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156-404: Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past , through the present , and into the future . It is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in

312-438: A n ) n = − ∞ ∞ {\textstyle {(a_{n})}_{n=-\infty }^{\infty }} is a bi-infinite sequence , and can also be written as ( … , a − 1 , a 0 , a 1 , a 2 , … ) {\textstyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )} . In cases where

468-464: A n ) . {\textstyle (a_{n}).} Here A is the domain, or index set, of the sequence. Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets . A net is a function from a (possibly uncountable ) directed set to a topological space. The notational conventions for sequences normally apply to nets as well. The length of

624-526: A causal relation . General relativity does not address the nature of time for extremely small intervals where quantum mechanics holds. In quantum mechanics, time is treated as a universal and absolute parameter, differing from general relativity's notion of independent clocks. The problem of time consists of reconciling these two theories. As of 2024, there is no generally accepted theory of quantum general relativity. Generally speaking, methods of temporal measurement, or chronometry , take two distinct forms:

780-417: A sodium flame (for alignment), or white light (for the actual observations), through a half-silvered mirror that was used to split it into two beams traveling at right angles to one another. After leaving the splitter, the beams traveled out to the ends of long arms where they were reflected back into the middle by small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing

936-617: A calendar based solely on twelve lunar months. Lunisolar calendars have a thirteenth month added to some years to make up for the difference between a full year (now known to be about 365.24 days) and a year of just twelve lunar months. The numbers twelve and thirteen came to feature prominently in many cultures, at least partly due to this relationship of months to years. Other early forms of calendars originated in Mesoamerica, particularly in ancient Mayan civilization. These calendars were religiously and astronomically based, with 18 months in

1092-614: A dimension. Isaac Newton said that we are merely occupying time, he also says that humans can only understand relative time . Relative time is a measurement of objects in motion. The anti-realists believed that time is merely a convenient intellectual concept for humans to understand events. This means that time was useless unless there were objects that it could interact with, this was called relational time . René Descartes , John Locke , and David Hume said that one's mind needs to acknowledge time, in order to understand what time is. Immanuel Kant believed that we can not know what something

1248-663: A distance from L {\displaystyle L} less than d {\displaystyle d} . For example, the sequence a n = n + 1 2 n 2 {\textstyle a_{n}={\frac {n+1}{2n^{2}}}} shown to the right converges to the value 0. On the other hand, the sequences b n = n 3 {\textstyle b_{n}=n^{3}} (which begins 1, 8, 27, ...) and c n = ( − 1 ) n {\displaystyle c_{n}=(-1)^{n}} (which begins −1, 1, −1, 1, ...) are both divergent. If

1404-599: A function of n . Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions. Not all sequences can be specified by a recurrence relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...). There are many different notions of sequences in mathematics, some of which ( e.g. , exact sequence ) are not covered by

1560-402: A head in the famous Leibniz–Clarke correspondence . Philosophers in the 17th and 18th century questioned if time was real and absolute, or if it was an intellectual concept that humans use to understand and sequence events. These questions lead to realism vs anti-realism; the realists believed that time is a fundamental part of the universe, and be perceived by events happening in a sequence, in

1716-405: A hundred minutes of a hundred seconds, which marked a deviation from the base 12 ( duodecimal ) system used in many other devices by many cultures. The system was abolished in 1806. A large variety of devices have been invented to measure time. The study of these devices is called horology . An Egyptian device that dates to c.  1500 BC , similar in shape to a bent T-square , measured

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1872-468: A letter to Lord Rayleigh in August 1887: The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. The expected deviation of the interference fringes from the zero should have been 0.40 of a fringe – the maximum displacement was 0.02 and the average much less than 0.01 – and then not in the right place. As displacement is proportional to squares of

2028-416: A limit if the elements of the sequence become closer and closer to some value L {\displaystyle L} (called the limit of the sequence), and they become and remain arbitrarily close to L {\displaystyle L} , meaning that given a real number d {\displaystyle d} greater than zero, all but a finite number of the elements of the sequence have

2184-483: A natural number N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} we have If ( a n ) {\displaystyle (a_{n})} is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that | ⋅ | {\displaystyle |\cdot |} denotes

2340-577: A number or calendar date to an instant (point in time), quantifying the duration of a time interval, and establishing a chronology (ordering of events). In modern times, several time specifications have been officially recognized as standards, where formerly they were matters of custom and practice. The invention in 1955 of the caesium atomic clock has led to the replacement of older and purely astronomical time standards such as sidereal time and ephemeris time , for most practical purposes, by newer time standards based wholly or partly on atomic time using

2496-409: A path difference. The path difference is zero only when the interferometer is aligned with or perpendicular to the aether wind, and it reaches a maximum when it is at a 45° angle. The path difference can be any fraction of the wavelength, depending on the angle and speed of the aether wind. To prove the existence of the aether, Michelson and Morley sought to find the "fringe shift". The idea was simple,

2652-461: A pattern of constructive and destructive interference whose transverse displacement would depend on the relative time it takes light to transit the longitudinal vs. the transverse arms. If the Earth is traveling through an aether medium, a light beam traveling parallel to the flow of that aether will take longer to reflect back and forth than would a beam traveling perpendicular to the aether, because

2808-426: A personal level, and the negative result haunted him for the rest of his life. If the same situation is described from the view of an observer co-moving with the interferometer, then the effect of aether wind is similar to the effect experienced by a swimmer, who tries to move with velocity c {\textstyle c} against a river flowing with velocity v {\textstyle v} . In

2964-485: A prime motivation in navigation and astronomy . Time is also of significant social importance, having economic value (" time is money ") as well as personal value, due to an awareness of the limited time in each day and in human life spans . The concept of time can be complex. Multiple notions exist and defining time in a manner applicable to all fields without circularity has consistently eluded scholars. Nevertheless, diverse fields such as business, industry, sports,

3120-415: A recurrence relation is Recamán's sequence , defined by the recurrence relation with initial term a 0 = 0. {\displaystyle a_{0}=0.} A linear recurrence with constant coefficients is a recurrence relation of the form where c 0 , … , c k {\displaystyle c_{0},\dots ,c_{k}} are constants . There

