Space Shuttle

Space Shuttle Discovery begins liftoff at the start of STS-120.
Function	Manned orbital launch and reentry
Manufacturer	United Space Alliance: Thiokol/Alliant Techsystems (SRBs) Lockheed Martin/Martin Marietta (ET) Boeing/Rockwell (orbiter)
Country of origin	United States of America
Size
Height	56.1 m (184.2 ft)
Diameter	8.7 m (28.5 ft)
Mass	2,030 t (4,470,000 lbm)
Capacity
Payload to LEO	24,400 kg (53,600 lb)
Payload to GTO	3,810 kg (8,390 lbm)
Payload to Polar orbit	12,700 kg (28,000 lb)
Payload to Landing[1]	14,400 kg (32,000 lb)[1] (Return Payload)
Launch history
Status	Retired
Launch sites	LC-39, Kennedy Space Center SLC-6, Vandenberg AFB (unused)
Total launches	135
Successes	134 successful launches 133 successful re-entries
Failures	2 (launch failure, Challenger; re-entry failure, Columbia)
First flight	April 12, 1981
Last flight	July 21, 2011
Notable payloads	Tracking and Data Relay Satellites Spacelab Great Observatories (including Hubble) Galileo, Magellan, Ulysses Mir Docking Module ISS components
Boosters (Stage 0) - Solid Rocket Boosters
No. boosters	2
Engines	1 solid
Thrust	12.5 MN each, sea level liftoff (2,800,000 lbf)
Specific impulse	269 s
Burn time	124 s
Fuel	solid
First stage - External Tank
Engines	3 SSMEs located on Orbiter
Thrust	5.45220 MN total, sea level liftoff (1,225,704 lbf)
Specific impulse	455 s
Burn time	480 s
Fuel	LOX/LH2
Second stage - Orbiter
Engines	2 OME
Thrust	53.4 kN combined total vacuum thrust (12,000 lbf)
Specific impulse	316 s
Burn time	1,250 s
Fuel	MMH / N2O4

The Space Shuttle was a partially reusable launch system and orbital spacecraft operated by the U.S. National Aeronautics and Space Administration (NASA) for human spaceflight missions. The system combined rocket launch, orbital spacecraft, and re-entry spaceplane with modular add-ons. The first of four orbital test flights occurred in 1981 leading to operational flights beginning in 1982. It was used on a total of 135 missions from 1981 to 2011 all launched from the Kennedy Space Center, Florida.
Major missions included launching numerous satellites, interplanetary probes, the Hubble Space Telescope (HST), conducting space science experiments, and constructing and servicing the International Space Station. Major components included the orbiters, recoverable boosters, external tanks, payloads, and supporting infrastructure. Five space-worthy orbiters were built; two were destroyed in mission accidents.
The Space Shuttle at launch consisted of the Orbiter Vehicle (OV), one external tank (ET), and two Solid Rocket Boosters (SRBs). It was launched vertically like a conventional rocket with thrust from the two SRBs and three main engines. During launch, the external tank provided fuel for the orbiter's main engines. The SRBs and ET were jettisoned before the orbiter reached orbit. At the conclusion of the orbiter's space mission, it fired its thrusters to drop out of orbit and re-enter the lower atmosphere. The orbiter decelerated in the atmosphere before flying like a glider but with reaction control system thrusters before landing on a long runway.

