[#19]  Teaching Reality: Starting with Gravity

By Piet Hut

In high school, students are taught 19th century physics, two centuries out of date.  Mechanics and gravitation appear as it was known in the late 17th century, and electricity and magnetism as it was known in the early 19th century.  This is a shame: it would be like teaching pre-Darwinian biology!

Darwin published his On the Origins of Species within a few years of Maxwell publishing his equations of electromagnetism. Yet electricity and magnetism are mostly taught on the level of Coulomb's and Faraday's laws, well before Maxwell's deeper insight.  And the early 20th century breakthroughs of relativity theory and quantum mechanics are typically not even mentioned in high school.

Teaching students the nature of physical reality in such an outmoded way is defended by the argument that students do not have the mathematical background to understand tensor calculus and other requirements, necessary to develop a full understanding of, say, Maxwell's, Einstein's, Heisenberg's or Schrödinger's equations.  But why not at least give them a real taste for the essence of the insights behind the equations?


Hiding our current understanding of reality for high school students would be akin to not letting them attend music festivals or professional sports events if they would not be able to play on at least a very advanced amateur level.  I feel sad and upset about this tradition of what could be called the "big lie", to pretend to young minds that physics has not progressed in the last couple centuries.  No wonder that many students never get a chance to get their curiosity triggered in physics class . . .

A similar sentiment is expressed by Edward Frenkel in his wonderful book Love & Math. He writes: "What if you had to take an art class in which you were taught only how to paint a fence, but were never shown the paintings of van Gogh or Picasso? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry."


The fact is, it is perfectly possible to give students a sense of general relativity, say, without teaching them Riemann tensors and Christoffel symbols and all the rest of the four-dimensional differential geometry toolkit.  Two beautiful examples can be found in the introductory chapter of a classic work in teaching general relativity, the book Gravitation by Misner, Thorne, and Wheeler.

The first example already occurs on the front cover, where there is a picture of an apple.  The lines drawn on the apple are the paths that are taken by ants crawling over the surface of the apple.  Each one follows their own nose, in a straight line, as far as they are concerned.  Now, imagine two ants walking towards the pit in the surface of the apple, near the stem, and crossing the stem from different sides.  Even though they try to go forwards in a straight line, due to the curvature of the dimple in the surface near the stem, they will meet each other again at the other side, as illustrated in the picture.


The conclusion for the ants will be that there is a mysterious force, emerging from the stem, pulling them towards each other. After their paths crossed, at one side of the stem, they first got further and further away from each other, as was to be expected.  But then their distance stopped growing, and after passing the stem, they would seem to be pulled towards each other.  And finally their paths would cross again.  A great mystery!

This is a very nice illustration in two dimensions of what happens in three dimensions in our world: any two bodies will attract each other gravitationally, and after passing each other with a speed that is smaller than their mutual escape velocity, they will then fall back towards each other.  Einstein's general relativity theory explains this phenomenon in a surprising way. Rather than postulating gravity as exerting an attractive force, as Newton did, he considered there to be no gravitational force at all.  Rather, he found the proper mathematical equations describing how the presence of mass curves spacetime in such a way as to give the illusion of there being a mysterious force, pulling trajectories of objects toward each other; as in the illustration of ants on an apple.

General relativity for elementary schools

This simple picture is something that can be explained to children in elementary school, when given enough time.  And perhaps one of the children will then ask where the curvature of the world can be found, which causes the force of gravity. For ants, it is the surface of the apple.  What is it for us?

The answer is not just curvature of the space we live in.  If that were the case, then it would make no difference whether you would throw a ball fast or slowly.  Here we hit on a limitation of the analogy between apple and gravitational field.  Whether two ants on an apple are crawling along at high speed or at leisure, they will wind up at the same spot at the other side of the stem of the apple.  But on Earth, a baseball player can score a home run, by hitting the ball over the outfield fence, only by hitting the ball hard enough; too soft a hit won't make it.

The true answer is that the presence of masses causes a curvature in spacetime, not just a curvature in space.  This may be more of a challenge to explain to elementary school students, but should be easy to grasp for high school students, as part of a physics course.  The situation is illustrated in the second diagram, from the same book.


The basic idea is to plot a trajectory in space and time, by leaving out one of the three spatial dimensions.  Taking only height and distance, together with time, the trajectories are plotted for a ball and a bullet, both leaving from the same spot, at the same time, and both traveling the same distance before falling to the ground.  Obviously, for this to happen, the bullet will need to be aimed very low over the ground, while the ball will be thrown much higher, in order to reach the same spot.

As illustrated, the surprising and delightful result is that in four-dimensional spacetime, the curvature of the path of the bullet and that of the ball are exactly the same !  Just like different ants in our earlier example live on the same apple that has one and the same spatial curvature, so all of us on Earth live in the same gravitational field, with the same spacetime curvature.  This is what Einstein hit upon, while searching for a way to describe gravitational fields purely in terms of geometry.  And this is what physics students in high school can verify for themselves, just by applying Newton's law of gravity, something they do learn, to orbits of objects traveling at different speeds in the same gravitational field.

Given that these two examples can be taught in high school in just a few hours, why teach general relativity only in university classes, and typically only for those graduate students or advanced undergraduates taking such a course?  Why not sharing these amazing insights with everybody getting any physics in high school, in the same way that we share Shakespeare or Mozart or other aspects of human cultural accomplishments?

The truth about reality is a wonderful thing.  Science may never find any ultimate truth about the nature of the physical world.  But at least our approximate models of the world are getting progressively more accurate.  What we know now to be true, to the extent of our best descriptions, comes closer to reality than what we knew one or two centuries ago.  So let us teach the truth, as best we can, to the extent that it can be transmitted, depending on the audience.

In future blog posts, I would like to extend this notion of teaching truth, as best we can, to other areas of physics, as well as other sciences.

Piet Hut is President of YHouse (where this blog is hosted), Professor of Astrophysics and Head of the Program in Interdisciplinary Studies at the Institute for Advanced Study in Princeton, and a Principal Investigator and Councilor of the Earth-Life Science Institute in the Tokyo Institute of Technology.