Originally introduced by Sir Isaac Newton in Philosophiæ Naturalis Principia Mathematica, the concepts of absolute time and space provided a theoretical foundation that facilitated Newtonian mechanics. According to Newton, absolute time and space respectively are independent aspects of objective reality:^[1]

Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ...

According to Newton, absolute time exists independently of any perceiver and progresses at a consistent pace throughout the universe. Unlike relative time, Newton believed absolute time was imperceptible and could only be understood mathematically. According to Newton, humans are only capable of perceiving relative time, which is a measurement of perceivable objects in motion (like the moon or sun). From these movements, we infer the passage of time.

To quote Newton again:

Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space ... Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another

These notions imply that absolute space and time do not depend upon physical events, but are a backdrop or stage setting within which physical phenomena occur. Thus, every object has an absolute state of motion relative to absolute space, so that an object must be either in a state of absolute rest, or moving at some absolute speed.^[2] To support his views, Newton provided some empirical examples: according to Newton, a solitary rotating sphere can be inferred to rotate about its axis relative to absolute space by observing the bulging of its equator, and a solitary pair of spheres tied by a rope can be inferred to be in absolute rotation about their center of gravity (barycenter) by observing the tension in the rope.

Absolute time and space continue to be used in classical mechanics, but modern formulations by authors such as Walter Noll and Clifford Truesdell go beyond the linear algebra of elastic moduli to use topology and functional analysis for non-linear field theories.^[3]

Two bodies orbiting around a common barycenter. Supposing the bodies tied by a rope to hold them together (rather than gravity, which is ignored in this example), the rope is under tension if the bodies are rotating relative to absolute space (according to Newton), or because they rotate relative to the Universe itself (according to Mach), or because they rotate relative to an inertial frame of reference according to modern ideas.

Historical controversy[link]

The idea of absolute space has proved particularly controversial from Newton's time to the present. For example, Leibniz was of the opinion that space made no sense except as the relative location of bodies, and time made no sense except as the relative movement of bodies.^[4] Bishop Berkeley suggested that, lacking any point of reference, a sphere in an otherwise empty Universe could not be conceived to rotate, and a pair of spheres could be conceived to rotate relative to one another, but not to rotate about their center of gravity.^[5] A more recent form of these objections was made by Ernst Mach. Mach's principle proposes that mechanics is entirely about relative motion of bodies and, in particular, mass is an expression of such relative motion. So, for example, a single particle in a Universe with no other bodies would have zero mass. According to Mach, Newton's examples simply illustrate relative rotation of spheres and the bulk of the Universe.^[6]

When, accordingly, we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.
—Ernst Mach; as quoted by Ciufolini and Wheeler: Gravitation and Inertia, p. 387

These views opposing absolute space and time may be seen from a modern stance as an attempt to introduce operational definitions for space and time, a perspective made explicit in the special theory of relativity.

Even within the context of Newtonian mechanics, the modern view is that absolute space is unnecessary. Instead, the notion of inertial frame of reference has taken precedence, that is, a preferred set of frames of reference that move uniformly with respect to one another. The laws of physics transform from one inertial frame to another according to Galilean relativity, leading to the following objections to absolute space, as outlined by Milutin Blagojević:^[7]

• The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out.
• Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.
• Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.

Newton himself recognized the role of inertial frames.^[8]

The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.

As a practical matter, inertial frames often are taken as frames moving uniformly with respect to the fixed stars.^[9] See Inertial frame of reference for more discussion on this.

In 1903 Bertrand Russell wrote a defense of absolute space and time in Principles of Mathematics, while admitting (page 465) that in the analysis of rational dynamics "non-Newtonian dynamics, like non-Euclidean geometry, must be as interesting to us as the orthodox system."

Impact of special relativity[link]

The concepts of space and time were separate in physical theory prior to the advent of special relativity theory, which connected the two and showed both to be dependent upon the observer's state of motion. In Einstein's theories, the ideas of absolute time and space were superseded by the notion of spacetime in special relativity, and by dynamically curved spacetime in general relativity.

The theory of relativity does not allow the existence of absolute time because of nonexistence of absolute simultaneity. Absolute simultaneity refers to the experimental establishment of coincidence of two or more events in time at different locations in space in a manner agreed upon by all observers in the universe. The theory of relativity postulates a maximum rate of transmission of information as the speed of light, and one consequence is that simultaneity at separated locations always is relative to the observer.^[10]

Einstein's later views[link]

Further information: Luminiferous Aether#Einstein's views on the aether

In some of his later papers (especially in 1920 and 1924), Einstein gave a new definition of the aether by identifying it with "properties of space", and this aether can be called absolute, as long its state cannot be influenced by matter. So he argued that Newton's absolute space can be considered as an "absolute aether", and also the four-dimensional spacetime of special relativity (which replaces the absolute space of Newton) would be some sort of "absolute aether", as its states cannot be influenced by matter as well. However, Einstein also said that in general relativity the "aether" is not absolute anymore, as the gravitational field and therefore the structure of spacetime depends on the presence of matter. (It also must be said that Einstein's terminology (i.e. aether = properties of space) was not accepted by the scientific community.)^[11]

1920: To deny the ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view. For the mechanical behaviour of a corporeal system hovering freely in empty space depends not only on relative positions (distances) and relative velocities, but also on its state of rotation, which physically may be taken as a characteristic not appertaining to the system in itself. In order to be able to look upon the rotation of the system, at least formally, as something real, Newton objectivises space. Since he classes his absolute space together with real things, for him rotation relative to an absolute space is also something real. Newton might no less well have called his absolute space “Ether”; what is essential is merely that besides observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real.^[12]

1924: Because it was no longer possible to speak, in any absolute sense, of simultaneous states at different locations in the aether, the aether became, as it were, four dimensional, since there was no objective way of ordering its states by time alone. According to special relativity too, the aether was absolute, since its influence on inertia and the propogation of light was thought of as being itself independent of physical influence....The theory of relativity resolved this problem by establishing the behaviour of the electrically neutral point-mass by the law of the geodetic line, according to which inertial and gravitational effects are no longer considered as separate. In doing so, it attached characteristics to the aether which vary from point to point, determining the metric and the dynamic behaviour of material points, and determined, in their turn, by physical factors, namely the distribution of mass/energy. Thus the aether of general relativity differs from those of classical mechanics and special relativity in that it is not ‘absolute’ but determined, in its locally variable characteristics, by ponderable matter.^[13]

See also[link]

Astronomy portal

• Absolute rotation
• Free space
• Inertial frame
• Luminiferous aether
• Mach's principle
• Philosophy of space and time

