User:Mpatel/sandbox/Spacetime



In physics, spacetime (alternatively space–time, space–time continuum) is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of the fourth dimension. According to Euclidean space perception, the universe has three dimensions of space and one of time.

Although formative ideas of spacetime were discussed before the 20th century, it was mathematician Hermann Minkowski who first formulated the concept mathematically and used it to simplify physical theory. Since then, the idea of representing space and time in a single manifold has led to vast simplification of many physical theories, as well as describing in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.

In classical mechanics, the use of spacetime over Euclidean space is optional, as time is treated as a constant, being independent of mechanical motion in three dimensions (which matches reality at low speeds). In relativistic contexts, however, time cannot be separated from the three dimensions of space because it depends on an object's velocity relative to the speed of light, and also the strength of intense gravitational fields which can slow the passage of time.

Historical origin
After discovering quaternions, 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." 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 in on the game. Edgar Allan Poe stated in his essay on cosmology titled Eureka (1848) that "Space and duration are one." This is the first known instance of suggesting space and time to be different perceptions of one thing. Poe arrived at this conclusion after approximately 90 pages of reasoning but employed no mathematics. In his 1895 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.” He added, “Scientific people…know very well that time is only a kind of space.”

The spacetime concept was first formulated mathematically by Hermann Minkowski in a 1908 essay. A mathematician and teacher of Albert Einstein, Minkowski reformulated Einstein's 1905 theory of special relativity four-dimensionally, placing space and time on an equal footing through the concept of Minkowski space. This concept emphasised the geometrical character of special relativity, this being important in general relativity too. (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 "space-time".

The beginning of his address delivered at the 80th Assembly of German Natural Scientists and Physicians (September 21, 1908) is now famous:

Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Basic concepts
Spacetimes are the arenas in which all physical phenomena take place. A spacetime is independent of any observer, but the description of events - points in spacetime specified by a unique time and location - requires each observer to choose a coordinate system. Events are specified by four real numbers in any coordinate system.

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. Examples of events include the explosion of a star or the single beat of a drum.

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 worldline 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.

The concept of spacetime combines space and time within a single coordinate system, typically with 4 dimensions: length, width, height, and time. Dimensions are components of a coordinate grid typically used to locate a point in space, or on the globe, such as by latitude, longitude and planet (Earth). However, with spacetime, the coordinate grid is used to locate "events" (rather than just points in space), so time is added as another dimension to the grid.

Formerly, from experiments at slow speeds, time was believed to be a constant, which progressed at a fixed rate; however, later high-speed experiments revealed that time slowed down at higher speeds (with such slowing called "time dilation"). Many experiments have confirmed the slowing from time dilation, such as atomic clocks onboard a Space Shuttle running slower than synchronized Earth-bound clocks. Since time varies, it is treated as a variable within the spacetime coordinate grid, and time is no longer assumed to be a constant, independent of the location in space.

Note that treating spacetime events with the 4 dimensions (including time) is the conventional view; however, other invented coordinate grids treat time as 3 additional dimensions, with length-time, width-time, and height-time, to accompany the 3 dimensions of space. 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, such as: latitude, longitude, plus Greenwich Mean Time (3 dimensions), or city, state, postal code, country, and UTC time (5 dimensions). The various dimensions are chosen, depending on the coordinate grid used.

The term spacetime has taken on a generalized meaning with the advent of higher-dimensional theories. 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.

Space-time 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 coordinates $$x, y, z, t$$):

$$s^2 = \, c^2t^2 - r^2$$

where $$c$$ is the speed of light, differences of the space and time coordinates of the two events are denoted by $$r$$ and $$t$$, respectively and $$r^2 = x^2 + y^2 + z^2$$. (Note that the choice of signs above follows the Landau-Lifshitz spacelike convention. Other treatments, including some within Wikipedia, reverse the order of the arguments on the right-hand side. If this alternate convention is chosen, the relationships in the next two paragraphs are reversed.)

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; there exists a reference frame such that the two events occur at the same place; $$s^2 > 0$$).
 * light-like (the space between the two events is exactly balanced by the time between the two events; $$s^2 = 0$$).
 * space-like (not enough time passes for there to be a cause-effect relationship between the two events; there exists a reference frame such that the two events occur at the same time; $$s^2 < 0$$).

