Spacetime: Difference between revisions
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* [http://www.scholarpedia.org/article/Encyclopedia_of_Space-time_and_gravitation Encyclopedia of Space-time and gravitation] [[Scholarpedia]] Expert articles |
* [http://www.scholarpedia.org/article/Encyclopedia_of_Space-time_and_gravitation Encyclopedia of Space-time and gravitation] [[Scholarpedia]] Expert articles |
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* [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/spacetime-iframes/ Space and Time: Inertial Frames]" by Robert DiSalle. |
* [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/spacetime-iframes/ Space and Time: Inertial Frames]" by Robert DiSalle. |
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* [http://blog.stephenwolfram.com/2015/12/what-is-spacetime-really/ What is Spacetime, Really?] by [Stephen Wolfram] |
* [http://blog.stephenwolfram.com/2015/12/what-is-spacetime-really/ What is Spacetime, Really?] by [[Stephen Wolfram]] |
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{{Dimension topics}} |
{{Dimension topics}} |
Revision as of 16:01, 26 October 2016
Part of a series on |
Spacetime |
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In
Explanation
In non-relativistic
In
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, following its prediction by special relativity, later experiments confirmed that time slows at higher speeds of the reference frame relative to another reference frame. Such slowing, called time dilation, is explained in special relativity theory. Many experiments have confirmed time dilation, such as the relativistic decay of muons from cosmic ray showers and the slowing of atomic clocks aboard a Space Shuttle relative to synchronized Earth-bound inertial clocks.[1] The duration of time can therefore vary according to events and 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
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
Spacetime in literature
The idea of a unified spacetime is stated by
Marcel Proust, in his novel Swann's Way (published 1913), describes the village church of his childhood's Combray as "a building which occupied, so to speak, four dimensions of space—the name of the fourth being Time".
Mathematical concept
In
Another early venture was by
The ancient idea of the cosmos gradually was described mathematically with differential equations, differential geometry, and abstract algebra. These mathematical articulations blossomed in the nineteenth century as electrical technology stimulated men like Michael Faraday and James Clerk Maxwell to describe the reciprocal relations of electric and magnetic fields. Daniel Siegel phrased Maxwell's role in relativity as follows:
[...] the idea of the propagation of forces at the velocity of light through the electromagnetic field as described by Maxwell's equations—rather than instantaneously at a distance—formed the necessary basis for relativity theory.[8]
Maxwell used vortex models in his papers on On Physical Lines of Force, but ultimately gave up on any substance but the electromagnetic field. Pierre Duhem wrote:
[Maxwell] was not able to create the theory that he envisaged except by giving up the use of any model, and by extending by means of analogy the abstract system of electrodynamics to displacement currents.[9]
In Siegel's estimation, "this very abstract view of the electromagnetic fields, involving no visualizable picture of what is going on out there in the field, is Maxwell's legacy."[10] Describing the behaviour of electric fields and magnetic fields led Maxwell to view the combination as an electromagnetic field. These fields have a value at every point of spacetime. It is the intermingling of electric and magnetic manifestations, described by Maxwell's equations, that give spacetime its structure. In particular, the rate of motion of an observer determines the electric and magnetic profiles of the electromagnetic field. The propagation of the field is determined by the electromagnetic wave equation, which requires spacetime for description.
Spacetime was described as an
The first inkling of general relativity in spacetime was articulated by W. K. Clifford. Description of the effect of gravitation on space and time was found to be most easily visualized as a "warp" or stretching in the geometrical fabric of space and time, in a smooth and continuous way that changed smoothly from point-to-point along the spacetime fabric. In 1947 James Jeans provided a concise summary of the development of spacetime theory in his book The Growth of Physical Science.[12]
Basic concepts
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 , 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 coordinate systems.