References and notes[link]

v t e Time Major concepts Time Eternity Arguments for eternity Immortality History Past Present Future Futures studies Deep time Time Portal Measurement and standards Chronometry UTC UT TAI Second Minute Hour Sidereal time Solar time Time zone Metric time Decimal time Hexadecimal time Clock Astrarium History of timekeeping devices Horology Marine chronometer Sundial Water clock Calendar Day Week Month Year Tropical year Gregorian Islamic Julian Intercalation Leap second Leap year Chronology Astronomical chronology Calendar era Chronicle Dating methodologies Geochronology Geologic Time Geological history Periodization Regnal year Timeline Religion and mythology Dreamtime Kāla Kalachakra Prophecy Time and fate deities Wheel of time Philosophy A-series and B-series B-Theory of time Causality Endurantism Eternal return Eternalism Event Perdurantism Presentism Temporal finitism Temporal parts The Unreality of Time Physical sciences Time in physics Absolute time and space Arrow of time Chronon Coordinate time Planck epoch Planck time Proper time Spacetime Theory of relativity Time dilation Gravitational time dilation Time domain T-symmetry Biology Chronobiology Circadian rhythms Psychology Mental chronometry Sense of time Specious present Sociology and anthropology Long Now Foundation Time discipline Time use research Economics Time value of money Time-based currency Time Banking Related topics Carpe diem Duration Space System time Tempus fugit Time capsule Time signature Time travel

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Two-dimensional analogy of spacetime distortion. Matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity. 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 (or space-time, space time, space-time continuum) is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. From a Euclidean space perspective, 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 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 slowed down at higher speeds of the reference frame relative to another reference frame (with such slowing called "time dilation" explained in the theory of "special relativity"). Many experiments have confirmed time dilation, such as atomic clocks onboard a Space Shuttle running slower than synchronized Earth-bound inertial clocks and the relativistic decay of muons from cosmic ray showers. The duration of time can therefore vary for various events and various 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. 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[link]

Incas regarded space and time as a single concept, named pacha (Quechua: pacha, Aymara: pacha).^[1][2] The peoples of the Andes have kept this understanding until now.^[3]

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."

Mathematical concept[link]

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".^[4]

After discovering quaternions,^[5] 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 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 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^[6] 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 Encyclopædia Britannica included an article by Einstein titled "Space-Time".^[7]

Basic concepts[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): (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 coordinates systems.

A spacetime is independent of any observer.^[8] 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.^[9]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): (x_0,x_1,x_2,x_3) = (ct,x,y,z)

(in the Lorentz metric) or Failed to parse (Missing texvc executable; please see math/README to configure.): (x_1,x_2,x_3,x_4) = (x,y,z,ict)
(in the original Minkowski metric)[10] where Failed to parse (Missing texvc executable; please see math/README to configure.): 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[link]

In a Euclidean space, the separation between two points is measured by the distance between the two points. A 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 between two events is defined as:

where c is the speed of light, and Δr and Δt denote differences of the space and time coordinates, respectively, between the events.

(Note that the choice of signs for Failed to parse (Missing texvc executable; please see math/README to configure.): s^2

above follows the space-like convention (-+++). Other treatments reverse the sign of Failed to parse (Missing texvc executable; please see math/README to configure.): s^2

.)

Spacetime intervals may be classified into three distinct types based on whether the temporal separation (Failed to parse (Missing texvc executable; please see math/README to configure.): c^2 \Delta t^2 ) or the spatial separation (Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta r^2 ) of the two events is greater.

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.

Time-like interval[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \\ c^2\Delta t^2 &> \Delta r^2 \\ s^2 &< 0 \\ \end{align}

For two events separated by a time-like interval, enough time passes between them for there to 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 (Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

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[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} c^2\Delta t^2 &= \Delta r^2 \\ s^2 &= 0 \\ \end{align}

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 (Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \\ c^2\Delta t^2 &< \Delta r^2 \\ s^2 &> 0 \\ \end{align}

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 (Failed to parse (Missing texvc executable; please see math/README to configure.): s^2 > 0 ), the measurement of space-like separation is the proper distance:

Like the proper time of time-like intervals, the proper distance (Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta\sigma ) of space-like spacetime intervals is a real number value.

Mathematics of spacetimes[link]

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold Failed to parse (Missing texvc executable; please see math/README to configure.): (M,g) . This means the smooth Lorentz metric Failed to parse (Missing texvc executable; please see math/README to configure.): g

has signature Failed to parse (Missing texvc executable; please see math/README to configure.): (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 Failed to parse (Missing texvc executable; please see math/README to configure.): (x, y, z, t)

are used. Moreover, for simplicity's sake, the speed of light Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): p . Another reference frame may be identified by a second coordinate chart about Failed to parse (Missing texvc executable; please see math/README to configure.): p . Two observers (one in each reference frame) may describe the same event Failed to parse (Missing texvc executable; please see math/README to configure.): p

but obtain different descriptions.

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing Failed to parse (Missing texvc executable; please see math/README to configure.): p

(representing an observer) and another containing Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): p ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples Failed to parse (Missing texvc executable; please see math/README to configure.): (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[link]

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:^{[citation needed]}

• 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[link]

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[link]

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[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \eta

and can be written as a four-by-four matrix:

Failed to parse (Missing texvc executable; please see math/README to configure.): \eta_{ab} \, = \operatorname{diag}(1, -1, -1, -1)

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[link]

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 3-dimensional "spacelike hypersurface", a continuous 3-dimensional "slice" through the 4-dimensional property with the property that every curve that is contained entirely within this hypersurface is a space-like curve.^[11]

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[link]

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[link]

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.

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 r^N−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.^[12] 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."^[13] Finally, Tangherlini^[14] 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^[15] 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.^[why?] (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.^[16] 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.^[15]

Hence anthropic and other arguments rule out all cases except N = 3 and T = 1—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.^{[citation needed]} 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;^[17] for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark.^[15] Barrow has repeatedly cited the work of Whitrow.^[18]

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 2-d 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 doesn't postulate but really explains observed number of spacetime dimensions.^[19]

See also[link]

	Time portal
	Space portal

Notes[link]

References[link]

