Time in physics

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Foucault's pendulum in the Pantheon of Paris can measure time as well as demonstrate the rotation of Earth.
Enlarge
Foucault's pendulum in the Pantheon of Paris can measure time as well as demonstrate the rotation of Earth.

In physics, the treatment of time is a central issue. It has been treated as a question of geometry. One can measure time and treat it as a geometrical dimension, such as length, and perform mathematical operations on it. It is a scalar quantity and, like length, mass, and charge, is usually listed in most physics books as a fundamental quantity. Time can be combined mathematically with other fundamental quantities to derive other concepts such as motion, energy and fields. Time is largely defined by its measurement in physics. Timekeeping is a complex of technology and science issues, and part of the foundation of recordkeeping.

Prerequisites
scientific notation
natural units
algebra
geometry
vector notation
optics
operators
differential equations
partial differential equations
electrical engineering
signal processing

Contents

[edit] Current definition of a second of time

Currently, the standard time interval (called conventional second, or simply second) is defined as 9,192,631,770 oscillations between the two hyperfine levels of the ground state in the 133Cs (caesium) atom.

[edit] The state of the art in timekeeping

Prerequisites
Measurement
Scientific notation
Natural units

The UTC timestamp in use worldwide is an atomic time standard. The accuracy of such a time standard is currently on the order of 10(-15) seconds. The smallest time step of quantum mechanics is called the Planck time, which is on the order of 5*10(-44) seconds, or 10(-33.304890836104) Caesium clock cycles.

[edit] Conceptions of time

Main article: Time
Andromeda galaxy (M31) is two million light-years away. Thus we are viewing M31's light from two million years ago, a time before humans existed.
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Andromeda galaxy (M31) is two million light-years away. Thus we are viewing M31's light from two million years ago[1], a time before humans existed.

Both Newton and Galileo and most people up until the 20th century thought that time was the same for everyone everywhere. This is the basis for timelines, where time is a parameter. Our modern conception of time is based on Einstein's theory of relativity, in which rates of time run differently everywhere, and space and time are merged into spacetime, where we live on a world line rather than a timeline. Thus time is part of a coordinate, in this view. Physicists believe the entire Universe and therefore time itself began about 13.7 billion years ago in the big bang. (See #Time in cosmology below) Whether it will ever come to an end is an open question. (See philosophy of physics.)

[edit] Regularities in nature

Main article: History of science

In order to measure time, one might record the number of times a phenomenon which is periodic have occurred. (In English usage, these occurrences are termed events -- event might now also refer to a coordinate in spacetime and not only a UTC timestamp.) The regular recurrences of the seasons, the motions of the sun, moon and stars were noted and tabulated for millennia, before the laws of physics were formulated. The sun was the arbiter of the flow of time, but time was known only to the hour, for millennia. The gnomon was known across Eurasia, at least as far southward as the jungles of Southeast Asia[2].

I farm the land from which I take my food.
I watch the sun rise and sun set.
Kings can ask no more.

-- as quoted by Joseph Needham Science and Civilisation in China

In particular, the astronomical observatories maintained for religious purposes became accurate enough to ascertain the regular motions of the stars, and even some of the planets.

At first, timekeeping was done by hand, by priests, and then for commerce, with watchmen to note time, as part of their duties. The tabulation of the equinoxes, the sandglass, and the water clock became more and more accurate, and finally reliable. For ships at sea, boys were used to turn the sandglasses, and to call the hours.

[edit] Mechanical clocks

Richard of Wallingford (1292–1336), abbot of St. Alban's abbey, famously built a mechanical clock as an astronomical orrery about 1330[3],[4].

By the time of Richard of Wallingford, the use of ratchets and gears allowed the towns of Europe to create mechanisms to display the time on their respective town clocks; by the time of the scientific revolution, the clocks became miniaturized enough for families to share a personal clock, or perhaps a pocket watch. At first, only kings could afford them. Pendulum clocks were widely used in the 18th and 19th century. They have largely been replaced in general use by quartz and digital clocks. Atomic clocks can theoretically keep accurate time for millions of years. They are appropriate for standards and scientific use.