Spacetime is independent of any
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 world line is a helix in spacetime.[14]
The unification of space and time is exemplified by the common practice of selecting a metric (the measure that specifies the
Spacetime intervals in flat space
In a
- (spacetime interval),
where c is the speed of light. The choice of signs for above follows the
Certain types of world lines are called geodesics of the spacetime – straight lines in the case of Minkowski space and their closest equivalent in the curved spacetime of general relativity. In the case of purely time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path between two events, whereas in Euclidean space and Riemannian manifolds, geodesics are paths of shortest distance between two points.[18][19] The concept of geodesics becomes central in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in spacetime, that is, free from any external influences.
Time-like interval
For two events separated by a time-like interval, enough time passes between them that there could 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 spacetime interval () 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 interval).
The proper time interval would be measured by an observer with a clock traveling between the two events in an
Light-like interval
In a light-like interval, the spatial distance between two events is exactly balanced by the time between the two events. The events define a spacetime interval of zero (). Light-like intervals are also known as "null" intervals.
Events which occur to or are initiated by a photon along its path (i.e., while traveling at , the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a light cone, and all the events which preceded from a light-like interval define a second (graphically inverted, which is to say "pastward") light cone.
Space-like interval
When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a
For these space-like event pairs with a positive spacetime interval (), the measurement of space-like separation is the
- (proper distance).
Like the proper time of time-like intervals, the proper distance of space-like spacetime intervals is a real number value.
Interval as area
The interval has been presented as the area of an oriented rectangle formed by two events and isotropic lines through them. Time-like or space-like separations correspond to oppositely oriented rectangles, one type considered to have rectangles of negative area. The case of two events separated by light corresponds to the rectangle degenerating to the segment between the events and zero area.[20] The transformations leaving interval-length invariant are the area-preserving squeeze mappings.
The parameters traditionally used rely on quadrature of the hyperbola, which is the
Mathematics of spacetimes
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected
A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event . Another reference frame may be identified by a second coordinate chart about . Two observers (one in each reference frame) may describe the same event but obtain different descriptions.
Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing (representing an observer) and another containing (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.
For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces
Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.
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
- A nonvanishing vector field.)
- Any non-compact 4-manifold can be turned into a spacetime.[21]
Spacetime symmetries
Often in relativity, spacetimes that have some form of symmetry are studied. As well as helping to classify spacetimes, these symmetries usually serve as a simplifying assumption in specialized work. Some of the most popular ones include:
- Axisymmetric spacetimes
- Spherically symmetric spacetimes
- Static spacetimes
- Stationary spacetimes
Causal structure
The causal structure of a spacetime describes causal relationships between pairs of points in the spacetime based on the existence of certain types of curves joining the points.
Spacetime in special relativity
The geometry of spacetime in special relativity is described by the
where the
Strictly speaking, one can also consider events in Newtonian physics as a single spacetime. This is
Spacetime in general relativity
General relativity |
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In
The earlier discussed notions of time-like, light-like and space-like intervals in special relativity can similarly be used to classify one-dimensional
Many spacetime continua have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compact spacetime 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.
Quantized spacetime
In general relativity, spacetime is assumed to be smooth and continuous—and not just in the mathematical sense. In the theory of
See also
- Anthropic principle § Applications of the principle § Spacetime
- Basic introduction to the mathematics of curved spacetime
- Four-vector
- Frame-dragging
- Global spacetime structure
- Hole argument
- List of mathematical topics in relativity
- Local spacetime structure
- Lorentz invariance
- Manifold
- Mathematics of general relativity
- Metric space
- Philosophy of space and time
- Relativity of simultaneity
- Strip photography
- World manifold
References
- .
- ^ Atuq Eusebio Manga Qespi, Instituto de lingüística y Cultura Amerindia de la Universidad de Valencia. Pacha: un concepto andino de espacio y tiempo. Revísta española de Antropología Americana, 24, p. 155–189. Edit. Complutense, Madrid. 1994
- ^ Paul Richard Steele, Catherine J. Allen, Handbook of Inca mythology, p. 86, (ISBN 1-57607-354-8)
- ^ Shirley Ardener, University of Oxford, Women and space: ground rules and social maps, p. 36 (ISBN 0-85496-728-1)
- Jean d'Alembert (1754) Dimensionfrom ARTFL Encyclopedie project
- ^ R.C. Archibald (1914) Time as a fourth dimension Bulletin of the American Mathematical Society 20:409.