• ^{[Edit this reference]}
  Barrow, John D.; Tipler, Frank J. (19 May 1988). The Anthropic Cosmological Principle. foreword by John A. Wheeler. Oxford: Oxford University Press. ISBN 9780192821478. LC 87-28148. http://books.google.com/books?id=uSykSbXklWEC&printsec=frontcover. Retrieved 31 December 2009. 
• Ehrenfest, Paul (1920) "How do the fundamental laws of physics make manifest that Space has 3 dimensions?" Annalen der Physik 61: 440.
• George F. Ellis and Ruth M. Williams (1992) Flat and curved space-times. Oxford Univ. Press. ISBN 0-19-851164-7
• Isenberg, J. A. (1981). "Wheeler-Einstein-Mach spacetimes". Phys. Rev. D 24 (2): 251–256. Bibcode 1981PhRvD..24..251I. DOI:10.1103/PhysRevD.24.251. 
• Kant, Immanuel (1929) "Thoughts on the true estimation of living forces" in J. Handyside, trans., Kant's Inaugural Dissertation and Early Writings on Space. Univ. of Chicago Press.
• Lorentz, H. A., Einstein, Albert, Minkowski, Hermann, and Weyl, Hermann (1952) The Principle of Relativity: A Collection of Original Memoirs. Dover.
• Lucas, John Randolph (1973) A Treatise on Time and Space. London: Methuen.
• Penrose, Roger (2004). The Road to Reality. Oxford: Oxford University Press. ISBN 0-679-45443-8.  Chpts. 17–18.
• Poe, Edgar A. (1848). Eureka; An Essay on the Material and Spiritual Universe. Hesperus Press Limited. ISBN 1-84391-009-8. 
• Robb, A. A. (1936). Geometry of Time and Space. University Press. 
• Erwin Schrödinger (1950) Space-time structure. Cambridge Univ. Press.
• Schutz, J. W. (1997). Independent axioms for Minkowski Space-time. Addison-Wesley Longman. ISBN 0-582-31760-6. 
• Tangherlini, F. R. (1963). "Atoms in Higher Dimensions". Nuovo Cimento 14 (27): 636. 
• Taylor, E. F.; Wheeler, John A. (1963). Spacetime Physics. W. H. Freeman. ISBN 0-7167-2327-1. 
• Wells, H.G. (2004). The Time Machine. New York: Pocket Books. ISBN 0-671-57554-6.  (pp. 5–6)

External links[link]

Wikibooks has a book on the topic of Special Relativity

• Stanford Encyclopedia of Philosophy: "Space and Time: Inertial Frames" by Robert DiSalle.

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Carl Sagan
Sagan in 1980
Born	(1934-11-09)November 9, 1934 Brooklyn, New York, U.S.
Died	December 20, 1996(1996-12-20) (aged 62) Seattle, Washington, U.S.
Residence	United States[1]
Nationality	United States
Fields	Astronomy, astrophysics, cosmology, astrobiology, space science, planetary science
Institutions	Cornell University Harvard University Smithsonian Astrophysical Observatory University of California, Berkeley
Alma mater	University of Chicago (B.A.), (B.Sc.), (M.Sc.), (PhD)
Known for	Search for Extra-Terrestrial Intelligence (SETI) Cosmos: A Personal Voyage Cosmos Voyager Golden Record Pioneer plaque Contact Pale Blue Dot
Notable awards	NASA Distinguished Public Service Medal (1977) Pulitzer Prize for General Non-Fiction (1978) Oersted Medal (1990) National Academy of Sciences Public Welfare Medal (1994)

Carl Edward Sagan ( /ˈseɪɡɪn/; November 9, 1934 – December 20, 1996) was an American astronomer, astrophysicist, cosmologist, author, science popularizer, and science communicator in astronomy and natural sciences. He published more than 600 scientific papers^[2] and articles and was author, co-author or editor of more than 20 books. He advocated scientifically skeptical inquiry and the scientific method, pioneered exobiology and promoted the Search for Extra-Terrestrial Intelligence (SETI).

Sagan is known for his popular science books and for the award-winning 1980 television series Cosmos: A Personal Voyage, which he narrated and co-wrote.^[3] The book Cosmos was published to accompany the series. Sagan wrote the novel Contact, the basis for a 1997 film of the same name.

Early life[link]

Carl Sagan was born in Brooklyn, New York,^[4] to a Ukrainian Jewish family. His father, Sam Sagan, was an immigrant garment worker from Kamenets-Podolsk, Ukraine;^[5] his mother, Rachel Molly Gruber, a housewife. Carl was named in honor of Rachel's biological mother, Chaiya Clara, in Sagan's words, "the mother she never knew." Sagan graduated from Rahway High School in Rahway, New Jersey, in 1951.^[6]

He had a sister, Carol, and the family lived in a modest apartment near the Atlantic Ocean, in Bensonhurst, a Brooklyn neighborhood. According to Sagan, they were Reform Jews, the most liberal of the three main Jewish groups. Both Sagan and his sister agree that their father was not especially religious, but that their mother "definitely believed in God, and was active in the temple ... and served only Kosher meat."^[6]:12 During the height of the Depression, his father had to accept a job as a theater usher.

According to biographer Keay Davidson, Sagan's "inner war" was a result of his close relations with both his parents, who were in many ways "opposites." Sagan traced his later analytical urges to his mother, a woman who had known "extreme poverty as a child," and had grown up almost homeless in New York City during World War I and the 1920s.^[6]:2 She had her own intellectual ambitions as a young woman, but they were blocked by social restrictions, because of her poverty, her being a woman and wife, and her Jewish ethnicity. Davidson notes that she therefore "worshiped her only son, Carl. He would fulfill her unfulfilled dreams."^[6]:2

However, his "sense of wonder" came from his father, who was a "quiet and soft-hearted escapee from the Czar." In his free time, he gave apples to the poor, or helped soothe labor-management tensions within New York's "tumultuous" garment industry.^[6]:2 Although he was "awed" by Carl's "brilliance, his boyish chatter about stars and dinosaurs," he took his son's inquisitiveness in stride, as part of his growing up.^[6]:2 In his later years as a writer and scientist, Sagan would often draw on his childhood memories to illustrate scientific points, as he did in his book, Shadows of Forgotten Ancestors.^[6]:9 Sagan describes his parents' influence on his later thinking:

My parents were not scientists. They knew almost nothing about science. But in introducing me simultaneously to skepticism and to wonder, they taught me the two uneasily cohabiting modes of thought that are central to the scientific method.^[7]

1939 World's Fair[link]

Sagan recalls that one of his best experiences was when he was four or five years old, his parents took him to the 1939 New York World's Fair. The exhibits became a turning point in his life. He later recalled the moving map of the "America of Tomorrow" exhibit: "It showed beautiful highways and cloverleaves and little General Motors cars all carrying people to skyscrapers, buildings with lovely spires, flying buttresses—and it looked great!"^[6]:14 At other exhibits, he remembered how a flashlight that shined on a photoelectric cell created a cracking sound, and how the sound from a tuning fork became a wave on an oscilloscope. He also witnessed the future media technology that would replace radio: television. Sagan wrote:

Plainly, the world held wonders of a kind I had never guessed. How could a tone become a picture and light become a noise?^[6]:14

