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108-481: Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However, space and time took on new meanings with the Lorentz transformation and special theory of relativity . In 1908, Hermann Minkowski presented

216-451: A . The pulse is reflected from a mirror situated a distance a from the light source (event Q), and returns to the light source at x ′ = 0,  ct ′ =  a (event R). The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have slopes = 1 and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to

324-402: A concept of neighbourhood is defined, frequently by means of a distance ( metric spaces ). The elements of a space are often called points , but they can have other names such as vectors in vector spaces and functions in function spaces . Space is one of the few fundamental quantities in physics , meaning that it cannot be defined via other quantities because nothing more fundamental

432-440: A flat surface. After a while, as the bucket continues to spin, the surface of the water becomes concave. If the bucket's spinning is stopped then the surface of the water remains concave as it continues to spin. The concave surface is therefore apparently not the result of relative motion between the bucket and the water. Instead, Newton argued, it must be a result of non-inertial motion relative to space itself. For several centuries

540-445: A fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space . The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation ) or

648-420: A geometric interpretation of special relativity that fused time and the three spatial dimensions into a single four-dimensional continuum now known as Minkowski space . This interpretation proved vital to the general theory of relativity , wherein spacetime is curved by mass and energy . Non-relativistic classical mechanics treats time as a universal quantity of measurement that is uniform throughout,

756-453: A great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor. He never made a priority claim and always gave Einstein his full share in the great discovery. Minkowski had been concerned with

864-574: A light signal in that same time interval Δ t {\displaystyle \Delta t} . If the event separation is due to a light signal, then this difference vanishes and Δ s = 0 {\displaystyle \Delta s=0} . When the event considered is infinitesimally close to each other, then we may write In a different inertial frame, say with coordinates ( t ′ , x ′ , y ′ , z ′ ) {\displaystyle (t',x',y',z')} ,

972-433: A location. In Fig. 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks is used to determine the time and position of events taking place within the whole frame. The term observer refers to

1080-441: A mere shadow, and only some sort of union of the two shall preserve independence." Space and Time included the first public presentation of spacetime diagrams (Fig. 1-4), and included a remarkable demonstration that the concept of the invariant interval ( discussed below ), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity. The spacetime concept and

1188-417: A person moving with respect to the first observer will see the two events occurring at different places, because the moving point of view sees itself as stationary, and the position of the event as receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events. In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as

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1296-443: A plane or sphere and, Poincaré argued, the same is true for the debate over whether real space is Euclidean or not. For him, which geometry was used to describe space was a matter of convention . Since Euclidean geometry is simpler than non-Euclidean geometry, he assumed the former would always be used to describe the 'true' geometry of the world. In 1905, Albert Einstein published his special theory of relativity , which led to

1404-498: A point P not on L 1 , there is exactly one straight line L 2 on the plane that passes through the point P and is parallel to the straight line L 1 . Until the 19th century, few doubted the truth of the postulate; instead debate centered over whether it was necessary as an axiom, or whether it was a theory that could be derived from the other axioms. Around 1830 though, the Hungarian János Bolyai and

1512-481: A similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people. Leibniz argued that space could not exist independently of objects in the world because that implies a difference between two universes exactly alike except for the location of the material world in each universe. But since there would be no observational way of telling these universes apart then, according to

1620-412: A single point in spacetime. Although it is possible to be in motion relative to the popping of a firecracker or a spark, it is not possible for an observer to be in motion relative to an event. The path of a particle through spacetime can be considered to be a sequence of events. The series of events can be linked together to form a curve that represents the particle's progress through spacetime. That path

1728-433: A spacetime diagram illustrating the world lines (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by c {\displaystyle c} so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at

1836-490: A spatial distance Δ x . {\displaystyle \Delta x.} Then the squared spacetime interval ( Δ s ) 2 {\displaystyle (\Delta {s})^{2}} between the two events that are separated by a distance Δ x {\displaystyle \Delta {x}} in space and by Δ c t = c Δ t {\displaystyle \Delta {ct}=c\Delta t} in

1944-544: A standard meter or simply meter, is defined as the distance traveled by light in vacuum during a time interval of exactly 1/299,792,458 of a second. This definition coupled with present definition of the second is based on the special theory of relativity in which the speed of light plays the role of a fundamental constant of nature. Geography is the branch of science concerned with identifying and describing places on Earth , utilizing spatial awareness to try to understand why things exist in specific locations. Cartography