Overview
The Space Shuttle was a partially reuseable^[2] launch system and orbital spacecraft operated by the U.S. National Aeronautics and Space Administration (NASA) for human spaceflight missions from 1981 to 2011. The system combined rocket launch, orbital spacecraft, and re-entry spaceplane with modular add-ons. The first of four orbital test flights occurred in 1981 leading to operational flights beginning in 1982, all launched from the Kennedy Space Center, Florida. The system was retired from service in 2011 after 135 missions;^[3] on July 8, 2011, with Space Shuttle Atlantis performing that 135th launch - the final launch of the three-decade Shuttle program.^[4] The program ended after Atlantis landed at the Kennedy Space Center on July 21, 2011. Major missions included launching numerous satellites and interplanetary probes,^[5] conducting space science experiments, and servicing and construction of space stations. Enterprise was a prototype orbiter used for atmospheric testing during development in the 1970s, and lacked engines and heat shield. Five space-worthy orbiters were built—two were destroyed in accidents and the others have been retired.
It was used for orbital space missions by NASA, the U.S. Department of Defense, the European Space Agency, Japan, and Germany.^[6][7] The United States funded Space Transportation System (STS) development and Shuttle operations except for Spacelab D1 and D2—sponsored by West Germany and reunified Germany respectively.^{[6][8][9][10][11]} In addition, SL-J was partially funded by Japan.^[7]
At launch, it consisted of the "stack", including a dark orange-colored external tank (ET);^[12][13] two white, slender Solid Rocket Boosters (SRBs); and the Orbiter Vehicle (OV), which contained the crew and payload. Some payloads were launched into higher orbits with either of two different booster stages developed for the STS (single-stage Payload Assist Module or two-stage Inertial Upper Stage). The Space Shuttle was stacked in the Vehicle Assembly Building and the stack mounted on a mobile launch platform held down by four explosive bolts on each SRB which are detonated at launch.^[14]
The Shuttle stack launched vertically like a conventional rocket. It lifted off under the power of its two SRBs and three main engines, which were fueled by liquid hydrogen and liquid oxygen from the external tank. The Space Shuttle had a two-stage ascent. The SRBs provided additional thrust during liftoff and first-stage flight. About two minutes after liftoff, explosive bolts were fired, releasing the SRBs, which then parachuted into the ocean, to be retrieved by ships for refurbishment and reuse. The Shuttle orbiter and external tank continued to ascend on an increasingly horizontal flight path under power from its main engines. Upon reaching 17,500 mph (7.8 km/s), necessary for low Earth orbit, the main engines were shut down. The external tank was then jettisoned to burn up in the atmosphere.^[15] After jettisoning the external tank, the orbital maneuvering system (OMS) engines were used to adjust the orbit.
The orbiter carried people and payloads such as satellites or space station parts into low Earth orbit, the Earth's upper atmosphere or thermosphere.^[16] Usually, five to seven crew members rode in the orbiter. Two crew members, the commander and pilot, were sufficient for a minimal flight, as in the first four "test" flights, STS-1 through STS-4. The typical payload capacity was about 22,700 kilograms (50,000 lb), but could be increased depending on the choice of launch configuration. The orbiter carried its payload in a large cargo bay with doors that opened along the length of its top, a feature which made the Space Shuttle unique among spacecraft. This feature made possible the deployment of large satellites such as the Hubble Space Telescope, and also the capture and return of large payloads back to Earth.
When the orbiter's space mission was complete, it fired its OMS thrusters to drop out of orbit and re-enter the lower atmosphere.^[16] During descent, the orbiter passed through different layers of the atmosphere and decelerated from hypersonic speed primarily by aerobraking. In the lower atmosphere and landing phase, it was more like a glider but with reaction control system (RCS) thrusters and fly-by wire-controlled hydraulically-actuated flight surfaces controlling its descent. It landed on a long runway as a spaceplane. The aerodynamic shape was a compromise between the demands of radically different speeds and air pressures during re-entry, hypersonic flight, and subsonic atmospheric flight. As a result, the orbiter had a relatively high sink rate at low altitudes, and it transitioned during re-entry from using RCS thrusters at very high altitudes to flight surfaces in the lower atmosphere.

