# Time in physics

In physics, the treatment of time is a central issue. It has been treated as a question of geometry. (See: philosophy of physics.)

 Contents

### Regularities in Nature

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.

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

### Astronomical observatories

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.

### Timekeeping technology by the advent of the scientific revolution

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.

### Galileo's water clock

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

### The flow of time

• 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

### Newtonian physics and linear time

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 (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:

[itex]
 \frac{d}{dt}
\frac{\partial L}{\partial \dot{\theta}}


- \frac{\partial L}{\partial \theta} = 0. [itex] The dotted quantities, [itex]{\dot{\theta}}[itex] denote a function which corresponds to a Newtonian fluxion, whereas [itex]{{\theta}}[itex] denote a function which corresponds to a Newtonian fluent. But linear time is the parameter for the relationship between the [itex]{\dot{\theta}}[itex] and the [itex]{{\theta}}[itex] 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:

[itex]\dot p = -\frac{\partial H}{\partial q} = \{p,H\} = -\{H,p\} [itex]
[itex]\dot q =~~\frac{\partial H}{\partial p} = \{q,H\} = -\{H,q\} [itex]

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.

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

Somewhere between 1831 and 1879, James Clerk Maxwell developed a combined theory of electricity and magnetism. These vector calculus equations which use the del operator ([itex]\nabla[itex]) are known as Maxwell's equations for electromagnetism. In free space, the equations take the form:

[itex]\nabla \times \mathbf{E} = - \frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}[itex]
[itex]\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t}[itex]
[itex]\nabla \cdot \mathbf{E} = 0[itex]
[itex]\nabla \cdot \mathbf{B} = 0[itex]

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

The solution to these equations is a wave, which propagates at speed c. 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 x-ray or cosmic ray. Thus 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.)

In free space, Maxwell's equations have a symmetry which was exploited by Einstein in the twentieth century.

### Einsteinian physics and time

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.

Einstein's theory of relativity uses Riemannian geometry, employing the metric tensor which describes Minkowski space:

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

to develop a geometric solution to Lorentz's transformation that preserves Maxwell's equations.

Einstein's theory was motivated by the assumption that no point in the universe can be a 'center', and that correspondingly, physics must act the same in all inertial 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.

[itex]E^2 = m^2c^4+p^2c^2 \ [itex] (atomic energy)

E = energy, m = mass, p = momentum, c = the speed of light

### Quantum physics and time

There is a time parameter in the equations of quantum mechanics. Currently, General relativity and quantum mechanics are inconsistent with each other. The Schrödinger equation 6

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

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.

### Dynamical systems

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.

### Notes

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