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In physics, spacetime (also space–time, space time or space–time continuum) is
any mathematical model that combines space and time into a single interwoven
continuum. The spacetime of our universe is usually interpreted from a Euclidean
space perspective, which regards space as consisting of three dimensions, and
time as consisting of one dimension (often termed Minkowski space or the fourth
dimension). By combining space and time into a single manifold, physicists have
significantly simplified a large number of physical theories, as well as
described in a more uniform way the workings of the universe at both the
supergalactic and subatomic levels.
In non-relativistic classical mechanics, the use of Euclidean space instead of
spacetime is appropriate, as time is treated as universal and constant, being
independent of the state of motion of an observer. In relativistic contexts,
time cannot be separated from the three dimensions of space, because the
observed rate at which time passes for an object depends on the object's
velocity relative to the observer and also on the strength of gravitational
fields, which can slow the passage of time.
In cosmology, the concept of spacetime combines space and time to a single
abstract universe. Mathematically it is a manifold consisting of "events" which
are described by some type of coordinate system. Typically three spatial
dimensions (length, width, height), and one temporal dimension (time) are
required. Dimensions are independent components of a coordinate grid needed to
locate a point in a certain defined "space". For example, on the globe the
latitude and longitude are two independent coordinates which together uniquely
determine a location. In spacetime, a coordinate grid that spans the 3+1
dimensions locates events (rather than just points in space), i.e., time is
added as another dimension to the coordinate grid. This way the coordinates
specify where and when events occur. However, the unified nature of spacetime
and the freedom of coordinate choice it allows imply that to express the
temporal coordinate in one coordinate system requires both temporal and spatial
coordinates in another coordinate system. Unlike in normal spatial coordinates,
there are still restrictions for how measurements can be made spatially and
temporally (see Spacetime intervals). These restrictions correspond roughly to a
particular mathematical model which differs from Euclidean space in its manifest
symmetry.
Until the beginning of the 20th century, time was believed to be independent of
motion, progressing at a fixed rate in all reference frames; however, later
experiments revealed that time slows at higher speeds of the reference frame
relative to another reference frame. Such slowing, called time dilation, is
explained in special relativity theory. Many experiments have confirmed time
dilation, such as the relativistic decay of muons from cosmic ray showers and
the slowing of atomic clocks aboard a Space Shuttle relative to synchronized
Earth-bound inertial clocks. The duration of time can therefore vary according
to events and reference frames.
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 as being simply the result of coordinate transformations.
The term spacetime has taken on a generalized meaning beyond treating spacetime
events with the normal 3+1 dimensions. It is really the combination of space and
time. Other proposed spacetime theories include additional dimensions—normally
spatial but there exist some speculative theories that include additional
temporal dimensions and even some that include dimensions that are neither
temporal nor spatial (e.g., superspace). 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.
Spacetime in literature
Incas regarded space and time as a single concept, referred to as pacha (Quechua:
pacha, Aymara: pacha). The peoples of the Andes maintain a similar
understanding.
Arthur Schopenhauer wrote in §18 of On the Fourfold Root of the Principle of
Sufficient Reason (1813): "the representation of coexistence is impossible in
Time alone; it depends, for its completion, upon the representation of Space;
because, in mere Time, all things follow one another, and in mere Space all
things are side by side; it is accordingly only by the combination of Time and
Space that the representation of coexistence arises".
The idea of a unified spacetime is stated by Edgar Allan Poe in his essay on
cosmology titled Eureka (1848) that "Space and duration are one". 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", and that "any real body must have extension in four directions:
it must have Length, Breadth, Thickness, and Duration".
Marcel Proust, in his novel Swann's Way (published 1913), describes the village
church of his childhood's Combray as "a building which occupied, so to speak,
four dimensions of space—the name of the fourth being Time".
Mathematical concept
The first reference to spacetime as a mathematical concept was in 1754 by Jean
le Rond d'Alembert in the article Dimension in Encyclopedie. Another early
venture was by Joseph Louis Lagrange in his Theory of Analytic Functions (1797,
1813). He said, "One may view mechanics as a geometry of four dimensions, and
mechanical analysis as an extension of geometric analysis".