2052-405: A way that all objects expand and contract in similar proportions in different places on the sphere. With a suitable falloff in temperature, if the scientists try to use measuring rods to determine the sum of the angles in a triangle, they can be deceived into thinking that they inhabit a plane, rather than a spherical surface. In fact, the scientists cannot in principle determine whether they inhabit

2160-641: A year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the Göttingen Mathematical society with the title, The Relativity Principle ( Das Relativitätsprinzip ). On 21 September 1908, Minkowski presented his talk, Space and Time ( Raum und Zeit ), to the German Society of Scientists and Physicians. The opening words of Space and Time include Minkowski's statement that "Henceforth, space for itself, and time for itself shall completely reduce to

2268-416: Is a priori because it belongs to the subjective constitution of our mind as the form or manner of our intuition of external objects. Euclid's Elements contained five postulates that form the basis for Euclidean geometry. One of these, the parallel postulate , has been the subject of debate among mathematicians for many centuries. It states that on any plane on which there is a straight line L 1 and

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2376-593: Is hyperbolic-orthogonal to each of the three spatial dimensions. Before Albert Einstein 's work on relativistic physics, time and space were viewed as independent dimensions. Einstein's discoveries showed that due to relativity of motion our space and time can be mathematically combined into one object– spacetime . It turns out that distances in space or in time separately are not invariant with respect to Lorentz coordinate transformations, but distances in Minkowski space along spacetime intervals are—which justifies

2484-408: Is "invariant". In special relativity, however, the distance between two points is no longer the same if measured by two different observers, when one of the observers is moving, because of Lorentz contraction . The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times,

2592-415: Is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects undergoing events, or because a single object in space is moving inertially between its events. The separation interval is the difference between the square of the spatial distance separating event B from event A and the square of the spatial distance traveled by

2700-417: Is actually what is indicated by moving clocks by applying an explicitly operational definition of clock synchronization assuming constant light speed. In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the principle of relativity . In 1905/1906 he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with

2808-407: Is called an event , and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). An event is represented by a set of coordinates x , y , z and t . Spacetime is thus four-dimensional . Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent

2916-547: Is called the particle's world line . Mathematically, spacetime is a manifold , which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, the surface of a globe appears to be flat. A scale factor, c {\displaystyle c} (conventionally called the speed-of-light ) relates distances measured in space to distances measured in time. The magnitude of this scale factor (nearly 300,000 kilometres or 190,000 miles in space being equivalent to one second in time), along with

3024-428: Is considered to be of fundamental importance to an understanding of the physical universe . However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework . In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean , in which space is conceived as curved , rather than flat , as in

3132-474: Is curved. Carl Friedrich Gauss , a German mathematician, was the first to consider an empirical investigation of the geometrical structure of space. He thought of making a test of the sum of the angles of an enormous stellar triangle, and there are reports that he actually carried out a test, on a small scale, by triangulating mountain tops in Germany. Henri Poincaré , a French mathematician and physicist of

3240-427: Is important due to its necessary relevance to survival, especially with regards to hunting and self preservation as well as simply one's idea of personal space . Several space-related phobias have been identified, including agoraphobia (the fear of open spaces), astrophobia (the fear of celestial space) and claustrophobia (the fear of enclosed spaces). The understanding of three-dimensional space in humans

3348-408: Is known at the present. On the other hand, it can be related to other fundamental quantities. Thus, similar to other fundamental quantities (like time and mass ), space can be explored via measurement and experiment. Today, our three-dimensional space is viewed as embedded in a four-dimensional spacetime , called Minkowski space (see special relativity ). The idea behind spacetime is that time

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3456-475: Is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together". Unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an idealised abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete . Space could be thought of in

3564-568: Is no preferred origin, single coordinate values have no essential meaning. The equation above is similar to the Pythagorean theorem, except with a minus sign between the ( c t ) 2 {\displaystyle (ct)^{2}} and the x 2 {\displaystyle x^{2}} terms. The spacetime interval is the quantity s 2 , {\displaystyle s^{2},} not s {\displaystyle s} itself. The reason

3672-481: Is not known, but space is known to be expanding very rapidly due to the cosmic inflation . The measurement of physical space has long been important. Although earlier societies had developed measuring systems, the International System of Units , (SI), is now the most common system of units used in the measuring of space, and is almost universally used. Currently, the standard space interval, called