[edit] Galileo : the flow of time

Main article: reproducibility

In 1583, Galileo Galilei (1564-1642) discovered that a pendulum's harmonic motion has a constant period, which he learned by timing the motion of a swaying lamp in harmonic motion at mass at the cathedral of Pisa, with his pulse[5].

In his Two New Sciences (1638), Galileo used a water clock to measure the time taken for a bronze ball to roll a known distance down an inclined plane; this clock was

"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.".[6]

Galileo's experimental setup to measure the literal flow of time, in order to describe the motion of a ball, preceded Isaac Newton's statement in his Principia:

I do not define time, space, place and motion, as being well known to all.[7]

The Galilean transformations assume that time is the same for all reference frames.

[edit] Newton's physics: linear time

Main article: classical physics

In or around 1665, when Isaac Newton (1643-1727) derived the motion of objects falling under gravity, the first clear formulation for mathematical physics of a treatment of time began: linear time, conceived as a universal clock.

Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.[8]

The water clock mechanism described by Galileo was engineered to provide laminar flow of the water during the experiments, thus providing a constant flow of water for the durations of the experiments, and embodying what Newton called duration.

In this section, the relationships listed below treat time as a parameter which serves as an index to the behavior of the physical system under consideration. Because Newton's fluents treat a linear flow of time (what he called mathematical time), time could be considered to be a linearly varying parameter, an abstraction of the march of the hours on the face of a clock. Calendars and ships logs could then be mapped to the march of the hours, days, months, years and centuries.

Prerequisites
differential equations
partial differential equations

Lagrange (1736-1813) would aid in the formulation of a simpler version[9] of Newton's equations. He started with an energy term, L, named the Lagrangian in his honor, and formulated Lagrange's equations:

\frac{d}{dt} \frac{\partial L}{\partial \dot{\theta}} - \frac{\partial L}{\partial \theta} = 0.

The dotted quantities, {\dot{\theta}} denote a function which corresponds to a Newtonian fluxion, whereas θ denote a function which corresponds to a Newtonian fluent. But linear time is the parameter for the relationship between the {\dot{\theta}} and the θ of the physical system under consideration. Some decades later, it was found that the second order equation of Lagrange or Newton can be more easily solved or visualized by suitable transformation to sets of first order differential equations.

Lagrange's equations can be transformed, under a Legendre transformation, to Hamilton's equations; the Hamiltonian formulation for the equations of motion of some conjugate variables p,q (for example, momentum p and position q) is:

Prerequisites
Operators
Poisson brackets
\dot p = -\frac{\partial H}{\partial q} = \{p,H\} = -\{H,p\}
\dot q =~~\frac{\partial H}{\partial p} = \{q,H\} = -\{H,q\}

in the Poisson bracket notation and clearly shows the dependence of the time variation of conjugate variables p,q on an energy expression.

This relationship, it was to be found, also has corresponding forms in quantum mechanics as well as in the classical mechanics shown above. These relationships bespeak a conception of time which is reversible.

[edit] Thermodynamics and the paradox of irreversibility

Main article: arrow of time

By 1798, Benjamin Thompson (1753-1814) had discovered that work could be transformed to heat without limit - a precursor of the conservation of energy or

In 1824 Sadi Carnot (1796-1832) scientifically analyzed the steam engines with his Carnot cycle, an abstract engine. Rudolf Clausius (1822–1888) noted a measure of disorder, or entropy, which affects the continually decreasing amount of free energy which is available to a Carnot engine in the:

Thus the continual march of a thermodynamic system, from lesser to greater entropy, at any given temperature, defines an arrow of time. In particular, Stephen Hawking identifies three arrows of time[10]:

  • Psychological arrow of time - our perception of an inexorable flow.
  • Thermodynamic arrow of time - distinguished by the growth of entropy.
  • Cosmological arrow of time - distinguished by the expansion of the universe.