Early history

Further information: Space Shuttle program and Space Shuttle design process

President Nixon (right) with NASA Administrator Fletcher in January 1972, three months before Congress approved funding for the Shuttle program

The formal design of what became the Space Shuttle began with "Phase A" contract design studies issued in the late 1960s. However, conceptualization began two decades earlier, before the Apollo program of the 1960s. One of the places the concept of a spacecraft returning from space to a horizontal landing originated was within NACA, in 1954, in the form of an aeronautics research experiment later named the X-15. The NACA proposal was submitted by Walter Dornberger.
In 1958, the X-15 concept further developed into proposal to launch an X-15 into space, and another X-series spaceplane proposal, called the X-20, which was not constructed, as well as variety of aerospace plane concepts and studies. Neil Armstrong was selected to pilot both the X-15 and the X-20. Though the X-20 was not built, another spaceplane similar to the X-20 was built several years later and delivered to NASA in January 1966 called the HL-10 ("HL" indicated "horizontal landing").
In the mid-1960s, the US Air Force conducted a series of classified studies on next-generation space transportation systems and concluded that semi-reusable designs were the cheapest choice. It proposed a development program with an immediate start on a "Class I" vehicle with expendable boosters, followed by slower development of a "Class II" semi-reusable design and perhaps a "Class III" fully reusable design later. In 1967, George Mueller held a one-day symposium at NASA headquarters to study the options. Eighty people attended and presented a wide variety of designs, including earlier Air Force designs as the Dyna-Soar (X-20).
In 1968, NASA officially began work on what was then known as the Integrated Launch and Re-entry Vehicle (ILRV). At the same time, NASA held a separate Space Shuttle Main Engine (SSME) competition. NASA offices in Houston and Huntsville jointly issued a Request for Proposal (RFP) for ILRV studies to design a spacecraft that could deliver a payload to orbit but also re-enter the atmosphere and fly back to Earth. For example, one of the responses was for a two-stage design, featuring a large booster and a small orbiter, called the DC-3, one of several Phase A Shuttle designs. After the aforementioned "Phase A" studies, B, C, and D phases progressively evaluated in-depth designs up to 1972. In the final design, the bottom stage was recoverable solid rocket boosters, and the top stage used an expendable external tank.^[17]
In 1969, President Richard Nixon decided to support proceeding with Space Shuttle development. A series of development programs and analysis refined the basic design, prior to full development and testing. In August 1973, the X-24B proved that an unpowered spaceplane could re-enter Earth's atmosphere for a horizontal landing.
Across the Atlantic, European ministers met in Belgium in 1973 to authorize Western Europe's manned orbital project and its main contribution to Space Shuttle—the Spacelab program.^[18] Spacelab would provide a multi-disciplinary orbital space laboratory and additional space equipment for the Shuttle.^[18]

Description

STS-1 on the launch pad, 1981

The Space Shuttle was the first operational orbital spacecraft designed for reuse. It carried different payloads to low Earth orbit, provided crew rotation for the International Space Station (ISS), and performed servicing missions. The orbiter could also recover satellites and other payloads from orbit and return them to Earth. Each Shuttle was designed for a projected lifespan of 100 launches or ten years of operational life, although this was later extended. The person in charge of designing the STS was Maxime Faget, who had also overseen the Mercury, Gemini, and Apollo spacecraft designs. The crucial factor in the size and shape of the Shuttle Orbiter was the requirement that it be able to accommodate the largest planned commercial and military satellites, and have the cross-range recovery range to meet the requirement for classified USAF missions for a once-around abort from a launch to a polar orbit. Factors involved in opting for solid rockets and an expendable fuel tank included the desire of the Pentagon to obtain a high-capacity payload vehicle for satellite deployment, and the desire of the Nixon administration to reduce the costs of space exploration by developing a spacecraft with reusable components.
Each Space Shuttle is a reusable launch system that is composed of three main assemblies: the reusable Orbiter Vehicle (OV), the expendable external tank (ET), and the two reusable solid rocket boosters (SRBs).^[19] Only the orbiter entered orbit shortly after the tank and boosters are jettisoned. The vehicle was launched vertically like a conventional rocket, and the orbiter glided to a horizontal landing like an airplane, after which it was refurbished for reuse. The SRBs parachuted to splashdown in the ocean where they were towed back to shore and refurbished for later Shuttle missions.

