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'' (, ). The peoples of the Andes have kept this understanding until now.

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".

After discovering quaternions, William Rowan Hamilton commented, "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." Hamilton's biquaternions, which have algebraic properties sufficient to model spacetime and its symmetry, were in play for more than a half-century before formal relativity. For instance, William Kingdon Clifford noted their relevance.

Another important antecedent to spacetime was the work of James Clerk Maxwell as he used partial differential equations to develop electrodynamics with the four parameters. Lorentz discovered some invariances of Maxwell's equations late in the 19th century which were to become the basis of 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 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".

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 $(x,y,z,t)$, 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. However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient metrical coordinate system. Events are specified by four real numbers in any such coordinate system. The trajectories of elementary (point-like) particles through space and time are thus a continuum of events called the world line of the particle. Extended or composite objects (consisting of many elementary particles) are thus a union of many world lines twisted together by virtue of their interactions through spacetime into a "world-braid".

However, in physics, it is common to treat an extended object as a "particle" or "field" with its own unique (e.g. center of mass) position at any given time, so that the world line of a particle or light beam is the path that this particle or beam takes in the spacetime and represents the history of the particle or beam. The world line of the orbit of the Earth (in such a description) is depicted in two spatial dimensions ''x'' and ''y'' (the plane of the Earth's orbit) and a time dimension orthogonal to ''x'' and ''y''. The orbit of the Earth is an ellipse in space alone, but its world line is a helix in spacetime.

The unification of space and time is exemplified by the common practice of selecting a metric (the measure that specifies the interval between two events in spacetime) such that all four dimensions are measured in terms of units of distance: representing an event as $(x\_0,x\_1,x\_2,x\_3)\; =\; (ct,x,y,z)$ (in the Lorentz metric) or $(x\_1,x\_2,x\_3,x\_4)\; =\; (x,y,z,ict)$ (in the original Minkowski metric) where $c$ is the speed of light. The metrical descriptions of Minkowski Space and spacelike, lightlike, and timelike intervals given below follow this convention, as do the conventional formulations of the Lorentz transformation.

===Spacetime intervals=== In 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:

$s^2\; =\; \backslash Delta\; r^2\; -\; c^2\backslash Delta\; t^2\; \backslash ,$   (spacetime interval),

where ''c'' is the speed of light, and Δ''r'' and Δ''t'' denote differences of the space and time coordinates, respectively, between the events.

(Note that the choice of signs for $s^2$ above follows the space-like convention (-+++). Other treatments reverse the sign of $s^2$.)

Spacetime intervals may be classified into three distinct types based on whether the temporal separation ($c^2\; \backslash Delta\; t^2$) or the spatial separation ($\backslash Delta\; r^2$) 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]

:''''''

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:

$\backslash Delta\backslash tau\; =\; \backslash sqrt\{\backslash Delta\; t^2\; -\; \backslash frac\{\backslash Delta\; r^2\}\{c^2\}\}$   (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]

:''' '''

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 ($s^2\; =\; 0$). Light-like intervals are also known as "null" intervals.

Events which occur to or are initiated by a photon along its path (i.e., while traveling at $c$, the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a light cone, and all the events which preceded from a light-like interval define a second (graphically inverted, which is to say "''pastward''") light cone.

Space-like interval[link]

:''''''

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 ($s^2\; >\; 0$), the measurement of space-like separation is the proper distance:

$\backslash Delta\backslash sigma\; =\; \backslash sqrt\{\backslash Delta\; r^2\; -\; c^2\backslash Delta\; t^2\}$   (proper distance).

Like the proper time of time-like intervals, the proper distance ($\backslash Delta\backslash sigma$) 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 $(M,g)$. This means the smooth Lorentz metric $g$ has signature $(3,1)$. The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates $(x,\; y,\; z,\; t)$ are used. Moreover, for simplicity's sake, the speed of light $c$ is usually assumed to be unity.

A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event $p$. Another reference frame may be identified by a second coordinate chart about $p$. Two observers (one in each reference frame) may describe the same event $p$ but obtain different descriptions.

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing $p$ (representing an observer) and another containing $q$ (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.

For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event $p$). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples $(x,\; y,\; z,\; t)$ (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.

Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.

Topology[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:

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 $\backslash eta$ and can be written as a four-by-four matrix:

:$\backslash eta\_\{ab\}\; \backslash ,\; =\; \backslash operatorname\{diag\}(1,\; -1,\; -1,\; -1)$

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.

