Nikodem Janusz Popławski is a Polish theoretical physicist, most noted for the hypothesis that every black hole could be a doorway to another universe and that the universe was formed within a black hole which itself exists in a larger universe. This hypothesis was listed by National Geographic and Science magazines among their top ten discoveries of 2010. Popławski appeared in a bonus episode of the TV show Through the Wormhole titled "Are There Parallel Universes?" and an episode of the Discovery Channel show Curiosity titled "Parallel Universes – Are They Real?", which were hosted by Morgan Freeman and aired in 2011. He was named by Forbes magazine in 2015 as one of five scientists in the world most to become the next Albert Einstein. Popławski's approach is based on the Einstein–Cartan theory of gravity which extends general relativity to matter with intrinsic angular momentum. Spin in curved spacetime requires that the affine connection cannot be constrained to zero and its antisymmetric part, the torsion tensor, must be a variable in Hamilton's principle of stationary action which gives the field equations.
Torsion gives the correct generalization of the conservation law for the total angular momentum to the presence of the gravitational field, but modifies the Dirac equation for fermions. Gravitational effects of torsion on fermionic matter are significant at high densities which exist inside black holes and at the beginning of the Universe. Popławski theorizes that torsion manifests itself as a repulsive force which causes fermions to be spatially extended and prevents the formation of a gravitational singularity within the black hole's event horizon; because of torsion, the collapsing matter on the other side of the horizon reaches an enormous but finite density and rebounds, forming an Einstein-Rosen bridge to a new, expanding universe. Analogously, the Big Bang is replaced by the Big Bounce before which the Universe was the interior of a black hole; this scenario explains why the present Universe at largest scales appears spatially flat and isotropic, providing a physical alternative to cosmic inflation.
It may explain the arrow of time, solve the black hole information paradox, explain the nature of dark matter. Torsion may be responsible for the observed asymmetry between matter and antimatter in the Universe; the rotation of a black hole could influence the spacetime on the other side of its event horizon and result in a preferred direction in the new universe. Popławski suggests that the observed fluctuations in the cosmic microwave background might provide evidence for his hypothesis. Popławski received his M. S. degree in astronomy from the University of Warsaw, his Ph. D. degree in physics from Indiana University, where he worked as a researcher and lecturer in physics. Since 2013, he has been a senior lecturer in the Department of Mathematics and Physics at the University of New Haven. Academic website
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object can always be rotated around an infinite number of imaginary lines called rotation axes. If the axis passes through the body's center of mass, the body is said to rotate upon itself, or spin. A rotation about an external point, e.g. the Earth about the Sun, is called a revolution or orbital revolution when it is produced by gravity. The axis is called a pole. Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed; this definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two. A rotation is a progressive radial orientation to a common point; that common point lies within the axis of that motion. The axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question the body is said to orbit. There is no fundamental difference between a “rotation” and an “orbit” and or "spin".
The key distinction is where the axis of the rotation lies, either within or outside of a body in question. This distinction can be demonstrated for "non rigid" bodies. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results; the reverse of a rotation is a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, followed by a rotation around the z axis; that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the principal rotations are known as yaw and roll; this terminology is used in computer graphics. In astronomy, rotation is a observed phenomenon.
Stars and similar bodies all spin around on their axes. The rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured by tracking active surface features; this rotation induces a centrifugal acceleration in the reference frame of the Earth which counteracts the effect of gravity the closer one is to the equator. One effect is that an object weighs less at the equator. Another is that the Earth is deformed into an oblate spheroid. Another consequence of the rotation of a planet is the phenomenon of precession. Like a gyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet; the tilt of the Earth's axis to its orbital plane is 23.44 degrees, but this angle changes slowly. While revolution is used as a synonym for rotation, in many fields astronomy and related fields, revolution referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis.
Moons revolve around their planet, planets revolve about their star. The motion of the components of galaxies is complex, but it includes a rotation component. Most planets in our solar system, including Earth, spin in the same direction; the exceptions are Uranus. Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. Venus may be thought of as rotating backwards; the dwarf planet Pluto is anomalous in other ways. The speed of rotation is given by period; the time-rate of change of angular frequency is angular acceleration, caused by torque. The ratio of the two is given by the moment of inertia; the angular velocity vector describes the direction of the axis of rotation. The torque is an axial vector; the physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw.
The laws of physics are believed to be invariant under any fixed rotation. In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, should, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field, laid down by the Big Bang. In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over ti
In mathematical physics, Minkowski space is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity. Minkowski space is associated with Einstein's theory of special relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events; because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.
