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the Planck

length

*convenient* mean in this case? Asking the greengrocer for a kilo of oranges is more convenient than asking him for 0.6×10^{27} proton masses of oranges, for example. Measuring the distance between Rome and Paris in kilometres is more convenient than doing it in millimetres. Depending on the phenomenon we have to measure, it is more convenient for us to choose certain units rather than others.

But if there were a community of intelligent living beings as big as microbes, they would surely measure the phenomena around them with different units from ours. Other units would be *convenient* for them. They would measure objects, that they would deal with in units comparable to our micrometres, weights of objects lifted in units comparable to our micrograms.

Therefore, the question now is: is it possible to invent units of measurement that are good for everyone? That can make people communicate with hypothetical intelligent life forms different from ours? Units that are somehow spontaneously suggested by Nature itself?

Planck’s starting point is to take seven fundamental constants of Nature and use them to derive universal units of measurement. Among these constants we have “c” which is the speed of light in a vacuum, “G” the universal gravitation constant, and “ħ” Planck’s constant. Each of them is associated with one of the great theories of physics of the twentieth century: Einstein’s special and general relativity and quantum mechanics.

By combining these three universal constants in different ways we are able to obtain Planck’s natural units. For example, taking *lp* = √(ħG/c³) we obtain the so-called Planck length. Similarly, *mp* = √(ħc/G) is called Planck mass and *tp* = √(ħG/c^{5}) Planck time.

Hence, any length, time or mass expressed in terms of Planck’s length is no longer a measurement on an arbitrary unit but has universal meaning. An alien from another galaxy, as big as a microbe or as a mountain, who is advanced enough to have measured the speed of light, the universal gravitation constant and the ratio between energy and frequency of a particle (given by ħ), would be able to communicate with us in lengths, times and masses of any phenomenon. It would certainly not be able to understand what a meter, a kilometre, a micron, a kilogram or a second are, since they are all arbitrarily chosen by man. But an object of length, 10 *lp* long and mass, 2.6 *mp* is unequivocally determined both for us and for it.

But what are, in Nature, the quantities obtained from Planck? To have an idea, let’s start with an object commonly considered very small: the thickness of a human hair. It has a length of about 100 micrometres. If we were to enlarge the thickness of the hair to the entire observable universe (93 billion light years or about 10^{27} meters), the length of Planck in ratio would be about 100 micrometres, that is the thickness of a hair. In other words, if the entire observable universe has the thickness of a hair, the thickness of a hair would have Planck’s length. Planck time, instead, is about 10^{-44 } seconds. The time between the Big Bang and the first Planck time is called the “Planck era” and it is the first stage of the birth of the universe in which the four fundamental forces of Nature are unified.

In any case, for lengths and times shorter than Planck’s ones, there is currently no physical theory capable of describing reality. The very sense of size and time loses meaning in current theories. Some scientists consider Planck’s units as the quanta of lengths and times, as tiny and indivisible bricks of which space-time is composed. In this interpretation there are no shorter times than Planck’s, nor shorter lengths than Planck’s. More reasonably, the physical theories we currently have are not able to describe the sub-Planckian world. The laws of physics can withstand no longer, they are no longer relevant. But this doesn’t preclude the fact that one day a new theory may exist. A theory able to look deeper and that can go beyond the understanding of our current limits today. A theory that is able to reveal what happened to our universe in its first 10^{-44 }seconds of life, for example.

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