After his discovery of 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". Hamilton's
biquaternions, which have algebraic properties sufficient to model spacetime and
its symmetry, were in play for more than a half-century before formal relativity.
For instance, William Kingdon Clifford noted their relevance.
Another important antecedent to spacetime was the work of James Clerk Maxwell as
he used partial differential equations to develop electrodynamics with the four
parameters. Lorentz discovered some invariances of Maxwell's equations late in
the 19th century which were to become the basis of Albert Einstein's theory of
special relativity. Fiction authors were also involved, as mentioned above. It
has always been the case that time and space are measured using real numbers,
and the suggestion that the dimensions of space and time are comparable could
have been raised by the first people to have formalized physics, but ultimately,
the contradictions between Maxwell's laws and Galilean relativity had to come to
a head with the realization of the import of finitude of the speed of light.
While spacetime can be viewed as a consequence of 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. (For an English translation of Minkowski's article, see
Lorentz et al. 1952.) The 1926 thirteenth edition of the Encyclopædia Britannica
included an article by Einstein titled "Space–Time". ) The idea of Minkowski
space led to special relativity being viewed in a more geometrical way.
However, the most important contribution of Minkowski's geometric viewpoint of
spacetime turned out to be in Einstein's later development of general relativity,
since the correct description of the effect of gravitation on space and time was
found to be most easily visualized as a "warp" or stretching in the geometrical
fabric of space and time, in a smooth and continuous way that changed smoothly
from point-to-point along the spacetime fabric.
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. Because events are spacetime
points, an example of an event in classical relativistic physics is , the
location of an elementary (point-like) particle at a particular time. A
spacetime itself can be viewed as the union of all events in the same way that a
line is the union of all of its points, formally organized into a manifold, a
space which can be described at small scales using coordinates systems.
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 metrical coordinate system. Events are
specified by four real numbers in any such coordinate system. The trajectories
of elementary (point-like) particles through space and time are thus a continuum
of events called the world line of the particle. Extended or composite objects (consisting
of many elementary particles) are thus a union of many world lines twisted
together by virtue of their interactions through spacetime into a "world-braid".
However, in physics, it is common to treat an extended object as a "particle" or
"field" with its own unique (e.g., center of mass) position at any given time,
so that the world line of a particle or light beam is the path that this
particle or beam takes in the spacetime and represents the history of the
particle or beam. The world line of the orbit of the Earth (in such a
description) is depicted in two spatial dimensions x and y (the plane of the
Earth's orbit) and a time dimension orthogonal to x and y. The orbit of the
Earth is an ellipse in space alone, but its world line is a helix in spacetime.
The unification of space and time is exemplified by the common practice of
selecting a metric (the measure that specifies the interval between two events
in spacetime) such that all four dimensions are measured in terms of units of
distance: representing an event as (in the Lorentz metric) or (in the original
Minkowski metric) where is the speed of light. The metrical descriptions of
Minkowski Space and spacelike, lightlike, and timelike intervals given below
follow this convention, as do the conventional formulations of the Lorentz
transformation.
Spacetime intervals
In a Euclidean space, the separation between two points is measured by the
distance between the two points. The distance is purely spatial, and is always
positive. In spacetime, the separation between two events is measured by the
invariant interval between the two events, which takes into account not only the
spatial separation between the events, but also their temporal separation. The
interval, s2, between two events is defined as:
(spacetime interval),
where c is the speed of light, and Δr and Δt denote differences of the space and
time coordinates, respectively, between the events. The choice of signs for
above follows the space-like convention (−+++). The reason is called the
interval and not is that the sign of is indefinite.
Certain types of world lines (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.
Spacetime intervals may be classified into three distinct types, based on
whether the temporal separation () or the spatial separation () of the two
events is greater.