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 4π''r'' 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. Ehrenfest also showed that if ''N'' is even, then the different parts of a wave impulse will travel at different speeds. If ''N'' > 3 and odd, then wave impulses become distorted. Only when ''N'' = 3 or 1 are both problems avoided. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only when ''N'' = 3 and ''T'' = 1, writing that this fact "...not only leads to a deeper understanding of Maxwell's theory, but also of the fact that the world is four dimensional, which has hitherto always been accepted as merely 'accidental,' become intelligible through it." Finally, Tangherlini showed in 1963 that when ''N'' > 3, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.

Max Tegmark expands on the preceding argument in the following anthropic manner. If ''T'' differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if ''T'' > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) If ''N'' > 3, Ehrenfest's argument above holds; atoms as we know them (and probably more complex structures as well) could not exist. If ''N'' < 3, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, when ''N'' < 3, nerves cannot cross without intersecting.

In general, it is not clear how physical law could function if ''T'' differed from 1. If ''T'' > 1, subatomic particles which decay after a fixed period would not behave predictably, because time-like geodesics would not be necessarily maximal. ''N'' = 1 and ''T'' = 3 has the peculiar property that the speed of light in a vacuum is a ''lower bound'' on the velocity of matter; all matter consists of tachyons.

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. ''N'' = 3 was the last case of the Poincaré conjecture to be proved.

For an elementary treatment of the privileged status of ''N'' = 3 and ''T'' = 1, see chpt. 10 (esp. Fig. 10.12) of Barrow; for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark. Barrow has repeatedly cited the work of Whitrow.

String theory hypothesizes that matter and energy are composed of tiny vibrating strings of various types, most of which are embedded in dimensions that exist only on a scale no larger than the Planck length. Hence ''N'' = 3 and ''T'' = 1 do not characterize string theory, which embeds vibrating strings in coordinate grids having 10, or even 26, dimensions.

The Causal dynamical triangulation (CDT) theory is a background independent theory which derives the observed 3+1 spacetime from a minimal set of assumptions, and needs no adjusting factors. It does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. It shows spacetime to be 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.

See also[link]

Notes[link]

References[link]

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

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.

Chpts. 17–18.

Erwin Schrödinger (1950) ''Space-time structure''. Cambridge Univ. Press.

(pp. 5–6)

External links[link]

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

Category:Fundamental physics concepts Category:Theory of relativity

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name	Karen Gillan
birth date	November 28, 1987
birth place	Inverness, Scotland
birthname	Karen Sheila Gillan
occupation	Actress, former model
yearsactive	2006–present }}

Karen Sheila Gillan (born 28 November 1987) is a Scottish actress and former model who is mostly known for her portrayal of Amy Pond in the British science fiction series ''Doctor Who''.

Early life[link]

Gillan was born and brought up in Inverness, Scotland. Her father, Raymond John Gillan, a Day Centre Manager, and mother, Marie Gillan née Paterson, live in Kinmylies. She learned to play the piano when she was seven. She has no brothers or sisters. At a young age, she developed a love for acting, joining several local youth theatre groups and taking part in a wide range of productions at her school, Charleston Academy. When she was sixteen, Gillan decided to use her interest in acting, studying at Edinburgh's Telford College. Later, she got a place at the Italia Conti Academy of Theatre Arts drama school in London on the BA (Hons) Acting degree course. Gillan moved to London when she was 18 to study at drama school.

Career[link]

Modelling [link]

Before Gillan became an actress, she was scouted by a modelling agency.

Gillan worked as a model in 2007's London Fashion Week for designer Allegra Hicks' autumn/winter catwalk show and the launch party of Nicola Roberts' Dainty Doll Make-Up Range. This was shown in Roberts' episode of the documentary, ''The Passions of Girls Aloud'' which premiered on 4 April 2008. Gillan has said she wouldn't give up her acting career to return back to modelling. She stated that she enjoyed modelling but acting was always her main interest and goal.

Acting [link]

Gillan's early television acting career included guest appearances on several drama series, with her first ongoing role being a two-year stint as a member of the ensemble cast of the sketch comedy series ''The Kevin Bishop Show'' in which she played multiple characters, as well as celebrities such as Katy Perry and Angelina Jolie. She appeared in Channel 4's 'Stacked'; she had the part of Ginny. Gillan also appeared on TV in a leading role in a horror project entitled ''The Well'', which was broadcast as a series of episodic short films on BBC Two, and then later also as a web series on BBC.co.uk. Part of the BBC's multimedia "switch" programming, the short episodes interconnect with online games that further explore the environments presented in the series.

She was cast for the role of the Eleventh Doctor's first companion, Amy Pond, on the British sci-fi series ''Doctor Who'' in May 2009. She made her first on-screen appearance as Amy Pond in "The Eleventh Hour" with her cousin Caitlin Blackwood portraying a young Amy Pond. She was confirmed in Issue 420 of the ''Doctor Who Magazine'' to be appearing in the 2011 series. Gillan had appeared in ''Doctor Who'' once before in the 2008 episode "The Fires of Pompeii" in the role of a Soothsayer.