He also saw one of the Fair's most publicized events, the burial of a time capsule at Flushing Meadows, which contained mementos of the 1930s to be recovered by Earth's descendants in a future millennium. "The time capsule thrilled Carl," writes Davidson. As an adult, Sagan and his colleagues created similar time capsules, but ones that would be sent out into the galaxy. These were the Pioneer plaque and the Voyager Golden Record records, all of which were spinoffs of Sagan's memories of the World Fair.^[6]:15

World War II[link]

During World War II, Sagan's family worried about the fate of their European relatives. Sagan, however, was generally unaware of the details of the ongoing war. He writes, "Sure, we had relatives who were caught up in the Holocaust. Hitler was not a popular fellow in our household ... But on the other hand, I was fairly insulated from the horrors of the war." His sister, Carol, said that their mother "above all wanted to protect Carl ... She had an extraordinarily difficult time dealing with World War II and the Holocaust."^[6]:15 Sagan's book, The Demon-Haunted World (1996), included his memories of this conflicted period, when his family dealt with the realities of the war in Europe, but tried to prevent it from undermining his optimistic spirit.^[7]

Inquisitiveness about nature[link]

Soon after entering elementary school, he began to express a strong inquisitiveness about nature. Sagan recalled taking his first trips to the public library alone, at the age of five, when his mother got him a library card. He wanted to learn what stars were, since none of his friends or their parents could give him a clear answer:

I went to the librarian and asked for a book about stars ... And the answer was stunning. It was that the Sun was a star but really close. The stars were suns, but so far away they were just little points of light ... The scale of the universe suddenly opened up to me. It was a kind of religious experience. There was a magnificence to it, a grandeur, a scale which has never left me. Never ever left me.^[6]:18

About the time he was six or seven, he and a close friend took trips to the American Museum of Natural History in New York City. While there, they went to the Hayden Planetarium and walked around the museum's exhibits of space objects, such as meteorites, and displays of dinosaurs and animals in natural settings. Sagan writes about those visits:

I was transfixed by the dioramas—lifelike representations of animals and their habitats all over the world. Penguins on the dimly lit Antarctic ice; ... a family of gorillas, the male beating his chest, ... an American grizzly bear standing on his hind legs, ten or twelve feet tall, and staring me right in the eye.^[6]:18

His parents helped nurture his growing interest in science by buying him chemistry sets and reading materials.^[8] His interest in space, however, was his primary focus, especially after reading science fiction stories by writers such as Edgar Rice Burroughs, which stirred his imagination about life on other planets, such as Mars. According to biographer Ray Spangenburg, these early years as Sagan tried to understand the mysteries of the planets, became a "driving force in his life, a continual spark to his intellect, and a quest that would never be forgotten."^[7]

Education and scientific career[link]

He attended the University of Chicago, where he participated in the Ryerson Astronomical Society,^[9] received a bachelor of arts in self-proclaimed "nothing" with general and special honors in 1954, a bachelor of science in physics in 1955, and a master of science in physics in 1956 before earning a PhD in astronomy and astrophysics in 1960.^[10][11][12] During his time as an honors program undergraduate, Sagan worked in the laboratory of the geneticist H. J. Muller and wrote a thesis on the origins of life with physical chemist H.C. Urey. From 1960 to 1962 Sagan was a Miller Fellow at the University of California, Berkeley. From 1962 to 1968, he worked at the Smithsonian Astrophysical Observatory in Cambridge, Massachusetts.

Sagan lectured and did research at Harvard University until 1968, when he moved to Cornell University in Ithaca, New York. He became a full Professor at Cornell in 1971, and he directed the Laboratory for Planetary Studies there. From 1972 to 1981, Sagan was the Associate Director of the Center for Radio Physics and Space Research at Cornell.

Sagan was associated with the US space program from its inception. From the 1950s onward, he worked as an advisor to NASA, where one of his duties included briefing the Apollo astronauts before their flights to the Moon. Sagan contributed to many of the robotic spacecraft missions that explored the solar system, arranging experiments on many of the expeditions. He conceived the idea of adding an unalterable and universal message on spacecraft destined to leave the solar system that could potentially be understood by any extraterrestrial intelligence that might find it. Sagan assembled the first physical message that was sent into space: a gold- anodized plaque, attached to the space probe Pioneer 10, launched in 1972. Pioneer 11, also carrying another copy of the plaque, was launched the following year. He continued to refine his designs; the most elaborate message he helped to develop and assemble was the Voyager Golden Record that was sent out with the Voyager space probes in 1977. Sagan often challenged the decisions to fund the Space Shuttle and Space Station at the expense of further robotic missions.^[13]

Sagan taught a course on critical thinking at Cornell University until he died in 1996 from pneumonia, a few months after finding that he was in remission of myelodysplastic syndrome.

Scientific achievements[link]

Sagan's contributions were central to the discovery of the high surface temperatures of the planet Venus. In the early 1960s no one knew for certain the basic conditions of that planet's surface, and Sagan listed the possibilities in a report later depicted for popularization in a Time-Life book, Planets. His own view was that Venus was dry and very hot as opposed to the balmy paradise others had imagined. He had investigated radio emissions from Venus and concluded that there was a surface temperature of 500 °C (900 °F). As a visiting scientist to NASA's Jet Propulsion Laboratory, he contributed to the first Mariner missions to Venus, working on the design and management of the project. Mariner 2 confirmed his conclusions on the surface conditions of Venus in 1962.

Sagan was among the first to hypothesize that Saturn's moon Titan might possess oceans of liquid compounds on its surface and that Jupiter's moon Europa might possess subsurface oceans of water. This would make Europa potentially habitable.^[14] Europa's subsurface ocean of water was later indirectly confirmed by the spacecraft Galileo. The mystery of Titan's reddish haze was also solved with Sagan's help. The reddish haze was revealed to be due to complex organic molecules constantly raining down onto Titan's surface.^[15]

He further contributed insights regarding the atmospheres of Venus and Jupiter as well as seasonal changes on Mars. Sagan established that the atmosphere of Venus is extremely hot and dense with pressures increasing steadily all the way down to the surface. He also perceived global warming as a growing, man-made danger and likened it to the natural development of Venus into a hot, life-hostile planet through a kind of runaway greenhouse effect. Sagan and his Cornell colleague Edwin Ernest Salpeter speculated about life in Jupiter's clouds, given the planet's dense atmospheric composition rich in organic molecules. He studied the observed color variations on Mars' surface and concluded that they were not seasonal or vegetational changes as most believed but shifts in surface dust caused by windstorms.

Sagan is best known, however, for his research on the possibilities of extraterrestrial life, including experimental demonstration of the production of amino acids from basic chemicals by radiation.^[16]

He is also the 1994 recipient of the Public Welfare Medal, the highest award of the National Academy of Sciences for "distinguished contributions in the application of science to the public welfare."^[17] He was denied membership in the Academy, reportedly because his media activities made him unpopular with many other scientists.^[18]

Scientific advocacy[link]

Planetary Society members at the organization's founding. Carl Sagan is seated on the right.