Time-like interval
For two events separated by a time-like interval, enough time passes between
them that there could be a cause–effect relationship between the two events. For
a particle traveling through space at less than the speed of light, any two
events which occur to or by the particle must be separated by a time-like
interval. Event pairs with time-like separation define a negative squared
spacetime interval () and may be said to occur in each other's future or past.
There exists a reference frame such that the two events are observed to occur in
the same spatial location, but there is no reference frame in which the two
events can occur at the same time.
The measure of a time-like spacetime interval is described by the proper time, :
(proper time).
The proper time interval would be measured by an observer with a clock traveling
between the two events in an inertial reference frame, when the observer's path
intersects each event as that event occurs. (The proper time defines a real
number, since the interior of the square root is positive.)
Light-like interval
In a light-like interval, the spatial distance between two events is exactly
balanced by the time between the two events. The events define a squared
spacetime interval of zero (). Light-like intervals are also known as "null"
intervals.
Events which occur to or are initiated by a photon along its path (i.e., while
traveling at , the speed of light) all have light-like separation. Given one
event, all those events which follow at light-like intervals define the
propagation of a light cone, and all the events which preceded from a light-like
interval define a second (graphically inverted, which is to say "pastward")
light cone.
Space-like interval
When a space-like interval separates two events, not enough time passes between
their occurrences for there to exist a causal relationship crossing the spatial
distance between the two events at the speed of light or slower. Generally, the
events are considered not to occur in each other's future or past. There exists
a reference frame such that the two events are observed to occur at the same
time, but there is no reference frame in which the two events can occur in the
same spatial location.
For these space-like event pairs with a positive squared spacetime interval (),
the measurement of space-like separation is the proper distance, :
(proper distance).
Like the proper time of time-like intervals, the proper distance of space-like
spacetime intervals is a real number value.
Mathematics of spacetimes
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional,
smooth, connected Lorentzian manifold . This means the smooth Lorentz metric has
signature . The metric determines the geometry of spacetime, as well as
determining the geodesics of particles and light beams. About each point (event)
on this manifold, coordinate charts are used to represent observers in reference
frames. Usually, Cartesian coordinates are used. Moreover, for simplicity's sake,
the speed of light 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 . Another reference frame may
be identified by a second coordinate chart about . Two observers (one in each
reference frame) may describe the same event but obtain different descriptions.
Usually, many overlapping coordinate charts are needed to cover a manifold.
Given two coordinate charts, one containing (representing an observer) and
another containing (representing 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 ). In general, they will disagree about the exact location and timing of
this impact, i.e., they will have different 4-tuples (as they are using
different coordinate systems). Although their kinematic descriptions will differ,
dynamical (physical) laws, such as momentum conservation and the first law of
thermodynamics, will still hold. In fact, relativity theory requires more than
this in the sense that it stipulates these (and all other physical) laws must
take the same form in all coordinate systems. This introduces tensors into
relativity, by which all physical quantities are represented.
Geodesics are said to be time-like, null, or space-like if the tangent vector to
one point of the geodesic is of this nature. Paths of particles and light beams
in spacetime are represented by time-like 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 properties 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. (Proof idea: the existence of a Lorentzian metric is shown
to be equivalent to the existence of a nonvanishing vector field.)
Any non-compact 4-manifold can be turned into a spacetime.
Spacetime symmetries
Often in relativity, spacetimes that have some form of symmetry are studied. As
well as helping to classify spacetimes, these symmetries usually serve as a
simplifying assumption in specialized work. Some of the most popular ones
include:
Axisymmetric spacetimes
Spherically symmetric spacetimes
Static spacetimes
Stationary spacetimes
Causal structure
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
Minkowski space
The geometry of spacetime in special relativity is described by the Minkowski
metric on R4. This spacetime is called Minkowski space. The Minkowski metric is
usually denoted by and can be written as a four-by-four matrix:
where the Landau–Lifshitz space-like convention is being used. A basic
assumption of relativity is that coordinate transformations must leave spacetime
intervals invariant. Intervals are invariant under Lorentz transformations. This
invariance property leads to the use of four-vectors (and other tensors) in
describing physics.