Gillan made her first theatre appearance playing the role of Shirley in John Osborne's play ''Inadmissible Evidence'' along with Douglas Hodge. The play debuted at the Donmar Warehouse on 16 October 2011.

During August 2011, it was announced Gillan will star in an indie comedy called ''Not Another Happy Ending'' along with Emun Elliott. Filming will take place some time in 2012.

During an interview at the 2011 San Diego Comic-Con, Gillan revealed she will be returning to ''Doctor Who'' for its seventh series. On 15 December 2011, it was announced Gillan and Arthur Darvill (Rory Williams) would be leaving ''Doctor Who'' during the seventh series in 2012. On 26 January 2012 Karen Gillan played the part of supermodel Jean Shrimpton in the BBC Four film, ''We'll Take Manhattan'' which told the story of Shrimpton's relationship with the photographer David Bailey.

She has also been cast as Brittney in a re-working of Shakespeare's ''Romeo and Juliet'' called ''Romeo & Brittney'' by David Baddiel.

Charity work [link]

In 2011 Gillan helped to promote ''Fashion Targets Breast Cancer (FTBC)'' and also helped celebrate the opening of ''Squirrel Ward'' in Great Ormond Street Hospital.

Filmography[link]

;Television ''The Kevin Bishop Show ''Coming Up (TV series) ''The Well (TV Series)'' ''Doctor Who'' ''We'll Take Manhattan (film)

Year
2006	Rebus (TV series)>Rebus''	Terry Cotter
2007–2009	|	Various
rowspan=3	2008	Coming Up'' ||	Anna
''Doctor Who''	Soothsayer
''Harley Street (TV series)	Harley Street	Holly
2010	|	Coll	Aired on BBC 2, then later a web series on the BBC website
2010-2012	|	Amy Pond	Series 5 (2010), Series 6 (2011) & Series 7 (5 episodes) (2012)
rowspan=2>2012	We'll Take Manhattan'' ||	Jean Shrimpton
''In Love With''	Laura

;Feature films

Year !! Title !! Role
rowspan=2	2008	''Stacked (TV film) ''''	Ginny Turner
''New Town Killers''	Girl at bus station
2010	Outcast (2010 film)>Outcast''
2012	''Not Another Happy Ending''

;Theatre

Shirley

Year !! Title !! Role !! Theatre !! Notes
2011	''Inadmissible Evidence''	|	Donmar Warehouse, London	Gillan's first theatre appearance

;Self

Year !! Title !! Notes
2010-2012	''Doctor Who Confidential''	Behind the scenes and making of Doctor Who
2010	''Doctor Who Prom''

;Video games

Year !! Title !! Role
rowspan=7	2010	''Doctor Who: The Adventure Games'': ''City of the Daleks''	Amy Pond
''Doctor Who: The Adventure Games'': ''Blood of the Cybermen''
''Doctor Who: Evacuation Earth''
''Doctor Who: Return to Earth''
''Doctor Who: The Mazes of Time''
2011	Doctor Who: The Adventure Games#The Gunpowder Plot>The Gunpowder Plot''

Awards[link]

+ List of awards and award nominations
! Year	! Award	! Category	! Nominated work	! Result
2010	TV Quick Award	Best Actress	''Doctor Who''
2010	Cosmopolitan 'Woman of the year' Awards	Best Actress	''Doctor Who''
2011	SFX Awards	Best Actress	''Doctor Who''
2011	TV Choice Awards	Best Actress	''Doctor Who''
2011	Scream Awards	Best Sci-Fi Actress	''Doctor Who''
2012	National Television Awards	Best Drama Performance: Female	''Doctor Who''

References[link]

External links[link]

Category:1987 births Category:Living people Category:Alumni of the Italia Conti Academy of Theatre Arts Category:People from Inverness Category:Scottish female models Category:Scottish film actors Category:Scottish television actors

cs:Karen Gillanová de:Karen Gillan es:Karen Gillan fr:Karen Gillan it:Karen Gillan nl:Karen Gillan ja:カレン・ギラン no:Karen Gillan pl:Karen Gillan pt:Karen Gillan ru:Гиллан, Карен simple:Karen Gillan tr:Karen Gillan zh:凯伦·吉兰

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'' (, ). The peoples of the Andes have kept this understanding until now.

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".

After discovering quaternions, William Rowan Hamilton commented, "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." Hamilton's biquaternions, which have algebraic properties sufficient to model spacetime and its symmetry, were in play for more than a half-century before formal relativity. For instance, William Kingdon Clifford noted their relevance.