Discovery rockets into orbit, seen here just after booster rocket (SRB) separation

Five space-worthy orbiters were built: Columbia (OV-102), Challenger (OV-099), Discovery (OV-103), Atlantis (OV-104), and Endeavour (OV-105). A mock-up, Inspiration (OV-100), currently stands at the entrance to the Astronaut Hall of Fame. An additional craft, Enterprise (OV-101), was not built for orbital space flight, and was used only for testing gliding and landing. Enterprise was originally intended to be made fully space-worthy after use for the approach and landing test (ALT) program, but it was found more economical to upgrade the structural test article STA-099 into orbiter Challenger (OV-099). Challenger disintegrated 73 seconds after launch in 1986, and Endeavour was built as a replacement for Challenger from structural spare components. Columbia broke apart during re-entry in 2003. Building Space Shuttle Endeavour cost about US$1.7 billion. One Space Shuttle launch costs around $450 million.^[20]
Roger A. Pielke, Jr. has estimated that the Space Shuttle program has cost about US$170 billion (2008 dollars) through early 2008. This works out to an average cost per flight of about US$1.5 billion.^[21] However, two missions were paid for by Germany, Spacelab D1 and D2 (D for Deutschland) with a payload control center in Oberpfaffenhofen, Germany.^[22][23] D1 was the first time that control of a manned STS mission payload was not in U.S. hands.^[6]
At times, the orbiter itself was referred to as the Space Shuttle. Technically, this was a slight misnomer, as the actual "Space Transportation System" (Space Shuttle) was the combination of the orbiter, the external tank, and the two solid rocket boosters. Combined, these were referred to as the "stack"; the components were assembled in the Vehicle Assembly Building, originally built to assemble the Apollo Saturn V rocket.
Responsibility for the Shuttle components was spread among multiple NASA field centers. The Kennedy Space Center was responsible for launch, landing and turnaround operations for equatorial orbits (the only orbit profile actually used in the program), the US Air Force at the Vandenberg Air Force Base was responsible for launch, landing and turnaround operations for polar orbits (though this was never used), the Johnson Space Center served as the central point for all Shuttle operations, the Marshall Space Flight Center was responsible for the main engines, external tank, and solid rocket boosters, the John C. Stennis Space Center handled main engine testing, and the Goddard Space Flight Center managed the global tracking network.^[24]

Orbiter vehicle

Main article: Space Shuttle orbiter

Atlantis deploys the landing gear before landing on a selected runway just like a common aircraft.

The orbiter resembles a conventional aircraft, with double-delta wings swept 81° at the inner leading edge and 45° at the outer leading edge. Its vertical stabilizer's leading edge is swept back at a 50° angle. The four elevons, mounted at the trailing edge of the wings, and the rudder/speed brake, attached at the trailing edge of the stabilizer, with the body flap, controlled the orbiter during descent and landing.
The orbiter's payload bay measures 15 by 60 feet (4.6 by 18 m), comprising most of the fuselage. Information declassified in 2011 showed that the payload bay was designed specifically to accommodate the KH-9 HEXAGON spy satellite operated by the National Reconnaissance Office.^[25] Two mostly symmetrical lengthwise payload bay doors hinged on either side of the bay comprise its entire top. Payloads are generally loaded horizontally into the bay while the orbiter is oriented vertically on the launch pad and unloaded vertically in the near-weightless orbital environment by the orbiter's robotic remote manipulator arm (under astronaut control), EVA astronauts, or under the payloads' own power (as for satellites attached to a rocket "upper stage" for deployment.)
Three Space Shuttle Main Engines (SSMEs) are mounted on the orbiter's aft fuselage in a triangular pattern. The engine nozzles can swivel 10.5 degrees up and down, and 8.5 degrees from side to side during ascent to change the direction of their thrust to steer the Shuttle. The orbiter structure is made primarily from aluminum alloy, although the engine structure is made primarily from titanium alloy.
The space-capable orbiters built were OV-102 Columbia, OV-099 Challenger, OV-103 Discovery, OV-104 Atlantis, and OV-105 Endeavour.^[26]