Strictly speaking, one can also consider events in Newtonian physics as a single
spacetime. This is Galilean–Newtonian relativity, and the coordinate systems are
related by Galilean transformations. However, since these preserve spatial and
temporal distances independently, such a spacetime can be decomposed into
spatial coordinates plus temporal coordinates, which is not possible in the
general case.
Spacetime in general relativity
Spacetime in General relativity General relativity
Introduction Mathematical formulation
Resources · Tests Fundamental concepts
Special relativity Equivalence principle
World line · Riemannian geometry Phenomena
Kepler problem · Lenses · Waves Frame-dragging · Geodetic effect
Event horizon · Singularity Black hole
Equations
Linearized gravity Post-Newtonian formalism
Einstein field equations Geodesic equation
Mathisson–Papapetrou–Dixon equations Friedmann equations
ADM formalism BSSN formalism
Hamilton–Jacobi–Einstein equation Advanced theories
Kaluza–Klein Quantum gravity
Solutions
Schwarzschild Reissner–Nordström · Gödel
Kerr · Kerr–Newman Kasner · Taub-NUT · Milne · Robertson–Walker
pp-wave · van Stockum dust Scientists
Einstein · Lorentz · Hilbert · Poincaré · Schwarzschild · Sitter · Reissner · Nordström
· Weyl · Eddington · Friedman · Milne · Zwicky · Lemaître · Gödel · Wheeler
· Robertson · Bardeen · Walker · Kerr · Chandrasekhar
· Ehlers · Penrose · Hawking · Taylor · Hulse · Stockum · Taub · Newman · Yau
· Thorne others
Spacetime
Spacetime Minkowski spacetime
Spacetime diagrams Spacetime in General relativity
v · t · e ·
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.
The earlier discussed notions of time-like, light-like and space-like intervals
in special relativity can similarly be used to classify one-dimensional curves
through curved spacetime. A time-like curve can be understood as one where the
interval between any two infinitesimally close events on the curve is time-like,
and likewise for light-like and space-like curves. Technically the three types
of curves are usually defined in terms of whether the tangent vector at each
point on the curve is time-like, light-like or space-like. The world line of a
slower-than-light object will always be a time-like curve, the world line of a
massless particle such as a photon will be a light-like curve, and a space-like
curve could be the world line of a hypothetical tachyon. In the local
neighborhood of any event, time-like curves that pass through the event will
remain inside that event's past and future light cones, light-like curves that
pass through the event will be on the surface of the light cones, and space-like
curves that pass through the event will be outside the light cones. One can also
define the notion of a three-dimensional "spacelike hypersurface", a continuous
three-dimensional "slice" through the four-dimensional property with the
property that every curve that is contained entirely within this hypersurface is
a space-like curve.
Many spacetime continua have physical interpretations which most physicists
would consider bizarre or unsettling. For example, a compact spacetime has
closed timelike 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 spacetime
In general relativity, spacetime is assumed to be smooth and continuous—and not
just in the mathematical sense. In the theory of quantum mechanics, there is an
inherent discreteness present in physics. In attempting to reconcile these two
theories, it is sometimes postulated that spacetime should be quantized at the
very smallest scales. Current theory is focused on the nature of spacetime at
the Planck scale. Causal sets, loop quantum gravity, string theory, and black
hole thermodynamics all predict a quantized spacetime 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
There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional).
Let the number of spatial dimensions be N and the number of temporal dimensions
be T. That N = 3 and T = 1, setting aside the compactified dimensions invoked by
string theory and undetectable to date, can be explained by appealing to the
physical consequences of letting N differ from 3 and T differ from 1. The
argument is often of an anthropic character.
