Spacetime

Introduction

Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, or indeed of anything external. Furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense.

In the context of special relativity, 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. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.

Mathematically, spacetime is a "manifold", which is to say, it is a "topological space" that locally resembles Euclidean space near each point. By analogy, at small enough scales, a globe appears flat. An extremely large scale factor, c (conventionally called the "speed of light") relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly 300000 km in space being equivalent to 1 s in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe which is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelson–Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space.

Einstein's theory of special relativity, introduced in 1905, provided a complete resolution to these and other major physics puzzles, and it made startling predictions that have since been repeatedly confirmed. Einstein performed his analyses in terms of kinematics (the study of moving bodies), and it would appear that he did not at first think geometrically about spacetime. It was Einstein's former mathematics professor, Hermann Minkowski, who was to provide a geometric interpretation of special relativity. The geometric interpretation of relativity was to prove vital to Einstein's later development of general relativity, and the spacetime of special relativity has since come to be known as Minkowski spacetime.

Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two "events" (because of time dilation) or the distance between the two events (because of length contraction). But special relativity provides a new invariant, called the "spacetime interval", which combines distances in space and in time. All observers who measure time and distance carefully will find the same spacetime interval between any two events. Suppose an observer measures two events as being separated by a time t and a spatial distance x . Then the spacetime interval between the two events is given by

s 2 = x 2 − c 2 t 2 .

The above is simply an expression of the Pythagorean theorem, except with a minus sign between the x 2 and the c 2 t 2 terms. Note also that the spacetime interval is the quantity s 2 , not s itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard s 2 as a distinct symbol in itself, rather than the square of something.

Because of the minus sign, the spacetime interval between two events can be zero. Spacetime intervals are zero when x = ± c t . In other words, the spacetime interval is zero for something moving at the speed of light. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage. Fig. 1 presents a spacetime diagram of two photons originating from the same event and going in opposite directions. The vertical time coordinate is scaled by c so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their "world lines" (i.e. their paths in spacetime) have a slope of 1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.

In Fig. 2, event O is centered at the origin of the spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called the "light cone", since as illustrated in Fig. 3, if another dimension is added to the diagram, the appearance would be that of two cones meeting with their apexes at the origin and centered around the time axis, one cone extending into the future, the other into the past. Since the spacetime interval is an invariant, all observers will assign the same events to the light cone of a given event. This statement is equivalent to the second postulate of special relativity: all observers measure the speed of light to be c .

The light cone divides spacetime into separate regions. The interior of the future light cone consists of events that are separated from the origin event by more time than there is space: these events comprise the "timelike future" of the origin event. Likewise, the "timelike past" comprises the interior events of the past light cone. Since in timelike intervals, the separation in time is greater than the separation in space, timelike intervals are negative. The region exterior to the light cone consists of events that are separated from the origin event by more space than there is time. These events comprise the spacelike "Elsewhere" of the origin event. Because of the invariance of spacetime, all observers will agree on this division of spacetime.

The light cone has an essential role in defining the concept of causality. Signals cannot travel faster than the speed of light. In Fig. 2, it is possible for a slower-than-or-equal-to-light-speed signal to travel from the position and time of O, to the position and time of D. It is hence possible for event O to have a causal influence on event D. The future light cone contains all of the events that could be causally influenced by O. Likewise, it is possible for a slower-than-or-equal-to-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all of the events that could have a causal influence on O. In contrast, in the Elsewhere region, event O cannot affect or be affected by event C, nor can event O affect or be affected by event B. There is no causal relationship between O and any events in the Elsewhere region.

All observers will agree that for any given event, an event within the given event's future light cone occurs after the given event. Likewise, for any given event, an event within the given event's past light cone occurs before the given event. The before-after relationship observed for timelike-separated events remains unchanged no matter what the reference frame of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. Fig. 2 was drawn from the reference frame of an observer moving at v = 0 . From this reference frame, event C is observed to occur after event O, and event B is observed to occur before event O. From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are necessarily separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the relativity of simultaneity.