3276-485: A recurring pattern of ages or cycles, where events and phenomena repeated themselves in a predictable manner. One of the most famous examples of this concept is found in Hindu philosophy , where time is depicted as a wheel called the " Kalachakra " or "Wheel of Time." According to this belief, the universe undergoes endless cycles of creation, preservation, and destruction. Similarly, in other ancient cultures such as those of

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3432-400: A sequence are discussed after the examples. The prime numbers are the natural numbers greater than 1 that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics , particularly in number theory where many results related to them exist. The Fibonacci numbers comprise

3588-440: A sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence ( a n ) {\displaystyle (a_{n})} is normally denoted lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} . If ( a n ) {\displaystyle (a_{n})}

3744-404: A sequence is defined as the number of terms in the sequence. A sequence of a finite length n is also called an n -tuple . Finite sequences include the empty sequence  ( ) that has no elements. Normally, the term infinite sequence refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence

3900-463: A sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting

4056-409: A sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways. Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be

4212-450: A sequence of sequences: ( ( a m , n ) n ∈ N ) m ∈ N {\textstyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }} denotes a sequence whose m th term is the sequence ( a m , n ) n ∈ N {\textstyle (a_{m,n})_{n\in \mathbb {N} }} . An alternative to writing

4368-485: A speed of around 30 km/s (18.64 mi/s), or 108,000 km/h (67,000 mph). The Earth is in motion, so two main possibilities were considered: (1) The aether is stationary and only partially dragged by Earth (proposed by Augustin-Jean Fresnel in 1818), or (2) the aether is completely dragged by Earth and thus shares its motion at Earth's surface (proposed by Sir George Stokes, 1st Baronet in 1844). In addition, James Clerk Maxwell (1865) recognized

4524-454: A vacuum, it was assumed that even a vacuum must be filled with aether. Because the speed of light is so great, and because material bodies pass through the aether without obvious friction or drag, it was assumed to have a highly unusual combination of properties. Designing experiments to investigate these properties was a high priority of 19th-century physics. Earth orbits around the Sun at

4680-524: A year and 20 days in a month, plus five epagomenal days at the end of the year. The reforms of Julius Caesar in 45 BC put the Roman world on a solar calendar . This Julian calendar was faulty in that its intercalation still allowed the astronomical solstices and equinoxes to advance against it by about 11 minutes per year. Pope Gregory XIII introduced a correction in 1582; the Gregorian calendar

4836-565: Is monotonically decreasing if each consecutive term is less than or equal to the previous one, and is strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function . The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing , respectively. If

Time - Misplaced Pages Continue

4992-474: Is a divergent sequence, then the expression lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} is meaningless. A sequence of real numbers ( a n ) {\displaystyle (a_{n})} converges to a real number L {\displaystyle L} if, for all ε > 0 {\displaystyle \varepsilon >0} , there exists

5148-764: Is a fundamental concept to define other quantities, such as velocity . To avoid a circular definition, time in physics is operationally defined as "what a clock reads", specifically a count of repeating events such as the SI second . Although this aids in practical measurements, it does not address the essence of time. Physicists developed the concept of the spacetime continuum, where events are assigned four coordinates: three for space and one for time. Events like particle collisions , supernovas , or rocket launches have coordinates that may vary for different observers, making concepts like "now" and "here" relative. In general relativity , these coordinates do not directly correspond to

5304-468: Is a general method for expressing the general term a n {\displaystyle a_{n}} of such a sequence as a function of n ; see Linear recurrence . In the case of the Fibonacci sequence, one has c 0 = 0 , c 1 = c 2 = 1 , {\displaystyle c_{0}=0,c_{1}=c_{2}=1,} and the resulting function of n

5460-403: Is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite , as in these examples, or infinite , such as the sequence of all even positive integers (2, 4, 6, ...). The position of an element in a sequence

5616-411: Is a simple classical example, defined by the recurrence relation with initial terms a 0 = 0 {\displaystyle a_{0}=0} and a 1 = 1 {\displaystyle a_{1}=1} . From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. A complicated example of a sequence defined by

5772-401: Is a strictly increasing sequence of positive integers. Some other types of sequences that are easy to define include: An important property of a sequence is convergence . If a sequence converges, it converges to a particular value known as the limit . If a sequence converges to some limit, then it is convergent . A sequence that does not converge is divergent . Informally, a sequence has

5928-458: Is a theoretical ideal scale realized by TAI. Geocentric Coordinate Time and Barycentric Coordinate Time are scales defined as coordinate times in the context of the general theory of relativity. Barycentric Dynamical Time is an older relativistic scale that is still in use. Many ancient cultures, particularly in the East, had a cyclical view of time. In these traditions, time was often seen as

6084-464: Is bi-infinite. This sequence could be denoted ( 2 n ) n = − ∞ ∞ {\textstyle {(2n)}_{n=-\infty }^{\infty }} . A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }}

6240-409: Is called a lower bound . If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded . A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of

6396-465: Is called a singly infinite sequence or a one-sided infinite sequence when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence , two-way infinite sequence , or doubly infinite sequence . A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ),

Time - Misplaced Pages Continue

6552-418: Is called an index , and the set of values that it can take is called the index set . It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , which denotes a sequence whose n th element

6708-429: Is credited to Egyptians because of their sundials, which operated on a duodecimal system. The importance of the number 12 is due to the number of lunar cycles in a year and the number of stars used to count the passage of night. The most precise timekeeping device of the ancient world was the water clock , or clepsydra , one of which was found in the tomb of Egyptian pharaoh Amenhotep I . They could be used to measure

6864-466: Is denoted by Δ λ {\displaystyle \Delta \lambda } because the beams are out of phase by a some number of wavelengths ( λ {\displaystyle \lambda } ). To visualise this, consider taking the two beam paths along the longitudinal and transverse plane, and lying them straight (an animation of this is shown at minute 11:00, The Mechanical Universe, episode 41 ). One path will be longer than

7020-499: Is dominated by temporality ( kala ), everything within time is subject to change and decay. Overcoming pain and death requires knowledge that transcends temporal existence and reveals its eternal foundation. Two contrasting viewpoints on time divide prominent philosophers. One view is that time is part of the fundamental structure of the universe  – a dimension independent of events, in which events occur in sequence . Isaac Newton subscribed to this realist view, and hence it

