GRIDS DESCRIBED FOR HUMANOIDS IN ENGLISH
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In physics, spacetime (or space-time, space time, space-time continuum) is any mathematical model that combines space and time into a single continuum.
Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions.
According to certain Euclidean space perceptions, the universe has three dimensions of space and one dimension of time.
By combining space and time into a single manifold, physicists have significantly simplified a large number ofphysical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.
In non-relativistic classical mechanics, the use of Euclidean space instead of spacetime is appropriate, as time is treated as universal and constant, being independent of the state of motion of an observer.
In relativistic contexts, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object’s velocity relative to the observer and also on the strength of gravitational fields, which can slow the passage of time.
In cosmology, the concept of spacetime combines space and time to a single abstract universe. Mathematically it is a manifold consisting of “events” which are described by some type of coordinate system.
Typically three spatial dimensions (length, width, height), and one temporal dimension (time) are required.
Dimensions are independent components of a coordinate grid needed to locate a point in a certain defined “space”.
For example, on the globe the latitude and longitude are two independent coordinates which together uniquely determine a location.
In spacetime, a coordinate grid that spans the 3+1 dimensions locates events (rather than just points in space), i.e. time is added as another dimension to the coordinate grid.
This way the coordinates specify where and when events occur.
However, the unified nature of spacetime and the freedom of coordinate choice it allows imply that to express the temporal coordinate in one coordinate system requires both temporal and spatial coordinates in another coordinate system.
Unlike in normal spatial coordinates, there are still restrictions for how measurements can be made spatially and temporally (see Spacetime intervals).
These restrictions correspond roughly to a particular mathematical modelwhich differs from Euclidean space in its manifest symmetry.
Until the beginning of the 20th century, time was believed to be independent of motion, progressing at a fixed rate in all reference frames; however, later experiments revealed that time slowed down at higher speeds of the reference frame relative to another reference frame (with such slowing called “time dilation” explained in the theory of “special relativity”).
Many experiments have confirmed time dilation, such asatomic clocks onboard a Space Shuttle running slower than synchronized Earth-bound inertial clocks and the relativistic decay of muons from cosmic ray showers.
The duration of time can therefore vary for various events and various reference frames.
When dimensions are understood as mere components of the grid system, rather than physical attributes of space, it is easier to understand the alternate dimensional views as being simply the result of coordinate transformations.
The term spacetime has taken on a generalized meaning beyond treating spacetime events with the normal 3+1 dimensions.
It is really the combination of space and time.
Other proposed spacetime theories include additional dimensions—normally spatial but there exist some speculative theories that include additional temporal dimensions and even some that include dimensions that are neither temporal nor spatial.
How many dimensions are needed to describe the universe is still an open question.
Speculative theories such as string theory predict 10 or 26 dimensions
(with M-theory predicting 11 dimensions: 10 spatial and 1 temporal), but the existence of more than four dimensions would only appear to make a difference at the subatomic level.
Spacetimes are the arenas in which all physical events take place—an event is a point in spacetime specified by its time and place.
For example, the motion of planets around the sun may be described in a particular type of spacetime, or the motion of light around a rotatingstar may be described in another type of spacetime.
The basic elements of spacetime are events.
In any given spacetime, an event is a unique position at a unique time.
Because events are spacetime points, an example of an event in classical relativistic physics is (x,y,z,t), the location of an elementary (point-like) particle at a particular time.
A spacetime itself can be viewed as the union of all events in the same way that a line is the union of all of its points, formally organized into a manifold, a space which can be described at small scales using coordinates systems.
A spacetime is independent of any observer.
However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient metrical coordinate system. Events are specified by four real numbers in any such coordinate system. The trajectories of elementary (point-like) particles through space and time are thus a continuum of events called the world line of the particle.
Extended or composite objects (consisting of many elementary particles) are thus a union of many worldlines twisted together by virtue of their interactions through spacetime into a “world-braid” (permitting a fascinating connection with the myth of theMoirae to be made).
However, in physics, it is common to treat an extended object as a “particle” or “field” with its own unique (e.g. center of mass) position at any given time, so that the world line of a particle or light beam is the path that this particle or beam takes in the spacetime and represents the history of the particle or beam. The world line of the orbit of the Earth (in such a description) is depicted in two spatial dimensions x and y(the plane of the Earth’s orbit) and a time dimension orthogonal to x and y. The orbit of the Earth is an ellipse in space alone, but its worldline is a helix in spacetime.