Fig. 4 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. (It is beyond the scope of this introduction to explain how to calculate the transformed axes of the spacetime diagram for different reference frames.) The three events (A, B, C) are simultaneous from the reference frame of an observer moving at v = 0 . From the reference frame of an observer moving at v = 0.3 c , the events appear to occur in the order C, B, A. From the reference frame of an observer moving at v = − 0.5 c , the events appear to occur in the order A, B, C. The white line represents a plane of simultaneity being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant.

The spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. The spacetime interval hence provides a measure of proper distance, i.e. the true distance in spacetime. Likewise, the spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. The spacetime interval hence provides a measure of proper time.

In ordinary Euclidean space, the set of points that are equidistant from an origin form a circle (in two dimensions) or a sphere (in three dimensions). In Minkowski spacetime, the points at some constant spacetime interval from the origin form a curve given by the equation

x 2 − c 2 t 2 = s o m e   c o n s t a n t

The above is the equation of a hyperbola in an x–ct spacetime diagram, and is termed the invariant hyperbola. (In a spacetime diagram of x–y–ct, the equivalent equation would define a hyperboloid.)

Fig. 5 illustrates the invariant hyperbola for all points that can be reached from the origin in a proper time of 5 meters (approximately 1.67×10⁻⁸s). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3c, the elapsed time measured by the observer is 5.24 meters (1.75×10⁻⁸s), while for a clock traveling at 0.7c, the elapsed time measured by the observer is 7.00 meters (2.34×10⁻⁸s). This illustrates the phenomenon known as "time dilation". Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis than they would have without time dilation.

This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena. A few of these phenomena are described in the later sections of this article.

In myth and 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.

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".

In math and physics

In Encyclopedie, published in 1754, under the term dimension Jean le Rond d'Alembert speculated that duration (time) might be considered a fourth dimension if the idea was not too novel.

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".

The ancient idea of the cosmos gradually was described mathematically with differential equations, differential geometry, and abstract algebra. These mathematical articulations blossomed in the nineteenth century as electrical technology stimulated men like Michael Faraday and James Clerk Maxwell to describe the reciprocal relations of electric and magnetic fields. Daniel Siegel phrased Maxwell's role in relativity as follows:

[...] the idea of the propagation of forces at the velocity of light through the electromagnetic field as described by Maxwell's equations—rather than instantaneously at a distance—formed the necessary basis for relativity theory.

Maxwell used vortex models in his papers on On Physical Lines of Force, but ultimately gave up on any substance but the electromagnetic field. Pierre Duhem wrote:

[Maxwell] was not able to create the theory that he envisaged except by giving up the use of any model, and by extending by means of analogy the abstract system of electrodynamics to displacement currents.

In Siegel's estimation, "this very abstract view of the electromagnetic fields, involving no visualizable picture of what is going on out there in the field, is Maxwell's legacy."

Describing the behaviour of electric fields and magnetic fields led Maxwell to view the combination as an electromagnetic field. These fields have a value at every point of spacetime. It is the intermingling of electric and magnetic manifestations, described by Maxwell's equations, that give spacetime its structure. In particular, the rate of motion of an observer determines the electric and magnetic profiles of the electromagnetic field. The propagation of the field is determined by the electromagnetic wave equation, which requires spacetime for description.

Various authors have credited W. K. Clifford with having anticipated the spacetime concept in general relativity as far back as 1870, although the actual extent to which Clifford anticipated spacetime is debatable: in an 1876 publication, Clifford speculated that "curvature of space is what really happens in that phenomenon which we call the motion of matter", i.e. he speculated on the possibility of curved space, not curved spacetime.

Basic concepts

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 ( x , y , z , t ) , 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 coordinate systems.

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 ( x 0 , x 1 , x 2 , x 3 ) = ( c t , x , y , z ) (in the Lorentz metric) or ( x 1 , x 2 , x 3 , x 4 ) = ( x , y , z , i c t ) (in the original Minkowski metric) where c 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 flat space

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 displacement four-vector ΔR is given by the space displacement vector Δr and the time difference Δt between the events. The spacetime interval, also called invariant interval, between the two events, s², is defined as:

s 2 = Δ r 2 − c 2 Δ t 2    (spacetime interval),

where c is the speed of light. The choice of signs for s 2 above follows the space-like convention (−+++). Spacetime intervals may be classified into three distinct types, based on whether the temporal separation ( c 2 Δ t 2 ) is greater than, equal to, or smaller than the spatial separation ( Δ r 2 ), corresponding to time-like, light-like, or space-like separated intervals, respectively.

