It’s been 120 years since Henry Cavendish measured the gravitational constant with a
pair of lead balls suspended by a wire.
The fundamental nature of gravity still eludes our best minds - but those secrets may be
revealed by turning back to the Cavendish experiment.
That steampunk contraption may even reveal the existence of extra dimensions of space.
Gravity is by far the most familiar of the four fundamental forces - after all, we wrestle
with it every day.
It’s no surprise that it was the first force to get a decent mathematical description
with Isaac Newton’s Law of Universal Gravitation back in the 17th century.
But what is surprising that nearly 500 years later, gravity remains the most mysterious
of the four forces.
It just doesn’t seem to gel with the other three forces.
One big difference is that the strength of gravity is vastly weaker than the other forces
- I’m talking 10^24 times weaker than even the weak nuclear force.
This mismatch is known as the hierarchy problem, and understanding the puniness of gravity
may help us figure out what gravity really IS, and to unify it with the rest of the forces.
But like I said, despite the mysteries about why gravity works the way it does, our current
theories still describe how gravity behaves remarkably well.
Einstein's general theory of relativity has a perfect track record predicting the behavior
of gravity in terms of the warping of the fabric of space and time.
But even Newton’s old theory works great in almost all situations.
And deviations from Newton’s description may point the way to an understanding deeper
than Einstein’s.
The key is in understanding the inverse square law.
Newton’s law of gravitation is an example of such a law - basically, the strength of
the gravitational force drops off with the square of the distance between two massive
objects.
If we have a massive object, we can depict the gravitational force (field) from this
object as little arrows pointing towards the object.
The density of the arrows at a given distance determines the strength of the gravitational
field.
So as you get further away, those arrows are more and more spread out - their density,
and so the strength of gravity drops.
The arrows get spread out over the surface of a sphere with the gravitating body in the
center.
So the density and the strength of the gravitational field drop proportional the surface area of
that sphere - that’s 4 pi r^2.
That’s where the r^2 comes from on the bottom of Newton’s law of gravitation.
So gravity sort of gets diluted in strength because it’s increasingly spread out over
a 2-D spherical surface within our 3-D space.
If our universe were 2-dimensional - say, a flat plain - then gravity would be spread
out over the circumference of concentric circles - circumference is 2 pi r - so gravity would
just be inversely proportional to radius - it would drop off more slowly.
And if our universe had more dimensions, say 4, then gravity would spread out over the
“surface” of a 4-D hypersphere - that surface would be 3-D, so gravity would follow
an inverse cube law - that’s a faster drop off.
In general, gravity drops off as 1/r to some power - and that power is the number of spatial
dimensions minus one.
Cool - so we have a way to test how many dimensions there are.
Newton’s inverse square law for gravity tells us there are 2+1 = 3 spatial dimensions.
I … guess we already knew that.
Except for the fact that some of the most promising approaches to unifying gravity with
the other forces actually propose that there are MORE than 3 dimensions of space, and that
these additional dimensions may explain the hierarchy problem - the relative weakness
of the gravitational force.
So let’s talk about dimensions a bit more.
We perceive a 3-D universe, but most of our movements from day to day are only in 2 of
them - walking across your 2-D floor, driving around the 2-D surface of your town or city.
We explore a 2-D landscape embedded in a 3-D world.
Now we perceive the 3rd dimension, but perhaps our 3 spatial dimensions are similarly embedded
in a higher dimensional space that we can’t perceive.
Well actually, that’s probably not the case.
In fact we saw in a previous episode how a particular gravitational wave detection from
LIGO seemed to rule out the possibility of extra spatial dimensions.
But there’s a caveat - that experiment only ruled out extra large dimensions.
And by large I mean they stretch out over astronomical scales like the familiar 3 spatial
dimensions.
But there’s another possibility - the extra dimensions could be compactified - in other
words, wound up very small.
In that case, gravity would only deviate from the inverse square law on the tiny scales
of these compactified dimensions.
Take another look at the surface you’re on right now.
It seems like a 2-dimensional surface, but we all know that the apparently 2-D ground
is made up of 3-D particles, and if we zoomed in on a seemingly flat floor to small enough
scale, we would see a wildly varying, 3-dimensional landscape.