7176-416: Is easily discernible by inspection. Other examples are sequences of functions , whose elements are functions instead of numbers. The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences. Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of π . One such notation

7332-540: Is given by To find the path difference, simply multiply by c {\displaystyle c} ; Δ λ 1 = 2 L ( 1 1 − v 2 c 2 − 1 1 − v 2 c 2 ) {\displaystyle \Delta {\lambda }_{1}=2L\left({\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}-{\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\right)} The path difference

7488-406: Is given by Binet's formula . A holonomic sequence is a sequence defined by a recurrence relation of the form where c 1 , … , c k {\displaystyle c_{1},\dots ,c_{k}} are polynomials in n . For most holonomic sequences, there is no explicit formula for expressing a n {\displaystyle a_{n}} as

7644-503: Is given by the variable a n {\displaystyle a_{n}} . For example: One can consider multiple sequences at the same time by using different variables; e.g. ( b n ) n ∈ N {\textstyle (b_{n})_{n\in \mathbb {N} }} could be a different sequence than ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} . One can even consider

7800-498: Is in Byrhtferth 's Enchiridion (a science text) of 1010–1012, where it was defined as 1/564 of a momentum (1 1 ⁄ 2 minutes), and thus equal to 15/94 of a second. It was used in the computus , the process of calculating the date of Easter. As of May 2010, the smallest time interval uncertainty in direct measurements is on the order of 12 attoseconds (1.2 × 10 seconds), about 3.7 × 10 Planck times . The second (s)

7956-431: Is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation. The Fibonacci sequence

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8112-548: Is inserted into the above formula for T ℓ {\textstyle T_{\ell }} , then the light propagation time in the longitudinal direction becomes equal to that in the transverse direction: However, length contraction is only a special case of the more general relation, according to which the transverse length is larger than the longitudinal length by the ratio γ {\textstyle \gamma } . This can be achieved in many ways. If L 1 {\textstyle L_{1}}

8268-456: Is its rank or index ; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis , a sequence is often denoted by letters in the form of a n {\displaystyle a_{n}} , b n {\displaystyle b_{n}} and c n {\displaystyle c_{n}} , where

8424-764: Is kept within 0.9 second of UT1 by the introduction of one-second steps to UTC, the leap second . The Global Positioning System broadcasts a very precise time signal based on UTC time. The surface of the Earth is split into a number of time zones . Standard time or civil time in a time zone deviates a fixed, round amount, usually a whole number of hours, from some form of Universal Time, usually UTC. Most time zones are exactly one hour apart, and by convention compute their local time as an offset from UTC. For example, time zones at sea are based on UTC. In many locations (but not at sea) these offsets vary twice yearly due to daylight saving time transitions. Some other time standards are used mainly for scientific work. Terrestrial Time

8580-404: Is monotonically increasing if and only if a n + 1 ≥ a n {\textstyle a_{n+1}\geq a_{n}} for all n ∈ N . {\displaystyle n\in \mathbf {N} .} If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing . A sequence

8736-406: Is neither an event nor a thing, and thus is not itself measurable nor can it be travelled. Furthermore, it may be that there is a subjective component to time, but whether or not time itself is "felt", as a sensation, or is a judgment, is a matter of debate. In Philosophy, time was questioned throughout the centuries; what time is and if it is real or not. Ancient Greek philosophers asked if time

8892-481: Is not rather than what it is, an approach similar to that taken in other negative definitions . However, Augustine ends up calling time a "distention" of the mind (Confessions 11.26) by which we simultaneously grasp the past in memory, the present by attention, and the future by expectation. Isaac Newton believed in absolute space and absolute time; Leibniz believed that time and space are relational. The differences between Leibniz's and Newton's interpretations came to

9048-549: Is now Case Western Reserve University in Cleveland , Ohio, and published in November of the same year. The experiment compared the speed of light in perpendicular directions in an attempt to detect the relative motion of matter, including their laboratory, through the luminiferous aether, or "aether wind" as it was sometimes called. The result was negative, in that Michelson and Morley found no significant difference between

9204-665: Is qualitative, as opposed to quantitative. In Greek mythology, Chronos (ancient Greek: Χρόνος) is identified as the Personification of Time. His name in Greek means "time" and is alternatively spelled Chronus (Latin spelling) or Khronos. Chronos is usually portrayed as an old, wise man with a long, gray beard, such as "Father Time". Some English words whose etymological root is khronos/chronos include chronology , chronometer , chronic , anachronism , synchronise , and chronicle . Rabbis sometimes saw time like "an accordion that

9360-500: Is replaced by the expression dist ⁡ ( a n , L ) {\displaystyle \operatorname {dist} (a_{n},L)} , which denotes the distance between a n {\displaystyle a_{n}} and L {\displaystyle L} . If ( a n ) {\displaystyle (a_{n})} and ( b n ) {\displaystyle (b_{n})} are convergent sequences, then

9516-596: Is seen as progressing in a straight line from past to future without repetition. In general, the Islamic and Judeo-Christian world-view regards time as linear and directional , beginning with the act of creation by God. The traditional Christian view sees time ending, teleologically, with the eschatological end of the present order of things, the " end time ". In the Old Testament book Ecclesiastes , traditionally ascribed to Solomon (970–928 BC), time (as

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9672-451: Is some number of wavelengths, and λ {\displaystyle \lambda } which is a single wavelength. As can be seen by this relation, fringe shift n is a unitless quantity. Since L  ≈ 11 meters and λ ≈ 500 nanometers , the expected fringe shift was n  ≈ 0.44. The negative result led Michelson to the conclusion that there is no measurable aether drift. However, he never accepted this on

9828-442: Is sometimes referred to as Newtonian time . The opposing view is that time does not refer to any kind of "container" that events and objects "move through", nor to any entity that "flows", but that it is instead part of a fundamental intellectual structure (together with space and number) within which humans sequence and compare events. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant , holds that time

9984-402: Is the SI base unit. A minute (min) is 60 seconds in length (or, rarely, 59 or 61 seconds when leap seconds are employed), and an hour is 60 minutes or 3600 seconds in length. A day is usually 24 hours or 86,400 seconds in length; however, the duration of a calendar day can vary due to Daylight saving time and Leap seconds . A time standard is a specification for measuring time: assigning

