Spacetime

For other uses of this term, see (disambiguation).

In physics, spacetime is a model that combines three-dimensional space and one-dimensional time into a single construct called the space-time 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 the two concepts into a single manifold, physicists are able to significantly simplify the form of most physical laws, as well as describe the workings of the universe at both supergalactic and subatomic levels in a more uniform way.

In Galilean mechanics, this is just a formal option, but in Einstein's special relativity, it is not possible to separate space and time. The notion of space depends on the observer, as instantaneous events depend on a reference frame.

Treating space and time as two aspects of a unified whole was an idea devised by Hermann Minkowski shortly after the theory of special relativity was developed by Albert Einstein. This unification is further exemplified by the common practice shown by some specialists of expressing time in the same units as space by multiplying time measurements by the speed of light. The concept of spacetime is vital to this theory and also to general relativity, an extension of special relativity, that takes into account gravitation. 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, usually put as the vertical axis. Note that the orbit of the Earth is an ellipse in space, but its worldline is a helix in spacetime.

Space-times 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 space-time, or the motion of light around a rotating star may be described in another type of space-time.

The problem of the actual number of dimensions of our universe is still open, as some theories (such as the string theory) predict as many as 26. In these theories, however, all the additional dimensions are such that the universe measured along them is subatomic in size. As a result, even if the universe had many more dimensions, we would only perceive 4 of them.

Historical origin

While the concept of spacetime is commonly associated with Albert Einstein, the discoverer of special relativity, the concept was in fact first proposed by Einstein's teacher Hermann Minkowski in an admiring 1908 essay building on and extending Einstein's work. Minkowski intended spacetime as a better way of thinking about special relativity. (For an English translation of Minkowski's article, see Lorentz et al. 1952.) The 1926 13th edition of the Encyclopedia Britannica included an article titled "space-time" by Einstein.

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

Basic concepts

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.

Space-time intervals

The new concept of spacetime brings with it 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 imaginary. The spacetime interval quantifies this new distance (in Cartesian coordinates

x, y, z, t

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

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 $r^2 = x^2 + y^2 + z^2$ .

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; $s^2 < 0$ ).
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; $s^2 > 0$ ).

Events with a negative 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.

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.

A reference frame (observer) being identified with one of these coordinate charts, any 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 continua and symmetry

For further details, see the article spacetime symmetries

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

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:

$\eta_{ab} \, = diag(-1, 1, 1, 1)$

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 space 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.

Is space-time quantized?

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. 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, 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.

Privileged character of 3x1 spacetime

That spacetime, speculative theories aside, consists of three spatial (bidirectional) and one temporal (and unidirectional) dimensions, is neither a chance matter, nor one lacking physical and mathematical significance. Quite the contrary; it can be shown that all other possible numbers of spatial and temporal dimensions lead to one or more of the following problematic situations:

The past does not determine the future, so that physical laws are impossible and natural phenomena are unpredictable. This holds in all cases where the number of spatial and temporal dimensions both exceed 2;
Gravitation does not result in stable orbits, and electromagnetism does not result in stable atoms and molecules. Protons and electrons have short half lives;
Of the 9 possible cases with no more than 2 spatial and 2 temporal dimensions, the 3 cases that are stable and predictable do not allow matter of any complexity, such as lifeforms with nervous systems;
With two exceptions, all cases with more than 1 time dimension are unstable or unpredictable. One exception does not allow complexity. The other exception, the case of 3 time and 1 space dimensions, requires that all matter have a velocity exceeding the speed of light in a vacuum.

It appears that complexity, life, and information processing are possible only in a universe subject to spacetime, i.e., 3 spatial and 1 temporal dimensions. This fact is an instance of anthropic reasoning.

Curiously, 3 dimensional space appears 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. Immanuel Kant thought that space has 3 dimensions because the law of universal gravitation between two objects is proportional to the inverse square of the distance separating them. Kant's argument is historically important but puts the cart before the horse. The law of gravitation follows from the dimensionality of space. More generally, in a space with N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of d is proportional to d^N-1.

Paul Ehrenfest showed in 1917 that if the number of spatial dimensions exceeds 3, the orbit of a planet about its sun cannot remain stable, and the same holds for a star's orbit around its galactic center. Likewise, electrons cannot have stable orbits around a nucleus; they either fall into the nucleus or disperse. He also noted 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 are both problems avoided. Tegmark (1997) makes the following anthropic argument. 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 manipulating technology could not emerge. If space had more than 3 dimensions, atoms as we know them (and probably more complex structures as well) could not exist. If space had fewer than 3 dimensions, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. Theories that propose that the universe has more than 3 spatial dimensions, such as Kaluza-Klein theory or string theory, do not overturn the privileged status of 3x1 spacetime, because the spatial dimensions in excess of 3 matter only for lengths on the order of the diameter of subatomic particles.

For an introduction to the privileged status of 3 spatial and 1 temporal dimensions, see Barrow (2002: chpt. 10, esp. Fig. 10.12); for a deeper treatment, see Barrow and Tipler (1986: 4.8). Barrow's repeated discussions of this topic owe much to Gerald James Whitrow (1959).

Historical origin

Basic concepts

Space-time intervals

Mathematics of space-times

Space-time topology

Space-time continua and symmetry

Spacetime in special relativity

Spacetime in general relativity

Is space-time quantized?

Other uses of the word 'spacetime'

Privileged character of 3x1 spacetime

Further reading

See also

Gourt Home::Gourt :: Spacetime

Historical origin

Basic concepts

Space-time intervals

Mathematics of space-times

Space-time topology

Space-time continua and symmetry

Spacetime in special relativity

Spacetime in general relativity

Is space-time quantized?

Other uses of the word 'spacetime'

Privileged character of 3x1 spacetime

Further reading

See also