We don’t notice this because of the vast difference in scales between us and that landscape.
Another example would be a long, thin tube.
It looks 1-dimensional from a distance, but up close you see that it’s really a 2-D
surface - extended in one dimension, but looped in the other.
And that’s the sense in which there may be more than 3 spatial dimensions in this universe.
3 extended dimension, but others that are looped - or “compactified” as we say in
physics.
The most famous example of compactified extra dimensions is in string theory, and in modern
superstring theory there are 6 additional spatial dimensions, coiled-up tight at every
point in the extended 3-D universe.
But the idea of compact extra dimensions is much older than string theory - first proposed
by Oscar Klein in the 1920s as part of what became Kaluza-Klein theory - one of the first
attempts to unify electromagnetism with gravity.
Although we’ve never detected these extra dimensions, they seem a promising potential
solution to unifying gravity with the other forces - and in the process explaining gravity’s
weakness.
There are different ways of thinking about how compactified extra dimensions can effect
the strength of gravity.
One is that you can imagine the gravity trying to propagate through multiple dimensions,
which would dilute it more quickly.
We can also think about it as gravity producing excitations within those extra dimensions
that sap its energy.
On large scales, gravity would still follow the inverse square law, but would be diluted
due to the hidden extra dimensions - and that could explain its relative weakness compared
compared to other forces.
And on small scales these extra dimensions would actually change the way gravity falls
off with distance - but to see that you’d need to test the strength of gravity on length
scales that are within spitting distance of those compactified dimensions.
So, to search for these hidden dimensions, we need a way to test gravity on very small
length scales.
Lots of different approaches have been tried, but today I want to tell you about one of
the most promising - because it’s also the oldest.
It’s actually the OG - the experiment first used to measure the strength of the force
of gravity - or the G in Newton’s law of gravitation.
It’s the Cavendish experiment, performed by Brit Henry Cavendish in the late 18th century.
The apparatus for the experiment was actually designed and built by another British scientist
John Mitchell - the same guy who first hypothesized the existence of black holes.
But Mitchell died before he could conduct the experiment, and so Cavendish inherited
and improved the device, and actually made the measurement.
The Cavendish experiment looked like this: a 6-ft rod suspended from its center by a
long, thin, metal wire.
A lead ball was attached to each end of the rod.
The whole contraption was then free to rotate on the wire.
This is an example of a torsion pendulum - where the restoring force of the twisted wire can
cause the whole apparatus to oscillate.
The amazing thing about the torsion pendulum is that it’s incredibly sensitive - if the
wire is very thin, then very tiny forces will induce a large twist before being halted by
the restoring force.
The amount of twist tells you precisely how much force was applied to rotate it.
So, Cavendish placed large masses near each of the lead balls at the ends of the rods..
The gravitational attraction between these masses and the balls would then rotate the
torsion pendulum ever so slightly, and by reading off the amount of rotation Cavendish
made the first measurement of the gravitational constant.
The device was so sensitive that it had to be protected from the slightest breeze, or
even the gravitational influence of other bodies - including Cavendish’s.
He measured the displacement through a telescope mounted into a hole in the device’s shed.
This allowed him to measure a force of gravity between the masses which was something like
10 million times smaller than the force the lead balls felt due to the earth below.
The meticulous care paid off—Cavendish measured a gravitational constant that agrees with
our modern value, with 1% uncertainty.
His measurement remained the most precise for over 100 years.
Not only did this contribute to our understanding of gravity, but also arguably marked the beginning
of a grand tradition in physics of making high-precision measurements to learn about
the fundamental nature of the universe.
Modern Cavendish experiments look quite a bit different than Mitchell and Cavendish’s
first apparatus.
The experiment is now typically done in a vacuum, with precise temperature control and
electrostatic and magnetic shielding.
The geometry of modern devices allow more precise positioning between the test masses,
and the displacement of the masses is measured with far more sophistication than with a telescope
through a hole in the wall.
In some cases, rotating masses produce oscillations in the torsion pendulum.
These incredible advances have led to a modern measurement with 6 digits of precision.
These advances have also allowed us to place the pendulum masses and the test masses extremely
close together, as close as 50 micrometers - less than the width of a human hair.