3780-767: Is on the multiple and overlapping social processes that produce space. In his book The Condition of Postmodernity, David Harvey describes what he terms the " time-space compression ." This is the effect of technological advances and capitalism on our perception of time, space and distance. Changes in the modes of production and consumption of capital affect and are affected by developments in transportation and technology. These advances create relationships across time and space, new markets and groups of wealthy elites in urban centers, all of which annihilate distances and affect our perception of linearity and distance. In his book Thirdspace, Edward Soja describes space and spatiality as an integral and neglected aspect of what he calls

3888-435: Is orders-of-magnitudes more than the downlink bandwidth of 550  kB/s . The on-board data reduction encompasses co-adding the raw frames for thirty minutes, reducing the bandwidth by a factor of 300. Furthermore, interesting targets are pre-selected and only the relevant pixels are processed, which is 6% of the total. This reduced data is then sent to Earth where it is processed further. Research has also been carried out on

3996-514: Is positive, the spacetime interval is referred to as timelike . Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, positive intervals are always timelike. If s 2 {\displaystyle s^{2}} is negative, the spacetime interval is said to be spacelike . Spacetime intervals are equal to zero when x = ± c t . {\displaystyle x=\pm ct.} In other words,

4104-531: Is postulated that spacetime is geometrically distorted – curved – near to gravitationally significant masses. One consequence of this postulate, which follows from the equations of general relativity, is the prediction of moving ripples of spacetime, called gravitational waves . While indirect evidence for these waves has been found (in the motions of the Hulse–Taylor binary system, for example) experiments attempting to directly measure these waves are ongoing at

4212-410: Is separate from space, and is agreed on by all observers. Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion , or anything external. It assumes that space is Euclidean : it assumes that space follows the geometry of common sense. In the context of special relativity , time cannot be separated from the three dimensions of space, because

4320-438: Is stationary with respect to them; and objects are measured to be shortened in the direction that they are moving with respect to the observer. Subsequently, Einstein worked on a general theory of relativity , which is a theory of how gravity interacts with spacetime. Instead of viewing gravity as a force field acting in spacetime, Einstein suggested that it modifies the geometric structure of spacetime itself. According to

4428-540: Is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard s 2 {\displaystyle s^{2}} as a distinct symbol in itself, rather than the square of something. In general s 2 {\displaystyle s^{2}} can assume any real number value. If s 2 {\displaystyle s^{2}}

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4536-415: Is the mapping of spaces to allow better navigation, for visualization purposes and to act as a locational device. Geostatistics apply statistical concepts to collected spatial data of Earth to create an estimate for unobserved phenomena. Geographical space is often considered as land, and can have a relation to ownership usage (in which space is seen as property or territory). While some cultures assert

4644-412: Is thought to be learned during infancy using unconscious inference , and is closely related to hand-eye coordination . The visual ability to perceive the world in three dimensions is called depth perception . Space has been studied in the social sciences from the perspectives of Marxism , feminism , postmodernism , postcolonialism , urban theory and critical geography . These theories account for

4752-399: The c t {\displaystyle ct} -coordinate is: or for three space dimensions, The constant c , {\displaystyle c,} the speed of light, converts time t {\displaystyle t} units (like seconds) into space units (like meters). The squared interval Δ s 2 {\displaystyle \Delta s^{2}}

4860-664: The Timaeus of Plato , or Socrates in his reflections on what the Greeks called khôra (i.e. "space"), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or in the later "geometrical conception of place" as "space qua extension" in the Discourse on Place ( Qawl fi al-Makan ) of the 11th-century Arab polymath Alhazen . Many of these classical philosophical questions were discussed in

4968-461: The distance Δ d {\displaystyle \Delta {d}} between two points can be defined using the Pythagorean theorem : Although two viewers may measure the x , y , and z position of the two points using different coordinate systems, the distance between the points will be the same for both, assuming that they are measuring using the same units. The distance

5076-472: The Euclidean space . According to Albert Einstein 's theory of general relativity , space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like

5184-510: The LIGO and Virgo collaborations. LIGO scientists reported the first such direct observation of gravitational waves on 14 September 2015. Relativity theory leads to the cosmological question of what shape the universe is, and where space came from. It appears that space was created in the Big Bang , 13.8 billion years ago and has been expanding ever since. The overall shape of space