The unification of space and time is exemplified by the common practice of selecting a metric (the measure that specifies the intervalbetween two events in spacetime) such that all four dimensions are measured in terms of units of distance: representing an event as
(x0,x1,x2,x3) = (ct,x,y,z) (in the Lorentz metric) or
(x1,x2,x3,x4) = (x,y,z,ict) (in the original Minkowski metric) where
c is the speed of light. The metrical descriptions of Minkowski Space and spacelike, lightlike, and timelike intervals given below follow this convention, as do the conventional formulations of the Lorentz transformation.
Ssee answer to this : Publication data
The Fabric of the Cosmos: Space, Time, and the Texture of Reality (2004). Alfred A. Knopf division, Random House, ISBN 0-375-41288-3
The Fabric of the Cosmos: Space, Time, and the Texture of Reality (2004) is the second book on theoretical physics, cosmology, and string theory written by Brian Greene, professor and co-director of Columbia’s Institute for Strings, Cosmology, and Astroparticle Physics (ISCAP).
The Fabric of the Cosmos: Space, Time, and the Texture of Reality &&&&&&&&
Author(s) Brian Greene
Publisher Alfred A. Knopf
Publication date 2004
Media type Print
OCLC Number 52854030
Dewey Decimal 523.1 22
LC Classification QB982 .G74 2004
Preceded by The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory
Followed by Icarus at the Edge of Time
Greene begins with the key question: What is reality? Or more specifically: What is spacetime? He sets out to describe the features he finds both exciting and essential to forming a full picture of the reality painted by modern science. In almost every chapter, Greene introduces its basic concepts and then slowly builds to a climax, which is usually a scientific breakthrough. Greene then attempts to connect with his reader by posing simple analogies to help explain the meaning of a scientific concept without oversimplifying the theory behind it.
In the preface, Greene acknowledges that some parts of the book are controversial among scientists. Greene discusses the leading viewpoints in the main text, and points of contention in the end notes. Greene has striven for balanced treatment of the controversial topics. In the end notes, the diligent reader will find more complete explanations relevant to points he has simplified in the main text.
Part 1: Reality’s Arena
The main focus of part one is space and time.
Chapter one is an introduction of what is to come later in the book, such as discussions revolving around classical physics, quantum mechanics and cosmological physics.
The second chapter, The Universe and the Bucket, features space as its key point. The question posed by Greene is this: “Is space a human abstraction, or is it a physical entity?”
The key thought experiment is a spinning bucket of water, designed to make one think about what creates the force felt inside the bucket when it is spinning. The ideas of Isaac Newton, Ernst Mach, and Gottfried Leibniz on this thought experiment are discussed in detail.
Chapter three, Relativity and the Absolute, makes spacetime its focal point. The question now becomes, “Is spacetime an Einsteinian abstraction or a physical entity?” In this chapter, concepts of both special relativity and general relativity are discussed as well as their importance to the meaning of spacetime.
In chapter four, Entangling Space, Greene explores the revolution of the quantum mechanical era, focusing on what it means for objects to be separate and distinct in a universe dictated by quantum laws. This chapter provides an in-depth study of quantum mechanics, including the concepts of probability waves and interference patterns, particle spin, the photon double slit experiment, and Heisenberg’s uncertainty principle. The reader will also be informed of the challenges posed to quantum mechanics that were compiled by Albert Einstein, Boris Podolsky, and Nathan Rosen.
Part 2: Time and Experience
Part two begins by addressing the issue that time is a very familiar concept, yet it is one of humanity’s least understood concepts.
Chapter five, The Frozen River, deals with the question, “Does time flow?” One of the key points in this chapter deals with special relativity. Observers moving relative to each other have different conceptions of what exists at a given moment, and hence they have different conceptions of reality. The conclusion is that time does not flow, as all things simultaneously exist at the same time.
Chapter six, Chance and the Arrow, asks the question, “Does time have an arrow?” The reader discovers that the laws of physics apply both moving forward in time and backwards in time. Such a law is called time-reversal symmetry. One of the major subjects of this chapter isentropy. Various analogies are given to illustrate how entropy works and its apparent paradoxes. The climax of the chapter is the co-relation between entropy and gravity, and that the beginning of the universe must be the state of minimum entropy.
In chapters five and six, time has been explained only in terms of pre-modern physics.