Another important antecedent to spacetime was the work of James Clerk Maxwell as he used partial differential equations to develop electrodynamics with the four parameters. Lorentz discovered some invariances of Maxwell's equations late in the 19th century which were to become the basis of 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 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".

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 $(x,y,z,t)$, 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. However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient metrical coordinate system. Events are specified by four real numbers in any such coordinate system. The trajectories of elementary (point-like) particles through space and time are thus a continuum of events called the world line of the particle. Extended or composite objects (consisting of many elementary particles) are thus a union of many world lines twisted together by virtue of their interactions through spacetime into a "world-braid".

However, in physics, it is common to treat an extended object as a "particle" or "field" with its own unique (e.g. center of mass) position at any given time, so that the world line of a particle or light beam is the path that this particle or beam takes in the spacetime and represents the history of the particle or beam. The world line of the orbit of the Earth (in such a description) is depicted in two spatial dimensions ''x'' and ''y'' (the plane of the Earth's orbit) and a time dimension orthogonal to ''x'' and ''y''. The orbit of the Earth is an ellipse in space alone, but its world line is a helix in spacetime.

The unification of space and time is exemplified by the common practice of selecting a metric (the measure that specifies the interval between two events in spacetime) such that all four dimensions are measured in terms of units of distance: representing an event as $(x\_0,x\_1,x\_2,x\_3)\; =\; (ct,x,y,z)$ (in the Lorentz metric) or $(x\_1,x\_2,x\_3,x\_4)\; =\; (x,y,z,ict)$ (in the original Minkowski metric) where $c$ is the speed of light. The metrical descriptions of Minkowski Space and spacelike, lightlike, and timelike intervals given below follow this convention, as do the conventional formulations of the Lorentz transformation.

===Spacetime intervals=== In 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:

$s^2\; =\; \backslash Delta\; r^2\; -\; c^2\backslash Delta\; t^2\; \backslash ,$   (spacetime interval),

where ''c'' is the speed of light, and Δ''r'' and Δ''t'' denote differences of the space and time coordinates, respectively, between the events.

(Note that the choice of signs for $s^2$ above follows the space-like convention (-+++). Other treatments reverse the sign of $s^2$.)

Spacetime intervals may be classified into three distinct types based on whether the temporal separation ($c^2\; \backslash Delta\; t^2$) or the spatial separation ($\backslash Delta\; r^2$) 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]

:''''''

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:

$\backslash Delta\backslash tau\; =\; \backslash sqrt\{\backslash Delta\; t^2\; -\; \backslash frac\{\backslash Delta\; r^2\}\{c^2\}\}$   (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]

:''' '''

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 ($s^2\; =\; 0$). Light-like intervals are also known as "null" intervals.

Events which occur to or are initiated by a photon along its path (i.e., while traveling at $c$, the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a light cone, and all the events which preceded from a light-like interval define a second (graphically inverted, which is to say "''pastward''") light cone.

Space-like interval[link]

:''''''

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 ($s^2\; >\; 0$), the measurement of space-like separation is the proper distance:

$\backslash Delta\backslash sigma\; =\; \backslash sqrt\{\backslash Delta\; r^2\; -\; c^2\backslash Delta\; t^2\}$   (proper distance).

Like the proper time of time-like intervals, the proper distance ($\backslash Delta\backslash sigma$) 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 $(M,g)$. This means the smooth Lorentz metric $g$ has signature $(3,1)$. The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates $(x,\; y,\; z,\; t)$ are used. Moreover, for simplicity's sake, the speed of light $c$ is usually assumed to be unity.

A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event $p$. Another reference frame may be identified by a second coordinate chart about $p$. Two observers (one in each reference frame) may describe the same event $p$ but obtain different descriptions.

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing $p$ (representing an observer) and another containing $q$ (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.

For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event $p$). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples $(x,\; y,\; z,\; t)$ (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.

Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.

Topology[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:

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 $\backslash eta$ and can be written as a four-by-four matrix:

:$\backslash eta\_\{ab\}\; \backslash ,\; =\; \backslash operatorname\{diag\}(1,\; -1,\; -1,\; -1)$

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.

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 4π''r'' 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. Ehrenfest also showed that if ''N'' is even, then the different parts of a wave impulse will travel at different speeds. If ''N'' > 3 and odd, then wave impulses become distorted. Only when ''N'' = 3 or 1 are both problems avoided. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only when ''N'' = 3 and ''T'' = 1, writing that this fact "...not only leads to a deeper understanding of Maxwell's theory, but also of the fact that the world is four dimensional, which has hitherto always been accepted as merely 'accidental,' become intelligible through it." Finally, Tangherlini showed in 1963 that when ''N'' > 3, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.

