Clocks
There are many ways to measure time. A time measuring device is commonly called a
clock (Wikipedia).
Atomic clocks are involved with modern definitions of time. However, there seems to
be a problem with this. The following is a quote from the book
"Relativity Reexamined" by
Léon Brillouin.
<quote>
We witnessed the invention of atomic clocks of incredible accuracy, whose physical
properties differ very much from the clocks Einstein imagined. This will be discussed
in some detail in Chapter 3. Let us mention here a real difficulty resulting from
internationally adopted definitions. The unit of length is based on the wavelength
of a spectral line of krypton-86 under carefully specified conditions with accuracy
$10^8$ and the unit of time is based on the frequency of a spectral line of cesium
with accuracy $10^{12}$. Hence, the same physical phenomenon, a spectral line, is used
for two different definitions: length and time, and the velocity $\,c\,$ of light remains
undefined and looks arbitrary. It should be stated, once and for all, whether a spectral
line should be used to define a frequency or a wavelength, but not both!
</quote>
Indeed, according to
Wikipedia: "in 1967 the Thirteenth General Conference on Weights
and Measures defined the SI second of atomic time". Therefore, whatever is right
or wrong, time period $T$ (= 1/frequency) and wavelength $\lambda$ are nowadays related
to each other via the speed of light: $\lambda = c\, T$ . So the behavior of time
intervals $\Delta t$ can be derived easily. They are simply inversely proportional
to (sub)atomic mass, just as it is the case with length, size and distances $\Delta L$ :
$$
\Delta L \sim 1/m_e \quad ; \quad \Delta t \sim 1/m_e
$$
It is clear that if size elongation applies to lengths to be measured with a measuring rod
and if the measuring rod itself has become longer as well, then it's impossible to tell
the difference between two measurements. But the same holds for empty space nowadays,
because the wavelength of light is used as a standard and we have seen that this wavelength
shows a redshift i.e. becomes larger in the past. It is evident herewith that size
elongation also extends to "empty" space. And the same holds with time elongation.
As a consequence of all this, according to Brillouin: the velocity $\,c\,$ of light [..]
looks arbitrary. Worse, the speed of light has been more or less defined nowadays
as being an absolute constant, thus leaving little or no room for a varying light speed
eventually. Together with this, there is little or no room left for other constants of nature
to vary with time, such as has been proposed by Setterfield for e.g. the Planck constant $\,h$ .
According to my humble opinion, these matters are not so serious as they seem to be. It's just
how decisions are made between what's supposed to be constant and what's supposed to be varying.
It seems that we have some freedom in doing this. The worst that can happen is that certain
phenomena - hitherto regular - will show up in the future with kind of anomalous behavior.
So far so good about atomic clocks. But there have been and still are mechanical
ways to measure time, as well. One very classical way to do it is with a
Pendulum (Wikipedia). The basic formula
for the period $T$ of an (ideal) pendulum is, according to the Wikipedia reference:
$$
T = 2\pi \sqrt{\frac{L}{g}}
$$
Where the gravity constant $g$ and the length $L$ are recognized, of course, immediately.
Let's do the time comparison again - it has become sort of routine meanwhile:
$$
T = \sqrt{\frac{L}{g}} \sim \sqrt{\frac{1/m_e}{m_e^3}} \sim \frac{1}{m_e^2}
$$
This would mean that pendulum clocks have been ($\mbox{quadratically}$) running slower
in the past!
Such is very much in contrast with the atomic clock, which runs only slower like this:
$$
T \sim 1/m_e \quad \mbox{: linear}
$$
Let's consider instead another time mechanism, an astronomical clock, namely the period of
a planet around the sun. Let $r$ be the radius of its circular orbit, then, according to a
Wikipedia reference:
$$
T=2\pi\sqrt{r^3\over{\mu}} \quad \mbox{with} \quad \mu = GM
\quad \Longrightarrow \quad
T \sim \sqrt{\frac{1/m_e^3}{m_e}} \sim \frac{1}{m_e^2}
$$
Exactly the same relationship as with the pendulum is found, thus enhancing confidence that
orbital clocks have been running slower than atomic clocks in the past, quadratically,
to be precise, as compared with linear for atomic clocks.