- ^ Daniel M. Siegel (2014) "Maxwell's contributions to electricity and magnetism", chapter 10 in James Clerk Maxwell: Perspectives on his Life and Work, Raymond Flood, Mark McCartney, Andrew Whitaker, editors, Oxford University Press ISBN 978-0-19-966437-5
- ^ Pierre Duhem (1954) The Aim and Structure of Physical Theory, page 98, Princeton University Press
- ^ Siegel 2014 p 191
- ^ Minkowski, Hermann (1909),
- Various English translations on Wikisource: Space and Time.
- ^ James Jeans (1947) The Growth of Physical Science, "Space-time", pp. 205–301, link from Internet Archive
- ^ Matolcsi, Tamás (1994). Spacetime Without Reference Frames. Budapest: Akadémiai Kiadó.
- ISBN 0-19-850657-0.
- ^ Note that the term spacetime interval is applied by several authors to the quantity s2 and not to s. The reason that the quantity s2 is used and not s is that s2 can be positive, zero or negative, and is a more generally convenient and useful quantity than the Minkowski norm with a timelike/null/spacelike distinguisher: the pair (√|s2|, sgn(s2)). Despite the notation, it should not be regarded as the square of a number, but as a symbol. The cost for this convenience is that this "interval" is quadratic in linear separation along a straight line.
- ^ More generally the spacetime interval in flat space can be written as with metric tensor g independent of spacetime position.
- ^ This characterization is not universal: both the arcs between two points of a great circle on a sphere are geodesics.
- MR520230
- ISBN 0521299284.
- ^ See "Quantum Spacetime and the Problem of Time in Quantum Gravity" by Leszek M. Sokolowski, where on this page he writes "Each of these hypersurfaces is spacelike, in the sense that every curve, which entirely lies on one of such hypersurfaces, is a spacelike curve." More commonly a space-like hypersurface is defined technically as a surface such that the normal vector at every point is time-like, but the definition above may be somewhat more intuitive.
Further reading
- LCCN 87028148.
- Ehrenfest, Paul (1920) "How do the fundamental laws of physics make manifest that Space has 3 dimensions?" Annalen der Physik 366: 440.
- George F. Ellis and Ruth M. Williams (1992) Flat and curved space–times. Oxford Univ. Press. ISBN 0-19-851164-7
- Isenberg, J. A. (1981). "Wheeler–Einstein–Mach spacetimes". Phys. Rev. D. 24 (2): 251–256. .
- Kant, Immanuel (1929) "Thoughts on the true estimation of living forces" in J. Handyside, trans., Kant's Inaugural Dissertation and Early Writings on Space. Univ. of Chicago Press.
- Lorentz, H. A., Einstein, Albert, Minkowski, Hermann, and Weyl, Hermann (1952) The Principle of Relativity: A Collection of Original Memoirs. Dover.
- Lucas, John Randolph (1973) A Treatise on Time and Space. London: Methuen.
- ISBN 0-679-45443-8. Chpts. 17–18.
- ISBN 1-84391-009-8.
- Robb, A. A. (1936). Geometry of Time and Space. University Press.
- Erwin Schrödinger (1950) Space–time structure. Cambridge Univ. Press.
- Schutz, J. W. (1997). Independent axioms for Minkowski Space–time. Addison-Wesley Longman. ISBN 0-582-31760-6.
- Tangherlini, F. R. (1963). "Schwarzschild Field in n Dimensions and the Dimensionality of Space Problem". Nuovo Cimento. 14 (27): 636.
- Taylor, E. F.; ISBN 0-7167-2327-1.
- ISBN 0-671-57554-6. (pp. 5–6)
External links
- Albert Einstein on space-time 13th edition Encyclopedia BritannicaHistorical: Albert Einstein's 1926 article
- Encyclopedia of Space-time and gravitation Scholarpedia Expert articles
- Stanford Encyclopedia of Philosophy: "Space and Time: Inertial Frames" by Robert DiSalle.
- What is Spacetime, Really? by Stephen Wolfram