Max Tegmark expands on the preceding argument in the following anthropic manner. If ''T'' differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if ''T'' > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) If ''N'' > 3, Ehrenfest's argument above holds; atoms as we know them (and probably more complex structures as well) could not exist. If ''N'' < 3, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, when ''N'' < 3, nerves cannot cross without intersecting.

In general, it is not clear how physical law could function if ''T'' differed from 1. If ''T'' > 1, subatomic particles which decay after a fixed period would not behave predictably, because time-like geodesics would not be necessarily maximal. ''N'' = 1 and ''T'' = 3 has the peculiar property that the speed of light in a vacuum is a ''lower bound'' on the velocity of matter; all matter consists of tachyons.

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. ''N'' = 3 was the last case of the Poincaré conjecture to be proved.

For an elementary treatment of the privileged status of ''N'' = 3 and ''T'' = 1, see chpt. 10 (esp. Fig. 10.12) of Barrow; for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark. Barrow has repeatedly cited the work of Whitrow.

String theory hypothesizes that matter and energy are composed of tiny vibrating strings of various types, most of which are embedded in dimensions that exist only on a scale no larger than the Planck length. Hence ''N'' = 3 and ''T'' = 1 do not characterize string theory, which embeds vibrating strings in coordinate grids having 10, or even 26, dimensions.

The Causal dynamical triangulation (CDT) theory is a background independent theory which derives the observed 3+1 spacetime from a minimal set of assumptions, and needs no adjusting factors. It does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. It shows spacetime to be 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.

See also[link]

Notes[link]

References[link]

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

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.

Chpts. 17–18.

Erwin Schrödinger (1950) ''Space-time structure''. Cambridge Univ. Press.

(pp. 5–6)

External links[link]

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

Category:Fundamental physics concepts Category:Theory of relativity

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Name	Roger Penrose
Birth date	August 08, 1931
Birth place	Colchester, Essex, England
Residence	United Kingdom Canada (During WWII)
Nationality	British
Fields	Mathematical physicsMathematical sciences
Workplaces	Bedford College, London St John's College, Cambridge Princeton University Syracuse University King’s College, London Birkbeck, University of London University of Oxford
Alma mater	University of Cambridge University College London University College School
Doctoral advisor	John A. Todd
Academic advisors	William Hodge
Doctoral students	Tristan NeedhamRichard JozsaRichard S. WardAndrew HodgesAsghar QadirGeorge Burnett-StuartMatthew GinsbergAdam HelferLane P. HughstonPeter LawClaude LeBrunRoss MooreDuncan StoneTim PostonGeorge SparlingK. Paul Tod
Known for	Twistor theoryGeometry of spacetime Cosmic censorshipWeyl curvature hypothesisPenrose inequalitiesPenrose interpretation of Quantum TheoryOrch-OR Moore–Penrose pseudoinverseNewman-Penrose formalismPenrose tilingPenrose stairsPenrose graphical notationSchrödinger–Newton equations
Influences	Dennis W. Sciama
Influenced	Michael Atiyah Stuart Hameroff
Awards	Wolf Prize (1988) Dirac Medal (1989) Copley Medal (2008)
Religion	Materialist Platonist with no doctrinal stance
Footnotes	He is the brother of Jonathan Penrose and Oliver Penrose, and son of Lionel Penrose. He is the nephew of Roland Penrose. }}

Sir Roger Penrose (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. He has received a number of prizes and awards, including the 1988 Wolf Prize for physics which he shared with Stephen Hawking for their contribution to our understanding of the universe. He is renowned for his work in mathematical physics, in particular his contributions to general relativity and cosmology. He is also a recreational mathematician and philosopher.

Career[link]

He was born in Colchester, Essex, England, Roger Penrose is a son of Lionel S. Penrose and Margaret Leathes. Penrose is the brother of mathematician Oliver Penrose and of chess Grandmaster Jonathan Penrose. Penrose attended University College School and University College, London, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalized matrix inverse, also known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar (1951). Penrose earned his Ph.D. at Cambridge (St John's College) in 1958, writing a thesis on "tensor methods in algebraic geometry" under algebraist and geometer John A. Todd. He devised and popularised the Penrose triangle in the 1950s, describing it as "impossibility in its purest form" and exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it. Escher's Waterfall, and Ascending and Descending were in turn inspired by Penrose. As reviewer Manjit Kumar puts it:

As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tri-bar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up and down. An article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces.

In 1965, at Cambridge, Penrose proved that singularities (such as black holes) could be formed from the gravitational collapse of immense, dying stars. This work was extended by Hawking to prove the Penrose–Hawking singularity theorems.