Sagan's ability to convey his ideas allowed many people to better understand the cosmos—simultaneously emphasizing the value and worthiness of the human race, and the relative insignificance of the Earth in comparison to the universe. He delivered the 1977 series of Royal Institution Christmas Lectures in London.^[19] He hosted and, with Ann Druyan, co-wrote and co-produced the highly popular thirteen-part PBS television series Cosmos: A Personal Voyage modeled on Jacob Bronowski's The Ascent of Man.

Cosmos covered a wide range of scientific subjects including the origin of life and a perspective of our place in the universe. The series was first broadcast by the Public Broadcasting Service in 1980, winning an Emmy and a Peabody Award. It has been broadcast in more than 60 countries and seen by over 500 million people,^[3][20] making it the most widely watched PBS program in history.^[21] In addition, Time magazine ran a cover story about Sagan soon after the show broadcast, referring to him as "creator, chief writer and host-narrator of the new public television series Cosmos, [and] takes the controls of his fantasy spaceship."^[22]

Sagan was a proponent of the search for extraterrestrial life. He urged the scientific community to listen with radio telescopes for signals from potential intelligent extraterrestrial life-forms. Sagan was so persuasive that by 1982 he was able to get a petition advocating SETI published in the journal Science and signed by 70 scientists including seven Nobel Prize winners. This was a tremendous increase in the respectability of this controversial field. Sagan also helped Dr. Frank Drake write the Arecibo message, a radio message beamed into space from the Arecibo radio telescope on November 16, 1974, aimed at informing potential extraterrestrials about Earth.

Sagan was chief technology officer of the professional planetary research journal Icarus for twelve years. He co-founded the Planetary Society, the largest space-interest group in the world, with over 100,000 members in more than 149 countries, and was a member of the SETI Institute Board of Trustees. Sagan served as Chairman of the Division for Planetary Science of the American Astronomical Society, as President of the Planetology Section of the American Geophysical Union, and as Chairman of the Astronomy Section of the American Association for the Advancement of Science.

At the height of the Cold War, Sagan became involved in public awareness efforts for the effects of nuclear war when a mathematical climate model suggested that a substantial nuclear exchange could upset the delicate balance of life on Earth. He was one of five authors—the "S" of the "TTAPS" report as the research paper came to be known. He eventually co-authored the scientific paper hypothesizing a global nuclear winter following nuclear war.^[23] He also co-authored the book A Path Where No Man Thought: Nuclear Winter and the End of the Arms Race, a comprehensive examination of the phenomenon of nuclear winter.

Sagan also wrote books to popularize science which reflected and expanded upon some of the themes of A Personal Voyage, and became the best-selling science book ever published in English;^[24] The Dragons of Eden: Speculations on the Evolution of Human Intelligence, which won a Pulitzer Prize; and Broca's Brain: Reflections on the Romance of Science. Sagan also wrote the best-selling science fiction novel Contact in 1985, based on a film treatment he wrote with his wife in 1979, but he did not live to see the book's 1997 motion picture adaptation, which starred Jodie Foster and won the 1998 Hugo Award for Best Dramatic Adaption.

Pale Blue Dot: Earth is a bright pixel when photographed from "Voyager 1" six billion kilometers out (past Pluto). Sagan encouraged NASA to generate this image.

He wrote a sequel to Cosmos, Pale Blue Dot: A Vision of the Human Future in Space, which was selected as a notable book of 1995 by The New York Times. He appeared on PBS' Charlie Rose program in January 1995.^[25] Sagan also wrote an introduction for the bestselling book by Stephen Hawking, A Brief History of Time. Sagan was also known for his popularization of science, his efforts to increase scientific understanding among the general public, and his positions in favor of scientific skepticism and against pseudoscience, such as his debunking of the Betty and Barney Hill abduction. To mark the tenth anniversary of Sagan's death, David Morrison, a former student of Sagan, recalled "Sagan's immense contributions to planetary research, the public understanding of science, and the skeptical movement" in Skeptical Inquirer.^[26]

Sagan hypothesized in January 1991 that enough smoke from the 1991 Kuwaiti oil fires "might get so high as to disrupt agriculture in much of South Asia ..." He later conceded in The Demon-Haunted World that this prediction did not turn out to be correct: "it was pitch black at noon and temperatures dropped 4°–6 °C over the Persian Gulf, but not much smoke reached stratospheric altitudes and Asia was spared."^[27] A 2007 study noted that modern computer models have been applied to the Kuwait oil fires, finding that individual smoke plumes are not able to loft smoke into the stratosphere, but that smoke from fires covering a large area, like some forest fires or the burning of cities that would be expected to follow a nuclear strike, would loft significant amounts of smoke into the stratosphere.^{[28][29][30][31]}

In his later years Sagan advocated the creation of an organized search for near Earth objects that might impact the Earth.^[32] When others suggested creating large nuclear bombs that could be used to alter the orbit of a NEO that was predicted to hit the Earth, Sagan proposed the Deflection Dilemma: If we create the ability to deflect an asteroid away from the Earth, then we also create the ability to deflect an asteroid towards the Earth—providing an evil power with a true doomsday bomb.^[33][34]

Billions and billions[link]

Sagan with a model of the Viking Lander probes which would land on Mars. Sagan examined possible landing sites for Viking along with Mike Carr and Hal Masursky.

From Cosmos and his frequent appearances on The Tonight Show Starring Johnny Carson, Sagan became associated with the catchphrase "billions and billions". Sagan stated that he never actually used the phrase in the Cosmos series.^[35] The closest that he ever came was in the book Cosmos, where he talked of "billions upon billions":^[36]

A galaxy is composed of gas and dust and stars—billions upon billions of stars.

—Carl Sagan, Cosmos, chapter 1, page 3^[37]

However, his frequent use of the word billions, and distinctive delivery emphasizing the "b" (which he did intentionally, in place of more cumbersome alternatives such as "billions with a 'b'", in order to distinguish the word from "millions" in viewers' minds),^[35] made him a favorite target of comic performers, including Johnny Carson,^[38] Gary Kroeger, Mike Myers, Bronson Pinchot, Penn Jillette, Harry Shearer, and others. Frank Zappa satirized the line in the song "Be In My Video", noting as well "atomic light". Sagan took this all in good humor, and his final book was entitled Billions and Billions, which opened with a tongue-in-cheek discussion of this catchphrase, observing that Carson was an amateur astronomer and that Carson's comic caricature often included real science.^[35]

Similarly he is also known for expressing wonderment at the vastness of space and time, as in his phrase "The total number of stars in the Universe is larger than all the grains of sand on all the beaches of the planet Earth."

