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Spacetime assumes another dimension exists in the universe: time itself.

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Spacetime was developed by Einstein. Spacetime considers time to be the fourth dimension. Spacetime is a mathematical model. Spacetime combines three-dimensional space and one-dimensional time into a single construct called the spacetime continuum, in which time plays the role of the 4th dimension. According to Euclidean space perception, our 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 good deal of physical theory, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.

The Image shows how spacetime works. For further info see special relativity.

In classical mechanics, the use of spacetime over Euclidean space is optional, as time is independent of mechanical motion in three dimensions. In relativistic contexts, however, time cannot be separated from the three dimensions of space as it depends on an object's velocity relative to the speed of light.

How many dimensions are needed to describe the universe is still an open question. Speculative theories (such as string theory) predict from 10 to 26 dimensions, but the existence of more than four dimensions would only appear to make a difference at the subatomic level.

Historical origin of spacetime.

While spacetime is a consequence of Albert Einstein's 1905 theory of special relativity, it was first explicitly proposed by one of his teachers, the mathematician Hermann Minkowski, in an admiring 1908 essay building on and extending Einstein's work. His Minkowski space is the earliest treatment of space and time as two aspects of a unified whole, the essence of special relativity. (For an English translation of Minkowski's article, see Lorentz et al. 1952.) The 1926 thirteenth edition of the Encyclopedia Britannica included an article by Einstein titled "spacetime".

H.G. Wells's 1895 novel The Time Machine refers to time as the "fourth dimension".

Basic concepts of spacetime.

Spacetimes are the arenas in which all physical events take 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. Examples of events include the explosion of a star or the single beat of a drum.

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 coordinate system. Events are specified by four real numbers in any coordinate system. The worldline 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 is depicted in two spatial dimensions x and y (the plane of the Earth 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 expressing distance in units of time, by dividing the distance measurement by the speed of light.

Spacetime intervals.

Spacetime entails a new concept of distance. Whereas distances are always positive in Euclidean spaces, the distance between any two events in spacetime (called an "interval") may be real, zero, or even imaginary. The spacetime interval quantifies this new distance (in Cartesian coordinatesx,y,z,t):

where c is the speed of light, differences of the space and time coordinates of the two events are denoted byr andt, respectively andr2 = x2 + y2 + z2.

Pairs of events in spacetime may be classified into 3 distinct types based on 'how far' apart they are:

  • time-like (more than enough time passes for there to be a cause-effect relationship between the two events;s2 < 0).
  • light-like (the space between the two events is exactly balanced by the time between the two events;s2 = 0).
  • space-like (not enough time passes for there to be a cause-effect relationship between the two events;s2 > 0).

Events with a negative spacetime interval are in each other's future or past, and the value of the interval defines the Proper time measured by an observer travelling between them. Events with a spacetime interval of zero are separated by the propagation of a light signal.

For special relativity, the spacetime interval is considered invariant across inertial reference frames.

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.

Mathematics of spacetimes

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected pseudo-Riemannian manifold together with a smooth, Lorentz metric of signature . 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 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 eventp. Another reference frame may be identified by a second coordinate chart aboutp. Two observers (one in each reference frame) may describe the same event p but obtain different descriptions. 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 tensors into relativity, by which all physical quantities are represented.

Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. The paths of particles and light beams in spacetime are represented by timelike and null (light-like) geodesics (respectively).

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 property 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.
  • Any non-compact 4-manifold can be turned into a spacetime.

Spacetime continua and symmetry

For further details, see the article spacetime symmetries

Often in general relativity, spacetime continua that have some form of symmetry are studied. Some of the most popular ones include:

  • Axially symmetric spacetimes.
  • Spherically symmetric spacetimes.
  • Static spacetimes.
  • Stationary spacetimes.

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? and can be written as a four-by-four matrix:

where the Landau-Lifshitz spacelike 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."

Many spacetime continua have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compact spacetime has closed, time-like 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.

Is spacetime quantized?