Entropy is maximum in an isolated thermodynamic system, and increases. In contrast, Erwin Schrödinger (1887–1961) pointed out that life depends on a "negative entropy flow"[11]. Ilya Prigogine (1917–2003) stated that other thermodynamic systems which, like life, are also far from equilibrium, can also exhibit stable spatio-temporal structures. Soon afterward, the Belousov-Zhabotinsky reactions [12] were reported, which demonstrate oscillating colors in a chemical solution[13]. These nonequilibrium thermodynamic branches reach a bifurcation point, which is unstable, and another thermodynamic branch becomes stable in its stead[14].

[edit] Electromagnetism and the speed of light

Main article: Maxwell's equations

In 1864, James Clerk Maxwell (1831-1879) presented a combined theory of electricity and magnetism. He combined all the laws then known relating to those two phenomenon into four equations. These vector calculus equations which use the del operator (\nabla) are known as Maxwell's equations for electromagnetism. In free space, the equations take the form:

Prerequisites
vector notation
partial differential equations
\nabla \times \mathbf{E} = - \frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}
\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t}
\nabla \cdot \mathbf{E} = 0
\nabla \cdot \mathbf{B} = 0

where

c is a constant that represents the speed of light in vacuum
E is the electric field
B is the magnetic field.

The solution to these equations is an electromagnetic wave, which always propagates at the 'speed of light' c, regardless of the speed of the electric charge that generated it. The wave is an oscillating electromagnetic field, intertwining E and B in a braid. This electromagnetic field is also often embodied as a photon which can be emitted by the acceleration of an electric charge. The frequency of the oscillation is variously a photon with a color, or a radio wave, or perhaps an x-ray or cosmic ray. The fact that light was predicted to always travel at speed c gave rise to the idea of the luminiferous aether and the detection of the absolute reference frame. The failure of the Michelson-Morley experiment to detect any motion of the Earth relative to light helped bring about relativity and the downfall of the idea of absolute time. In free space, Maxwell's equations have a symmetry which was exploited by Einstein in the twentieth century. See: #Signalling, below.

By the twentieth century, Einstein, on a train leaving the Bern railway station[15], could ask himself:

Would a train see the clock on the Bern station if it were moving at speed c?.
The answer was yes, but the clock would not be moving until the train slowed down from speed c.
Thus Einstein on the train would enjoy a longer interval than those at Bern station.

The equitable flow of time could not be universal among all observers[16].

[edit] Einstein's physics : spacetime

Main articles: special relativity (1905), general relativity (1915).

Einstein's 1905 special relativity challenged the notion of an absolute definition for times, and could only formulate a definition of synchronization for clocks that mark a linear flow of time:

If at the point A of space there is a clock ... If there is at the point B of space there is another clock in all respects resembling the one at A ... it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. ... We assume that ...
1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B, and also with the clock at C, the clocks at B and C also synchronize with each other.[17]
Stylized light cone to celebrate the centennial of Einstein's annus mirabilis
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Stylized light cone to celebrate the centennial of Einstein's annus mirabilis

In 1875, Hendrik Lorentz (1853-1928) discovered the Lorentz transformation, upon which Einstein's theory of relativity, published in 1915, is based. The Lorentz transformation states that the speed of light is constant in all inertial frames, frames with a constant velocity. Velocity is defined by space and time:

\textbf{V}={d\over t}

where

d is distance
t is time

Henri Poincaré (1854-1912) noted the importance of Lorentz' transformation and popularized it. In particular, the railroad car description can be found in Science and Hypothesis[18], which was published before Einstein's articles of 1905.

Einstein showed that if the speed of light is not changing between reference frames, space and time must be so that the moving observer will measure the same speed of light as the stationary one. Time in a moving reference frame is shown to run more slowly than in a stationary one by the following relation:

Prerequisites
algebra
trigonometry
\textbf{T}={{t}\over\sqrt{1 - v^2/c^2}}

where

T is the time in the moving reference frame
t is the time in the stationary reference frame
v is the velocity of the moving reference frame relative to the stationary one.
c is the speed of light

Moving objects therefore experience a slower passage of time. This is known as time dilation.

One may ask which reference frame is really the moving one, since observers in both would "feel" as if they were standing still and assume the other frame is the one in motion. This gives rise to such paradoxes as the Twin paradox.