10140-425: Is the moving longitudinal length and L 2 {\textstyle L_{2}} the moving transverse length, L 1 ′ = L 2 ′ {\textstyle L'_{1}=L'_{2}} being the rest lengths, then it is given: φ {\textstyle \varphi } can be arbitrarily chosen, so there are infinitely many combinations to explain

10296-432: Is the primary framework for understanding how spacetime works. Through advances in both theoretical and experimental investigations of spacetime, it has been shown that time can be distorted and dilated , particularly at the edges of black holes . Throughout history, time has been an important subject of study in religion, philosophy, and science. Temporal measurement has occupied scientists and technologists and has been

10452-568: Is to write down a general formula for computing the n th term as a function of n , enclose it in parentheses, and include a subscript indicating the set of values that n can take. For example, in this notation the sequence of even numbers could be written as ( 2 n ) n ∈ N {\textstyle (2n)_{n\in \mathbb {N} }} . The sequence of squares could be written as ( n 2 ) n ∈ N {\textstyle (n^{2})_{n\in \mathbb {N} }} . The variable n

10608-1610: Is true (if the velocity of the aether is small relative to the speed of light), then the expression can be simplified using a first order binomial expansion; ( 1 − x ) n ≈ 1 − n x {\displaystyle (1-x)^{n}\approx {1-nx}} So, rewriting the above in terms of powers; Δ λ 1 = 2 L ( ( 1 − v 2 c 2 ) − 1 − ( 1 − v 2 c 2 ) − 1 / 2 ) {\displaystyle \Delta {\lambda }_{1}=2L\left(\left({1-{\frac {v^{2}}{c^{2}}}}\right)^{-1}-\left(1-{\frac {v^{2}}{c^{2}}}\right)^{-1/2}\right)} Applying binomial simplification; Δ λ 1 = 2 L ( ( 1 + v 2 c 2 ) − ( 1 + v 2 2 c 2 ) ) = 2 L v 2 2 c 2 {\displaystyle \Delta {\lambda }_{1}=2L\left((1+{\frac {v^{2}}{c^{2}}})-(1+{\frac {v^{2}}{2c^{2}}})\right)={2L}{\frac {v^{2}}{2c^{2}}}} Therefore; Δ λ 1 = L v 2 c 2 {\displaystyle \Delta {\lambda }_{1}={L}{\frac {v^{2}}{c^{2}}}} It can be seen from this derivation that aether wind manifests as

10764-446: Is unless we experience it first hand. Time is not an empirical concept. For neither co-existence nor succession would be perceived by us, if the representation of time did not exist as a foundation a priori . Without this presupposition, we could not represent to ourselves that things exist together at one and the same time, or at different times, that is, contemporaneously, or in succession. Sequence For example, (M, A, R, Y)

10920-577: The Clock of the Long Now . They can be driven by a variety of means, including gravity, springs, and various forms of electrical power, and regulated by a variety of means such as a pendulum . Alarm clocks first appeared in ancient Greece around 250 BC with a water clock that would set off a whistle. This idea was later mechanized by Levi Hutchins and Seth E. Thomas . A chronometer is a portable timekeeper that meets certain precision standards. Initially,

11076-494: The Lorentz factor . This hypothesis was partly motivated by Oliver Heaviside 's discovery in 1888 that electrostatic fields are contracting in the line of motion. But since there was no reason at that time to assume that binding forces in matter are of electric origin, length contraction of matter in motion with respect to the aether was considered an ad hoc hypothesis . If length contraction of L {\textstyle L}

11232-615: The United States Naval Academy in Annapolis, Michelson conducted his first known light speed experiments as a part of a classroom demonstration. In 1881, he left active U.S. Naval service while in Germany concluding his studies. In that year, Michelson used a prototype experimental device to make several more measurements. The device he designed, later known as a Michelson interferometer , sent yellow light from

11388-424: The calendar , a mathematical tool for organising intervals of time, and the clock , a physical mechanism that counts the passage of time. In day-to-day life, the clock is consulted for periods less than a day, whereas the calendar is consulted for periods longer than a day. Increasingly, personal electronic devices display both calendars and clocks simultaneously. The number (as on a clock dial or calendar) that marks

11544-441: The codomain of the sequence is fixed by context, for example by requiring it to be the set R of real numbers, the set C of complex numbers, or a topological space . Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, a n rather than a ( n ) . There are terminological differences as well:

11700-496: The conscious experience . Time is often referred to as a fourth dimension , along with three spatial dimensions . Time is one of the seven fundamental physical quantities in both the International System of Units (SI) and International System of Quantities . The SI base unit of time is the second , which is defined by measuring the electronic transition frequency of caesium atoms. General relativity

11856-427: The convergence properties of sequences. In particular, sequences are the basis for series , which are important in differential equations and analysis . Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers . There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify

12012-473: The electromagnetic nature of light and developed what are now called Maxwell's equations , but these equations were still interpreted as describing the motion of waves through an aether, whose state of motion was unknown. Eventually, Fresnel's idea of an (almost) stationary aether was preferred because it appeared to be confirmed by the Fizeau experiment (1851) and the aberration of star light . According to

12168-411: The limit of a sequence of rational numbers (e.g. via its decimal expansion , also see completeness of the real numbers ). As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π , that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that

12324-420: The natural numbers . In the second and third bullets, there is a well-defined sequence ( a k ) k = 1 ∞ {\textstyle {(a_{k})}_{k=1}^{\infty }} , but it is not the same as the sequence denoted by the expression. Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion . This

12480-774: The y direction (assuming equal-length arms) and v T 3 {\textstyle vT_{3}} in the x direction. This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame. Therefore, the Pythagorean theorem gives the actual beam travel distance of L 2 + ( v T 3 ) 2 {\textstyle {\sqrt {L^{2}+\left(vT_{3}\right)^{2}}}} . Thus c T 3 = L 2 + ( v T 3 ) 2 {\textstyle cT_{3}={\sqrt {L^{2}+\left(vT_{3}\right)^{2}}}} and consequently

