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, 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
Theoretical physics, as well as described in a more uniform way the workings of the universe at both the Physical cosmology and
quantum mechanics levels.
In
classical mechanics, the use of spacetime over Euclidean space is optional, as time is independent of mechanical motion in three dimensions. In
theory of relativity contexts, however, time cannot be separated from the three dimensions of space as it depends on an object's
velocity relative to the Light speed.
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 20th century scientific concept began in the 19th century with fiction writers. Edgar Allan Poe stated in his essay on cosmology titled
Eureka (Edgar Allan Poe) (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 Hermann Minkowski, "Raum und Zeit", 80. VersammlungDeutscher 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). 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".
Albert Einstein, 1926, " Space-Time,"
Encyclopedia Britannica, 13th ed.
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 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 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 measurement 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 coordinate system 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
sign convention#relativity. 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 cones.
For special relativity, the spacetime interval is considered
invariant (physics) across inertial reference frames.
Certain types of World line (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" (Fictitious force) 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).
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 space 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:
Causal structure
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
sign convention#relativity is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are Lorentz invariance 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 space 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 interaction 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
compactification (physics), so that they loop around over a very short distance (usually around the Planck length).
Privileged character of 3+1 spacetime
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 principle 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 4π
r2. 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
atomic orbital around nuclei; they would either fall into the atomic 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 principle 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
N1, individual subatomic particles which decay after a fixed period would not have much predictability because timelike geodesics would not be necessarily maximal.Dorling, J. (1970) "The Dimensionality of Time"
American Journal of Physics '38'(4): 539-40.
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 (chpt. 6, esp. Fig. 10.12); for a deeper treatment, see Barrow and Tipler. (4.8) Chpt. 6 is a good survey of modern cosmology, which builds on spacetime. Barrow regularly cites Whitrow.
See also
References
- Paul Ehrenfest, 1920, "How do the fundamental laws of physics make manifest that space has 3 dimensions?" Annalen der Physik 61: 440.
- Immanuel Kant, 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.
- Hendrik Lorentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl, 1952. The Principle of Relativity: A Collection of Original Memoirs. Dover.
- John Lucas (philosopher), 1973. A Treatise on Time and Space. London: Methuen.
- Chpts. 17,18.
-
External links
- Spacetime - Summary and collection of links to academic sites.
, 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 Theoretical physics, as well as described in a more uniform way the workings of the universe at both the Physical cosmology and quantum mechanics levels.
In classical mechanics, the use of spacetime over Euclidean space is optional, as time is independent of mechanical motion in three dimensions. In
theory of relativity contexts, however, time cannot be separated from the three dimensions of space as it depends on an object's velocity relative to the
Light speed.
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 20th century scientific concept began in the 19th century with fiction writers.
Edgar Allan Poe stated in his essay on cosmology titled
Eureka (Edgar Allan Poe) (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 Hermann Minkowski, "Raum und Zeit", 80. VersammlungDeutscher 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). 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".Albert Einstein, 1926, " Space-Time,"
Encyclopedia Britannica, 13th ed.
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 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 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 measurement 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 coordinate system 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 sign convention#relativity. 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 cones.
For special relativity, the spacetime interval is considered invariant (physics) across
inertial reference frames.
Certain types of World line (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" (Fictitious force) 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).
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 space 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:
Causal structure
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 sign convention#relativity is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are
Lorentz invariance 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 space 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 interaction 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
compactification (physics), so that they loop around over a very short distance (usually around the Planck length).
Privileged character of 3+1 spacetime
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 principle 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 4π
r2. 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
atomic orbital around nuclei; they would either fall into the
atomic 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 principle 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
N1, individual subatomic particles which decay after a fixed period would not have much predictability because timelike
geodesics would not be necessarily maximal.Dorling, J. (1970) "The Dimensionality of Time"
American Journal of Physics '38'(4): 539-40.
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 (chpt. 6, esp. Fig. 10.12); for a deeper treatment, see Barrow and Tipler. (4.8) Chpt. 6 is a good survey of modern cosmology, which builds on spacetime. Barrow regularly cites Whitrow.
See also
References
- Paul Ehrenfest, 1920, "How do the fundamental laws of physics make manifest that space has 3 dimensions?" Annalen der Physik 61: 440.
- Immanuel Kant, 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.
- Hendrik Lorentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl, 1952. The Principle of Relativity: A Collection of Original Memoirs. Dover.
- John Lucas (philosopher), 1973. A Treatise on Time and Space. London: Methuen.
- Chpts. 17,18.
-
External links
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