That paradox can be resolved using Einstein's General theory of relativity, which uses Riemannian geometry, geometry in accelerated, noninertial reference frames. Employing the metric tensor which describes Minkowski space:

\left[(dx^1)^2+(dx^2)^2+(dx^3)^2-c(dt)^2)\right],

Einstein developed a geometric solution to Lorentz's transformation that preserves Maxwell's equations. His field equations give an exact relationship between the measurements of space and time in a given region of spacetime and the energy density of that region.

Einstein's equations predict that time should be altered by the presence of gravitational fields (see the Schwarzschild metric):

T=\frac{dt}{\sqrt{\left( 1 - \frac{2GM}{rc^2} \right ) dt^2 - \frac{1}{c^2}\left ( 1 - \frac{2GM}{rc^2} \right )^{-1} dr^2 - \frac{r^2}{c^2} d\theta^2 - \frac{r^2}{c^2} \sin^2 \theta \; d\phi^2}}

Where:

T is the gravitational time dilation of an object at a distance of r.
dt is the change in coordinate time, or the interval of coordinate time.
G is the gravitational constant
M is the mass generating the field
\sqrt{\left( 1 - \frac{2GM}{rc^2} \right ) dt^2 - \frac{1}{c^2}\left ( 1 - \frac{2GM}{rc^2} \right )^{-1} dr^2 - \frac{r^2}{c^2} d\theta^2 - \frac{r^2}{c^2} \sin^2 \theta \; d\phi^2} is the change in proper time dτ, or the interval of proper time.

Or one could use the following simpler approximation:

\frac{dt}{d\tau} = \frac{1}{ \sqrt{1 - \left( \frac{2GM}{rc^2} \right)}}

Time runs slower the stronger the gravitational field, and hence acceleration, is. The predictions of time dilation are confirmed by particle acceleration experiments and cosmic ray evidence, where moving particles decay slower than their less energetic counterparts. Gravitational time dilation gives rise to the phenomenon of gravitational redshift and delays in signal travel time near massive objects such as the sun. The Global Positioning System must also adjust signals to account for this effect.

Einstein's theory was motivated by the assumption that every point in the universe can be treated as a 'center', and that correspondingly, physics must act the same in all reference frames. His simple and elegant theory shows that time is relative to an inertial frame. In an inertial frame, Newton's first law holds; it has its own local geometry, and therefore its own measurements of space and time; there is no 'universal clock'. An act of synchronization must be performed between two systems, at the least.

[edit] Time in quantum mechanics

See also: quantum mechanics

There is a time parameter in the equations of quantum mechanics. The Schrödinger equation [19] is

Prerequisites
physics
quantum mechanics
H(t) \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle

One solution can be

| \psi_e(t) \rangle = e^{-iHt / \hbar} | \psi_e(0) \rangle.

where e^{-iHt / \hbar} is a Wick rotation (in the complex plane), and H is the scalar Hamiltonian.

But the Schrödinger picture shown above is equivalent to the Heisenberg picture, which enjoys a similarity to the Poisson brackets of classical mechanics. The Poisson brackets are superseded by a nonzero commutator, say [H,A] for observable A, and Hamiltonian H:

\frac{d}{dt}A=(i\hbar)^{-1}[A,H]+\left(\frac{\partial A}{\partial t}\right)_\mathrm{classical}.

This equation denotes an uncertainty relation in quantum physics. For example, with time (the observable A), the energy E (from the Hamiltonian H) gives:

\Delta E \Delta T \ge \frac{\hbar}{2}
where
ΔE is the uncertainty in energy
ΔT is the uncertainty in time
\hbar is Planck's constant

The more precisely one measures the duration of an event the less precisely one can measure the energy of the event and vice versa. This equation is different from the standard uncertainty principle because time is not an operator in quantum mechanics. Energy and time are canonical conjugate variables of each other.

Corresponding commutator relations also hold for momentum p and position q, which are conjugate variables of each other, along with a corresponding uncertainty principle in momentum and position, similar to the energy and time relation above.