In 3-dimensional Euclidean space, the isometry group is the Euclidean group. It is generated by rotations and translations; when time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, the group of all these transformations is called the Galilean group. All Galilean transformations preserve the 3-dimensional Euclidean distance; this distance is purely spatial. Time differences are separately preserved as well; this changes in the spacetime of special relativity, where time are interwoven. Spacetime is equipped with an indefinite non-degenerate bilinear form, variously called the Minkowski metric, the Minkowski norm squared or Minkowski inner product depending on the context; the Minkowski inner product is defined as to yield the spacetime interval between two events when given their coordinate difference vector as argument. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space; the analogue of the Galilean group for Minkowski space, preserving the spacetime interval is the Poincaré group.
In summary, Galilean spacetime and Minkowski spacetime are, when viewed as manifolds the same. They differ in; the former has the Euclidean distance function and time together with inertial frames whose coordinates are related by Galilean transformations, while the latter has the Minkowski metric together with inertial frames whose coordinates are related by Poincaré transformations. In 1905–06 Henri Poincaré showed that by taking time to be an imaginary fourth spacetime coordinate ict, where c is the speed of light and i is the imaginary unit, a Lorentz transformation can formally be regarded as a rotation of coordinates in a four-dimensional space with three real coordinates representing space, one imaginary coordinate representing time, as the fourth dimension. In physical spacetime special relativity stipulates that the quantity − t 2 + x 2 + y 2 + z 2 is invariant under coordinate changes from one inertial frame to another, i. e. under Lorentz transformations. Here the speed of light c is, following Poincaré, set to unity.
In the space suggested by him where physical spacetime is coordinatized by ↦, call it coordinate space, Lorentz transformations appear as ordinary rotations preserving the quadratic form x 2 + y 2 + z 2 + t 2 on coordinate space. The naming and ordering of coordinates, with the same labels for space coordinates, but with the imaginary time coordinate as the fourth coordinate, is conventional; the above expression, while making the former expression more familiar, may be confusing because it is not the same t that appears in the latter as in the former. Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime appear as Euclidean rotations and are interpreted in the ordinary sense; the "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a Lorentz boost in physical spacetime with real inertial coordinates. The analogy with Euclidean rotations is thus only partial.
This idea was elaborated by Hermann Minkowski, who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation. He further reformulated in four dimensions the then-recent theory of special relativity of Einstein. From this he concluded that time and space should be treated and so arose his concept of events taking place in a unified four-dimensional spacetime continuum. In a further development in his 1908 "Space and Time" lecture, Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables of space and time in coordinate form in a four dimensional real vector space. Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point, events not on the light-cone are classified by their relation to the apex as spacelike or timel
In geometry and physics, spinors are elements of a vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight rotation. However, when a sequence of such small rotations is composed to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used: unlike vectors and tensors, a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°; this property characterizes spinors. It is possible to associate a similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. Spinors are characterized by the specific way.
They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved. There are two topologically distinguishable classes of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle; these two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class, it doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class. In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO. Although spinors can be defined purely as elements of a representation space of the spin group, they are defined as elements of a vector space that carries a linear representation of the Clifford algebra.
The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, in applications the Clifford algebra is the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations; the spinors are the column vectors. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what constitutes a "column vector", involves the choice of basis and gamma matrices in an essential way; as a representation of the spin group, this realization of spinors as column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.
What characterizes spinors and distinguishes them from geometric vectors and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any tensor associated with the system has coordinate descriptions that adjust to compensate for changes to the coordinate system itself. Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is rotated between some initial and final configuration. For any of the familiar and intuitive quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration.
Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: they exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are two inequivalent gradual rotations of the coordinate system that result in this same configuration; this ambiguity is called the homotopy class of the gradual rotation. The belt trick puzzle famously demonstrates two different rotations, one through an angle of 2π and the other through an angle of 4π, having the same final configurations but different classes. Spinors exhibit a sign-reversal that genuinely depends on this homotopy class; this distinguishes them from other tensors, none of which can feel the class. Spinors can be exhibited as concrete objects using a choice of Cartesian coordinates. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to the thre
Dennis W. Sciama
Dennis William Siahou Sciama, was a British physicist who, through his own work and that of his students, played a major role in developing British physics after the Second World War. He was the Ph. D supervisor including Stephen Hawking and Martin Rees. Sciama was born in Manchester, the son of Nelly Ades and Abraham Sciama, he was of Syrian-Jewish ancestry—his father born in Manchester and his mother born in Egypt both traced their roots back to Aleppo, Syria. Sciama earned his PhD in 1953 at the University of Cambridge supervised by Paul Dirac, with a dissertation on Mach's principle and inertia, his work influenced the formulation of scalar-tensor theories of gravity. Sciama taught at Cornell University, King's College London, Harvard University and the University of Texas at Austin, but spent most of his career at the University of Cambridge and the University of Oxford as a Senior Research Fellow in All Souls College, Oxford. In 1983, he moved from Oxford to Trieste, becoming Professor of Astrophysics at the International School of Advanced Studies, a consultant with the International Centre for Theoretical Physics.