Sagan units[link]

As a humorous tribute to him, a sagan has been defined as a unit of measurement equal to at least four billion, since the smallest number that may be described as billions and billions must be two billion plus two billion.^[39][40]

Social concerns[link]

Sagan believed that the Drake equation, on substitution of reasonable estimates, suggested that a large number of extraterrestrial civilizations would form, but that the lack of evidence of such civilizations highlighted by the Fermi paradox suggests technological civilizations tend to self-destruct. This stimulated his interest in identifying and publicizing ways that humanity could destroy itself, with the hope of avoiding such a cataclysm and eventually becoming a spacefaring species. Sagan's deep concern regarding the potential destruction of human civilization in a nuclear holocaust was conveyed in a memorable cinematic sequence in the final episode of Cosmos, called "Who Speaks for Earth?" Sagan had already resigned from the Air Force Scientific Advisory Board and voluntarily surrendered his top secret clearance in protest over the Vietnam War.^[41] Following his marriage to his third wife (novelist Ann Druyan) in June 1981, Sagan became more politically active—particularly in opposing escalation of the nuclear arms race under President Ronald Reagan.

In March 1983, Reagan announced the Strategic Defense Initiative—a multi-billion dollar project to develop a comprehensive defense against attack by nuclear missiles, which was quickly dubbed the "Star Wars" program. Sagan spoke out against the project, arguing that it was technically impossible to develop a system with the level of perfection required, and far more expensive to build than for an enemy to defeat through decoys and other means—and that its construction would seriously destabilize the nuclear balance between the United States and the Soviet Union, making further progress toward nuclear disarmament impossible.

When Soviet leader Mikhail Gorbachev declared a unilateral moratorium on the testing of nuclear weapons, which would begin on August 6, 1985—the 40th anniversary of the atomic bombing of Hiroshima—the Reagan administration dismissed the dramatic move as nothing more than propaganda, and refused to follow suit. In response, US anti-nuclear and peace activists staged a series of protest actions at the Nevada Test Site, beginning on Easter Sunday in 1986 and continuing through 1987. Hundreds of people were arrested, including Sagan, who was arrested on two separate occasions as he climbed over a chain-link fence at the test site.^[42]

Personal life and beliefs[link]

Sagan was married three times—in 1957, to biologist Lynn Margulis, mother of Dorion Sagan and Jeremy Sagan; in 1968, to artist Linda Salzman, mother of Nick Sagan; and in 1981, to author Ann Druyan, mother of Alexandra Rachel (Sasha) Sagan and Samuel Democritus Sagan. His marriage to Druyan continued until his death in 1996.

Isaac Asimov described Sagan as one of only two people he ever met whose intellect surpassed his own. The other, he claimed, was the computer scientist and artificial intelligence expert Marvin Minsky.^[43]

Sagan wrote frequently about religion and the relationship between religion and science, expressing his skepticism about the conventional conceptualization of God as a sapient being. For example:

"Some people think God is an outsized, light-skinned male with a long white beard, sitting on a throne somewhere up there in the sky, busily tallying the fall of every sparrow. Others—for example Baruch Spinoza and Albert Einstein—considered God to be essentially the sum total of the physical laws which describe the universe. I do not know of any compelling evidence for anthropomorphic patriarchs controlling human destiny from some hidden celestial vantage point, but it would be madness to deny the existence of physical laws."^[44]

In another description of his view of God, Sagan emphatically writes:

"The idea that God is an oversized white male with a flowing beard who sits in the sky and tallies the fall of every sparrow is ludicrous. But if by God one means the set of physical laws that govern the universe, then clearly there is such a God. This God is emotionally unsatisfying... it does not make much sense to pray to the law of gravity."^[45]

On atheism, Sagan commented in 1981:

"An atheist is someone who is certain that God does not exist, someone who has compelling evidence against the existence of God. I know of no such compelling evidence. Because God can be relegated to remote times and places and to ultimate causes, we would have to know a great deal more about the universe than we do now to be sure that no such God exists. To be certain of the existence of God and to be certain of the nonexistence of God seem to me to be the confident extremes in a subject so riddled with doubt and uncertainty as to inspire very little confidence indeed".^[46]

In reply to a question in 1996 about his religious beliefs, Sagan answered, "I'm agnostic."^[47] Sagan's views on religion have been interpreted as a form of pantheism comparable to Einstein's belief in Spinoza's God.^[48] Sagan maintained that the idea of a creator of the universe was difficult to prove or disprove and that the only conceivable scientific discovery that could challenge it would be an infinitely old universe.^[49] According to his last wife, Ann Druyan, he was not a believer:

When my husband died, because he was so famous and known for not being a believer, many people would come up to me—it still sometimes happens—and ask me if Carl changed at the end and converted to a belief in an afterlife. They also frequently ask me if I think I will see him again. Carl faced his death with unflagging courage and never sought refuge in illusions. The tragedy was that we knew we would never see each other again. I don't ever expect to be reunited with Carl.^[50]

In 2006, Ann Druyan edited Sagan's 1985 Glasgow Gifford Lectures in Natural Theology into a book, The Varieties of Scientific Experience: A Personal View of the Search for God, in which he elaborates on his views of divinity in the natural world.

Carl Sagan (center) speaks with CDC employees in 1988.

Sagan is also widely regarded as a freethinker or skeptic; one of his most famous quotations, in Cosmos, was, "Extraordinary claims require extraordinary evidence"^[51] (called the "Sagan Standard" by some^[52]). This was based on a nearly identical statement by fellow founder of the Committee for the Scientific Investigation of Claims of the Paranormal, Marcello Truzzi, "An extraordinary claim requires extraordinary proof."^[53][54] This idea originated with Pierre-Simon Laplace (1749–1827), a French mathematician and astronomer who said, "The weight of evidence for an extraordinary claim must be proportioned to its strangeness."^[55]

Late in his life, Sagan's books elaborated on his skeptical, naturalistic view of the world. In The Demon-Haunted World: Science as a Candle in the Dark, he presented tools for testing arguments and detecting fallacious or fraudulent ones, essentially advocating wide use of critical thinking and the scientific method. The compilation Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium, published in 1997 after Sagan's death, contains essays written by Sagan, such as his views on abortion, and his widow Ann Druyan's account of his death as a skeptic, agnostic, and freethinker.