Quantum mechanics explains the properties of the periodic table of the elements. Starting with Otto Stern's and Walter Gerlach's experiment with molecular beams in a magnetic field, Isidor Rabi (1898-1988), was able to modulate the magnetic resonance of the beam. In 1945 Rabi then suggested that this technique be the basis of a clock[20] using the resonant frequency of an atomic beam.

John Cramer is preparing an experiment to determine whether quantum entanglement is also nonlocal in time as it is in space. This can also be stated as 'sending a signal back in time'. The experiment is still in preparation as of 10:57, 16 November 2006 (UTC).

[edit] Dynamical systems

See dynamical systems and chaos theory, dissipative structures

One could say that time is a parameterization of a dynamical system that allows the geometry of the system to be manifested and operated on. It has been asserted that time is an implicit consequence of chaos (i.e. nonlinearity/irreversibility): the characteristic time, or rate of information entropy production, of a system. Mandelbrot introduces intrinsic time in his book Multifractals and 1/f noise.

[edit] Signalling

Prerequisites
electrical engineering
signal processing

Signalling is one application of the electromagnetic waves described above. In general, a signal is part of communication between parties and places. One example might be a yellow ribbon tied to a tree, or the ringing of a church bell. A signal can be part of a conversation, which involves a protocol. Another signal might be the position of the hour hand on a town clock or a railway station. An interested party might wish to view that clock, to learn the time. See: Time ball, an early form of Time signal.

Evolution of a world line. The light cones, top (future) and bottom (past), contain the worldline. The top and bottom sections are spacelike. The left and right sections, outside the cones are timelike.
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Evolution of a world line. The light cones, top (future) and bottom (past), contain the worldline. The top and bottom sections are spacelike. The left and right sections, outside the cones are timelike.

We as observers can still signal different parties and places as long as we live within their past light cone. But we cannot receive signals from those parties and places outside our past light cone.

Along with the formulation of the equations for the electromagnetic wave, the field of telecommunication could be founded. In 19th century telegraphy, electrical circuits, some spanning continents and oceans, could transmit codes - simple dots, dashes and spaces. From this, a series of technical issues have emerged; see Category:Synchronization. But it is safe to say that our signalling systems can be only approximately synchronized, a plesiochronous condition, from which jitter need be eliminated.

That said, systems can be synchronized (at an engineering approximation), using technologies like GPS. The GPS satellites must account for the effects of gravitation and other relativistic factors in their circuitry. See: Self-clocking signal.

[edit] Technology for timekeeping standards

The primary time standard in the U.S. is currently NIST-F1, a laser-cooled Cs fountain[21], the latest in a series of time and frequency standards, from the ammonia-based atomic clock (1949) to the caesium-based NBS-1 (1952) to NIST-7 (1993). The respective clock uncertainty declined from 10,000 nanoseconds/day to 0.5 nanoseconds/day in 5 decades[22]. In 2001 the clock uncertainty for NIST-F1 was 0.1 nanoseconds/day. Development of increasingly accurate frequency standards is underway.

In this time and frequency standard, a population of caesium atoms is laser-cooled to temperatures of one-millionth Kelvin. The atoms collect in a ball shaped by six lasers, two for each spatial dimension, vertical (up/down), horizontal (left/right), and back/forth. The vertical lasers push the caesium ball through a microwave cavity. As the ball is cooled, the caesium population cools to its ground state and emits light at its natural frequency, stated in the definition of second above. Eleven physical effects are accounted for in the emissions from the caesium population, which are then controlled for in the NIST-F1 clock. These results are reported to BIPM.

Additionally, a reference hydrogen maser is also reported to BIPM as a frequency standard for TAI (international atomic time).

The measurement of time is overseen by BIPM (Bureau International des Poids et Mesures), located in Sèvres, France, which ensures uniformity of measurements and their traceability to the International System of Units (SI) world-wide. BIPM operates under authority of the Convention du Metre, a diplomatic treaty between fifty-one nations, the Member States of the Convention, through a series of Consultative Committees, whose members are the respective national metrology laboratories.