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  Space Shuttle Atlantis transported by a Boeing 747 Shuttle Carrier Aircraft (SCA), 1998 (NASA)
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  Space Shuttle Endeavour being transported by a Boeing 747
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  An overhead view of Atlantis as it sits atop the Mobile Launcher Platform (MLP) before STS-79. Two Tail Service Masts (TSMs) to either side of the orbiter's tail provide umbilical connections for propellant loading and electrical power.
• 

  Water is released onto the mobile launcher platform on Launch Pad 39A at the start of a sound suppression system test in 2004. During launch, 350,000 US gallons (1,300,000 L) of water are poured onto the pad in 41 seconds.^[27]
  
  External tank
  

  Main article: Space Shuttle external tank

  The main function of the Space Shuttle external tank was to supply the liquid oxygen and hydrogen fuel to the main engines. It was also the backbone of the launch vehicle, providing attachment points for the two Solid Rocket Boosters and the Orbiter. The external tank was the only part of the Shuttle system that was not reused. Although the external tanks were always discarded, it was possible to take them into orbit and re-use them (such as for incorporation into a space station).^[15][28]
  

  [edit] Solid rocket boosters

  Main article: Space Shuttle Solid Rocket Booster

  Two solid rocket boosters (SRBs) each provided 12.5 million newtons (2.8 million lbf) of thrust at liftoff,^[29] which was 83% of the total thrust needed for liftoff. The SRBs were jettisoned two minutes after launch at a height of about 150,000 feet (46 km), and then deployed parachutes and landed in the ocean to be recovered.^[30] The SRB cases were made of steel about ½ inch (13 mm) thick.^[31] The Solid Rocket Boosters were re-used many times; the casing used in Ares I engine testing in 2009 consisted of motor cases that had been flown, collectively, on 48 Shuttle missions, including STS-1.^[32]
  

  [edit] Orbiter add-ons

  The orbiter could be used in conjunction with a variety of add-ons depending on the mission. This has included orbital laboratories (Spacelab, Spacehab), boosters for launching payloads farther into space (Inertial Upper Stage, Payload Assist Module), and other functions, such as provided by Extended Duration Orbiter, Multi-Purpose Logistics Modules, or Canadarm (RMS). An upper-stage called Transfer Orbit Stage (Orbital Science Corp. TOS-21) was also used once.^[33] Other types of systems and racks were part of the modular Spacelab system —pallets, igloo, IPS, etc., which also supported special missions such as SRTM.^[34]
  

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    MPLM Leonardo
  • 

    IUS deploying with Galileo
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    PAM-D with satellite
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    EDO being installed
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    Spacelab in orbit
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    RMS (Canadarm)
  • 

    Spacehab
     
    
    Spacelab
    

    Main article: Spacelab

    

    

    Spacelab LM2

    A major component of the Space Shuttle Program was Spacelab, primarily contributed by a consortium of European countries, and operated in conjunction with the United States and international partners.^[34] Supported by a modular system of pressurized modules, pallets, and systems, Spacelab missions executed on multidisciplinary science, orbital logistics, international cooperation.^[34] Over 29 missions flew on subjects ranging from astronomy, microgravity, radar, and life sciences, to name a few.^[34] Spacelab hardware also supported missions such as Hubble (HST) servicing and space station resupply.^[34] STS-2 and STS-3 provided testing, and the first full mission was Spacelab-1 (STS-9) launched on November 28, 1983.^[34]
    Spacelab formally began in 1973, after a meeting in Brussels, Belgium, by European heads of state.^[18] Within the decade, Spacelab would go into orbit and provide not only Europe, but also the United States, with an orbital workshop                                                                                            

Spacetime

In physics, spacetime (or space-time, space time), 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.

Two-dimensional analogy of spacetime distortion. Matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity. The grid 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.

 

Spacetime in literature

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

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. (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]) The idea of Minkowski space led to special relativity being viewed in a more geometrical way.

However, the most important contribution of Minkowski's geometric viewpoint of spacetime turned out to be in Einstein's later development of general relativity, since the correct description of the effect of gravitation on space and time was found to be most easily visualized as a "warp" or stretching in the geometical fabric of space and time, in a smooth and continuous way that changed smoothly from point-to-point along the spacetime fabric.