In 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature (2,2). In 1969, he conjectured the cosmic censorship hypothesis. This proposes (rather informally) that the universe protects us from the inherent unpredictability of singularities (such as the one in the centre of a black hole) by hiding them from our view behind an event horizon. This form is now known as the "weak censorship hypothesis"; in 1979, Penrose formulated a stronger version called the "strong censorship hypothesis". Together with the BKL conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. Also from 1979 dates Penrose's influential Weyl curvature hypothesis on the initial conditions of the observable part of the Universe and the origin of the second law of thermodynamics. Penrose and James Terrell independently realized that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation. This effect has come to be called the Terrell rotation or Penrose–Terrell rotation.

Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, and are the first tilings to exhibit fivefold rotational symmetry. Penrose developed these ideas based on the article ''Deux types fondamentaux de distribution statistique'' (1938; an English translation ''Two Basic Types of Statistical Distribution'') by Czech geographer, demographer and statistician Jaromír Korčák. In 1984, such patterns were observed in the arrangement of atoms in quasicrystals. Another noteworthy contribution is his 1971 invention of spin networks, which later came to form the geometry of spacetime in loop quantum gravity. He was influential in popularizing what are commonly known as Penrose diagrams (causal diagrams). In 2004 Penrose released ''The Road to Reality: A Complete Guide to the Laws of the Universe'', a 1,099-page book aimed at giving a comprehensive guide to the laws of physics. He has proposed a novel interpretation of quantum mechanics. In 2010, Penrose reported possible evidence, based on concentric circles found in WMAP data of the CMB sky, of an earlier universe existing before the Big Bang of our own present universe.

Penrose is the Francis and Helen Pentz Distinguished (visiting) Professor of Physics and Mathematics at Pennsylvania State University. Penrose is married to Vanessa Thomas, head of mathematics at Abingdon School, with whom he has one son. He has three sons from a previous marriage to American Joan Isabel Wedge, whom he married in 1959.

Physics and consciousness[link]

Penrose has written books on the connection between fundamental physics and human (or animal) consciousness. In ''The Emperor's New Mind'' (1989), he argues that known laws of physics are inadequate to explain the phenomenon of consciousness. Penrose proposes the characteristics this new physics may have and specifies the requirements for a bridge between classical and quantum mechanics (what he calls ''correct quantum gravity''). Penrose uses a variant of Turing's halting theorem to demonstrate that a system can be deterministic without being algorithmic. (E.g., imagine a system with only two states, ON and OFF. The system's state is ON if a given Turing machine halts, and OFF if the Turing machine does not halt, then the system's state is completely determined by the Turing machine, however there is no algorithmic way to determine whether the Turing machine stops.) Penrose believes that such deterministic non-algorithmic processes may come in play in the quantum mechanical wave function reduction, and may be harnessed by the brain. He argues that the present computer is unable to have intelligence because it is an algorithmically deterministic system. He argues against the viewpoint that the rational processes of the mind are completely algorithmic and can thus be duplicated by a sufficiently complex computer. This contrasts with supporters of strong artificial intelligence, who contend that thought can be simulated algorithmically. He bases this on claims that consciousness transcends formal logic because things such as the insolubility of the halting problem and Gödel's incompleteness theorem prevent an algorithmically based system of logic from reproducing such traits of human intelligence as mathematical insight. These claims were originally espoused by the philosopher John Lucas of Merton College, Oxford. The Penrose/Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence has been widely criticized by mathematicians, computer scientists and philosophers, and the consensus among experts in these fields seems to be that the argument fails, though different authors may choose different aspects of the argument to attack. Marvin Minsky, a leading proponent of artificial intelligence, was particularly critical, stating that Penrose "tries to show, in chapter after chapter, that human thought cannot be based on any known scientific principle." Minsky's position is exactly the opposite - he believes that humans are, in fact, machines, whose functioning, although complex, is fully explainable by current physics. Minsky maintains that "one can carry that quest [for scientific explanation] too far by only seeking new basic principles instead of attacking the real detail. This is what I see in Penrose's quest for a new basic principle of physics that will account for consciousness."

Penrose responded to criticism of ''The Emperor's New Mind'' with his follow up 1994 book ''Shadows of the Mind'', and in 1997 with ''The Large, the Small and the Human Mind''. In those works, he also combined his observations with that of anesthesiologist Stuart Hameroff.

Penrose and Hameroff have argued that consciousness is the result of quantum gravity effects in microtubules, which they dubbed Orch-OR (orchestrated objective reduction). Max Tegmark, in a paper in ''Physical Review E'', calculated that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 10,000,000,000. The reception of the paper is summed up by this statement in Tegmark's support: "Physicists outside the fray, such as IBM's John A. Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior'". Tegmark's paper has been widely cited by critics of the Penrose–Hameroff position.