Chapter seven, Time and the Quantum, gives insights into time’s nature in the quantum realm. Probability plays a major role in this chapter because it is an inescapable part of quantum mechanics. The double slit experiment is revisited in a stunning way that reveals both interesting and shocking things about the past. Many other experiments are presented in this chapter, such as the delayed choice quantum eraser experiment. Other major issues are brought to the reader’s attention, such as quantum mechanics and experience, as well as quantum mechanics and the measurement problem. Finally, this chapter thoroughly addresses the important subject of decoherence and its relevance towards the macroscopic world.
Part 3: Spacetime and Cosmology
Part three deals with the macroscopic realm of the cosmos.
Chapter eight, Of Snowflakes and Spacetime, tells the reader that the history of the universe is in fact the history of symmetry. Symmetry and its importance to cosmic evolution becomes the focus of this chapter. Again, general relativity is addressed as a stretching fabric of spacetime. Cosmology, symmetry, and the shape of space are put together in a unique way.
Chapter nine, Vaporizing the Vacuum, introduces the theoretical idea of the Higgs boson. This chapter focuses on the critical first fraction of a second after the big bang, when the amount of symmetry in the universe was thought to have changed abruptly by a process known assymmetry breaking. This chapter also brings into play the theory of grand unification and entropy is also revisited.
Chapter ten, Deconstructing the Bang, makes inflationary cosmology the main point. General relativity and the discovery of dark energy(repulsive gravity) are taken into account, as well as the cosmological constant. Certain problems that arise due to the standard big bang theory are addressed and new answers are given using inflationary cosmology. Such problems include the horizon problem and the flatness problem. Matter distribution throughout the cosmos is also discussed, and the concepts of dark matter and dark energy come full circle.
Chapter eleven, Quanta in the Sky with Diamonds, continues with the topic of inflation, and the arrow of time is also discussed again. The chapter addresses three main developments, the formation of structures such as galaxies, the amount of energy required to spawn the universe we now see, and, of prime importance, the origin of time’s arrow.
Part 4: Origins and Unification
Part four deals with new theoretical aspects of physics, particularly in the field of the author.
Chapter twelve, The World on a String, informs the reader of the structure of the fabric of space according to string theory. New concepts are introduced, including the Planck length and the Planck time, and ideas from The Elegant Universe are revisited. The reader will learn how string theory fills the gaps between general relativity and quantum mechanics.
Chapter thirteen, The Universe on a Brane, expands on ideas from chapter twelve, particularly, a theory called M-theory, of which string theory is a branch. This chapter is devoted to speculations on space and time according to M-theory. The collective insights of a number of physicists are presented, including those of Edward Witten and Paul Dirac. The focal point of the chapter becomes gravity and its involvement with extra dimensions. Near the end of the chapter, a brief section is devoted to cyclic cosmology, otherwise known as the cyclic model.
Part 5: Reality and Imagination
Part five deals with many theoretical concepts, including space and time travel.
Chapter fourteen, Up in the Heavens and Down on the Earth, is about various experiments with space and time. Previous theories are brought back from previous chapters, such as Higgs theory, supersymmetry, and string theory. Future planned experiments are described in an attempt to verify many of the theoretical concepts discussed, including the constituents of dark matter and dark energy, the existence of the Higgs boson, and the verification of extra spacial dimensions.
Chapter fifteen, Teleporters and Time Machines, is about travelling through space and time using intriguing methods. Quantum mechanics is brought back into the picture when the reader comes across teleportation. Puzzles of time travel are posed, such as the idea of time travel to the past being a possibility. The end of the chapter focuses on worm holes and the theory behind them.
Chapter sixteen, The Future of an Allusion, focuses on black holes and their relationship to entropy. The main idea of this chapter is that spacetime may not be the fundamental make up of the universe’s fabric.
The Fabric of the Cosmos became the most popular science book among Amazon.com customers in 2005 and was on the New York Times bestseller list – from its publication on February 10, 2004 it appeared 10 times in the Non-Fiction top 15, peaking at number 3 on April 4th, before dropping off the list on May 9.  With a first printing of 125,000 and as a main selection of the Book of the Month Club, Knopf expected it to do well. It currently (October 2011) has an average reviewer rating of 4½ stars (out of 5), with 175 out of the 254 reviewers having given it 5 stars.
NOVA has made a sequel to the popular Elegant Universe adaptation based on The Fabric of the Cosmos and retaining the same name.
In a Euclidean space, the separation between two points is measured by the distance
between the two points.
A distance is purely spatial, and is always positive.
In spacetime, the separation between two events is measured by the interval between the two events, which takes into account not only the spatial separation between the events, but also their temporal separation.
The interval between two events is defined as:
where c is the speed of light, and ?r and ?t denote differences of the space and time coordinates, respectively, between the events.