Sagan warned against humans' tendency towards anthropocentrism. He was the faculty adviser for the Cornell Students for the Ethical Treatment of Animals. In the Cosmos chapter "Blues For a Red Planet", Sagan wrote, "If there is life on Mars, I believe we should do nothing with Mars. Mars then belongs to the Martians, even if the Martians are only microbes."^[56]

Sagan was a user and advocate of marijuana. Under the pseudonym "Mr. X", he contributed an essay about smoking cannabis to the 1971 book Marihuana Reconsidered.^[57][58] The essay explained that marijuana use had helped to inspire some of Sagan's works and enhance sensual and intellectual experiences. After Sagan's death, his friend Lester Grinspoon disclosed this information to Sagan's biographer, Keay Davidson. The publishing of the biography, Carl Sagan: A Life, in 1999 brought media attention to this aspect of Sagan's life.^[59][60][61] Not long after his death, widow Ann Druyan had gone on to preside over the board of directors of NORML, a foundation dedicated to reforming cannabis laws.^[62]

In 1994, engineers at Apple Computer code-named the Power Macintosh 7100 "Carl Sagan" in the hope that Apple would make "billions and billions" with the sale of the PowerMac 7100.^[4] The name was only used internally, but Sagan was concerned that it would become a product endorsement and sent Apple a cease and desist letter. Apple complied, but engineers retaliated by changing the internal codename to "BHA" for "Butt-Head Astronomer".^[63][64] Sagan then sued Apple for libel, a form of defamation, in federal court. The court granted Apple's motion to dismiss Sagan's claims and opined in dicta that a reader aware of the context would understand Apple was "clearly attempting to retaliate in a humorous and satirical way", and that "It strains reason to conclude that Defendant was attempting to criticize Plaintiff's reputation or competency as an astronomer. One does not seriously attack the expertise of a scientist using the undefined phrase 'butt-head'."^[63][65] Sagan then sued for Apple's original use of his name and likeness, but again lost.^[66] Sagan appealed the ruling.^[66] In November 1995, an out of court settlement was reached and Apple's office of trademarks and patents released a conciliatory statement that "Apple has always had great respect for Dr. Sagan. It was never Apple's intention to cause Dr. Sagan or his family any embarrassment or concern."^[67]

Sagan briefly served as an adviser on Stanley Kubrick's film 2001: A Space Odyssey.^[6] Sagan proposed that the film would suggest, rather than depict, extraterrestrial superintelligence.^[68]

Sagan and UFOs[link]

Sagan had some interest in UFO reports from at least August 3, 1952 when he wrote a letter to U.S. Secretary of State Dean Acheson to ask how the U.S. would respond if flying saucers turned out to be extraterrestrial.^[69] He later had several conversations on the subject in 1964 with Jacques Vallee.^[70] Though quite skeptical of any extraordinary answer to the UFO question, Sagan thought scientists should study the phenomenon, at least because there was widespread public interest in UFO reports.

Stuart Appelle notes that Sagan "wrote frequently on what he perceived as the logical and empirical fallacies regarding UFOs and the abduction experience. Sagan rejected an extraterrestrial explanation for the phenomenon but felt there were both empirical and pedagogical benefits for examining UFO reports and that the subject was, therefore, a legitimate topic of study."^[71]

In 1966, Sagan was a member of the Ad Hoc Committee to Review Project Blue Book, the U.S. Air Force's UFO investigation project. The committee concluded Blue Book had been lacking as a scientific study, and recommended a university-based project to give the UFO phenomenon closer scientific scrutiny. The result was the Condon Committee (1966–1968), led by physicist Edward Condon, and in their final report they formally concluded that UFOs, regardless of what any of them actually were, did not behave in a manner consistent with a threat to national security.

Ron Westrum writes that "The high point of Sagan's treatment of the UFO question was the AAAS's symposium in 1969. A wide range of educated opinions on the subject were offered by participants, including not only proponents such as James McDonald and J. Allen Hynek but also skeptics like astronomers William Hartmann and Donald Menzel. The roster of speakers was balanced, and it is to Sagan's credit that this event was presented in spite of pressure from Edward Condon".^[70] With physicist Thornton Page, Sagan edited the lectures and discussions given at the symposium; these were published in 1972 as UFOs: A Scientific Debate. Some of Sagan's many books examine UFOs (as did one episode of Cosmos) and he claimed a religious undercurrent to the phenomenon.

Sagan again revealed his views on interstellar travel in his 1980 Cosmos series. In one of his last written works, Sagan argued that the chances of extraterrestrial spacecraft visiting Earth are vanishingly small. However, Sagan did think it plausible that Cold War concerns contributed to governments misleading their citizens about UFOs, and that "some UFO reports and analyses, and perhaps voluminous files, have been made inaccessible to the public which pays the bills ... It's time for the files to be declassified and made generally available." He cautioned against jumping to conclusions about suppressed UFO data and stressed that there was no strong evidence that aliens were visiting the Earth either in the past or present.^[72]

Death[link]

Stone dedicated to Carl Sagan in the Celebrity Path of the Brooklyn Botanic Garden

After a long and difficult fight with myelodysplasia, which included three bone marrow transplants, Sagan died of pneumonia at the age of 62 at the Fred Hutchinson Cancer Research Center in Seattle, Washington, on December 20, 1996.^[73] He was buried at Lakeview Cemetery in Ithaca, New York.^[74]

Posthumous recognition[link]

The 1997 movie Contact, based on Sagan's novel of the same name and finished after his death, ends with the dedication "For Carl".

In 1997, the Sagan Planet Walk was opened in Ithaca, New York. It is a walking scale model of the solar system, extending 1.2 km from the center of The Commons in downtown Ithaca to the Sciencenter, a hands-on museum. The exhibition was created in memory of Carl Sagan, who was an Ithaca resident and Cornell Professor. Professor Sagan had been a founding member of the museum's advisory board.^[75]

The landing site of the unmanned Mars Pathfinder spacecraft was renamed the Carl Sagan Memorial Station on July 5, 1997. Asteroid 2709 Sagan is named in his honor.

Sagan's son, Nick Sagan, wrote several episodes in the Star Trek franchise. In an episode of Star Trek: Enterprise entitled "Terra Prime", a quick shot is shown of the relic rover Sojourner, part of the Mars Pathfinder mission, placed by a historical marker at Carl Sagan Memorial Station on the Martian surface. The marker displays a quote from Sagan: "Whatever the reason you're on Mars, I'm glad you're there, and I wish I was with you." Sagan's student Steve Squyres led the team that landed the Spirit Rover and Opportunity Rover successfully on Mars in 2004.

On November 9, 2001, on what would have been Sagan's 67th birthday, the NASA Ames Research Center dedicated the site for the Carl Sagan Center for the Study of Life in the Cosmos. "Carl was an incredible visionary, and now his legacy can be preserved and advanced by a 21st century research and education laboratory committed to enhancing our understanding of life in the universe and furthering the cause of space exploration for all time", said NASA Administrator Daniel Goldin. Ann Druyan was at the Center as it opened its doors on October 22, 2006.