12636-676: The Hebrew word עידן, זמן iddan (age, as in "Ice age") zĕman(time) is often translated) is a medium for the passage of predestined events. (Another word, زمان" זמן" zamān , meant time fit for an event , and is used as the modern Arabic , Persian , and Hebrew equivalent to the English word "time".) The Greek language denotes two distinct principles, Chronos and Kairos . The former refers to numeric, or chronological, time. The latter, literally "the right or opportune moment", relates specifically to metaphysical or Divine time. In theology, Kairos

12792-528: The Mayans, Aztecs, and Chinese, there were also beliefs in cyclical time, often associated with astronomical observations and calendars. These cultures developed complex systems to track time, seasons, and celestial movements, reflecting their understanding of cyclical patterns in nature and the universe. The cyclical view of time contrasts with the linear concept of time more common in Western thought, where time

12948-534: The Michelson–Morley null result. For instance, if φ = 1 {\textstyle \varphi =1} the relativistic value of length contraction of L 1 {\textstyle L_{1}} occurs, but if φ = 1 / γ {\textstyle \varphi =1/\gamma } then no length contraction but an elongation of L 2 {\textstyle L_{2}} occurs. This hypothesis

13104-480: The Middle Dutch word klocke which, in turn, derives from the medieval Latin word clocca , which ultimately derives from Celtic and is cognate with French, Latin, and German words that mean bell . The passage of the hours at sea was marked by bells and denoted the time (see ship's bell ). The hours were marked by bells in abbeys as well as at sea. Clocks can range from watches to more exotic varieties such as

13260-517: The SI second. International Atomic Time (TAI) is the primary international time standard from which other time standards are calculated. Universal Time (UT1) is mean solar time at 0° longitude, computed from astronomical observations. It varies from TAI because of the irregularities in Earth's rotation. Coordinated Universal Time (UTC) is an atomic time scale designed to approximate Universal Time. UTC differs from TAI by an integral number of seconds. UTC

13416-592: The amount necessary to compensate for the angle discrepancy of the two beams. A first step to explaining the Michelson and Morley experiment's null result was found in the FitzGerald–Lorentz contraction hypothesis , now simply called length contraction or Lorentz contraction, first proposed by George FitzGerald (1889) in a letter to same journal that published the Michelson-Morley paper, as "almost

13572-544: The apparatus was assembled in a closed room in the basement of the heavy stone dormitory, eliminating most thermal and vibrational effects. Vibrations were further reduced by building the apparatus on top of a large block of sandstone (Fig. 1), about a foot thick and five feet (1.5 m) square, which was then floated in a circular trough of mercury. They estimated that effects of about 0.01 fringe would be detectable. Michelson and Morley and other early experimentalists using interferometric techniques in an attempt to measure

13728-428: The beam travel time T 3 {\textstyle T_{3}} as mentioned above. The classical analysis predicted a relative phase shift between the longitudinal and transverse beams which in Michelson and Morley's apparatus should have been readily measurable. What is not often appreciated (since there was no means of measuring it), is that motion through the hypothetical aether should also have caused

13884-522: The causal structure of events. Instead, the spacetime interval is calculated and classified as either space-like or time-like, depending on whether an observer exists that would say the events are separated by space or by time. Since the time required for light to travel a specific distance is the same for all observers—a fact first publicly demonstrated by the Michelson–Morley experiment —all observers will consistently agree on this definition of time as

14040-441: The complex modulus, i.e. | z | = z ∗ z {\displaystyle |z|={\sqrt {z^{*}z}}} . If ( a n ) {\displaystyle (a_{n})} is a sequence of points in a metric space , then the formula can be used to define convergence, if the expression | a n − L | {\displaystyle |a_{n}-L|}

14196-432: The definitions and notations introduced below. In this article, a sequence is formally defined as a function whose domain is an interval of integers . This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring

14352-413: The development of the Lorentz transformation and special relativity . After the "failed" experiment Michelson and Morley ceased their aether drift measurements and started to use their newly developed technique to establish the wavelength of light as a standard of length . The beam travel time in the longitudinal direction can be derived as follows: Light is sent from the source and propagates with

14508-399: The device to turn with close to zero friction, so that once having given the sandstone block a single push it would slowly rotate through the entire range of possible angles to the "aether wind", while measurements were continuously observed by looking through the eyepiece. The hypothesis of aether drift implies that because one of the arms would inevitably turn into the direction of the wind at

14664-409: The direction and the speed of the motion. At any given point on the Earth's surface, the magnitude and direction of the wind would vary with time of day and season. By analyzing the return speed of light in different directions at various different times, it was thought to be possible to measure the motion of the Earth relative to the aether. The expected relative difference in the measured speed of light

14820-697: The domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation ( k 2 ) ) k = 1 10 {\textstyle (k^{2}){\vphantom {)}}_{k=1}^{10}} denotes the ten-term sequence of squares ( 1 , 4 , 9 , … , 100 ) {\displaystyle (1,4,9,\ldots ,100)} . The limits ∞ {\displaystyle \infty } and − ∞ {\displaystyle -\infty } are allowed, but they do not represent valid values for

14976-433: The domain of a sequence to be the set of natural numbers . This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition,

15132-520: The events of the abbeys and monasteries of the Middle Ages. Richard of Wallingford (1292–1336), abbot of St. Alban's abbey, famously built a mechanical clock as an astronomical orrery about 1330. Great advances in accurate time-keeping were made by Galileo Galilei and especially Christiaan Huygens with the invention of pendulum-driven clocks along with the invention of the minute hand by Jost Burgi. The English word clock probably comes from

15288-414: The expected velocity of the Earth's motion in orbit and "certainly less than one-fourth". Although this small "velocity" was measured, it was considered far too small to be used as evidence of speed relative to the aether, and it was understood to be within the range of an experimental error that would allow the speed to actually be zero. For instance, Michelson wrote about the "decidedly negative result" in

15444-438: The first mechanical clocks driven by an escapement mechanism. The hourglass uses the flow of sand to measure the flow of time. They were used in navigation. Ferdinand Magellan used 18 glasses on each ship for his circumnavigation of the globe (1522). Incense sticks and candles were, and are, commonly used to measure time in temples and churches across the globe. Water clocks, and, later, mechanical clocks, were used to mark

15600-507: The following limits exist, and can be computed as follows: Michelson%E2%80%93Morley experiment The Michelson–Morley experiment was an attempt to measure the motion of the Earth relative to the luminiferous aether , a supposed medium permeating space that was thought to be the carrier of light waves . The experiment was performed between April and July 1887 by American physicists Albert A. Michelson and Edward W. Morley at what