He taught at the Scuola Normale Superiore of Pisa. During the 1990s, he divided his time between Trieste and his main residence at Oxford, where he was a visiting professor until the end of his life. Sciama made connections among some topics in astronomy and astrophysics, he wrote on radio astronomy, X-ray astronomy, the anisotropies of the cosmic microwave radiation, the interstellar and intergalactic medium, astroparticle physics and the nature of dark matter. Most significant was his work in general relativity and without quantum theory, black holes, he helped revitalize the classical relativistic alternative to general relativity known as Einstein-Cartan gravity. Early in his career, he supported Fred Hoyle's steady state cosmology, interacted with Hoyle, Hermann Bondi, Thomas Gold; when evidence against the steady state theory, e.g. the cosmic microwave radiation, mounted in the 1960s, Sciama abandoned it and worked on the Big Bang cosmology. During his last years, Sciama became interested in the issue of dark matter in galaxies.
Among other aspects he pursued a theory of dark matter that consists of a heavy neutrino disfavored in his realization, but still possible in a more complicated scenario. Several leading astrophysicists and cosmologists of the modern era completed their doctorates under Sciama's supervision, notably: Sciama strongly influenced Roger Penrose, who dedicated his The Road to Reality to Sciama's memory; the 1960s group he led in Cambridge, has proved of lasting influence. Sciama, Dennis; the Unity of the Universe. London: Faber & Faber. Sciama, Dennis. "The Physical Foundations of General Relativity". Science Study Series. New York: Doubleday. 58. Short and written non-mathematical book on the physical and conceptual foundations of General Relativity. Could be read with profit by physics students before immersing themselves in more technical studies of General Relativity. Sciama, Dennis. Modern Cosmology. Cambridge University Press. ISBN 9780521080699. Sciama, Dennis. Modern Cosmology and the Dark Matter Problem.
Cambridge University Press. ISBN 9780521438483. Sciama was elected a Fellow of the Royal Society in 1983, he was an honorary member of the American Academy of Arts and Sciences, the American Philosophical Society and the Academia Lincei of Rome. He served as president of the International Society of General Relativity and Gravitation, 1980–84, his work at SISSA and the University of Oxford led to the creation of a lecture series in his honour, the Dennis Sciama Memorial Lectures. In 2009, the Institute of Cosmology and Gravitation at the University of Portsmouth elected to name their new building, their supercomputer in 2011, in his honour. Sciama has been portrayed in a number of biographical projects about his most famous student, Stephen Hawking. In the 2004 BBC TV movie Hawking, Sciama was played by John Sessions. In the 2014 film The Theory of Everything, Sciama was played by David Thewlis. Physicist Adrian Melott criticized the portrayal of Sciama in the film. Sciama was of an avowed atheist.
In 1959, Sciama married Lidia Dina, a social anthropologist, who survived him, along with their two daughters
Sir Thomas Walter Bannerman Kibble, was a British theoretical physicist, senior research investigator at the Blackett Laboratory and Emeritus Professor of Theoretical Physics at Imperial College London. His research interests were in quantum field theory the interface between high-energy particle physics and cosmology, he is best known as one of the first to describe the Higgs mechanism, for his research on topological defects. From the 1950s he was concerned about the nuclear arms race and from 1970 took leading roles in promoting the social responsibility of the scientist. Kibble was born in Madras, British India, on 23 December 1932, he was the son of the statistician Walter F. Kibble, the grandson of William Bannerman, an officer in the Indian Medical Service, the author Helen Bannerman, he was educated at Doveton Corrie School in Madras and in Edinburgh, Scotland, at Melville College and at the University of Edinburgh. He graduated from the University of Edinburgh with a BSc in 1955, MA in 1956 and a PhD in 1958.
Kibble worked on mechanisms of symmetry breaking, phase transitions and the topological defects that can be formed. He is most noted for his co-discovery of the Higgs mechanism and Higgs boson with Gerald Guralnik and C. R. Hagen; as part of Physical Review Letters 50th anniversary celebration, the journal recognised this discovery as one of the milestone papers in PRL history. For this discovery Kibble was awarded the American Physical Society's 2010 J. J. Sakurai Prize for Theoretical Particle Physics. While Guralnik and Kibble are considered to have authored the most complete of the early papers on the Higgs theory, they were controversially not included in the 2013 Nobel Prize in Physics. In 2014, Nobel Laureate Peter Higgs expressed disappointment that Kibble had not been chosen to share the Nobel Prize with François Englert and himself. Kibble pioneered the study of topological defect generation in the early universe; the paradigmatic mechanism of defect formation across a second-order phase transition is known as the Kibble-Zurek mechanism.
