How you are actually ignorant about gravity

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It’s well known that gravity pulls two objects together with a force proportional to the mass of one, times the mass of the other, divided by the square of the distance between them. . Anyway, this equation is called Newton’s law of universal gravitation, it’s taught to school children the world over, and it predicts the motions of the planets and moons and asteroids in our solar system with incredible precision. 

However, Newton’s law of universal gravitation isn’t actually a universal law: first, we know that when the gravitational force in question is really strong, Newton’s law is just wrong . And second, we know that when the gravitational force in question is really weak , we don’t know whether it’s right or wrong because gravity gets too weak to measure. Only in between (like, on the scale of the solar system), do we know that the “law” of gravitation reasonably applies. 

Okay, but if Newton’s Law of gravitation has been confirmed so accurately by the motions of planets and moons, how could it be wrong at a different scale? Well, the earth looks flat when you’re relatively close to the ground, but zoom out and it looks round, or zoom in and it looks bumpy; the “law” describing the shape of the earth is different at different scales. Similarly, when the force of gravity is really strong (like near a black hole), gravity is better described by the mathematics of general relativity, and only when the forces in question get a bit weaker (for things farther apart or with less mass) does gravity start to match up with Newton’s law of gravitation. 

But when you go even weaker (with objects even farther away or even less massive ), we get to a point where we don’t know whether Newton’s law of gravitation applies any more. And yet, even many physicists appear to be ignorant about our ignorance about how gravity works when it’s weak – or at least, they ignore our ignorance. It’s common to blindly apply -G m M/r^2 to decidedly non-astronomical objects, for which we haven’t tested gravitational attraction very well at all. 

if you have two pieces of tape, you can calculate the gravitational force that they in principle exert on each other according to the law of universal gravitation – but it’s far too ridiculously ridiculously small for you to ever have the remotest chance of noticing any effect whatsoever, let alone actually checking that the attraction between them follows the law of universal gravitation as you move the bits of tape farther apart. In contrast, if you stick the two pieces of tape together and then pull them apart, they’ll exchange some electric charge and then measurably attract each other; an electrical attraction which is a million billion times stronger than the predicted gravitational attraction , and whose strength has allowed us to confirm Coulomb’s law of electrical attraction to a very high degree of accuracy. 

So it makes sense to apply Coulomb’s law of electrical attraction to objects at normal human scales. But testing Newton’s law of gravitation at these scales requires very delicate experiments, like very very sensitive oscillating pendulums that oscillate slightly differently if there’s a heavy mass nearby (and can thus measure the gravitational force with great precision), or incredibly finely-controlled lasers that simultaneously levitate and measure the positions and forces on tiny little beads of glass – these can measure ridiculously faint forces, like zepto newtons. 

And so far, for objects a meter apart, we’ve only confirmed that the gravitational attraction between them follows the law of universal gravitation to within around one one hundredth of a percent . Which is a trillion times less precise than our knowledge of the equivalent law for electricity. And our grasp on gravity gets worse the smaller you go – Here’s a plot showing how our uncertainty about Newton’s gravitational law varies across a whole range of distances – small distances on the left, big distances on the right; and the higher the line the higher the uncertainty. Which, you will notice, is very high on the left. 

Our existing experimental understanding of short-distance gravity is so bad that gravity at the scale of the atomic nucleus could actually be as much as a quadrillion quadrillion times stronger than Newton’s law of gravitation predicts! That’s a HUGE range; it would be like not knowing whether the moon pulls on us with the force of a hundred billion billion tons of rock , or the force of a fruit fly . Or, put another way, at the scale of an atomic nucleus, the law of gravitation could depend instead on the square of one or both masses, or the square root, or the inverse cube of the distance, or G could be a million billion times bigger, or a bunch of other possibilities, and we wouldn’t even know it. 

The fact that there’s so much uncertainty about gravity at short distances means that a lot of interesting truths about our universe could be hiding under our very noses! One possibility, for example, is that there are not just 3 dimensions of space, but an extra one that only the gravitational force can travel through, which loops back on itself at the scale of micrometers or smaller . Just like how the surface of a hair is technically 2-dimensional but hairs look one dimensional from afar, this would mean that at distances much much longer than a micrometer, gravity would act as if space had 3 dimensions and follow a roughly inverse square law (which is what Newton’s law of gravitation is), while at distances much shorter, it would behave as if space had 4 dimensions and follow more of an inverse cube law (which we haven’t ruled out at particularly small scales) . 

However, as we’ve made increasingly precise measurements of the gravitational attraction between small things, we haven’t yet discovered any gravitational forces inconsistent with Newton’s law of gravitation . so it may be that -G m M over r squared does describe the strength of gravity for very very short distances; but our uncertainty is still very big, and it remains pretty crazy to blindly apply Newton’s law of gravitation to things like an electron and a proton in a hydrogen atom. To find out more, please consult a physics tutor.