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 .

Ok, 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 very 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

zeptonewtons.

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 proton in a hydrogen atom.

This video was made with the support of the Heising-Simons Foundation, which also supports

research in precision measurement of the strength of gravity at short distances.

These experiments are super cool, because they're small, simple, clever, and are testing

our fundamental understanding of physics without the need for giant multi-billion-dollar particle

accelerators.

Thanks again to the Heising-Simons Foundation for supporting MinutePhysics, and for supporting

fundamental physics research.