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Spacetime
In physics, spacetime (also space–time, space time or space–time continuum) is any mathematical model that combines space and time into a single interwoven continuum. The spacetime of our universe is usually interpreted from a Euclidean space perspective, which regards space as consisting of three dimensions, and time as consisting of one dimension, the "fourth dimension". By combining space and time into a single manifold called Minkowski space, physicists have significantly simplified a large number of physical 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, because time is treated as universal with a constant rate of passage that is 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 for an object as seen by an observer outside the field.
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 model which 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 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. 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 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 (e.g., superspace). 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.[1]
Spacetime in literature
Incas regarded space and time as a single concept, referred to as pacha (Quechua: pacha, Aymara: pacha).[2][3] The peoples of the Andes maintain a similar understanding.[4]
Arthur Schopenhauer wrote in §18 of On the Fourfold Root of the Principle of Sufficient Reason (1813): "the representation of coexistence is impossible in Time alone; it depends, for its completion, upon the representation of Space; because, in mere Time, all things follow one another, and in mere Space all things are side by side; it is accordingly only by the combination of Time and Space that the representation of coexistence arises".
The idea of a unified spacetime is stated by Edgar Allan Poe in his essay on cosmology titled Eureka (1848) that "Space and duration are one". In 1895, in his novel The Time Machine, H. G. Wells wrote, "There is no difference between time and any of the three dimensions of space except that our consciousness moves along it", and that "any real body must have extension in four directions: it must have Length, Breadth, Thickness, and Duration".
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 Encyclopedie under the term dimension Jean le Rond d'Alembert speculated that duration (time) might be considered a fourth dimension if the idea was not too novel.[5]
Another early venture was by Joseph Louis Lagrange in his Theory of Analytic Functions (1797, 1813). He said, "One may view mechanics as a geometry of four dimensions, and mechanical analysis as an extension of geometric analysis".[6]
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.[7]
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.[8]
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."[9] Describing the behaviour of electric fields and magnetic fields led Maxwell to a unified view of an electromagnetic field. Being functions, these fields took values on a domain, a piece 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 also requires spacetime for description.
Spacetime was described as an affine space with quadratic form in Minkowski space of 1908.[10] In his 1914 textbook The Theory of Relativity, Ludwik Silberstein used biquaternions to represent events in Minkowski space. He also exhibited the Lorentz transformations between observers of differing velocities as biquaternion mappings. Biquaternions were described in 1853 by W. R. Hamilton, so while the physical interpretation was new, the mathematics was well known in English literature, making relativity an instance of applied mathematics.
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.[11]
Basic concepts
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 rotating star 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 coordinate systems.
A spacetime is independent of any observer.[12] 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 world lines twisted together by virtue of their interactions through spacetime into a "world-braid".
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.[13]
The unification of space and time is exemplified by the common practice of selecting a metric (the measure that specifies the interval between two events in spacetime) such that all four dimensions are measured in terms of units of distance: representing an event as \( (x_0,x_1,x_2,x_3) = (ct,x,y,z) \) (in the Lorentz metric) or \( (x_1,x_2,x_3,x_4) = (x,y,z,ict) \) (in the original Minkowski metric) where c is the speed of light.[14] 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.
Spacetime intervals in flat space
In a Euclidean space, the separation between two points is measured by the distance between the two points. The distance is purely spatial, and is always positive. In spacetime, the displacement four-vector ΔR is given by the space displacement vector Δr and the time difference Δt between the events. The spacetime interval, also called invariant interval, between the two events, s2,[15] is defined as:
\( s^2 = \Delta r^2 - c^2\Delta t^2 \, \) (spacetime interval),
where c is the speed of light. The choice of signs for \( s^2 \) above follows the space-like convention (−+++).[16] Spacetime intervals may be classified into three distinct types, based on whether the temporal separation (c^2 \Delta t^2) or the spatial separation \( (\Delta r^2) \) of the two events is greater: time-like, light-like or space-like.
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.[17][18] 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
\( \begin{align} \\ c^2\Delta t^2 &> \Delta r^2\\ s^2 &< 0 \\ \end{align} \)
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\( (s^2 < 0) \)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 interval, \Delta\tau:
\( \Delta\tau = \sqrt{\Delta t^2 - \frac{\Delta r^2}{c^2}} \) (proper time interval).
The proper time interval would be measured by an observer with a clock traveling between the two events in an inertial reference frame, when the observer's path intersects each event as that event occurs. (The proper time interval defines a real number, since the interior of the square root is positive.)
Light-like interval
\( \begin{align} c^2\Delta t^2 &= \Delta r^2 \\ s^2 &= 0 \\ \end{align} ])
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 \( (s^2 = 0)\). 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 c, 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
\( \begin{align} \\ c^2\Delta t^2 &< \Delta r^2 \\ s^2 &> 0 \\ \end{align} \)
When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a causal relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur at the same time, but there is no reference frame in which the two events can occur in the same spatial location.