15756-480: The frequency of electronic transitions in certain atoms to measure the second. One of the atoms used is caesium ; most modern atomic clocks probe caesium with microwaves to determine the frequency of these electron vibrations. Since 1967, the International System of Measurements bases its unit of time, the second, on the properties of caesium atoms. SI defines the second as 9,192,631,770 cycles of

15912-454: The fringes of the interference pattern should shift when rotating it by 90° as the two beams have exchanged roles. To find the fringe shift, subtract the path difference in first orientation by the path difference in the second, then divide by the wavelength , λ {\displaystyle \lambda } , of light; Note the difference between Δ λ {\displaystyle \Delta \lambda } , which

16068-436: The fundamental tests of special relativity . Physics theories of the 19th century assumed that just as surface water waves must have a supporting substance, i.e., a "medium", to move across (in this case water), and audible sound requires a medium to transmit its wave motions (such as air or water), so light must also require a medium, the " luminiferous aether ", to transmit its wave motions. Because light can travel through

16224-437: The hours even at night but required manual upkeep to replenish the flow of water. The ancient Greeks and the people from Chaldea (southeastern Mesopotamia) regularly maintained timekeeping records as an essential part of their astronomical observations. Arab inventors and engineers, in particular, made improvements on the use of water clocks up to the Middle Ages. In the 11th century, Chinese inventors and engineers invented

16380-421: The incorrect expression because he overlooked the increase in path length in the rest frame of the aether. This was corrected by Alfred Potier (1882) and Hendrik Lorentz (1886). The derivation in the transverse direction can be given as follows (analogous to the derivation of time dilation using a light clock ): The beam is propagating at the speed of light c {\textstyle c} and hits

16536-407: The increase in elapsed time from traveling against the aether wind is more than the time saved by traveling with the aether wind. Michelson expected that the Earth's motion would produce a fringe shift equal to 0.04 fringes—that is, of the separation between areas of the same intensity. He did not observe the expected shift; the greatest average deviation that he measured (in the northwest direction)

16692-474: The index, only the supremum or infimum of such values, respectively. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }} is the same as the sequence ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , and does not contain an additional term "at infinity". The sequence (

16848-434: The integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). Other examples of sequences include those made up of rational numbers , real numbers and complex numbers . The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as

17004-464: The intense work of Michelson during the preparation of the experiments. In 1886, Michelson and Morley successfully confirmed Fresnel's drag coefficient – this result was also considered as a confirmation of the stationary aether concept. This result strengthened their hope of finding the aether wind. Michelson and Morley created an improved version of the Michelson experiment with more than enough accuracy to detect this hypothetical effect. The experiment

17160-497: The interferometer was set up in a basement. Because the fringes would occasionally disappear due to vibrations caused by passing horse traffic, distant thunderstorms and the like, an observer could easily "get lost" when the fringes returned to visibility. The advantages of white light, which produced a distinctive colored fringe pattern, far outweighed the difficulties of aligning the apparatus due to its low coherence length . As Dayton Miller wrote, "White light fringes were chosen for

17316-482: The longitudinal direction the swimmer first moves upstream, so his velocity is diminished due to the river flow to c − v {\textstyle c-v} . On his way back moving downstream, his velocity is increased to c + v {\textstyle c+v} . This gives the beam travel times T 1 {\textstyle T_{1}} and T 2 {\textstyle T_{2}} as mentioned above. In

17472-428: The mirror at time T 1 {\textstyle T_{1}} and thus travels the distance c T 1 {\textstyle cT_{1}} . At this time, the mirror has traveled the distance v T 1 {\textstyle vT_{1}} . Thus c T 1 = L + v T 1 {\textstyle cT_{1}=L+vT_{1}} and consequently

17628-420: The mirror at time T 3 {\textstyle T_{3}} , traveling the distance c T 3 {\textstyle cT_{3}} . At the same time, the mirror has traveled the distance v T 3 {\textstyle vT_{3}} in the x direction. So in order to hit the mirror, the travel path of the beam is L {\textstyle L} in

17784-500: The most famous failed experiment in history. Instead of providing insight into the properties of the aether, Michelson and Morley's article in the American Journal of Science reported the measurement to be as small as one-fortieth of the expected displacement (Fig. 7), but "since the displacement is proportional to the square of the velocity" they concluded that the measured velocity was "probably less than one-sixth" of

17940-581: The nature of time. Plato , in the Timaeus , identified time with the period of motion of the heavenly bodies. Aristotle , in Book IV of his Physica defined time as 'number of movement in respect of the before and after'. In Book 11 of his Confessions , St. Augustine of Hippo ruminates on the nature of time, asking, "What then is time? If no one asks me, I know: if I wish to explain it to one that asketh, I know not." He begins to define time by what it

18096-414: The observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all readings." Use of partially monochromatic light (yellow sodium light) during initial alignment enabled the researchers to locate the position of equal path length, more or less easily, before switching to white light. The mercury trough allowed

18252-510: The occurrence of a specified event as to hour or date is obtained by counting from a fiducial epoch – a central reference point. Artifacts from the Paleolithic suggest that the moon was used to reckon time as early as 6,000 years ago. Lunar calendars were among the first to appear, with years of either 12 or 13 lunar months (either 354 or 384 days). Without intercalation to add days or months to some years, seasons quickly drift in

18408-518: The only hypothesis that can reconcile" the apparent contradictions. It was independently also proposed by Hendrik Lorentz (1892). According to this law all objects physically contract by L / γ {\textstyle L/\gamma } along the line of motion (originally thought to be relative to the aether), γ = 1 / 1 − v 2 / c 2 {\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}} being

18564-499: The other hand, the much more precise Michelson–Morley experiment (1887) apparently confirmed complete aether dragging and refuted the stationary aether. In addition, the Michelson–Morley null result was further substantiated by the null results of other second-order experiments of different kind, namely the Trouton–Noble experiment (1903) and the experiments of Rayleigh and Brace (1902–1904). These problems and their solution led to