5292-519: The Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics . Isaac Newton viewed space as absolute, existing permanently and independently of whether there was any matter in the. In contrast, other natural philosophers , notably Gottfried Leibniz , thought that space was in fact a collection of relations between objects, given by their distance and direction from one another. In

5400-441: The body mass index reduces two dimensions (body and mass) into a single measure, without any information being lost in the process. When information is derived from instrument readings there may also be a transformation from analog to digital form. When the data are already in digital form the 'reduction' of the data typically involves some editing, scaling , encoding , sorting , collating, and producing tabular summaries. When

5508-427: The ct ′ axis is tilted with respect to the ct axis by an angle θ given by The x ′ axis is also tilted with respect to the x axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at x ′ = 0, ct ′ = −

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5616-585: The data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks. In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted. Physicists distinguish between what one measures or observes , after one has factored out signal propagation delays, versus what one visually sees without such corrections. Failing to understand

5724-460: The identity of indiscernibles , there would be no real difference between them. According to the principle of sufficient reason , any theory of space that implied that there could be these two possible universes must therefore be wrong. Newton took space to be more than relations between material objects and based his position on observation and experimentation. For a relationist there can be no real difference between inertial motion , in which

5832-464: The x and ct axes. Since OP = OQ = OR, the angle between x ′ and x must also be θ . Space Space is a three-dimensional continuum containing positions and directions . In classical physics , physical space is often conceived in three linear dimensions . Modern physicists usually consider it, with time , to be part of a boundless four-dimensional continuum known as spacetime . The concept of space

5940-576: The " trialectics of being ," the three modes that determine how we inhabit, experience and understand the world. He argues that critical theories in the Humanities and Social Sciences study the historical and social dimensions of our lived experience, neglecting the spatial dimension. He builds on Henri Lefebvre's work to address the dualistic way in which humans understand space—as either material/physical or as represented/imagined. Lefebvre's "lived space" and Soja's "thirdspace" are terms that account for

6048-424: The 1850s, Bernhard Riemann developed an equivalent theory of elliptical geometry , in which no parallel lines pass through P . In this geometry, triangles have more than 180° and circles have a ratio of circumference-to-diameter that is less than pi . Although there was a prevailing Kantian consensus at the time, once non-Euclidean geometries had been formalised, some began to wonder whether or not physical space

6156-454: The 18th century, the philosopher and theologian George Berkeley attempted to refute the "visibility of spatial depth" in his Essay Towards a New Theory of Vision . Later, the metaphysician Immanuel Kant said that the concepts of space and time are not empirical ones derived from experiences of the outside world—they are elements of an already given systematic framework that humans possess and use to structure all experiences. Kant referred to

6264-477: The Aristotelian worldview with a theory about space and motion as determined by natural laws . In other words, he sought a metaphysical foundation or a mechanical explanation for his theories about matter and motion. Cartesian space was Euclidean in structure—infinite, uniform and flat. It was defined as that which contained matter; conversely, matter by definition had a spatial extension so that there

6372-584: The Lorentz group are closely connected to certain types of sphere , hyperbolic , or conformal geometries and their transformation groups already developed in the 19th century, in which invariant intervals analogous to the spacetime interval are used. Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907,

6480-505: The Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry that does not include the parallel postulate, called hyperbolic geometry . In this geometry, an infinite number of parallel lines pass through the point P . Consequently, the sum of angles in a triangle is less than 180° and the ratio of a circle 's circumference to its diameter is greater than pi . In

6588-432: The best suited to the description of our world". Even as late as 1909, Poincaré continued to describe the dynamical interpretation of the Lorentz transform. In 1905, Albert Einstein analyzed special relativity in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His results were mathematically equivalent to those of Lorentz and Poincaré. He obtained them by recognizing that

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6696-528: The bucket argument was considered decisive in showing that space must exist independently of matter. In the eighteenth century the German philosopher Immanuel Kant published his theory of space as "a property of our mind" by which "we represent to ourselves objects as outside us, and all as in space" in the Critique of Pure Reason On his view the nature of spatial predicates are "relations that only attach to

6804-477: The community, and managed in their name by delegated bodies; such spaces are open to all, while private property is the land culturally owned by an individual or company, for their own use and pleasure. Abstract space is a term used in geography to refer to a hypothetical space characterized by complete homogeneity. When modeling activity or behavior, it is a conceptual tool used to limit extraneous variables such as terrain. Psychologists first began to study