[edit] Time in computational physics

Computational physics uses models of physical systems which are implemented in software, providing a simulation of the system. In the case of Monte Carlo simulations the model 'changes' on the bases of the input of many random numbers and the behavior of the system is studied to obtain knowledge of the real system (provided that the model simulates the real system adequately). Unlike in theoretical physics, where time may be represented as a variable in a mathematical equation, it is not obvious how time is to be represented adequately in a model which is basically a static structure of values combined with rules as to how those values should change in response to numerical input.

This problem is encountered in the study of magnetism by means of Ising and Potts spin models. Spins located in a lattice structure are changed from one step (or 'state' of the system) to the next according to a set of rules (known as a dynamics algorithm) formulated on the basis of thermodynamic principles. One might expect that time can be incorporated into such a model simply as the linear succession of its states, but in some cases this leads to behavior of the model which is inconsistent with what is observed in real systems (this, and how to define a unit of time in such a model, is discussed in some detail in the section on Time in Spin Models).

[edit] Time in cosmology

Main article: physical cosmology

The equations of general relativity predict a non-static universe. However, Einstein accepted only a static universe, and modified the Einstein field equation to reflect this by adding the cosmological constant, which he later described as the biggest mistake of his life. But in 1927, Georges LeMaître (1894-1966) argued, on the basis of general relativity, that the universe originated in a primordial explosion. At the fifth Solvay conference, that year, Einstein brushed him off with "Vos calculs sont corrects, mais votre physique est abominable"[23]. In 1929, Edwin Hubble (1889-1953) announced his discovery of the expanding universe. The current generally accepted cosmological model, the Lambda-CDM model, has a positive cosmological constant and thus not only an expanding universe but an accelerating expanding universe.

If the universe were expanding, then it must have been much smaller and therefore hotter and denser in the past. George Gamow (1904-1968) hypothesized that the abundance of the elements in the Periodic Table of the Elements, might be accounted for by nuclear reactions in a hot dense universe. He was disputed by Fred Hoyle (1915-2001), who invented the term 'Big Bang' to disparage it. Fermi and others noted that this process would have stopped after only the light elements were created, and thus did not account for the abundance of heavier elements.

Gamow's prediction was a 5–10 kelvin black body radiation temperature for the universe, after it cooled during the expansion. This was corroborated by Penzias and Wilson in 1965. Subsequent experiments arrived at a 2.7 kelvin temperature, corresponding to an age of the universe of 13.7 billion years after the Big Bang.

This dramatic result has raised issues: what happened between the singularity of the Big Bang and the Planck time, which, after all, is the smallest observable time. When might have time separated out from the spacetime foam[25]; there are only hints based on broken symmetries (see Spontaneous symmetry breaking, Timeline of the Big Bang, and the articles in Category:Physical cosmology).

General relativity gave us our modern notion of the expanding universe that started in the big bang. Using relativity and quantum theory we have been able to roughly reconstruct the history of the universe. In our epoch, during which electromagnetic waves can propagate without being disturbed by conductors or charges, we can see the stars, at great distances from us, in the night sky. (Before this epoch, there was a time, 300,000 years after the big bang, during which starlight would not have been visible.)

[edit] Reprise

Ilya Prigogine's reprise is "Time precedes existence". He contrasts the views of Newton, Einstein and quantum physics which offer a symmetric view of time (as discussed above) with his own views, which point out that statistical and thermodynamic physics can explain irreversible phenomena[26] as well as the arrow of time and the Big Bang.

