Logarithmic time may explain the beginning
Isaac Asimov once proposed that we should measure time logarithmically. You are familiar with logarithmic scales from the Richter scale for…
Isaac Asimov once proposed that we should measure time logarithmically. You are familiar with logarithmic scales from the Richter scale for earthquake severity where a 7 is 10 times more powerful than a 6. We also use it for measuring sound amplitude. Decibels are tenths of Bels which is a logarithmic scale. A sound that is 5 Bels is 10 times louder than a sound that is 4 Bels.
We typically think of time as linear. One second now is the same as one second last year. This view of time, however, implicitly assumes that time doesn’t speed up or slow down with, well, time.
That makes sense for anything within the last 13 or so billion years, but the closer we get to the beginning of time, the Big Bang, the less it makes sense. Indeed, the Big Bang time itself may be like absolute zero, a theoretical beginning of time that is unreachable by any measurement and may not exist.
Logarithmic scales are useful because they make it easy to understand the ratio of one measurement to another. For example, when Richter invented his scale in the 1930s, he set 0 as the minimum value that seismographs can measure. That is not “no earthquake” it is just no measurable earthquake.
Seismographs are so sensitive now, however, they can detect tremors with negative Richter values.
Likewise, 0 Bels doesn’t mean “no sound”. It means a human can’t hear anything quieter. Anything with negative Bels (or deciBels) cannot be heard. Of course, sensitive sound equipment can easily pick up sounds with negative deciBels.
If you are familiar with sound recording, you may notice that the sound is all measured in negative deciBels and that anything above 0 deciBels is considered too loud for the equipment. That is because the sound is given by the ratio of the sound intensity to the system being “saturated”, which for digital sound equipment may mean that it has run out of bits to represent the sound intensity. For analog, it could be a maximum voltage in its circuits. It’s all relative.
This is different from linear measuring scales that have a genuine zero value like distance.
So, the question is whether time is more like deciBels, in that it is a ratio to something else, or is it more like distance with a zero at some point in the past?
Asimov believed time should be a ratio because as one moves back further in time objects get closer together. As they get closer together, more things happen. Thus, any thermal “clock” used to measure time would run faster. I say thermal clock because when we talk about things moving faster what we really mean is temperature is increasing. A clock that measures the frequency of interaction of particles would be considered a thermal clock.
If we take our thermal clock and put it further back in time, it will speed up. At the Big Bang, which has theoretically infinite temperature, the clock would run infinitely fast.
Many physicists believe that there was no Big Bang singularity. It seems as if quantum theory should prevent singularities by the uncertainty principle.
So, when I say I want to measure time logarithmically, I really mean relative to the speed of time at that point where the Big Bang was at maximal temperature, perhaps at the end of its inflationary phase.
The reason is that this is the furthest back in time we can measure. Thus, like the Richter scales and deciBel scales, we are measuring time logarithmically relative to our instruments.
The relationship between time and temperature goes even deeper. Models of cosmology use a standard formulation of General Relativity called the Arnowitt-Deser-Misner (ADM) formalism that describes how a three dimensional universe evolves in a time dimension. This is a formal mathematical description of geometrodynamics, the dynamics of spacetime geometry.
Spacetime stretches in the time direction as space expands, and this stretching acts like the inverse of temperature. The more stretched time is, the fewer things can happen and the longer they take, so the cooler spacetime is.
This inverse temperature time has to do with the random fluctuations in spacetime itself, similar to the jiggling atoms. What’s going on is that as spacetime evolves, little waves happen, interference between those waves occurs which becomes chaotic at very small spacetime scales. The energy of matter feeds this turbulence.
The universe at cosmological scale resembles a big heat engine and as such follows simple thermodynamical laws. Energy is pumped into the fabric of spacetime from matter. Spacetime responds by expanding its volume in a gas-like way. Pressure of matter couples to that volume expansion the same way as a gas.
The first law of thermodynamics applies: The change in energy is the internal energy plus the work done. That work is the negative of pressure times the change in volume for a gas or fluid. In the case of the universe, matter is doing work on spacetime, increasing its volume. Since the change in volume is positive, spacetime is getting bigger, it means that matter is losing energy to spacetime.
The second law of thermodynamics comes into play as well. The entropy density of the universe is given by the difference between the internal energy of matter and the work done on spacetime times the inverse temperature. You can show, in the case of the universe, entropy is conserved if you assume that the inverse temperature is a scaling of the arrow of time. This suggests that we could create time coordinates that are in units of this scaling, and hence we can regard temperature and time to be the same.
Unlike arbitrary time coordinates, this inverse temperature time has a physical meaning. While the time we measure is quite arbitrary, a thermal time tells us exactly how disorder relates to energy and work done in an expanding universe.
So, if time is like temperature, why not measure it on a linear scale like Kelvin?
This is where that pesky problem of beginnings comes from. At infinite temperature and density, the temperature as given by time would be infinite as well. (Inverse temperature, which is our time parameter, is zero at this point.) At the other end of time, towards the heat death of the universe, zero temperature is a limit as the universe becomes infinitely old and infinitely expanded to the point where no particle can even communicate with another, and all light is redshifted to oblivion.
We know that, from our observations, these limits are extrapolations just as infinite and zero temperature are extrapolations. They are likely never reached. The Big Bang was no singularity with infinite density. A logarithm captures this behavior because the log of zero is minus infinity, an unreachable number, while the log of infinity is also infinity, another unreachable number. Zero, meanwhile, on a log scale is a finite number that you know you can reach.
Temperature, while normally linear, has a logarithmic scale as well. John Dalton, one of the fathers of modern chemistry, actually invented such as scale back in the early 19th century.
There was a practical reason for needing such a scale because hot things, like pans of boiled water, cool according to an exponential law, Newton’s law of cooling. In order to measure something that varies exponentially, you need a logarithmic scale. Otherwise, the tick marks on your thermometer will be too narrow as things are changing rapidly to be useful and too wide when they are changing slowly to be able to measure actual change at regular time intervals. Thus, you are wasting precious real estate on the thermometer. It was precisely this problem that Dalton wanted to solve so that they could confirm Newton’s law.
The same applies to the universe as a whole, and time is central to understanding that exponential cooling. If we want to understand how things change in the universe, we need a logarithmic time scale.
Logarithmic time scales are typically used to represent the declining amount of information we have about the past. This means that we represent time going backwards with 0 being the present. As we go back, the distance between events gets larger, but, because there is less information to fill in, we need about the same amount of space to indicate what happens.
When it comes to things the universe is doing rather than what human beings are aware of, the situation is the precise opposite and looks more like this:
Both the time after Big Bang and the Temperature in Kelvin are logarithmic and form a straight line.
Thus, while for the number of events we know about declines as we move further back in the past, the total number of events, given by particle collisions for example, increases exponentially the closer you get to the Big Bang. This requires that we have a logarithmic sense of time.
So, since logarithmic time is relative to some zero, it seems only reasonable that our scale be relative to the scale of time back to which we can see or measure just as the Richter scale. If we ever end up seeing further back, well, we can just have negative time.
Logarithmic time is not only a useful way to plot or measure but how the universe actually “experiences” time. Indeed, it may have experienced an infinite amount of time going back to the initial point. Thus we would have an infinitely old universe with a finite beginning.
Aoki, Sinya, Tetsuya Onogi, and Shuichi Yokoyama. “Charge conservation, entropy current and gravitation.” International Journal of Modern Physics A 36.29 (2021): 2150201.
Besson, Ugo. “The cooling law and the search for a good temperature scale, from Newton to Dalton.” European journal of physics 32.2 (2011): 343.