Basic concepts

Spacetimes are the arenas in which all physical events take place—an event is a point in spacetime specified by its time and place. For example, the motion of planets around the sun may be described in a particular type of spacetime, or the motion of light around a rotating star may be described in another type of spacetime. The basic elements of spacetime are events. In any given spacetime, an event is a unique position at a unique time. Because events are spacetime points, an example of an event in classical relativistic physics is , the location of an elementary (point-like) particle at a particular time. A spacetime itself can be viewed as the union of all events in the same way that a line is the union of all of its points, formally organized into a manifold, a space which can be described at small scales using coordinates systems.

A spacetime is independent of any observer.^[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 (in the Lorentz metric) or (in the original Minkowski metric)^[10] where is the speed of light. The metrical descriptions of Minkowski Space and spacelike, lightlike, and timelike intervals given below follow this convention, as do the conventional formulations of the Lorentz transformation

Spacetime intervals

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

(spacetime interval),

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

(Note that the choice of signs for above follows the space-like convention (-+++). Other treatments reverse the sign of .)

Spacetime intervals may be classified into three distinct types based on whether the temporal separation () or the spatial separation () of the two events is greater.

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

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 () and may be said to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur in the same spatial location, but there is no reference frame in which the two events can occur at the same time.
The measure of a time-like spacetime interval is described by the proper time:
(proper time).The proper time interval would be measured by an observer with a clock traveling between the two events in an inertial reference frame, when the observer's path intersects each event as that event occurs. (The proper time defines a real number, since the interior of the square root is positive.)

Light-like interval

In a light-like interval, the spatial distance between two events is exactly balanced by the time between the two events. The events define a squared spacetime interval of zero (). Light-like intervals are also known as "null" intervals.
Events which occur to or are initiated by a photon along its path (i.e., while traveling at , the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a light cone, and all the events which preceded from a light-like interval define a second (graphically inverted, which is to say "pastward") light cone.

Space-like interval

When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a causal relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur at the same time, but there is no reference frame in which the two events can occur in the same spatial location.
For these space-like event pairs with a positive squared spacetime interval (), the measurement of space-like separation is the proper distance:
(proper distance).Like the proper time of time-like intervals, the proper distance () of space-like spacetime intervals is a real number value.

Mathematics of spacetimes
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold . This means the smooth Lorentz metric has signature . The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates are used. Moreover, for simplicity's sake, the speed of light is usually assumed to be unity.
A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event . Another reference frame may be identified by a second coordinate chart about . Two observers (one in each reference frame) may describe the same event but obtain different descriptions.
Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing (representing an observer) and another containing (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.
For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.
Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.

[edit] Topology

Main article: Spacetime 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:^{[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 in special relativity

Main article: Minkowski space

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

where the Landau–Lifshitz space-like convention is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are invariant under Lorentz transformations. This invariance property leads to the use of four-vectors (and other tensors) in describing physics.
Strictly speaking, one can also consider events in Newtonian physics as a single spacetime. This is Galilean-Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a spacetime can be decomposed into spatial coordinates plus temporal coordinates, which is not possible in the general case.


Spacetime in general relativity
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

Main article: Quantum spacetime

In general relativity, spacetime is assumed to be smooth and continuous—and not just in the mathematical sense. In the theory of quantum mechanics, there is an inherent discreteness present in physics. In attempting to reconcile these two theories, it is sometimes postulated that spacetime should be quantized at the very smallest scales. Current theory is focused on the nature of spacetime at the Planck scale. Causal sets, loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized spacetime with agreement on the order of magnitude. Loop quantum gravity makes precise predictions about the geometry of spacetime at the Planck scale.

Privileged character of 3+1 spacetime
There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional). Let the number of spatial dimensions be N and the number of temporal dimensions be T. That N = 3 and T = 1, setting aside the compactified dimensions invoked by string theory and undetectable to date, can be explained by appealing to the physical consequences of letting N differ from 3 and T differ from 1. The argument is often of an anthropic character.
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]