In their reply to Tegmark's paper, also published in ''Physical Review E'', the physicists Scott Hagan, Jack Tuszynski and Hameroff claimed that Tegmark did not address the Orch-OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch-OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, but still well short of the 25 ms required if the quantum processing in the theory was to be linked to the 40 Hz gamma synchrony, as Orch-OR suggested. To bridge this gap, the group made a series of proposals. It was supposed that the interiors of neurons could alternate between liquid and gel states. In the gel state, it was further hypothesized that the water electrical dipoles are oriented in the same direction, along the outer edge of the microtubule tubulin subunits. Hameroff et al. proposed that this ordered water could screen any quantum coherence within the tubulin of the microtubules from the environment of the rest of the brain. Each tubulin also has a tail extending out from the microtubules, which is negatively charged, and therefore attracts positively charged ions. It is suggested that this could provide further screening. Further to this, there was a suggestion that the microtubules could be pumped into a coherent state by biochemical energy. Finally, it is suggested that the configuration of the microtubule lattice might be suitable for quantum error correction, a means of holding together quantum coherence in the face of environmental interaction. In the last decade, some researchers who are sympathetic to Penrose's ideas have proposed an alternative scheme for quantum processing in microtubules based on the interaction of tubulin tails with microtubule-associated proteins, motor proteins and presynaptic scaffold proteins. These proposed alternative processes have the advantage of taking place within Tegmark's time to decoherence.

Hameroff, in a lecture in part of a Google Tech talks series exploring Quantum biology, gave an overview of current research in the area, and responded to subsequent criticisms of the Orch-OR model. In addition to this, a recent 2011 paper by Roger Penrose and Stuart Hameroff gives an updated model of their Orch-OR theory, in light of criticisms, and discusses the place of consciousness within the universe.

Phillip Tetlow, although himself supportive of Penrose's views, acknowledges that Penrose's ideas about the human thought process are at present a minority view in scientific circles, citing Minsky's criticisms and quoting science journalist Charles Seife's description of Penrose as "one of a handful of scientists" who believe that the nature of consciousness suggests a quantum process.

Religious views[link]

Penrose does not hold to any religious doctrine, and refers to himself as an atheist. In the film ''A Brief History of Time'', he said, "I think I would say that the universe has a purpose, it's not somehow just there by chance ... some people, I think, take the view that the universe is just there and it runs along–it's a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it." Penrose is a Distinguished Supporter of the British Humanist Association.

Awards and honours[link]

Penrose has been awarded many prizes for his contributions to science. He was elected a Fellow of the Royal Society of London in 1972. In 1975, Stephen Hawking and Penrose were jointly awarded the Eddington Medal of the Royal Astronomical Society. In 1985, he was awarded the Royal Society Royal Medal. Along with Stephen Hawking, he was awarded the prestigious Wolf Foundation Prize for Physics in 1988. In 1989 he was awarded the Dirac Medal and Prize of the British Institute of Physics. In 1990 Penrose was awarded the Albert Einstein Medal for outstanding work related to the work of Albert Einstein by the Albert Einstein Society. In 1991, he was awarded the Naylor Prize of the London Mathematical Society. From 1992 to 1995 he served as President of the International Society on General Relativity and Gravitation. In 1994, Penrose was knighted for services to science. In the same year he was also awarded an Honorary Degree (Doctor of Science) by the University of Bath. In 1998, he was elected Foreign Associate of the United States National Academy of Sciences. In 2000 he was appointed to the Order of Merit. In 2004 he was awarded the De Morgan Medal for his wide and original contributions to mathematical physics. To quote the citation from the London Mathematical Society: :''His deep work on General Relativity has been a major factor in our understanding of black holes. His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.'' In 2005 Penrose was awarded an honorary doctorate (Honoris Causa) by Warsaw University and Katholieke Universiteit Leuven (Belgium), and in 2006 by the University of York. In 2008 Penrose was awarded the Copley Medal. He is also a Distinguished Supporter of the British Humanist Association and one of the patrons of the Oxford University Scientific Society. In 2011, Penrose was awarded the Fonseca prize by the University of Santiago de Compostela. In 2012 Penrose was awarded the Richard R. Ernst Medal by ETHZ for his contributions to science and strengthening the connection between science and society.