For these space-like event pairs with a positive spacetime interval \((s^2 > 0) \), the measurement of space-like separation is the proper distance, \( \Delta\sigma \):
\( \Delta\sigma = \sqrt{s^2} = \sqrt{\Delta r^2 - c^2\Delta t^2} \) (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.[19] 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 natural logarithm. This transcendental function is essential in mathematical analysis as its inverse unites circular functions and hyperbolic functions: The exponential function, et, t a real number, used in the hyperbola (et, e–t ), generates hyperbolic sectors and the hyperbolic angle parameter. The functions cosh and sinh, used with rapidity as hyperbolic angle, provide the common representation of squeeze in the form \(\begin{pmatrix}\cosh \phi & \sinh \phi \\ \sinh \phi & \cosh \phi \end{pmatrix}, or as the split-complex unit e^{j \phi} = \cosh \phi \ + j \ \sinh \phi . \)
Mathematics of spacetimes
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold (M,g). This means the smooth Lorentz metric g has signature (3,1). The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates (x, y, z, t) are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light c is equal to 1.
A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event p. Another reference frame may be identified by a second coordinate chart about p. Two observers (one in each reference frame) may describe the same event p but obtain different descriptions.
Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing p (representing an observer) and another containing q (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 p). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (x, y, z, t) (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 tensors into relativity, by which all physical quantities are represented.
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.
T
opology
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.[20]
Spacetime symmetries
Main article: 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
Main article: Causal structure
See also: Causality (physics) and Causality
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
Main article: Minkowski space
The geometry of spacetime in special relativity is described by the Minkowski metric on R4. This spacetime is called Minkowski space. The Minkowski metric is usually denoted by \eta and can be written as a four-by-four matrix:
\( \eta_{ab} \, = \operatorname{diag}(1, -1, -1, -1) \)
where the Landau–Lifshitz space-like convention is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are invariant under Lorentz transformations. This invariance property leads to the use of four-vectors (and other tensors) in describing physics.
Strictly speaking, one can also consider events in Newtonian physics as a single spacetime. This is Galilean–Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a spacetime can be decomposed into spatial coordinates plus temporal coordinates, which is not possible in the general case.
Spacetime in general relativity
General relativity
Spacetime curvature schematic
\( G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu} \)
Introduction History
Mathematical formulation
Resources Tests
Fundamental concepts
Phenomena
Kepler problem Gravitational lensing Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole
Spacetime
Spacetime diagrams Minkowski spacetime Wormhole
Equations Formalisms
In general relativity, it is assumed that spacetime is curved by the presence of matter (energy), this curvature being represented by the Riemann 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 infinitesimally close 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 three-dimensional "spacelike hypersurface", a continuous three-dimensional "slice" through the four-dimensional property with the property that every curve that is contained entirely within this hypersurface is a space-like curve.[21]
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
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, causal dynamical triangulation, 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.
See also
Portal icon Time portal
Portal icon Physics portal
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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
Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011). Relativistic Celestial Mechanics of the Solar System. John Wiley & Sons. p. 157. ISBN 3527634576., Extract of page 157
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) Dimension from 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), "Raum und Zeit", Physikalische Zeitschrift 10: 75–88
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.
Ellis, G. F. R.; Williams, Ruth M. (2000). Flat and curved space–times (2nd ed.). Oxford University Press. p. 9. ISBN 0-19-850657-0.
Petkov, Vesselin (2010). Minkowski Spacetime: A Hundred Years Later. Springer. p. 70. ISBN 90-481-3474-9., Section 3.4, p. 70
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 \( s^2= g_{\alpha\beta}\Delta x^\alpha \Delta x^\beta \) 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.
Berry, Michael V. (1989). Principles of Cosmology and Gravitation. CRC Press. p. 58. ISBN 0-85274-037-9., Extract of page 58, caption of Fig. 25
I. M. Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, page 178, Springer, ISBN 0387-90332-1, MR 520230
Geroch, Robert; Horowitz, Gary T. (1979). "Chapter 5. Global structure of spacetimes". In Hawking, S.W.; Israel, W. General Relativity An Einstein Centenary Survey. Cambridge University Press. p. 219. 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.
External links
Wikibooks has a book on the topic of: Special Relativity
http://universaltheory.org
Barrow, John D.; Tipler, Frank J. (1988). The Anthropic Cosmological Principle. Oxford University Press. ISBN 978-0-19-282147-8. 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. Bibcode:1981PhRvD..24..251I. doi:10.1103/PhysRevD.24.251.
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.
Penrose, Roger (2004). The Road to Reality. Oxford: Oxford University Press. ISBN 0-679-45443-8. Chpts. 17–18.
Poe, Edgar A. (1848). Eureka; An Essay on the Material and Spiritual Universe. Hesperus Press Limited. 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.; Wheeler, John A. (1963). Spacetime Physics. W. H. Freeman. ISBN 0-7167-2327-1.
Wells, H.G. (2004). The Time Machine. New York: Pocket Books. ISBN 0-671-57554-6. (pp. 5–6)
Stanford Encyclopedia of Philosophy: "Space and Time: Inertial Frames" by Robert DiSalle.
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