(Note that the choice of signs for
s2 above follows the space-like convention (-+++). Other treatments reverse the sign of
Spacetime intervals may be classified into three distinct types based on whether the temporal separation (
c2?t2) or the spatial separation (
?r2) of the two events is greater.
Certain types of worldlines (called geodesics of the spacetime) are the shortest paths between any two events, with distance being defined in terms of spacetime intervals. The concept of geodesics becomes critical in general relativity, since geodesic motion may be thought of as “pure motion” (inertial motion) in spacetime, that is, free from any external influences.
For two events separated by a time-like interval, enough time passes between them for there to be a cause-effect relationship between the two events. For a particle traveling through space at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval. Event pairs with time-like separation define a negative squared spacetime interval (
s2 < 0) and may be said to occur in each other’s future or past. There exists a reference frame such that the two events are observed to occur in the same spatial location, but there is no reference frame in which the two events can occur at the same time.
The measure of a time-like spacetime interval is described by the proper time:
Main article: Spacetime topology
The assumptions contained in the definition of a spacetime are usually justified by the following considerations.
The connectedness assumption serves two main purposes. First, different observers making measurements (represented by coordinate charts) should be able to compare their observations on the non-empty intersection of the charts. If the connectedness assumption were dropped, this would not be possible. Second, for a manifold, the properties of connectedness and path-connectedness are equivalent and, one requires the existence of paths (in particular, geodesics) in the spacetime to represent the motion of particles and radiation.
Every spacetime is paracompact. This property, allied with the smoothness of the spacetime, gives rise to a smooth linear connection, an important structure in general relativity. Some important theorems on constructing spacetimes from compact and non-compact manifolds include the following:
A compact manifold can be turned into a spacetime if, and only if, its Euler characteristic is 0. (Proof idea: the existence of a Lorentzian metric is shown to be equivalent to the existence of a nonvanishing vector field.)
Any non-compact 4-manifold can be turned into a spacetime.
n general relativity, it is assumed that spacetime is curved by the presence of matter (energy), this curvature being represented by theRiemann tensor. In special relativity, the Riemann tensor is identically zero, and so this concept of “non-curvedness” is sometimes expressed by the statement Minkowski spacetime is flat.
The earlier discussed notions of time-like, light-like and space-like intervals in special relativity can similarly be used to classify one-dimensional curves through curved spacetime. A time-like curve can be understood as one where the interval between any two infinitesimallyclose events on the curve is time-like, and likewise for light-like and space-like curves. Technically the three types of curves are usually defined in terms of whether the tangent vector at each point on the curve is time-like, light-like or space-like.
The world line of a slower-than-light object will always be a time-like curve, the world line of a massless particle such as a photon will be a light-like curve, and a space-like curve could be the world line of a hypothetical tachyon.
In the local neighborhood of any event, time-like curves that pass through the event will remain inside that event’s past and future light cones, light-like curves that pass through the event will be on the surface of the light cones, and space-like curves that pass through the event will be outside the light cones. One can also define the notion of a 3-dimensional “spacelike hypersurface”, a continuous 3-dimensional “slice” through the 4-dimensional property with the property that every curve that is contained entirely within this hypersurface is a space-like curve.
Many spacetime continua have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compactspacetime has closed timelike curves, which violate our usual ideas of causality (that is, future events could affect past ones). For this reason, mathematical physicists usually consider only restricted subsets of all the possible spacetimes.
One way to do this is to study “realistic” solutions of the equations of general relativity. Another way is to add some additional “physically reasonable” but still fairly general geometric restrictions and try to prove interesting things about the resulting spacetimes.
The latter approach has led to some important results, most notably the Penrose–Hawking singularity theorems.
Main article: Quantum spacetime
In general relativity, spacetime is assumed to be smooth and continuous—and not just in the mathematical sense. In the theory of quantum mechanics, there is an inherent discreteness present in physics. In attempting to reconcile these two theories, it is sometimes postulated that spacetime should be quantized at the very smallest scales.
Current theory is focused on the nature of spacetime at the Planck scale.Causal sets, loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized spacetime with agreement on the order of magnitude. Loop quantum gravity makes precise predictions about the geometry of spacetime at the Planck scale.
In general, it is not clear how physical law could function if T differed from 1. If T > 1, subatomic particles which decay after a fixed period would not behave predictably, because time-like geodesics would not be necessarily maximal.
N = 1 and T = 3 has the peculiar property that the speed of light in a vacuum is a lower bound on the velocity of matter; all matter consists of tachyons.