6912-476: The complex ways in which humans understand and navigate place, which "firstspace" and "Secondspace" (Soja's terms for material and imagined spaces respectively) do not fully encompass. Postcolonial theorist Homi Bhabha 's concept of Third Space is different from Soja's Thirdspace, even though both terms offer a way to think outside the terms of a binary logic. Bhabha's Third Space is the space in which hybrid cultural forms and identities exist. In his theories,

7020-449: The concept that space and time can be viewed as a single construct known as spacetime . In this theory, the speed of light in vacuum is the same for all observers—which has the result that two events that appear simultaneous to one particular observer will not be simultaneous to another observer if the observers are moving with respect to one another. Moreover, an observer will measure a moving clock to tick more slowly than one that

7128-408: The data and allows for easier visualization, such as the example below where 3-dimensional data is transformed into 2 dimensions to show hidden parts. One method of dimensionality reduction is wavelet transform , in which data is transformed to preserve relative distance between objects at different levels of resolution, and is often used for image compression . This method of data reduction reduces

7236-402: The data volume by choosing alternate, smaller forms of data representation. Numerosity reduction can be split into 2 groups: parametric and non-parametric methods. Parametric methods (regression, for example) assume the data fits some model, estimate model parameters, store only the parameters, and discard the data. One example of this is in the image below, where the volume of data to be processed

7344-406: The design of buildings and structures, and on farming. Ownership of space is not restricted to land. Ownership of airspace and of waters is decided internationally. Other forms of ownership have been recently asserted to other spaces—for example to the radio bands of the electromagnetic spectrum or to cyberspace . Public space is a term used to define areas of land as collectively owned by

7452-476: The difference between what one measures and what one sees is the source of much confusion among students of relativity. By the mid-1800s, various experiments such as the observation of the Arago spot and differential measurements of the speed of light in air versus water were considered to have proven the wave nature of light as opposed to a corpuscular theory . Propagation of waves was then assumed to require

7560-457: The distance between the two events (because of length contraction ). Special relativity provides a new invariant, called the spacetime interval , which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by Δ t {\displaystyle \Delta t} and

7668-416: The effect of the history of colonialism, transatlantic slavery and globalization on our understanding and experience of space and place. The topic has garnered attention since the 1980s, after the publication of Henri Lefebvre 's The Production of Space . In this book, Lefebvre applies Marxist ideas about the production of commodities and accumulation of capital to discuss space as a social product. His focus

7776-420: The entire theory can be built upon two postulates: the principle of relativity and the principle of the constancy of light speed. His work was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples. Einstein in 1905 superseded previous attempts of an electromagnetic mass –energy relation by introducing

7884-451: The existence of a waving medium; in the case of light waves, this was considered to be a hypothetical luminiferous aether . The various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the Fizeau experiment of 1851, conducted by French physicist Hippolyte Fizeau , demonstrated that the speed of light in flowing water was less than

7992-485: The experience of "space" in his Critique of Pure Reason as being a subjective "pure a priori form of intuition". Galilean and Cartesian theories about space, matter, and motion are at the foundation of the Scientific Revolution , which is understood to have culminated with the publication of Newton 's Principia Mathematica in 1687. Newton's theories about space and time helped him explain

8100-563: The fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe that is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelson–Morley experiment , that puzzling discrepancies began to be noted between observation versus predictions based on

8208-406: The form of intuition alone, and thus to the subjective constitution of our mind, without which these predicates could not be attached to anything at all." This develops his theory of knowledge in which knowledge about space itself can be both a priori and synthetic . According to Kant, knowledge about space is synthetic because any proposition about space cannot be true merely in virtue of

8316-555: The further development of general relativity, Einstein fully incorporated the spacetime formalism. When Einstein published in 1905, another of his competitors, his former mathematics professor Hermann Minkowski , had also arrived at most of the basic elements of special relativity. Max Born recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator: I went to Cologne, met Minkowski and heard his celebrated lecture 'Space and Time' delivered on 2 September 1908. [...] He told me later that it came to him as

8424-495: The general equivalence of mass and energy , which was instrumental for his subsequent formulation of the equivalence principle in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass–energy equivalence, Einstein showed that the gravitational mass of a body is proportional to its energy content, which was one of the early results in developing general relativity . While it would appear that he did not at first think geometrically about spacetime, in