[edit] See also

[edit] Further reading

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[edit] References

  1. ^ Fred Adams and Greg Laughlin (1999), Five Ages of the Universe ISBN 0-684-86576-9 p.35.
  2. ^ Fred Hoyle (1955), Frontiers of Astronomy. New York: Harper & Brothers
  3. ^ North, J. (2004) God's Clockmaker: Richard of Wallingford and the Invention of Time. Oxbow Books. ISBN 1-85285-451-0
  4. ^ Watson, E (1979) "The St Albans Clock of Richard of Wallingford". Antiquarian Horology 372-384.
  5. ^ Jo Ellen Barnett, Time's Pendulum ISBN 0-306-45787-3 p.99.
  6. ^ Galileo 1638 Discorsi e dimostrazioni matematiche, intorno á due nuoue scienze 213, Leida, Appresso gli Elsevirii (Louis Elsevier), or Mathematical discourses and demonstrations, relating to Two New Sciences, English translation by Henry Crew and Alfonso de Salvio 1914. Section 213 is reprinted on pages 534-535 of On the Shoulders of Giants:The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4
  7. ^ Newton 1687 Philosophiae Naturalis Principia Mathematica, Londini, Jussu Societatis Regiae ac Typis J. Streater, or The Mathematical Principles of Natural Philosophy, London, English translation by Andrew Motte 1700s. From part of the Scholium, reprinted on page 737 of On the Shoulders of Giants:The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4
  8. ^ Newton 1687 page 738.
  9. ^ "Dynamics is a four-dimensional geometry." --Lagrange (1796), Thèorie des fonctions analytiques, as quoted by Ilya Prigogine (1996), The End of Certainty ISBN 0-684-83705-6 p.58
  10. ^ pp. 182-195. Stephen Hawking 1996. The Illustrated Brief History of Time: updated and expanded edition ISBN 0-553-10374-1
  11. ^ Erwin Schrödinger (1945) What is Life?
  12. ^ G. Nicolis and I. Prigogine (1989), Exploring Complexity
  13. ^ R. Kapral and K. Showalter, eds. (1995), Chemical Waves and Patterns
  14. ^ Ilya Prigogine (1996) The End of Certainty pp. 63-71
  15. ^ A picture of the coordinated clocks of Bern railway station can be seen in Einstein's Clocks, Poincaré's Maps: Empires of Time, ISBN 0-393-02001-0 by Peter Galison (2003) p.31.
  16. ^ "From the well-worn statement that the speed of light is constant we conclude that space and time are in the eye of the beholder."--Brian Greene (2004) The Fabric of the Cosmos: space, time and the texture of reality ISBN 0-375-41288-3 p.47
  17. ^ Einstein 1905, Zur Elektrodynamik bewegter Körper [On the electrodynamics of moving bodies] reprinted 1922 in Das Relativitätsprinzip, B.G. Teubner, Leipzig. The Principles of Relativity: A Collection of Original Papers on the Special Theory of Relativity, by H.A. Lorentz, A. Einstein, H. Minkowski, and W. H. Weyl, is part of Fortschritte der mathematischen Wissenschaften in Monographien, Heft 2. The English translation is by W. Perrett and G.B. Jeffrey, reprinted on page 1169 of On the Shoulders of Giants:The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4
  18. ^ Henri Poincaré, (1902). Science and Hypothesis Eprint
  19. ^ E. Schrödinger, Phys. Rev. 28 1049 (1926)
  20. ^ [http://tf.nist.gov/timefreq/cesium/atomichistory.htm A Brief History of Atomic Clocks at NIST]
  21. ^ D. M. Meekhof, S. R. Jefferts, M. Stepanovíc, and T. E. Parker (2001) "Accuracy Evaluation of a Cesium Fountain Primary Frequency Standard at NIST", IEEE Transactions on Instrumentation and Measurement. 50, no. 2, (April 2001) pp. 507-509
  22. ^ James Jespersen and Jane Fitz-Randolph (1999). From sundials to atomic clocks : understanding time and frequency. Washington, D.C. : U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology. 308 p. : ill. ; 28 cm. ISBN 0160500109
  23. ^ John C. Mather and John Boslough (1996), The Very First Light ISBN 0-465-01575-1 p.41.
  24. ^ George Smoot and Keay Davidson (1993) Wrinkles in Time ISBN 0-688-12330-9 A memoir of the experiment program for detecting the predicted fluctuations in the cosmic microwave background radiation
  25. ^ Martin Rees (1997), Before the Beginning ISBN 0-201-15142-1 p.210
  26. ^ Prigogine, Ilya (1996), The End of Certainty: Time, Chaos and the New Laws of Nature. ISBN 0-684-83705-6 On pages 163 and 182.
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