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Professor of Astronomy, Harvard University; Director, Harvard Origins of Life Initiative; Author, The Life of Super-Earths

Frames of Reference

Deep and elegant explanations relate to natural or social phenomena and the observer often has no place in them. As a young student I was fascinated to understand how frames of reference work, i.e., to learn what it means to be an observer.

The reference frame concept is central in physics and astronomy. For example, the study of flows relies most often on two basic frames: one in which the flow is described as it moves through space, called an Eulerian frame, and another—called a Lagrangian frame, which moves with the flow, stretching and bending as it goes. The equations of motion in the Eulerian frame seemed intuitively obvious to me, but I felt exhilarated when I understood the same flow described by equations in the Lagrangian frame.

It is beautiful: let's think of a flow of water—a winding river. You are perched on a hill by the riverbank observing the water flow marked by a multitude of floating tree leaves. The banks of the river, the details of the surroundings—they provide a natural coordinate system, just as you would on a geographical map—you could almost create a mental image of fixed criss crossing lines—your frame of reference. The river flow of water moves through that fixed map: you are able to describe the twists and turns of the currents and their changing speed, all thanks to this fixed Eulerian frame of reference, named after Leonhard Euler (1707-1783).

It turns out that you could describe the flow with equal success if, instead of standing safely on the top of the hill, you plunged into the river and floated downstream, observing the whirling motions of the tree leaves all around you. Your frame of reference—the one named after Joseph-Louis Lagrange (1736-1813), is no longer fixed; instead you are describing all motions as relative to you and to each other. Your description of the flow will match exactly the description you achieved by observing from the hill, although the mathematical equations appear unrecognizably different.

To younger me, back then, the transformation between the two frames looked like magic. It was not deep perhaps, but it was elegant, and extremely helpful. However, it was also just the easy start of a journey—a journey that would pull the old frames of reference out from under me. It started with the naïve unmoving Earth as the absolute frame of Aristotle, soon to be rejected and replaced by Galileo with a frame of reference in which motion is not absolute—oh, how I loved floating with Lagrange down Euler's river!, only to be unsettled again by the special relativity of Einstein and trying to comprehend the loss of simultaneity. And a loss it was.

A fundamental shift in our frame of reference, especially the one that defines our place in the world, affects deeply each and every one of us personally. We live and learn, the next generation is born into the new with no attachment to the old. In science it is easy. Human frames of reference go beyond mathematics, physics, and astronomy. Do we know how to transform between human frames of reference successfully? Are they more often than not "Lagrangian" and relative? Perhaps we could take a cue from science and find an elegant solution. Or at least—an elegant explanation.