gauge fields 'acquire mass' despite gauge-invariance enforcing that such fields be massless. This is because the SSB of gauge symmetry breaks gauge-invariance, and such a break allows for the existence of massive gauge fields. This is an important exemption from Goldstone's Theorem, where a Nambu-Goldstone Boson can gain mass, becoming a Higgs Boson in the process.

Further, in this context the usage of 'symmetry breaking' while standard, is a misnomer, as gauge 'symmetry' is not really a symmetry but a redundancy in the description of the system. Mathematically, this redundancy is a choice of trivialization, somewhat analogous to redundancy arising from a choice of basis.

Spontaneous symmetry breaking is also associated with phase transitions. For example in the Ising model, as the temperature of the system falls below the critical temperature the ${\displaystyle {\mathbb{Z}}_2}$ symmetry of the vacuum is broken, giving a phase transition of the system.

Explicit symmetry breaking

Main article: Explicit symmetry breaking

In explicit symmetry breaking (ESB), the equations of motion describing a system are variant under the broken symmetry. In Hamiltonian mechanics or Lagrangian Mechanics, this happens when there is at least one term in the Hamiltonian (or Lagrangian) that explicitly breaks the given symmetry.

In the Hamiltonian setting, this is often studied when the Hamiltonian can be written ${\displaystyle H=H_0+H_{\text{int}}}$.

Here ${\displaystyle H_0}$ is a 'base Hamiltonian', which has some manifest symmetry. More explicitly, it is symmetric under the action of a (Lie) group ${\displaystyle G}$. Often this is an integrable Hamiltonian.

The ${\displaystyle H_{\text{int}}}$ is a perturbation or interaction Hamiltonian. This is not invariant under the action of ${\displaystyle G}$. It is often proportional to a small, perturbative parameter.

This is essentially the paradigm for perturbation theory in quantum mechanics. An example of its use is in finding the fine structure of atomic spectra.

Examples

Symmetry breaking can cover any of the following scenarios:

• The breaking of an exact symmetry of the underlying laws of physics by the apparently random formation of some structure;
• A situation in physics in which a minimal energy state has less symmetry than the system itself;
• Situations where the actual state of the system does not reflect the underlying symmetries of the dynamics because the manifestly symmetric state is unstable (stability is gained at the cost of local asymmetry);
• Situations where the equations of a theory may have certain symmetries, though their solutions may not (the symmetries are "hidden").

One of the first cases of broken symmetry discussed in the physics literature is related to the form taken by a uniformly rotating body of incompressible fluid in gravitational and hydrostatic equilibrium. Jacobi and soon later Liouville, in 1834, discussed the fact that a tri-axial ellipsoid was an equilibrium solution for this problem when the kinetic energy compared to the gravitational energy of the rotating body exceeded a certain critical value. The axial symmetry presented by the McLaurin spheroids is broken at this bifurcation point. Furthermore, above this bifurcation point, and for constant angular momentum, the solutions that minimize the kinetic energy are the non-axially symmetric Jacobi ellipsoids instead of the Maclaurin spheroids.