8532-462: The general theory, time goes more slowly at places with lower gravitational potentials and rays of light bend in the presence of a gravitational field. Scientists have studied the behaviour of binary pulsars , confirming the predictions of Einstein's theories, and non-Euclidean geometry is usually used to describe spacetime. In modern mathematics spaces are defined as sets with some added structure. They are typically topological spaces , in which

8640-406: The geometric interpretation of relativity proved to be vital. In 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity. Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as Minkowski spacetime. In three dimensions,

8748-416: The hypothetical aether on the speed of light, and the most likely explanation, complete aether dragging, was in conflict with the observation of stellar aberration . George Francis FitzGerald in 1889, and Hendrik Lorentz in 1892, independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that

8856-418: The implicit assumption of Euclidean space. In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events is being measured. This usage differs significantly from the ordinary English meaning of the term. Reference frames are inherently nonlocal constructs, and according to this usage of the term, it does not make sense to speak of an observer as having

8964-413: The late 19th century, introduced an important insight in which he attempted to demonstrate the futility of any attempt to discover which geometry applies to space by experiment. He considered the predicament that would face scientists if they were confined to the surface of an imaginary large sphere with particular properties, known as a sphere-world . In this world, the temperature is taken to vary in such

9072-438: The math with no loss of generality in the conclusions that are reached. In Fig. 2-2, two Galilean reference frames (i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime") belongs to a second observer O′. Fig. 2-3a redraws Fig. 2-2 in a different orientation. Fig. 2-3b illustrates a relativistic spacetime diagram from

9180-430: The meaning of the terms contained in the proposition. In the counter-example, the proposition "all unmarried men are bachelors" is true by virtue of each term's meaning. Further, space is a priori because it is the form of our receptive abilities to receive information about the external world. For example, someone without sight can still perceive spatial attributes via touch, hearing, and smell. Knowledge of space itself

9288-468: The movement of objects. While his theory of space is considered the most influential in physics, it emerged from his predecessors' ideas about the same. As one of the pioneers of modern science , Galileo revised the established Aristotelian and Ptolemaic ideas about a geocentric cosmos. He backed the Copernican theory that the universe was heliocentric , with a stationary Sun at the center and

9396-546: The name. In addition, time and space dimensions should not be viewed as exactly equivalent in Minkowski space. One can freely move in space but not in time. Thus, time and space coordinates are treated differently both in special relativity (where time is sometimes considered an imaginary coordinate) and in general relativity (where different signs are assigned to time and space components of spacetime metric ). Furthermore, in Einstein's general theory of relativity , it

9504-433: The object travels with constant velocity , and non-inertial motion , in which the velocity changes with time, since all spatial measurements are relative to other objects and their motions. But Newton argued that since non-inertial motion generates forces , it must be absolute. He used the example of water in a spinning bucket to demonstrate his argument. Water in a bucket is hung from a rope and set to spin, starts with

9612-571: The observations are discrete but the underlying phenomenon is continuous then smoothing and interpolation are often needed. The data reduction is often undertaken in the presence of reading or measurement errors . Some idea of the nature of these errors is needed before the most likely value may be determined. An example in astronomy is the data reduction in the Kepler satellite. This satellite records 95-megapixel images once every six seconds, generating dozens of megabytes of data per second, which

9720-515: The observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. In ordinary space, a position is specified by three numbers, known as dimensions . In the Cartesian coordinate system , these are often called x , y and z . A point in spacetime

9828-464: The physical constituents of matter. Lorentz's equations predicted a quantity that he called local time , with which he could explain the aberration of light , the Fizeau experiment and other phenomena. Henri Poincaré was the first to combine space and time into spacetime. He argued in 1898 that the simultaneity of two events is a matter of convention. In 1900, he recognized that Lorentz's "local time"

9936-728: The planets—including the Earth—revolving around the Sun. If the Earth moved, the Aristotelian belief that its natural tendency was to remain at rest was in question. Galileo wanted to prove instead that the Sun moved around its axis, that motion was as natural to an object as the state of rest. In other words, for Galileo, celestial bodies, including the Earth, were naturally inclined to move in circles. This view displaced another Aristotelian idea—that all objects gravitated towards their designated natural place-of-belonging. Descartes set out to replace

10044-484: The postulate of relativity. While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional spacetime by defining various four vectors , namely four-position , four-velocity , and four-force . He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems