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. loop quantum gravity, string theory, and Black hole thermodynamics all predict a quantized spacetime with agreement on the order of magnitude. Loop quantum gravity even makes precise predictions about the geometry of spacetime at the Planck scale.

Other uses of the word 'spacetime'.

Spacetime has taken on meanings different from the four-dimensional one given above. For example, when drawing a graph of the distance a car has travelled for a certain time, it is natural to draw a two-dimensional spacetime diagram. As drawing four-dimensional spacetime diagrams is impossible, physicists often resort to drawing three-dimensional spacetime diagrams. For example, the Earth orbiting the Sun is a helical shape traced out in the direction of the time axis.

In higher-dimensional theories of physics, for example, string theory, the assumption that our universe has more than four dimensions is frequently made. For example, Kaluza-Klein theory was an attempt to unify the two fundamental forces of gravitation and electromagnetism and used four space dimensions with one of time. Modern theories use as many as ten or more spacetime dimensions. These theories are highly speculative, as there has been no experimental evidence to support them. To explain why the extra dimensions are not observed, it is assumed that they are compactified, so that they loop around over a very short distance (usually around the Planck length).

Privileged character of 3+1 spacetime.

Dimensions are of two kinds: spatial and temporal. That spacetime, ignoring any undetectable compactified dimensions, consists of 3+1 dimensions (ie three spatial (bidirectional) and one temporal (unidirectional)), is often explained by appeal to the mathematical and physical effects of differing numbers of dimensions. Most often this takes the form of an anthropic argument.

Immanuel Kant argued that space having 3 dimensions followed from the inverse square Law of universal gravitation. Kant's argument is historically important, but John D. Barrow has stated that "we would regard this as getting 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) This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the fact that space has 3 dimensions and 3-dimensional solid objects have surface area proportional to the square of their size in one chosen dimension (particularly a sphere has area of 4pr2 with r as the radius of the sphere). More generally, in a space with N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to rN-1.

Fixing the number of temporal dimensions at 1 and letting the number of spatial dimensions exceed 3, Paul Ehrenfest (1920) showed that the orbit of a planet about its sun cannot remain stable, and that the same holds for a star's orbit around its galactic center. Likewise, in 1963, F. R. Tangherlini showed that electrons would not form stable orbitals around nuclei; they would either fall into the nucleus or disperse. Ehrenfest also showed that if space has an even number of dimensions, then the different parts of a Wave impulse will travel at different speeds. If the number of dimensions is odd and greater than 3, wave impulses become distorted. Only with three dimensions (or one dimension) are both problems avoided.

Another anthropic argument, expanding upon the preceding one, is due to Tegmark (1997). If the number of time dimensions differed from 1, Tegmark argues, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life manipulating technology could not emerge. In addition, he argues that protons and Electrons would be unstable in a universe with more than one time dimension, as they can decay into more massive particles. However, he also argues that this phenomenon would be suppressed if the temperature is sufficiently low. If space had more than 3 dimensions, atoms as we know them (and probably more complex structures as well) could not exist (following Ehrenfest's argument). If space had fewer than 3 dimensions, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, nerves cannot overlap; they must intersect.

In general, it is not clear how physical laws should operate in the presence of more than one temporal dimension, or in the absence of time. But 3 time and 1 space dimensions has the peculiar property that that the speed of light in a vacuum is a lower bound on the velocity of matter. The only remaining case, 3 spatial and 1 temporal dimensions, is the world we live in. Hence anthropic arguments require a universe with 3 spatial and 1 temporal dimensions.

Curiously, 3 and 4 dimensional spaces appear to be the mathematically richest. For example, there are geometric statements whose truth or falsity is known for any number of spatial dimensions except 3, 4, or both.

For a more detailed introduction to the privileged status of 3 spatial and 1 temporal dimensions, see Barrow (2002: chpt. 6, esp. Fig. 10.12); for a deeper treatment, see Barrow and Tipler (1986: 4.8). Barrow regularly cites Gerald James Whitrow (1959).

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