Certain types of world lines are called geodesics of the spacetime – straight lines in the case of Minkowski space and their closest equivalent in the curved spacetime of general relativity. In the case of purely time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path between two events, whereas in Euclidean space and Riemannian manifolds, geodesics are paths of shortest distance between two points. The concept of geodesics becomes central in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in spacetime, that is, free from any external influences.

Time-like interval

c 2 Δ t 2 > Δ r 2 s 2 < 0

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 spacetime interval ( s 2 < 0 ) 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 interval, Δ τ :

Δ τ = Δ t 2 − Δ r 2 c 2    (proper time interval).

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 interval defines a real number, since the interior of the square root is positive.)

Light-like interval

c 2 Δ t 2 = Δ r 2 s 2 = 0

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 spacetime interval of zero ( s 2 = 0 ). 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 c , 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

c 2 Δ t 2 < Δ r 2 s 2 > 0

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 spacetime interval ( s 2 > 0 ), the measurement of space-like separation is the proper distance, Δ σ :

Δ σ = s 2 = Δ r 2 − c 2 Δ t 2    (proper distance).

Like the proper time of time-like intervals, the proper distance of space-like spacetime intervals is a real number value.

Interval as area

The interval has been presented as the area of an oriented rectangle formed by two events and isotropic lines through them. Time-like or space-like separations correspond to oppositely oriented rectangles, one type considered to have rectangles of negative area. The case of two events separated by light corresponds to the rectangle degenerating to the segment between the events and zero area. The transformations leaving interval-length invariant are the area-preserving squeeze mappings.

The parameters traditionally used rely on quadrature of the hyperbola, which is the natural logarithm. This transcendental function is essential in mathematical analysis as its inverse unites circular functions and hyperbolic functions: The exponential function, e^t,  t a real number, used in the hyperbola (e^t, e^–t ), generates hyperbolic sectors and the hyperbolic angle parameter. The functions cosh and sinh, used with rapidity as hyperbolic angle, provide the common representation of squeeze in the form ( cosh ⁡ ϕ sinh ⁡ ϕ sinh ⁡ ϕ cosh ⁡ ϕ ) , or as the split-complex unit e j ϕ = cosh ⁡ ϕ   + j   sinh ⁡ ϕ .

Generalized spacetime (extra dimensions)

The term spacetime has taken on a generalized meaning beyond treating spacetime events with the normal 3+1 dimensions. Other proposed spacetime theories include additional dimensions—normally spatial but there exist some theories that include additional temporal dimensions, and even some, such as superspace of supersymmetric theories, that include dimensions that are neither temporal nor spatial.

The number of dimensions required to describe the universe is still an open question. Theories such as string theory predict 10 or 26 dimensions (with M-theory predicting 11 dimensions: 10 spatial and 1 temporal). The existence of more than four dimensions only appears to make a difference at subatomic scales.

Mathematics of spacetimes

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold ( M , g ) . This means the smooth Lorentz metric g has signature ( 3 , 1 ) . 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 ( x , y , z , t ) are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light c is equal to 1.

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 (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 p ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples ( x , y , z , t ) (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

The causal structure of a spacetime describes causal relationships between pairs of points in the spacetime based on the existence of certain types of curves joining the points.

Spacetime in special relativity

The geometry of spacetime in special relativity is described by the Minkowski metric on R⁴. This spacetime is called Minkowski space. The Minkowski metric is usually denoted by η and can be written as a four-by-four matrix:

η a b = diag ⁡ ( 1 , − 1 , − 1 , − 1 )

where the Landau–Lifshitz time-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 always be decomposed into spatial coordinates plus temporal coordinates, which is not possible for general spacetimes.

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.

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, causal dynamical triangulation, 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.

Spin networks provide a language to describe quantum geometry of space. Spin foam does the same job on spacetime. A spin network is a one-dimensional graph, together with labels on its vertices and edges which encodes aspects of a spatial geometry.