In physics, the treatment of time is
a central issue. It has been treated as a question of geometry.
(See: philosophy of physics.) 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 measure and 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. 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 133 caesium atom.
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. There is also a theoretical smallest time,
the Planck time. 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 thirteen billion years ago in the big bang. Whether
it will ever come to an end is an open question.
Regularities in Nature
In order to measure time, one must record
the number of times a phenomenon which is periodic had 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.
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.
Measuring Time
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.
The use of the pendulum, 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.
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.
Mechanical pendulums clocks were widely used
in the 18th and 19th century, and have largely been replaced
by quartz and digital clocks in general use and atomic clocks,
which can theoretically keep accurate time for millions of
years, in scientific use.
The current smallest measurable times are
on the order of 10 ^{15} seconds. It is theorized
that there is a smallest possible time, called the Planck
time, which is on the order of 10 ^{44} seconds.
Galilean Time
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.".1
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.2
The Galilean transformations assume that time is the same
for all reference frames.
Newtonian physics and linear time
See 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.3
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 (17361813) 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:
The dotted quantities, denote
a function which corresponds to a Newtonian fluxion, whereas
? denote a function which corresponds to a Newtonian fluent.
But linear time is the parameter for the relationship between
the and the ?
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:
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.
Thermodynamics and the paradox of irreversibility
1824  Sadi Carnot scientifically analyzed
the steam engines with his Carnot cycle, an abstract engine.
Along with the conservation of energy, which was enunciated
in the nineteenth century, the second law of thermodynamics
noted a measure of disorder, or entropy.
1st law of thermodynamics
2nd law of thermodynamics
See the arrow of time for the relationship
between irreversible processes and the laws of thermodynamics.
In particular, Stephen Hawking identifies three arrows of
time5:
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 ()
are known as Maxwell's equations for electromagnetism. In free
space, the equations take the form:
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 a wave, which always propagates
at speed c, regardless of the speed of the electric charge
that generated it. The wave is an oscillating electromagnetic
field, 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 xray 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.
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 time4:
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:
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:
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:
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 by the following relation:
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
Or one could use the following simpler approximation:
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.)
Quantum physics and time
See quantum mechanics
There is a time parameter in the equations
of quantum mechanics. The Schrodinger equation 6
can be transformed by the Wick rotation, into
the diffusion equation (Schrodinger himself noted this). The
meaning of this transformation is not understood, and highly
controversial.
It is also theorized that time obeys an uncertainty
relation in quantum physics with energy:
can be transformed by the Wick rotation, into
the diffusion equation (Schrodinger himself noted this). The
meaning of this transformation is not understood, and highly
controversial.
It is also theorized that time obeys an uncertainty
relation in quantum physics with energy:
where
?E is the uncertainty in energy
?T is the uncertainty in time
where
?E is the uncertainty in energy
?T is the uncertainty in time
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.
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 consquence 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.
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).
