Time in physics

File:Foucaultspendulum.250px.jpg

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. Physicists use theories to predict measurements of time. What exactly time "is" and how it works is still largely undefined, except in relation to the other fundamental quantities.

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 of a hyperfine transition in the ¹³³Cs (caesium) atom.

The state of the art in timekeeping

The current smallest measurable times are on the order of 10^(-15) seconds. The hypothesised smallest possible time that can be ever be theoretically measured using scattering of point particles is called the Planck time, which is on the order of 5*10^(-44) seconds, or 10^{(-33.304890836104)} Caesium clock cycles.

Conceptions of time

Both Newton and Galileo and most people up until the 20th century thought that time was the same for everyone everywhere. 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. Physicists, based on Einstein's general relativity as well as the redshift of the light from receding distant galaxies, believe the entire Universe and therefore time itself began about 13.7 billion years ago in the big bang. Whether it will ever come to an end is an open question. (See: philosophy of physics.)

Regularities in nature

In order to measure time, one must record the number of times a phenomenon which is periodic have occurred. 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.

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.

Mechanical clocks

Richard of Wallingford, abbot of St. Alban's abbey, famously built a mechanical clock as an astronomical orrery about 1330,.

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 are appropriate for standards and scientific use.

Galilean time

Galileo Galilei 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, with his pulse.

In his Two New Sciences, 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.".

Galileo's experimental setup to measure the literal flow of time (see above), 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.

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

Newtonian physics and linear time

Main article: classical physics

In or around 1665, when Isaac Newton 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.

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.

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

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

The dotted quantities, ${\dot {\theta }}$ denote a function which corresponds to a Newtonian fluxion, whereas ${\theta }$ denote a function which corresponds to a Newtonian fluent. But linear time is the parameter for the relationship between the ${\dot {\theta }}$ and the ${\theta }$ of the physical system under consideration. Some decades later, it was found that, under a Legendre transformation, Lagrange's equations can be transformed 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:

${\dot {p}}=-{\frac {\partial H}{\partial q}}=\{p,H\}=-\{H,p\}$

${\dot {q}}=~~{\frac {\partial H}{\partial p}}=\{q,H\}=-\{H,q\}$

in the Poisson bracket notation. Thus by transformation to suitable functions, the solutions to sets of these first order differential equations can be more easily implemented or visualized than the second order equation of Lagrange or Newton, and clearly show 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.

The relationships 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, 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.

Thermodynamics and the paradox of irreversibility

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

• 1st law of thermodynamics

1824 - Sadi Carnot scientifically analyzed the steam engines with his Carnot cycle, an abstract engine. Rudolf Clausius 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:

• 2nd law of thermodynamics

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:

• 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.

Electromagnetism and the speed of light

In 1864, James Clerk Maxwell 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:

$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$

$\nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}$

$\nabla \cdot \mathbf {E} =0$

$\nabla \cdot \mathbf {B} =0$

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.

Signalling is one application of the electromagnetic wave 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. 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.

By the twentieth century, Einstein, on a train leaving the Bern railway station, 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.

Einsteinian physics and time

See 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.

In 1875, Hendrik Lorentz 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

From this one can show 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:

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

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 $d\tau$, 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 the inertial frame, i.e. that there is no 'universal clock'. Each inertial frame has its own local geometry, and therefore it's own measurements of space and time. This geometry is related to the energy of the reference frame.

Einstein's theory 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.)

Time in quantum mechanics

See also: quantum mechanics

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

$H(t)\left|\psi (t)\right\rangle =i\hbar {\partial \over \partial t}\left|\psi (t)\right\rangle$

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:

$\Delta E\Delta T\geq {\frac {\hbar }{2}}$

where

$\Delta E$ is the uncertainty in energy

$\Delta T$ 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. E and T are canonical conjugate variables of each other.

Corresponding commutator relations hold for momentum and position, 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.

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.

Signalling

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.

That said, systems can be synchronized, using technologies like GPS. The GPS satellites must account for the effects of gravitation and other relativistic factors in their circuitry.

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).

Time in cosmology

One solution to the equations of general relativity predicts non-static solutions. However, Einstein accepted only the single static universe, and modified the Einstein field equation to reflect this. But in 1927, Georges LeMaître 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". In 1929, Hubble announced his discovery of the expanding universe.

If the universe were expanding, then it must have been much smaller and therefore hotter and denser in the past. George Gamow 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, 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.

See also

• Arrow of time
• Relativistic dynamics
• Time domain
• Chronon
• Category:systems of units
• Category:Timekeeping

Further reading

• Boorstein, Daniel J., "The Discoverers". Vintage. February 12, 1985. ISBN 0-394-72625-1
• Prigogine, Ilya, "Order out of Chaos". ISBN 0-394-54204-5
• Stengers, Isabelle, and Ilya Prigogine, "Theory Out of Bounds". University of Minnesota Press. November 1997. ISBN 0-8166-2517-4
• Mandelbrot, Benoît, "Multifractals and 1/f noise". Springer Verlag. February 1999. ISBN 0-387-98539-5
• Serres, Michel, et al., "Conversations on Science, Culture, and Time (Studies in Literature and Science)". March, 1995. ISBN 0-472-06548-3
• Kuhn, Thomas S., "The Structure of Scientific Revolutions". ISBN 0-226-45808-3