18720-452: The other, this distance is Δ λ {\displaystyle \Delta \lambda } . Alternatively, consider the rearrangement of the speed of light formula c Δ T = Δ λ {\displaystyle c{\Delta }T=\Delta \lambda } . If the relation v 2 / c 2 << 1 {\displaystyle {v^{2}}/{c^{2}}<<1}

18876-418: The passage of time from the shadow cast by its crossbar on a nonlinear rule. The T was oriented eastward in the mornings. At noon, the device was turned around so that it could cast its shadow in the evening direction. A sundial uses a gnomon to cast a shadow on a set of markings calibrated to the hour. The position of the shadow marks the hour in local time . The idea to separate the day into smaller parts

19032-628: The positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved. Formally, a subsequence of the sequence ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} is any sequence of the form ( a n k ) k ∈ N {\textstyle (a_{n_{k}})_{k\in \mathbb {N} }} , where ( n k ) k ∈ N {\displaystyle (n_{k})_{k\in \mathbb {N} }}

19188-438: The properties of the luminiferous aether, used (partially) monochromatic light only for initially setting up their equipment, always switching to white light for the actual measurements. The reason is that measurements were recorded visually. Purely monochromatic light would result in a uniform fringe pattern. Lacking modern means of environmental temperature control , experimentalists struggled with continual fringe drift even when

19344-641: The radiation that corresponds to the transition between two electron spin energy levels of the ground state of the Cs atom. Today, the Global Positioning System in coordination with the Network Time Protocol can be used to synchronize timekeeping systems across the globe. In medieval philosophical writings, the atom was a unit of time referred to as the smallest possible division of time. The earliest known occurrence in English

19500-422: The relative velocities it follows that if the ether does slip past the relative velocity is less than one sixth of the earth’s velocity. From the standpoint of the then current aether models, the experimental results were conflicting. The Fizeau experiment and its 1886 repetition by Michelson and Morley apparently confirmed the stationary aether with partial aether dragging, and refuted complete aether dragging. On

19656-531: The relativity theory as a (halfway) redemption." Michelson–Morley type experiments have been repeated many times with steadily increasing sensitivity. These include experiments from 1902 to 1905, and a series of experiments in the 1920s. More recently, in 2009, optical resonator experiments confirmed the absence of any aether wind at the 10 level. Together with the Ives–Stilwell and Kennedy–Thorndike experiments , Michelson–Morley type experiments form one of

19812-408: The same time that another arm was turning perpendicularly to the wind, an effect should be noticeable even over a period of minutes. The expectation was that the effect would be graphable as a sine wave with two peaks and two troughs per rotation of the device. This result could have been expected because during each full rotation, each arm would be parallel to the wind twice (facing into and away from

19968-573: The sciences, and the performing arts all incorporate some notion of time into their respective measuring systems . Traditional definitions of time involved the observation of periodic motion such as the apparent motion of the sun across the sky, the phases of the moon, and the passage of a free-swinging pendulum. More modern systems include the Global Positioning System , other satellite systems, Coordinated Universal Time and mean solar time . Although these systems differ from one another, with careful measurements they can be synchronized. In physics, time

20124-426: The sequence of real numbers ( a n ) is such that all the terms are less than some real number M , then the sequence is said to be bounded from above . In other words, this means that there exists M such that for all n , a n ≤ M . Any such M is called an upper bound . Likewise, if, for some real m , a n ≥ m for all n greater than some N , then the sequence is bounded from below and any such m

20280-460: The set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes ( a k ) {\textstyle (a_{k})} for an arbitrary sequence. Often, the index k is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in In some cases, the elements of the sequence are related naturally to

20436-413: The speed of light c {\textstyle c} in the aether. It passes through the half-silvered mirror at the origin at T = 0 {\textstyle T=0} . The reflecting mirror is at that moment at distance L {\textstyle L} (the length of the interferometer arm) and is moving with velocity v {\textstyle v} . The beam hits

20592-491: The speed of light in the direction of movement through the presumed aether, and the speed at right angles. This result is generally considered to be the first strong evidence against some aether theories , as well as initiating a line of research that eventually led to special relativity , which rules out motion against an aether. Of this experiment, Albert Einstein wrote, "If the Michelson–Morley experiment had not brought us into serious embarrassment, no one would have regarded

20748-473: The speed of light. As pointed out by Maxwell (1878), only experimental arrangements capable of measuring second order effects would have any hope of detecting aether drift, i.e., effects proportional to v / c . Existing experimental setups, however, were not sensitive enough to measure effects of that size. Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect aether flow. In 1877, while teaching at his alma mater,

20904-416: The stationary and the partially dragged aether hypotheses, Earth and the aether are in relative motion, implying that a so-called "aether wind" (Fig. 2) should exist. Although it would be theoretically possible for the Earth's motion to match that of the aether at one moment in time, it was not possible for the Earth to remain at rest with respect to the aether at all times, because of the variation in both

21060-482: The subscript n refers to the n th element of the sequence; for example, the n th element of the Fibonacci sequence F {\displaystyle F} is generally denoted as F n {\displaystyle F_{n}} . In computing and computer science , finite sequences are usually called strings , words or lists , with the specific technical term chosen depending on

21216-583: The term was used to refer to the marine chronometer , a timepiece used to determine longitude by means of celestial navigation , a precision first achieved by John Harrison . More recently, the term has also been applied to the chronometer watch , a watch that meets precision standards set by the Swiss agency COSC . The most accurate timekeeping devices are atomic clocks , which are accurate to seconds in many millions of years, and are used to calibrate other clocks and timekeeping instruments. Atomic clocks use

21372-413: The transverse direction, the swimmer has to compensate for the river flow by moving at a certain angle against the flow direction, in order to sustain his exact transverse direction of motion and to reach the other side of the river at the correct location. This diminishes his speed to c 2 − v 2 {\textstyle {\sqrt {c^{2}-v^{2}}}} , and gives

21528-770: The travel time T 1 = L / ( c − v ) {\textstyle T_{1}=L/(c-v)} . The same consideration applies to the backward journey, with the sign of v {\textstyle v} reversed, resulting in c T 2 = L − v T 2 {\textstyle cT_{2}=L-vT_{2}} and T 2 = L / ( c + v ) {\textstyle T_{2}=L/(c+v)} . The total travel time T ℓ = T 1 + T 2 {\textstyle T_{\ell }=T_{1}+T_{2}} is: Michelson obtained this expression correctly in 1881, however, in transverse direction he obtained