Hence anthropic and other arguments rule out all cases except N = 3 and T = 1—which happens to describe the world about us. Curiously, the cases N = 3 or 4 have the richest and most difficult geometry and topology. There are, for example, geometric statements whose truth or falsity is known for all Nexcept one or both of 3 and 4.
N = 3 was the last case of the Poincaré conjecture to be proved.
For an elementary treatment of the privileged status of N = 3 and T = 1, see chpt. 10 (esp. Fig. 10.12) of Barrow; for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark.
Barrow has repeatedly cited the work of Whitrow.
String theory hypothesizes that matter and energy are composed of tiny vibrating strings of various types, most of which are embedded in dimensions that exist only on a scale no larger than the Planck length. Hence N = 3 and T = 1 do not characterize string theory, which embeds vibrating strings in coordinate grids having 10, or even 26, dimensions.
The Causal dynamical triangulation (CDT) theory is a background independent theory which derives the observed 3+1 spacetime from a minimal set of assumptions, and needs no adjusting factors. It does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. It shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time, but spacetime becomes 3+1-d in scales significantly larger than Planck. So, CDT may become the first theory which doesn’t postulate but really explains observed number of spacetime dimensions.
Privileged character of 3+1 spacetime
There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional). Let the number of spatial dimensions be N and the number of temporal dimensions be T.
That N = 3 and T = 1, setting aside the compactified dimensions invoked by string theory and undetectable to date, can be explained by appealing to the physical consequences of letting N differ from 3 and T differ from 1. The argument is often of an anthropic character.
Immanuel Kant argued that 3-dimensional space was a consequence of the inverse square law of universal gravitation.
While Kant’s argument is historically important, John D. Barrow says that it “…gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa.” (Barrow 2002: 204).
This is because the law of gravitation (or any otherinverse-square law) follows from the concept of flux and the proportional relationship of flux density and the strength of field. If N = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius r has area of 4pr ².
More generally, in a space of N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to rN-1.
In 1920, Paul Ehrenfest showed that if we fix T = 1 and let N > 3, the orbit of a planet about its sun cannot remain stable. The same is true of a star’s orbit around the center of its galaxy.
Ehrenfest also showed that if N is even, then the different parts of a wave impulse will travel at different speeds. If N > 3 and odd, then wave impulses become distorted. Only when N = 3 or 1 are both problems avoided. In 1922, Hermann Weyl showed that Maxwell’s theory of electromagnetism works only when N = 3 and T = 1, writing that this fact “…not only leads to a deeper understanding of Maxwell’s theory, but also of the fact that the world is four dimensional, which has hitherto always been accepted as merely ‘accidental,’ become intelligible through it.”
Finally, Tangherlini showed in 1963 that when N > 3, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.
￼Properties of n+m-dimensional spacetimes.
Max Tegmark expands on the preceding argument in the following anthropicmanner. If T differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations.
In such a universe, intelligent life capable of manipulating technology could not emerge.
Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.)
If N > 3, Ehrenfest’s argument above holds; atoms as we know them (and probably more complex structures as well) could not exist. If N < 3, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, when N < 3, nerves cannot cross without intersecting.
Two-dimensional analogy of spacetime distortion. Matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity.
White lines do not represent the curvature of space but instead represent the coordinate system imposed on the curved spacetime, which would be rectilinear in a flat spacetime.
A regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Grids of this type appear on graph paper and may be used in finite element analysis as well as finite volume methods and finite difference methods. Since the derivatives of field variables can be conveniently expressed as finite differences, structured grids mainly appear in finite difference methods. Unstructured grids offer more flexibility than structured grids and hence are very useful in finite element and finite volume methods.
Each cell in the grid can be addressed by index (i, j) in two dimensions or (i, j, k) in three dimensions, and each vertex has coordinates in 2D or in 3D for some real numbers dx, dy, and dz representing the grid spacing.
A Cartesian grid is a special case where the elements are unit squares or unit cubes, and the vertices are integer points.
A rectilinear grid is a tessellation by rectangles or parallelepipeds that are not, in general, all congruent to each other. The cells may still be indexed by integers as above, but the mapping from indexes to vertex coordinates is less uniform than in a regular grid. An example of a rectilinear grid that is not regular appears on logarithmic scale graph paper.
A curvilinear grid or structured grid is a grid with the same combinatorial structure as a regular grid, in which the cells are quadrilaterals or cuboids rather than rectangles or rectangular parallelepipeds.
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