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The Same Only Different: The Role and Reach of Analogies in Physics

Writer's picture: Mishkat BhattacharyaMishkat Bhattacharya

Analogies form an important part of human thought. They seem to be an evolutionary mechanism which leverage previous experience in the understanding of newly encountered phenomena. Not surprisingly, analogical learning is a big deal in cognitive science and - wait for the buzzword - AI.


In the Greek language, ana logos translates to 'same logic'. In physics, analogies are plentiful. A few examples: "light is like sound", "the atom is like a planetary system", "the atomic nucleus is like a drop of liquid". There is a mechanical aspect to these analogies, and maybe even a pictorial aspect.


Another type of analogy is mathematical (I mean formulaic). This kind of analogy does not have a mechanical component. Instead it has a strong pictorial flavor as the analogy rests on identifying similar symbols.


For example, if we write down the energy of a pendulum, we find there are two contributions. One contribution, the potential energy, is proportional to the square of position of the pendulum. The second contribution, the kinetic energy, is proportional to the square of the pendulum's momentum. Now if we also write down the energy carried by an electromagnetic wave, there are again two contributions, this time proportional to the squares of the electric and magnetic fields, respectively. The energy of the wave thus has the same mathematical form as that of the pendulum, with the electric (magnetic) field being analogous to the pendulum position (momentum).


What the analogy implies: an electromagnetic wave (read light) is like a pendulum. How that turns out in practice: the electric and magnetic fields oscillate at the frequency of the wave. This is a useful insight, as a wave is somewhat more complicated - and less tangible - than a pendulum. Since the behavior of a pendulum is familiar and easier to understand, using the analogy, we can gain insight into the behavior of electromagnetic waves. In fact, the analogy allows us to connect mechanics to optics in far-reaching ways. A humble contribution to this area from my research group has been the realization of the mechanical analog of an optical laser.


At a higher level of mathematical sophistication, is another example: Bill Unruh's discovery that the equations for sound waves in a moving fluid resemble those for light waves in curved spacetime. This insight sparked the field of analog gravity, which simulates e.g. black holes and Hawking radiation in the laboratory. These phenomena are otherwise not accessible to experimentalists.


Why does Nature repeat herself, and give us the benefit of these insights into apparently unrelated systems? Why do such varied and different phenomena follow similar mathematical models? No one knows.


We may ask how much novelty can be derived from an analogy. It may be suspected that the solution to any problem which can be found using analogy cannot teach us anything fundamentally new. It merely allows us to understand the new in terms of the old. But then the new simply becomes the latest manifestation of the old paradigm. It does not change our thinking fundamentally.


A counterexample to this argument is Bohr's atom. Although it was solved using the analogy of the solar system, it ushered in quantum mechanics, which - though similar in some aspects - is fundamentally different from classical mechanics. This is because Bohr departed from the model for a classical planet going around the sun, by quantizing the angular momentum for the electron orbiting the nucleus. This assumption went beyond the analogy. In other words, the analogy was not complete. Incomplete or - in a fruitful sense - superficial analogies are therefore fertile sources of new paradigms. They give us access to the fundamentally new by allowing a grip on the part of the problem that is similar to what we already know.


For those who wish to take a highly detailed, entertaining and informative five hundred page dive into the topic, including delicious subjects such as 'banalogies' (page 143) and Einstein as a superb analogizer (page 452):



The first author is well known for his cult book Gödel, Escher, Bach.



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