21684-505: The travel time T 3 = L / c 2 − v 2 {\textstyle T_{3}=L/{\sqrt {c^{2}-v^{2}}}} , which is the same for the backward journey. The total travel time T t = 2 T 3 {\textstyle T_{t}=2T_{3}} is: The time difference between T ℓ {\displaystyle T_{\ell }} and T t {\displaystyle T_{t}}

21840-463: The two beams to diverge as they emerged from the interferometer by about 10 radians. For an apparatus in motion, the classical analysis requires that the beam-splitting mirror be slightly offset from an exact 45° if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed. In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely

21996-494: The type of object the sequence enumerates and the different ways to represent the sequence in computer memory . Infinite sequences are called streams . The empty sequence ( ) is included in most notions of sequence. It may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions , spaces , and other mathematical structures using

22152-546: The value of a sequence at the lowest input (often 1) is called the "first element" of the sequence, the value at the second smallest input (often 2) is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. f , a sequence abstracted from its input is usually written by a notation such as ( a n ) n ∈ A {\textstyle (a_{n})_{n\in A}} , or just as (

22308-544: The wavelength of light as a standard of length . At this time Michelson was professor of physics at the Case School of Applied Science, and Morley was professor of chemistry at Western Reserve University (WRU), which shared a campus with the Case School on the eastern edge of Cleveland. Michelson suffered a mental health crisis in September 1885, from which he recovered by October 1885. Morley ascribed this breakdown to

22464-484: The wind giving identical readings) and perpendicular to the wind twice. Additionally, due to the Earth's rotation, the wind would be expected to show periodic changes in direction and magnitude during the course of a sidereal day . Because of the motion of the Earth around the Sun, the measured data were also expected to show annual variations. After all this thought and preparation, the experiment became what has been called

22620-481: Was an illusion to humans. Plato believed that time was made by the Creator at the same instant as the heavens. He also says that time is a period of motion of the heavenly bodies . Aristotle believed that time correlated to movement, that time did not exist on its own but was relative to motion of objects. He also believed that time was related to the motion of celestial bodies ; the reason that humans can tell time

22776-464: Was because of orbital periods and therefore there was a duration on time. The Vedas , the earliest texts on Indian philosophy and Hindu philosophy dating to the late 2nd millennium BC , describe ancient Hindu cosmology , in which the universe goes through repeated cycles of creation, destruction and rebirth, with each cycle lasting 4,320 million years. Ancient Greek philosophers , including Parmenides and Heraclitus , wrote essays on

22932-478: Was expanded and collapsed at will." According to Kabbalists , "time" is a paradox and an illusion . According to Advaita Vedanta , time is integral to the phenomenal world, which lacks independent reality. Time and the phenomenal world are products of maya , influenced by our senses, concepts, and imaginations. The phenomenal world, including time, is seen as impermanent and characterized by plurality, suffering, conflict, and division. Since phenomenal existence

23088-429: Was intended to detect interferometric fringe shifts due to speed differences of oppositely propagating light waves through water at rest. The results of such experiments were all negative. This could be explained by using Fresnel's dragging coefficient , according to which the aether and thus light are partially dragged by moving matter. Partial aether-dragging would thwart attempts to measure any first order change in

23244-467: Was later extended by Joseph Larmor (1897), Lorentz (1904) and Henri Poincaré (1905), who developed the complete Lorentz transformation including time dilation in order to explain the Trouton–Noble experiment , the Experiments of Rayleigh and Brace , and Kaufmann's experiments . It has the form It remained to define the value of φ {\textstyle \varphi } , which

23400-463: Was linear or cyclical and if time was endless or finite . These philosophers had different ways of explaining time; for instance, ancient Indian philosophers had something called the Wheel of Time. It is believed that there was repeating ages over the lifespan of the universe. This led to beliefs like cycles of rebirth and reincarnation . The Greek philosophers believe that the universe was infinite, and

23556-477: Was only 0.018 fringes; most of his measurements were much less. His conclusion was that Fresnel's hypothesis of a stationary aether with partial aether dragging would have to be rejected, and thus he confirmed Stokes' hypothesis of complete aether dragging. However, Alfred Potier (and later Hendrik Lorentz ) pointed out to Michelson that he had made an error of calculation, and that the expected fringe shift should have been only 0.02 fringes. Michelson's apparatus

23712-533: Was only slowly adopted by different nations over a period of centuries, but it is now by far the most commonly used calendar around the world. During the French Revolution , a new clock and calendar were invented as part of the dechristianization of France and to create a more rational system in order to replace the Gregorian calendar. The French Republican Calendar 's days consisted of ten hours of

23868-454: Was performed in several periods of concentrated observations between April and July 1887, in the basement of Adelbert Dormitory of WRU (later renamed Pierce Hall, demolished in 1962). As shown in the diagram to the right, the light was repeatedly reflected back and forth along the arms of the interferometer, increasing the path length to 11 m (36 ft). At this length, the drift would be about 0.4 fringes. To make that easily detectable,

24024-482: Was possible with the accuracy required. For instance, the Fizeau wheel could measure the speed of light to perhaps 5% accuracy, which was quite inadequate for measuring directly a first-order 0.01% change in the speed of light. A number of physicists therefore attempted to make measurements of indirect first-order effects not of the speed of light itself, but of variations in the speed of light (see First order aether-drift experiments ). The Hoek experiment , for example,

24180-411: Was quite small, given that the velocity of the Earth in its orbit around the Sun has a magnitude of about one hundredth of one percent of the speed of light. During the mid-19th century, measurements of aether wind effects of first order, i.e., effects proportional to v / c ( v being Earth's velocity, c the speed of light) were thought to be possible, but no direct measurement of the speed of light

24336-581: Was subject to experimental errors far too large to say anything conclusive about the aether wind. Definitive measurement of the aether wind would require an experiment with greater accuracy and better controls than the original. Nevertheless, the prototype was successful in demonstrating that the basic method was feasible. In 1885, Michelson began a collaboration with Edward Morley , spending considerable time and money to confirm with higher accuracy Fizeau's 1851 experiment on Fresnel's drag coefficient, to improve on Michelson's 1881 experiment, and to establish

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