A mathematical definition of time shows that time is a system of calibrating motion, devised to synchronize events or to order their sequence. Measurement of what we call time is invariably based on motion, whether of planets or of clocks. What clocks actually measure is precisely calibrated motion, and we choose to call it time. The following shows that a unit of time is derived from fundamental units of mass, energy, and space. A mathematical definition of time can be derived from the equation for the kinetic energy of a body in motion,
e = mv2 / 2
where e is kinetic energy, m is
mass, and v is velocity, defined as
v = d / t
where d is distance and t is time.
Substituting d/t for v gives
e = m(d/t)2 / 2
which we can solve for t, giving us a mathematical definition of time in
terms of space, mass, and energy:
t = d√m / √(2e)
This equation tells us that for a body in motion, time is proportional to
distance traveled, proportional to the square root of mass, and inversely
proportional to the square root of two times kinetic energy. The relationship
of these variables is apparent. A given value of t does not require a
unique set of values of d, m, and e, but any set that yields the
same value of t. Therefore, the interval t it takes a body of
mass m, having kinetic energy e, to travel in space a distance d
can be measured by a clock, which correlates in its mechanism a set of values for d, m,
and e to give the same value of t. This correlation is illustrated by considering that not only can the clock measure the interval of time for the body to travel a given distance after receiving an impulse of a given energy, but the accuracy of the clock can be checked against the moving body, if the values of d, m, and e for the body are precisely known. What we call time is a means to correlate the motions of two or more systems. Time is essentially a system of calibrating the parameters of motion, according to the above equation, by which the standard unit can be defined.
Using SI units of 1 meter, 1 kilogram, and 1 joule in the above mathematical definition of time gives a value of 0.707 seconds for the value of time represented by t. To make that equation a definition of the second, a value of 1.414 meters can be used for distance. A second is therefore the time it takes a body of 1 kilogram mass, having 1 joule of kinetic energy, to travel 1.414 meters. It is important to note two things. First, that is what a second is, a result of three variables: mass, distance, and kinetic energy. Second, those quantities give us one possible instance of a second. Other values for distance, mass, and energy can also give one second, in accordance with the mathematical definition. For example, a projectile having a mass of 10 grams, kinetic energy of 5000 joules, moving a distance of 1000 meters in a vacuum is also an instance of a second. Mass, space, and energy are real,
and what we call time is a mathematical derivation of those three aspects of reality.
The mathematical definition of time is based on the motion of macroscopic bodies such
as planets, mechanical clocks, and electromechanical clocks which use electronics to
count the physical vibrations of crystals. While the second can be defined mathematically,
it has traditionally been defined in terms of the day or the year, based on the rotational
or orbital motions of Earth. In 1956, the second was defined as "the fraction 1/31,556,925.9747
of the tropical year..." in a curious mixing of decimal and fractional notation. In that
definition, the duration of the second was a function of the above equation, in terms of
the distance traveled by Earth in one orbit, Earth's mass, and its kinetic energy, divided
by that unwieldy denominator. As Earth's orbit is variable, this definition proved unreliable,
and in 1967, the second was defined as "the duration of 9,192,631,770 periods of the
radiation corresponding to the transition between the two hyperfine levels of the
ground state of the caesium-133 atom."
Using the cesium atomic clock shifts the basis of measuring time from the kinetic energy of a mass
in motion to the energy of the photon. That energy is derived from the transistion of the electron to a lower energy level in the atom, and the frequency of the emitted photon is that energy divided by Planck's constant. The atomic clock defines a unit of time according to a count of a specified number of cycles of electromagnetic oscillation of the photon. The duration of that count is inversely proportional to the frequency, and because the frequency is proportional to the energy of the photon, the unit of time is inversely proportional to the energy of the photon. The energy of the photon is given by
E = hf
where E is the energy, h is Planck's constant, and f is the frequency of the electromagnetic oscillation of the photon. By transposition,
f = E/h
The time for a given number of cycles is the number of cycles divided by the frequency. In the case of the cesium
atomic clock it is
t = 9,192,631,770 / f
Substituting E/h for f,
t = 9,192,631,770 / (E/h)
t = 9,192,631,770*h / E
The last equation defines the second in terms of a fixed number of cycles times Planck's constant, divided by the energy
of the photon emitted by the hyperfine transition of the cesium-133 atom. The units of h are joules*seconds, and the units of E are joules. The joules in numerator and denominator cancel out, so the resulting unit for t is seconds. From that definition of the second, we derive a mathematical definition of time that allows the number of cycles to be variable:
t = nh / E
where t is time in seconds, n is the number of cycles counted, h is Planck's constant, and E is the energy of the photon. In this case, time is defined in terms of two variables: the number of electromagnetic oscillations of the photon and the energy of the photon, which is not constrained to a fixed value as in the cesium atomic clock. In this mathematical definition of time, the energy of the photon depends on the atom used and on which energy level transition generated the photon. A photon having greater energy than that used in the cesium atomic clock will have a higher frequency, and the result will be a shorter interval of time for the same number of cycles counted. An atomic clock based on a higher energy photon needs to specify a greater number of cycles in order to define the standard unit of time.
The previous mathematical definition of time, based on the kinetic energy and distance traveled of a mass in motion, differs from the definition based on the number of oscillations of a photon and its energy. The obvious difference is
that the latter substitutes the count of oscillations of the photon for distance traveled and mass.
It is possible to substitute one variable for two because the photon has zero rest mass, so its energy alone is the determining factor in the frequency of oscillation. The count of cycles of oscillation is analogous to distance traveled in the former definition. The atomic clock is an extremely precise analog of the mechanical clock.
How is a mathematical definition of time to be reconciled with traditional notions
of time, specifically that time is a dimension? All that is required is to
acknowledge that it always has and always will be the present. Past and future
are abstract. When the future arrives, it is always the present. When
dinosaurs roamed the Earth, they did so in the present. The present persists;
it is the realm of possibility in which motion and events may occur. If one
recognizes that events always occur in the present, it becomes unnecessary to
regard time as anything more than a mathematical relation of variables that quantify
space, mass, and energy in mechanical systems, or energy and number of oscillations
in the case of atomic clocks. In the definition of time concerning mechanical systems,
the equivalence of mass and energy may reduce time to a more fundamental relation
between space and energy.
Steven A. Brown