Events with a positive spacetime interval are in each other's future or past, and the value of the interval defines the proper time measured by an observer traveling 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 space-times
For physical reasons, a space-time continuum is mathematically defined as a four-dimensional, smooth, connected pseudo-Riemannian manifold together with a smooth Lorentz metric of signature $$\left(3,1\right)$$. 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 $$\left(x, y, z, t\right)$$ 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 $$q$$ (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 $$\left(x, y, z, t\right)$$ (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).

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.

Space-time symmetries
Often in relativity, space-times that have some form of symmetry are studied. As well as helping to classify spacetimes, these symmetries usually serve as a simplifying assumption in specialised work. Some of the most popular ones include:


 * Axially symmetric 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 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 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 space-time can be decomposed into spatial coordinates plus temporal coordinates, which is not possible in the general case.

Spacetime in general relativity
In general relativity, spacetime is assumed 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 space-time continua have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compact spacetime has closed timelike curves, which violate the usual ideas of causality (that is, future events could affect past ones). For this reason, mathematical physicists usually consider only restricted subsets of all possible spacetimes. One way to do this is to study "realistic" solutions of the field 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 space-time
In general relativity, space-time 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 space-time at the Planck scale. Causal sets, loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized space-time 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
A number of scientists and philosophers have written about spacetime, and concepts have evolved as more theories have been deduced and tested by mathematical analysis or experimentation. Also the subject of spacetime has been a topic for pseudo-intellectuals, espousing their views in attempts to impress others, regardless of matching any experimental data.

Other writers have been limited by the scientific evidence available at the time. For example, in the latter 20th century, experiments with "atom-smasher" particle accelerators had revealed that individual protons accelerated to high speeds were gaining the mass equivalent to a car at rest, requiring ever-increasing amounts of energy to accelerate the protons even faster. While the passage of Time slowed at high speeds, the mass of the particles increased. Writers from previous eras were not aware of that evidence, so fanciful views are sometimes expressed in the writings that are described below.

Let dimensions be of two kinds: spatial and temporal. That spacetime, ignoring any undetectable compactified dimensions, consists of three spatial (bidirectional) and one temporal (unidirectional) dimensions can be explained by appealing to the physical consequences of differing numbers of dimensions. The argument is often of an anthropic nature.

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 of it 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, from Space having 3 dimensions, and from 3-dimensional solid objects having surface area proportional to the square of their size in one chosen dimension. In particular, a sphere of radius r has area of 4πr2. 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.

Fixing the number of temporal dimensions at 1 and letting the number of spatial dimensions N exceed 3, Paul Ehrenfest showed in 1920 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, F. R. Tangherlini showed in 1963 that when N>3, electrons would not form stable orbitals around nuclei; they would either fall into the nucleus or disperse. Ehrenfest also showed that if N is even, then the different parts of a wave impulse will travel at different speeds. If N is odd and greater than 3, wave impulses become distorted. Only when N=3 or 1 are both problems avoided.

Tegmark expands on the preceding argument in the following anthropic manner. If the number of Time dimensions differed 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. In addition, Tegmark maintains that protons and electrons would be unstable in a universe with more than one Time dimension, as they can decay into more massive particles (this is not a problem if the temperature is sufficiently low). If N>3, Ehrenfest's above argument 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, nerves must intersect and cannot overlap.

In general, it is not clear how physical laws could operate if the number of Time dimensions T differed from 1. If T>1, individual subatomic particles which decay after a fixed period would not have much predictability because timelike geodesics would not be necessarily maximal. N=1 and T=3 has the peculiar property that that the speed of light in a vacuum is a lower bound on the velocity of matter. Hence anthropic arguments rule out all cases except 3 spatial and 1 temporal dimensions, which describes the world we live in.

Curiously, 3 and 4 dimensional spaces appear richest geometrically and topologically. 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; for a deeper treatment, see Barrow and Tipler. Barrow regularly cites Whitrow.

In string theory, physicists are not constrained by notions limited to 3+1 dimensions, so coordinate grids of 10, or perhaps 26 dimensions, are used to describe the types and locations of the vibrating strings. String theory follows the notion that the "universe is wiggly" and considers matter and energy to be composed of tiny vibrating strings of various types, specified by some of the dimensions.