The implicit notion that the dimensionality of the universe is special is first
attributed to Gottfried Wilhelm Leibniz, who in the Discourse on Metaphysics
suggested that the world is "the one which is at the same time the simplest
in hypothesis and the richest in phenomena." 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 that it "...gets
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: 204). This is because the law of gravitation (or any other inverse-square
law) follows from the concept of flux and the proportional relationship of flux
density and the strength of field. If N = 3, then 3-dimensional solid objects
have surface areas proportional to the square of their size in any selected
spatial dimension. In particular, a sphere of radius r has area of 4πr ². 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.
In 1920, Paul Ehrenfest showed that if we fix T = 1 and let N > 3, the orbit of
a planet about its sun cannot remain stable. The same is true of a star's orbit
around the center of its galaxy. Ehrenfest also showed that if N is even,
then the different parts of a wave impulse will travel at different speeds. If N
> 3 and odd, then wave impulses become distorted. Only when N = 3 or 1 are both
problems avoided. In 1922, Hermann Weyl showed that Maxwell's theory of
electromagnetism works only when N = 3 and T = 1, writing that this fact "not
only leads to a deeper understanding of Maxwell's theory, but also of the fact
that the world is four dimensional, which has hitherto always been accepted as
merely 'accidental,' become intelligible through it". Finally, Tangherlini
showed in 1963 that when N > 3, electron orbitals around nuclei cannot be stable;
electrons would either fall into the nucleus or disperse.
Properties of n+m-dimensional spacetimes
Max Tegmark expands on the preceding argument in the following anthropic manner.
If T differs 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.
Moreover, if T > 1, Tegmark maintains that protons and electrons would be
unstable and could decay into particles having greater mass than themselves. (This
is not a problem if the particles have a sufficiently low temperature.) If N > 3,
Ehrenfest's argument above 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, when N < 3, nerves cannot cross without intersecting.
In general, it is not clear how physical law could function if T differed from 1.
If T > 1, subatomic particles which decay after a fixed period would not behave
predictably, because time-like geodesics would not be necessarily maximal. N
= 1 and T = 3 has the peculiar property that the speed of light in a vacuum is a
lower bound on the velocity of matter; all matter consists of tachyons.
However, signature (1,3) and (3,1) are physically equivalent. To call vectors
with positive Minkowski "length" timelike is just a convention that depends on
the convention for the sign of the metric tensor. Indeed, particle phyicists
tend to use a metric with signature (+−−−) that results in positive Minkowski "length"
for timelike intervals and energies while spatial separations have negative
Minkowski "length". Relativists, however, tend to use the opposite convention (−+++)
so that spatial separations have positive Minkowski length.
Hence anthropic and other arguments rule out all cases except N = 3 and T = 1 (or
N = 1 and T = 3 in different conventions)—which happens to describe the world
about us. Curiously, the cases N = 3 or 4 have the richest and most difficult
geometry and topology. There are, for example, geometric statements whose truth
or falsity is known for all N except one or both of 3 and 4. N
= 3 was the last case of the Poincaré conjecture to be proved.
For an elementary treatment of the privileged status of N = 3 and T = 1, see
chpt. 10 (esp. Fig. 10.12) of Barrow; for deeper treatments, see §4.8 of
Barrow and Tipler (1986) and Tegmark. Barrow has repeatedly cited the work
of Whitrow.
String theory hypothesizes that matter and energy are composed of tiny,
vibrating strings of various types, most of which are embedded in dimensions
that exist only on a scale no larger than the Planck length. Hence N = 3 and T =
1 do not characterize string theory, which embeds vibrating strings in
coordinate grids having 10, or even 26, dimensions.
The Causal dynamical triangulation (CDT) theory is a background independent
theory which derives the observed 3+1 spacetime from a minimal set of
assumptions, and needs no adjusting factors. It does not assume any pre-existing
arena (dimensional space), but rather attempts to show how the spacetime fabric
itself evolves. It shows spacetime to be two-dimensional near the Planck scale,
and reveals a fractal structure on slices of constant time, but spacetime
becomes 3+1-d in scales significantly larger than Planck. So, CDT may become the
first theory which does not postulate but really explains observed number of
spacetime dimensions.