If seconds have been longer in the past and if time is measured in these longer seconds too,
then perhaps things in the universe are not as old as they are assumed to be with nowadays
standard cosmology. Is it possible to utter some wishful thinking now without being ridiculous?
Whatever. It's not too much of a difficult experiment to figure out whether the distinction
between orbital clocks and atomic clocks makes sense. "Common" time is measured with atomic
clocks these days. So the only thing we have to do is observe a pendulum clock under strict
conditions, such as a (temperature) controlled environment. If our theory is allright, then
it shall be observed that the pendulum clock will be ticking faster and faster, because due
to increasing mass, time intervals are decreasing (quadratically in comparison with linear).
So our pendumum will be running ahead of common time in due time. Such a simple test could
even be attempted at the NCCI in Urk.
Zero Point Energy. One further thing we can say is about energy, especially
about the Zero Point Energy of (ideal quantum mechanical) harmonic oscillators:
$$
E = \frac{1}{2} h\,\nu = \frac{1}{2} h\, \frac{1}{T} \quad \Longrightarrow \quad
E \sim m_e
$$
Here $\nu = $ frequency and $h = $ Planck's constant $= 6.62607004 \times
10^{-34} m^2 kg / s$ .
The radiation energy of photons is, of course, increasing as well:
$$
E = h\,\nu = h\, \frac{1}{T} \quad \Longrightarrow \quad E \sim m_e
$$
All this could have been seen immediately, though. Because, according to Albert Einstein's
most famous formula, energy is proportional to (rest) mass:
$$
E = m_e c^2 \sim m_e
$$
We conclude that all energy in the cosmos is increasing with time.
Setterfield's idea that
only the ZPE would be increasing while $\,m c^2\,$ would remain constant is in blatant
contradiction with the very meaning of $\,E = m c^2$ .
So far so good about clocks varying with varying (rest) mass. Now the question may be raised
if there exist clocks that are independent of any varying mass. It could turn out that
the answer is affirmative. Such clocks may be called nuclear clocks and the associated
time might be called numclear time. Read the section on Radioactive
decay for details.
The second flavor of the theory,
with varying mass of electrons alone, gives for both orbital clocks a somewhat different result:
$$
g = GM/R^2 \sim m_e^2 \quad \Longrightarrow \quad \left\{ \begin{array}{ll}
T = \sqrt{L/g} \sim \sqrt{(1/m_e)/(m_e^2)} & \sim 1/m_e^{3/2} \\
T = 2\pi\sqrt{r^3/GM} \sim \sqrt{1/m_e^3} & \sim 1/m_e^{3/2} \end{array} \right.
$$
The third flavor of the theory,
with varying mass of protons / neutrons alone, gives for both orbital clocks:
$$
g = GM/R^2 \sim m_e \quad \Longrightarrow \quad \left\{ \begin{array}{ll}
T = \sqrt{L/g} & \sim 1/\sqrt{m_e} \\ T = 2\pi\sqrt{r^3/GM} & \sim 1/\sqrt{m_e} \end{array} \right.
$$
Obviously, the product of the proportionalities of flavor (2) and flavor (3) is the proportionality
of flavor (1) : $1/m_e^{3/2} \times 1/\sqrt{m_e} = 1/m_e^2$ .
It may be questioned if the relationships $L \sim 1/m_e$ and $T \sim 1/m_e^2$ also hold for empty
space (still supposed that they hold water for rigid rods and clocks).
The answer must be affirmative, because, according to the General theory of Relativity:
- Inertial mass = Gravitational mass
Consequently:
$$
m\cdot a = G\frac{m\cdot M}{R^2} = m\cdot g \sim \frac{m_e^2}{1/m_e^2} = m_e^4 \quad \Longrightarrow \quad
a = \frac{d^2 L}{dT^2} \sim [m/s^2] \sim m_e^3 = \frac{1/m_e}{\left(1/m_e^2\right)^2}
$$