Now that’s still quite a bit larger than any likely compactified extra dimensions,
but we’d still expect a tiny and perhaps detectable deviation from the inverse square
law several orders of magnitude larger than the size-scale of those dimensions - depending
on other factors - most importantly depending the strength with which gravity interacts
with these extra dimensions.
So, how many dimensions are there?
Well, still just three.
The inverse square law seems to hold down to the current limits of our measurements.
And while that may seem anticlimactic, it’s actually extremely useful, ruling out some
theories and constraining the space of possibility from others.
We are still nowhere near being able to probe the size-scale of the extra dimensions from
string theory though, and so string theorists can keep on theorizing.
So what's next?
Well, to build better Cavendish-style experiments.
Things get complicated below about 1 micrometer in separation, that’s where the Casimir
force comes into play.
This is the negative pressure due to the exclusion of quantum vacuum modes, or virtual particles,
between two very closely separated plates.
But there are other ingenious approaches that we can come back to another time.
For now, we'll have to content ourselves with our merely 4-dimensional universe - four...
what were they again? space, space space, ... time.
So this is pretty amazingly unbelievable.
This little show about hardcore physics and astrophsyics just hit 2 million subscribers.
Honestly - it warms all our hearts to know there are so many kindred nerdy spirits out
there.
We should, like, all hang out some time - you know, after all this is over.
OF course if we did hang out we’d be a whole city.
In fact we looked it up.
We’d be the 238th largest city in the world.
And growing.
That’s right, we’re coming for you Maltan, Pakistan - you’ve been chillin in rank 237
far too long.
For now, I just want to thank you from the bottom of my heart for your incredible support.
And speaking of support, I want to give an extra big thank you to our Patreon supporters
and most especially to Christina Oegren, who’s supporting us at the quasar level.
Christina, with your great generosity and excellent taste in youtube shows, I hereby
appoint you mayor of spacetime city.
Because no one said spacetime city was a democracy, and I think you’d do an amazing job.
In last week’s episode we talked about the strange regions that lie beneath the event
horizon of a rotating black hole - regions like the Carter time machine and the infinite
string of parallel universes.
Unsurprisingly, thoughts were had and comments were made.
Zoltan asks for clarification - really, confirmation of his hunch which is that the “multiverse” you
get from traveling through black holes has nothing to do with the multiverse predicted
by quantum physics.
Zoltan, you have it right.
The quantum multiverse is what you get when you accept Hugh Everett's many worlds interpretation,
which implies that the universe splits at every quantum event.
That multiverse may or may not be real.
Now the multiverse through the black hole arises from tracing the paths of spacetime through
the black hole mathematically.
I'd say the second multiverse is less likely to be real than the quantum multiverse - it's
almost certainly a mathematical figment, arising from idealized types of black hole that don't
really exist.
On a related note: Polygondwanaland asks how much of this weirdness is actually weirdness
and how much is broken math.
Well, the universe on the other side of the black hole is probably a figment in the math
- which I suppose you can call broken math if you like.
But really, the math is just fine - the problem is assigning physical reality after pushing
the math too far.
After all, the universe may be incredibly well described and modeled by the math, but
that doesn't mean the universe is the math.
The quality of the predictions of any model depend on the assumptions that go into it.
And in the case of the black holes, several unreasonable assumptions allow us to trace
geodesics into alternate universes.
For example, the parallel universe through the non-rotating Schwarzschild black hole
requires the assumption that the black hole has been there forever.
And the string of universes through the rotating Kerr black hole requires the assumption that
the impossibly unstable inner structure of the Kerr spacetime doesn't collapse immediately.
But you're totally right, Polygondwanaland: in reality, impossible-seeming things like
an infinite string of universes or a time machine indicate that the math may have led
our physical understanding astray, and that's even more true when we see that the universe
seems to conspire to prevent these things from existing.
And to wrap up - my favourite joke in the comments that I wish I'd said in the episode
was this.
Instead of "You're in for a weird time.”
I should have said "I'm in for a weird space!" because space and time switched places inside
the blackhole.
At least I can live vicariously through the superior wit of the brilliant Dr Poopstick.