Sagan has at least three awards named in his honor:

• The Carl Sagan Memorial Award presented jointly since 1997 by the American Astronomical Society (AAS) and the Planetary Society,
• The Carl Sagan Medal for Excellence in Public Communication in Planetary Science presented since 1998 by the American Astronomical Society's Division for Planetary Sciences (AAS/DPS) for outstanding communication by an active planetary scientist to the general public—Carl Sagan was one of the original organizing committee members of the DPS, and
• The Carl Sagan Award for Public Understanding of Science presented by the Council of Scientific Society Presidents (CSSP)—Sagan was the first recipient of the CSSP award in 1993.^[76]

In 2006, the Carl Sagan Medal was awarded to astrobiologist and author David Grinspoon, the son of Sagan's friend Lester Grinspoon.

On December 20, 2006, the tenth anniversary of Sagan's death, a blogger, Joel Schlosberg, organized a Carl Sagan "blog-a-thon" to commemorate Sagan's death, and the idea was supported by Nick Sagan.^[77] Many members of the blogging community participated.

August 2007 the Independent Investigative Group IIG awarded Sagan posthumously a Lifetime Achievement Award. This honor has also been awarded to Harry Houdini and James Randi.^[78]

Beginning in 2009, a musical project known as Symphony of Science sampled several excerpts of Sagan from his series Cosmos and remixed them to electronic music. To date, the videos have received over twenty-million views worldwide on the popular user generated website Youtube, and have exposed a new generation to the interconnected grandeur of the universe as Sagan described them in his documentary series.^[79]

Awards and honors[link]

NASA Distinguished Public Service Medal

• Annual Award for Television Excellence—1981—Ohio State University—PBS series Cosmos
• Apollo Achievement Award—National Aeronautics and Space Administration
• NASA Distinguished Public Service Medal—National Aeronautics and Space Administration (twice)
• Emmy—Outstanding Individual Achievement—1981—PBS series Cosmos
• Emmy—Outstanding Informational Series—1981—PBS series Cosmos
• Exceptional Scientific Achievement Medal—National Aeronautics and Space Administration
• Helen Caldicott Leadership Award – Awarded by Women's Action for Nuclear Disarmament
• Hugo Award—1981—Cosmos
• Humanist of the Year—1981—Awarded by the American Humanist Association
• In Praise of Reason Award—1987—Committee for the Scientific Investigation of Claims of the Paranormal
• Isaac Asimov Award—1994—Committee for the Scientific Investigation of Claims of the Paranormal
• John F. Kennedy Astronautics Award—American Astronautical Society
• John W. Campbell Memorial Award—1974—The Cosmic Connection: An Extraterrestrial Perspective
• Joseph Priestley Award—"For distinguished contributions to the welfare of mankind"
• Klumpke-Roberts Award of the Astronomical Society of the Pacific—1974
• Konstantin Tsiolkovsky Medal—Awarded by the Soviet Cosmonauts Federation
• Locus Award 1986—Contact
• Lowell Thomas Award—Explorers Club—75th Anniversary
• Masursky Award—American Astronomical Society
• Miller Research Fellowship—Miller Institute (1960–1962)
• Oersted Medal—1990—American Association of Physics Teachers
• Peabody Award—1980—PBS series Cosmos
• Prix Galbert—The international prize of Astronautics
• Public Welfare Medal—1994—National Academy of Sciences^[80]
• Pulitzer Prize for General Non-Fiction—1978—The Dragons of Eden
• SF Chronicle Award—1998—Contact
• Named the "99th Greatest American" on the June 5, 2005, Greatest American show on the Discovery Channel
• Named an honorary member of the Demosthenian Literary Society on November 10, 2011
• 2009 New Jersey Hall of Fame inductee.^[81]

Publications[link]

• Planets (LIFE Science Library), Sagan, Carl, Jonathan Norton Leonard and editors of Life, Time, Inc., 1966
• Intelligent Life in the Universe, I.S. Shklovskii coauthor, Random House, 1966, ISBN 1-892803-02-X, 509 pgs (note: republished in 1998 by Emerson-Adams Press)
• UFOs: A Scientific Debate, Thornton Page coauthor, Cornell University Press, 1972, 310 pgs
• Communication with Extraterrestrial Intelligence. MIT Press, 1973, 428 pgs
• Mars and the Mind of Man, Sagan, Carl, et al., Harper & Row, 1973, 143 pgs
• The Cosmic Connection: An Extraterrestrial Perspective, Jerome Agel coauthor, Anchor Press, 1973, ISBN 0-521-78303-8, 301 pgs
• Other Worlds. Bantam Books, 1975
• Murmurs of Earth: The Voyager Interstellar Record, Sagan, Carl, et al., Random House, ISBN 0-394-41047-5, 1978
• The Dragons of Eden: Speculations on the Evolution of Human Intelligence. Ballantine Books, 1978, ISBN 0-345-34629-7, 288 pgs
• Broca's Brain: Reflections on the Romance of Science. Ballantine Books, 1979, ISBN 0-345-33689-5, 416 pgs
• Cosmos. Random House, 1980. Random House New Edition, May 7, 2002, ISBN 0-375-50832-5, 384 pgs
• The Cold and the Dark: The World after Nuclear War, Sagan, Carl et al., Sidgwick & Jackson, 1985
• Comet, Ann Druyan coauthor, Ballantine Books, 1985, ISBN 0-345-41222-2, 496 pgs
• Contact. Simon and Schuster, 1985; Reissued August 1997 by Doubleday Books, ISBN 1-56865-424-3, 352 pgs
• A Path Where No Man Thought: Nuclear Winter and the End of the Arms Race, Richard Turco coauthor, Random House, 1990, ISBN 0-394-58307-8, 499 pgs
• Shadows of Forgotten Ancestors: A Search for Who We Are, Ann Druyan coauthor, Ballantine Books, October 1993, ISBN 0-345-38472-5, 528 pgs
• Pale Blue Dot: A Vision of the Human Future in Space. Random House, November 1994, ISBN 0-679-43841-6, 429 pgs
• The Demon-Haunted World: Science as a Candle in the Dark. Ballantine Books, March 1996, ISBN 0-345-40946-9, 480 pgs (note: the book was first published and copyrighted in 1995 with an errata slip inserted)
• Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium, Ann Druyan coauthor, Ballantine Books, June 1997, ISBN 0-345-37918-7, 320 pgs
• The Varieties of Scientific Experience: A Personal View of the Search for God, Carl Sagan (writer) & Ann Druyan (editor), 1985 Gifford lectures, Penguin Press HC, November 2006, ISBN 1-59420-107-2, 304 pgs

References[link]