## Sunday, May 6, 2012

### Spacetime

We are building to the most important equation in Science, The Dirac Equation, that which united Special Relativity and Quantum Mechanics. Read this and back through the posts of this month May 2012, and be prepared.

In physicsspacetime (or space-timespace timespace-time continuum) is any mathematical model that combinesspace 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. From a Euclidean spaceperspective, 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 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, 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 amanifold 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 whereand 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 manifestsymmetry.
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 as atomic clocks onboard a Space Shuttle running slower than synchronized Earth-bound inertial clocks and the relativisticdecay 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 asstring 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.

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## Spacetime in literature

Incas regarded space and time as a single concept, named pacha (QuechuapachaAymarapacha).[1][2] The peoples of the Andes have kept this understanding until now.[3]
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 thecombination 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 MachineH.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."

### Mathematical concept

The first reference to spacetime as a mathematical concept was in 1754 by Jean le Rond d'Alembert in the article Dimension in Encyclopedie. 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".[4]
After discovering quaternions,[5] William Rowan Hamilton commented, "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." Hamilton's biquaternions, which have algebraic properties sufficient to model spacetime and its symmetry, were in play for more than a half-century before formal relativity. For instance, William Kingdon Clifford noted their relevance.
Another important antecedent to spacetime was the work of James Clerk Maxwell as he used partial differential equations to develop electrodynamics with the four parameters.Lorentz discovered some invariances of Maxwell's equations late in the 19th century which were to become the basis of Einstein's theory of special relativity. Fiction authors were also involved, as mentioned above. It has always been the case that time and space are measured using real numbers, and the suggestion that the dimensions of space and time are comparable could have been raised by the first people to have formalized physics, but ultimately, the contradictions between Maxwell's laws and Galilean relativity had to come to a head with the realization of the import of finitude of the speed of light.
While spacetime can be viewed as a consequence of Albert Einstein's 1905 theory of special relativity, it was first explicitly proposed mathematically by one of his teachers, the mathematician Hermann Minkowski, in a 1908 essay[6] building on and extending Einstein's work. His concept of Minkowski space is the earliest treatment of space and time as two aspects of a unified whole, the essence of special relativity. The idea of Minkowski space also led to special relativity being viewed in a more geometrical way, this geometric viewpoint of spacetime being important in general relativity too. (For an English translation of Minkowski's article, see Lorentz et al. 1952.) The 1926 thirteenth edition of theEncyclopædia Britannica included an article by Einstein titled "Space-Time".[7]

## 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 thesun 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 coordinates systems.
A spacetime is independent of any observer.[8] 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.[9]
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)[10] 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.

### Spacetime intervals

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 invariant 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:
$s^2 = \Delta r^2 - c^2\Delta t^2 \,$   (spacetime interval),
where c is the speed of light, and Δr and Δt denote differences of the space and time coordinates, respectively, between the events.
(Note that the choice of signs for $s^2$ above follows the space-like convention (-+++). Other treatments reverse the sign of $s^2$.)
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.
Certain types of world lines (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.

#### 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 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 ($s^2 < 0$) 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:
$\Delta\tau = \sqrt{\Delta t^2 - \frac{\Delta r^2}{c^2}}$   (proper time).
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 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 squared 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 squared spacetime interval ($s^2 > 0$), the measurement of space-like separation is the proper distance:
$\Delta\sigma = \sqrt{\Delta r^2 - c^2\Delta t^2}$   (proper distance).
Like the proper time of time-like intervals, the proper distance ($\Delta\sigma$) of space-like spacetime intervals is a real number value.

## 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, the speed of light $c$ is usually assumed to be unity.
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.

### 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:[citation needed]
• 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.

### 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:

### 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 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

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 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.[11]
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 thePenrose–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 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 scaleCausal setsloop quantum gravitystring 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.

## 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 other inverse-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 4πr 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 itsgalaxy.[12] 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."[13] Finally, Tangherlini[14] 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[15] expands on the preceding argument in the following anthropic manner. 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 thatprotons 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.
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.[16] 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.[15]
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 N except one or both of 3 and 4.[citation needed] 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;[17] for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark.[15] Barrow has repeatedly cited the work ofWhitrow.[18]
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.[19]

## Notes

1. ^ 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
2. ^ Paul Richard Steele, Catherine J. Allen, Handbook of Inca mythology, p. 86, (ISBN 1-57607-354-8)
3. ^ Shirley Ardener, University of Oxford, Women and space: ground rules and social maps, p. 36 (ISBN 0-85496-728-1)
4. ^ R.C. Archibald (1914) Time as a fourth dimension Bulletin of the American Mathematical Society 20:409.
5. ^ Gallier, Jean H. (2001). Geometric methods and applications: for computer science and engineering. Springer. p. 249. ISBN 0-387-95044-3.Chapter 8, page 249
6. ^ Hermann Minkowski"Raum und Zeit", 80. Versammlung Deutscher Naturforscher (Köln, 1908). Published in Physikalische Zeitschrift 10 104–111 (1909) and Jahresbericht der Deutschen Mathematiker-Vereinigung 18 75-88 (1909). For an English translation, see Lorentz et al. (1952).
7. ^ Einstein, Albert, 1926, "Space-Time,Encyclopædia Britannica, 13th ed.
8. ^ Matolcsi, Tamás (1994). Spacetime Without Reference Frames. Budapest: Akadémiai Kiadó.
9. ^ Ellis, G. F. R.; Williams, Ruth M. (2000). Flat and curved space-times (2nd ed.). Oxford University Press. p. 9. ISBN 0-198-50657-0.
10. ^ Petkov, Vesselin (2010). Minkowski Spacetime: A Hundred Years Later. Springer. p. 70. ISBN 9-048-13474-9.Section 3.4, p. 70
11. ^ 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.
12. ^ Ehrenfest, Paul (1920). "How do the fundamental laws of physics make manifest that Space has 3 dimensions?". Annalen der Physik 61 (5): 440. Bibcode1920AnP...366..440Edoi:10.1002/andp.19203660503.. Also see Ehrenfest, P. (1917) "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?" Proceedings of the Amsterdam Academy 20: 200.
13. ^ Weyl, H. (1922) Space, time, and matter. Dover reprint: 284.
14. ^ Tangherlini, F. R. (1963). "Atoms in Higher Dimensions". Nuovo Cimento 14 (27): 636.
15. a b c Tegmark, Max (April 1997). "On the dimensionality of spacetime"Classical and Quantum Gravity 14 (4): L69–L75. arXiv:gr-qc/9702052Bibcode1997CQGra..14L..69Tdoi:10.1088/0264-9381/14/4/002. Retrieved 2006-12-16.
16. ^ Dorling, J. (1970). "The Dimensionality of Time"American Journal of Physics 38 (4): 539–40. Bibcode 1970AmJPh..38..539Ddoi:10.1119/1.1976386.
17. ^ Barrow, J. D. (2002). The Constants of Nature. Pantheon Books. ISBN 0-375-42221-8.
18. ^ Whitrow, G. J. (1955) " ," British Journal of the Philosophy of Science 6: 13. Also see his (1959) The Structure and Evolution of the Universe. London: Hutchinson.
19. ^