10152-577: The rights of the individual in terms of ownership, other cultures will identify with a communal approach to land ownership, while still other cultures such as Australian Aboriginals , rather than asserting ownership rights to land, invert the relationship and consider that they are in fact owned by the land. Spatial planning is a method of regulating the use of space at land-level, with decisions made at regional, national and international levels. Space can also impact on human and cultural behavior, being an important factor in architecture, where it will impact on

10260-731: The spacetime interval d s ′ {\displaystyle ds'} can be written in a same form as above. Because of the constancy of speed of light, the light events in all inertial frames belong to zero interval, d s = d s ′ = 0 {\displaystyle ds=ds'=0} . For any other infinitesimal event where d s ≠ 0 {\displaystyle ds\neq 0} , one can prove that d s 2 = d s ′ 2 {\displaystyle ds^{2}=ds'^{2}} which in turn upon integration leads to s = s ′ {\displaystyle s=s'} . The invariance of

10368-584: The spacetime interval between the same events for all inertial frames of reference is one of the fundamental results of special theory of relativity. Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, x {\displaystyle x} means Δ x {\displaystyle \Delta {x}} , etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there

10476-414: The spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed lightlike or null . A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage. A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2-1 presents

10584-434: The speed of light, their world lines have a slope of ±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time. To gain insight in how spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration. With care, this allows simplification of

10692-401: The state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, Poincaré et al. Minkowski saw Einstein's work as an extension of Lorentz's, and was most directly influenced by Poincaré. On 5 November 1907 (a little more than

10800-520: The sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction. Among other issues, the dependence of the partial aether-dragging implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether simultaneously flows at different speeds for different colors of light. The Michelson–Morley experiment of 1887 (Fig. 1-2) showed no differential influence of Earth's motions through

10908-602: The term hybrid describes new cultural forms that emerge through the interaction between colonizer and colonized. Data reduction Data reduction is the transformation of numerical or alphabetical digital information derived empirically or experimentally into a corrected, ordered, and simplified form. The purpose of data reduction can be two-fold: reduce the number of data records by eliminating invalid data or produce summary data and statistics at different aggregation levels for various applications. Data reduction does not necessarily mean loss of information. For example,

11016-586: The use of data reduction in wearable (wireless) devices for health monitoring and diagnosis applications. For example, in the context of epilepsy diagnosis, data reduction has been used to increase the battery lifetime of a wearable EEG device by selecting and only transmitting EEG data that is relevant for diagnosis and discarding background activity. When dimensionality increases, data becomes increasingly sparse while density and distance between points, critical to clustering and outlier analysis, becomes less meaningful. Dimensionality reduction helps reduce noise in

11124-447: The viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times t  = 0 in frame S and t ′ = 0 in frame S′. The ct ′ axis passes through the events in frame S′ which have x ′ = 0. But the points with x ′ = 0 are moving in the x -direction of frame S with velocity v , so that they are not coincident with the ct axis at any time other than zero. Therefore,

11232-461: The way space is perceived in the middle of the 19th century. Those now concerned with such studies regard it as a distinct branch of psychology . Psychologists analyzing the perception of space are concerned with how recognition of an object's physical appearance or its interactions are perceived, see, for example, visual space . Other, more specialized topics studied include amodal perception and object permanence . The perception of surroundings

11340-404: The whole ensemble of clocks associated with one inertial frame of reference. In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, will see a delay between the emission of a signal and its detection due to the speed of light. To synchronize the clocks, in

11448-569: The world to our ability to think rather than to our experiences, as the empiricists believe. He posited a clear distinction between the body and mind, which is referred to as the Cartesian dualism . Following Galileo and Descartes, during the seventeenth century the philosophy of space and time revolved around the ideas of Gottfried Leibniz , a German philosopher–mathematician, and Isaac Newton , who set out two opposing theories of what space is. Rather than being an entity that independently exists over and above other matter, Leibniz held that space

11556-554: Was exactly what was necessary to explain the negative results of the Michelson–Morley experiment. No length changes occur in directions transverse to the direction of motion. By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein was to derive later, i.e. the Lorentz transformation . As a theory of dynamics (the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of

11664-417: Was no such thing as empty space. The Cartesian notion of space is closely linked to his theories about the nature of the body, mind and matter. He is famously known for his "cogito ergo sum" (I think therefore I am), or the idea that we can only be certain of the fact that we can doubt, and therefore think and therefore exist. His theories belong to the rationalist tradition, which attributes knowledge about

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