Spacetime

Two-dimensional analogy of space-time distortion. The presence of matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity. Note that the white lines do not represent the curvature of space, but instead represent the coordinate system imposed on the curved spacetime which would be rectilinear in a flat spacetime.

In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time 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 dimension of time. By combining space and time into a single manifold, physicists have significantly simplified a large amount of physical theory, as well as described 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 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.

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.

Historical origin

The origins of this twentieth-century scientific theory began in the nineteenth century with fiction writers. 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 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.” He added, “Scientific people…know very well that Time is only a kind of Space.”

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 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 the Encyclopedia Britannica included an article by Einstein titled "space-time".

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

A space-time 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 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.

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 $x,y,z,t$):

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

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 space-time 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 space-time interval is considered invariant across inertial reference frames.

Certain types of worldlines (called geodesics of the space-time) are the shortest paths between any two events, with distance being defined in terms of space-time intervals. The concept of geodesics becomes critical in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in space-time, 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).

Space-time 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

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, 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 space-time 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.

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 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 such as 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 4πr² 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 r^N-1.

Fixing the number of temporal dimensions at 1 and letting the number of spatial dimensions 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, 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. 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. If there was more than one time dimension, individual subatomic particles which decay after a fixed period would not have such predictability because timelike geodesics would not be necessarily maximal. Three time and one 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 ; for a deeper treatment, see Barrow and Tipler. Barrow regularly cites Whitrow.

See also

• Dimensional analysis
• Four-vector
• Fourth dimension
• Global spacetime structure
• Local spacetime structure
• Lorentz invariance
• Manifold
• Mathematics of general relativity
• Metric space
• Simultaneity
• Basic introduction to the mathematics of curved spacetime
• Frame-dragging

References

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• Ehrenfest, Paul, 1920, "How do the fundamental laws of physics make manifest that space has 3 dimensions?" Annalen der Physik 61: 440.
• 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, Minkowksi, 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.  Chpts. 17,18.
• Robb, A. A. (1936). Geometry of time and space. University Press. 

• Poe, Edgar A. (1848). Eureka; An Essay on the Material and Spiritual Universe. Hesperus Press Limited. ISBN 1-84391-009-8. 

• Schutz, J. W. (1997). Independent axioms for Minkowski Space-time. Addison-Wesley Longman. 

• Tangherlini, F. R. (1963). Atoms in Higher Dimensions. Nuovo Cimento 14 (27): 636.

• Taylor, E. F. and Wheeler, John A. (1963). Spacetime Physics. W. H. Freeman. 

• Wells, H.G. (2004). The Time Machine. New York: Pocket Books.  (pp. 5; 6)

External links

• Wikibooks: Special Relativity
• Spacetime - Summary and collection of links to academic sites.

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