His paper on cosmic strings introduced the phenomenon into modern cosmology. He was one of the two co-chairs of an interdisciplinary research programme funded by the European Science Foundation on Cosmology in the Laboratory which ran from 2001 to 2005, he was the coordinator of an ESF Network on Topological Defects in Particle Physics, Condensed Matter & Cosmology. Kibble was an elected Fellow of the Royal Society in 1980, of the Institute of Physics, of Imperial College London, he was a member of the American Physical Society, the European Physical Society and the Academia Europaea. In 2008, Kibble was named an Outstanding Referee by the American Physical Society. In addition to the Sakurai Prize, Kibble has been awarded the Hughes Medal of the Royal Society, the Rutherford and Guthrie Medals of the Institute of Physics, the Dirac Medal, the Albert Einstein Medal and the Royal Medal of the Royal Society of Edinburgh, he was appointed a CBE in the 1998 Birthday Honours and was knighted in the 2014 Birthday Honours for services to physics.
Kibble was posthumously awarded the Isaac Newton Medal by the Institute of Physics for his outstanding lifelong commitment to the field. In 1966 Kibble co-authored, with Frank H. Berkshire, a textbook, Classical Mechanics, which as of 2016 is still in print and is now in its 5th edition. Kibble was married to Anne Allan from 1957 until her death in 2005. Kibble had three children. In the 1950s and 1960s, Kibble became concerned about the nuclear arms race and from 1970 he took leading roles in several organisations promoting scientists' social responsibility. In the period 1970–1977, he was a national committee member treasurer chair of the British Society for Social Responsibility in Science, he was chair of the organising committee of the Second International Scientists' Congress, held at Imperial College in 1988, was a co-editor of the proceedings. In retirement, Kibble chaired the Richmond branch of the Ramblers Association, he died in London on 2 June 2016 at the age of 83. Imperial College People Official website 2010 J. J. Sakurai Prize for Theoretical Particle Physics Recipient Papers written by T. Kibble in the INSPIRE-HEP database T. W. B.
Papers written by T. Kibble on the Mathematical Reviews website Papers written by T. Kibble in Physical Review Physical Review Letters – 50th Anniversary Milestone Papers Imperial College London on PRL 50th Anniversary Milestone Papers
In mathematics, topology, a fiber bundle is a space, locally a product space, but globally may have a different topological structure. The similarity between a space E and a product space B × F is defined using a continuous surjective map π: E → B that in small regions of E behaves just like a projection from corresponding regions of B × F to B; the map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, F the fiber. In the trivial case, E is just B × F, the map π is just the projection from the product space to the first factor; this is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself to E is called a section of E. Fiber bundles can be specialized in a number of ways, the most common of, requiring that the transitions between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber F. In topology, the terms fiber and fiber space appeared for the first time in a paper by Herbert Seifert in 1933, but his definitions are limited to a special case; the main difference from the present day conception of a fiber space, was that for Seifert what is now called the base space of a fiber space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space was given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.
The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau, Norman Steenrod, Charles Ehresmann, Jean-Pierre Serre, others. Fiber bundles became their own object of study in the period 1935–1940; the first general definition appeared in the works of Whitney. Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, a fiber bundle whose fiber is a sphere of arbitrary dimension. A fiber bundle is a structure, where E, B, F are topological spaces and π: E → B is a continuous surjection satisfying a local triviality condition outlined below; the space B is called the base space of the bundle, E the total space, F the fiber. The map π is called the projection map. We shall assume in. We require that for every x in E, there is an open neighborhood U ⊂ B of π such that there is a homeomorphism φ: π−1 → U × F in such a way that π agrees with the projection onto the first factor.
That is, the following diagram should commute: where proj1: U × F → U is the natural projection and φ: π−1 → U × F is a homeomorphism. The set of all is called a local trivialization of the bundle, thus for any p in B, the preimage π−1 is homeomorphic to F and is called the fiber over p. Every fiber bundle π: E → B is an open map, since projections of products are open maps; therefore B carries the quotient topology determined by the map π. A fiber bundle is denoted that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a fiber bundle in the category of smooth manifolds; that is, E, B, F are required to be smooth manifolds and all the functions above are required to be smooth maps. Let E = B × F and let π: E → B be the projection onto the first factor. E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle.
Any fiber bundle over a contractible CW-complex is trivial. The simplest example of a nontrivial bundle E is the Möbius strip, it has the circle that runs lengthwise along the center of the strip as a base B and a line segment for the fiber F, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood U of π ∈ B is an arc; the preimage π − 1 in the picture is a slice of the strip one long. A homeomorphism exists that maps the preimage of U to a slice of a cylinder: curved, but not twisted; this pair locally trivializes the strip. The corresponding trivial bundle B × F would be a cylinder, but the Möbius strip has an overall "twist". Note that this twist is visible only globally.