Works[link]

''Techniques of Differential Topology in Relativity'' (1972, ISBN 0-89871-005-7)

''Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields'' (with Wolfgang Rindler, 1987) ISBN 0-521-33707-0 (paperback)

''Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry'' (with Wolfgang Rindler, 1988) (reprint), ISBN 0-521-34786-6 (paperback)

''The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics'' (1989, ISBN 0-14-014534-6 (paperback); it received the Rhone-Poulenc science book prize in 1990)

''Shadows of the Mind: A Search for the Missing Science of Consciousness'' (1994, ISBN 0-19-853978-9 (hardback))

''The Nature of Space and Time'' (with Stephen Hawking, 1996, ISBN 0-691-03791-4 (hardback), ISBN 0-691-05084-8 (paperback))

''The Large, the Small, and the Human Mind'' (with Abner Shimony, Nancy Cartwright, and Stephen Hawking, 1997, ISBN 0-521-56330-5 (hardback), ISBN 0-521-65538-2 (paperback), Canto edition: ISBN 0-521-78572-3)

''White Mars or, The Mind Set Free'' (with Brian W. Aldiss, 1999, ISBN 978-0-316-85243-2 (hardback))

''The Road to Reality: A Complete Guide to the Laws of the Universe'' (2004, ISBN 0-224-04447-8 (hardcover), ISBN 0-09-944068-7 (paperback))

''Cycles of Time: An Extraordinary New View of the Universe'' (Bodley Head (23 Sep 2010) ISBN 978-0-224-08036-1)

Penrose also wrote forewords to ''Quantum Aspects of Life'' and Zee's book ''Fearful Symmetry''.

See also[link]

Conformal Cyclic Cosmology

Orch-OR

Quantum mind

Multiverse

''Quantum Aspects of Life'' (book)

Illumination problem

References[link]

Further reading[link]

Ferguson, Kitty (1991). ''Stephen Hawking: Quest For A Theory of Everything''. Franklin Watts. ISBN 0-553-29895-X.

; see ''Box 34.2''.

External links[link]

Dangerous Knowledge – Penrose was one of the principal interviewees in a BBC documentary about the mathematics of infinity directed by David Malone

Penrose's new theory "Aeons Before the Big Bang?":

*Original 2005 lecture: "Before the Big Bang? A new perspective on the Weyl curvature hypothesis" (Isaac Newton Institute for Mathematical Sciences, Cambridge, Nov 11, 2005).

*Original publication: "Before the Big Bang: an outrageous new perspective and its implications for particle physics". ''Proceedings of EPAC 2006''. Edinburgh. 2759–2762 (cf. also Hill, C.D. & Nurowski, P. (2007) "On Penrose's 'Before the Big Bang' ideas". Ithaca)

* Revised 2009 lecture: "Aeons Before the Big Bang?" (Georgia Institute of Technology, Center for Relativistic Astrophysics)

* BBC interview on the new theory

Roger Penrose on The Forum

Penrose on sidestepping reason

Shaking Up Foundations of Math: Roger Penrose on Kurt Gödel's Groundbreaking Work

Hilary Putnam's review of Penrose's 'Shadows of the Mind' claiming that Penrose's use of Godel's Incompleteness Theorem is fallacious

*Beyond the Doubting of a Shadow: A Reply to Commentaries on Shadows of the Mind

Penrose Tiling found in Islamic Architecture

Two theories for the formation of quasicrystals resembling Penrose tilings

Tegmark, Max. 2000. "The importance of quantum decoherence in brain processes". ''Physical Review E''. vol 61. pp. 4194–4206.

*"Biological feasibility of quantum states in the brain" – (a disputation of Tegmark's result by Hagan, Hameroff, and Tuszyński)

**Tegmarks's rejoinder to Hagan ''et al.''

"Toilet Paper Plagiarism" – D. Trull about Penrose's lawsuit concerning the use of his Penrose tilings on toilet paper

Roger Penrose: A Knight on the tiles (Plus magazine)

Penrose's Gifford Lecture biography

Quantum-Mind

Audio: Roger Penrose in conversation on the BBC World Service discussion show

''The Forum''

Roger Penrose speaking about Hawking's new book on Premier Christian Radio

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Helen Fay Dowker (born 1965 in Manchester, Lancashire) is a British Theoretical Physicist at Imperial College London.

As a student, she was interested in wormholes and quantum cosmology. Having studied at the University of Cambridge, Dowker was awarded the Tyson Medal in 1987 and completed her Doctor of Philosophy under the supervision of Stephen Hawking in 1990.

Fay Dowker was later a Postdoctoral Researcher at the Fermi National Accelerator Laboratory, at the University of California at Santa Barbara and also the California Institute of Technology.

She is currently a Professor of Theoretical Physics and a member of the Theoretical Physics Group at Imperial College London and an Affiliate of the Institute for Quantum Computing. She conducts research in a number of areas of theoretical physics including quantum gravity and causal set theory.

Dowker was previously a Lecturer at Queen Mary, University of London.

References[link]

External links[link]

Theoretical Physics Group at Imperial College London

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