Why the minimum length is the most fascinating aspect of nature ever (part 2)
As explained in the first part, the minimum length in nature is (twice) the Planck length. It has the tiny value of
As explained in the first part, the minimum length in nature is (twice) the Planck length. It has the tiny value of
l ≥ √(4Gℏ / c³) ≈ 3·10⁻³⁵ m .
In everyday life we do not notice this small value, and we can approximate the world around us by stating that space, time and all measurement quantities are continuous.
But here we want to go the limits. And there, space and time are not continuous. We also saw that at these limits, there are no sets and no equations. We saw that all our habits of thought are challenged.
In fact, the minimum length has important consequences:
Space is an approximation.
Black holes imply that space is due to smallest constituents.
Black holes imply that the constituents also form matter and radiation.
The constituents are Planck-sized, are similar to tiny ropes, and do not form sets.
The individual constituents do not follow equations.
The only math and the only equations possible describe the average motion of many constituents.
There is much more to tell, but for today, this list of consequences is sufficient. It is explained in detail this text.
The first consequence is almost self-evident. A smallest length contradicts our usual concept of space. Our usual concept of space is an approximation by idealization.
The second consequence starts from a discovery made in 1973 by Bekenstein and Hawking. Black holes are not black, but have a tiny glow due to their entropy and temperature. And the expressions for both quantities contain the Planck length. But everything that has temperature and entropy is made of a large number of constituents.
Black holes are curved space. Therefore, the constituents of black holes are also the constituents of space.
Black holes are compressed matter and radiation. Therefore, the constituents of black holes are also the constituents of matter and radiation.
Black holes have a finite entropy and temperature. Thus they are made of a finite number of constituents. And the value of the entropy and the temperature implies that the constituents have Planck size.
More precisely, the entropy and the temperature depend on the smallest size squared. This means that the constituents are of Planck size in two dimensions. If they had Planck size in three dimensions, the dependence of the entropy and the temperature would be on the Planck length cubed. (Many theories of quantum gravity are based on constituents that have zero size in two directions. That is wrong.)
In fact, the limit in just two dimensions makes sense. The constituents of black holes also make up space. Space is extended and reaches until the cosmological horizon. Thus, also the constituents of space must be extended (in one dimension) and reach the cosmological horizon.
This extension also explains why there are no sets. Whatever region of nature we consider, the constituents of space, matter and radiation in that region reach up to the “border of the universe”. Thus we cannot have sets in nature.
Now come the next surprise. There is no way to reach the Planck length. No experiment, no measurement, no observation can reach it. The best achieved so far (the measurement of the electron dipole moment) is a bit more than 1000 times the Planck length. In other words, we cannot observe individual constituents. And if we cannot observe them, we cannot find an equation for their motion.
Thus we found that we know how large numbers of constituents behave, for example when space bends (as described by general relativity) or when a wave function evolves (as described by quantum gravity). But we do not know and we cannot know how a single, individual constituent behaves.
This is the fascination of quantum gravity.
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