Brian Cox: The Planck scale is where physics breaks

I think it is legitimate to make the argument that given what we know about the universe

given the measurement we make of the strength of gravity, the measurement we make of Planck's

constant, and the measurement we make of the speed of light, then there is something fundamental

about this very tiny length, 10 to the minus 35 metres. Just one caveat though, one caveat

The Planck length itself does seem to be a fundamental property of the universe

But actually what it is, is a measurement. And at the moment, our measurement is, given what we measure, then it's 10 to the minus 35 meters

But you could imagine configurations of the universe where it's rather bigger than that

What kind of insights does the Planck scale reveal? When we think about the size of things, of course, we tend to think of the size of things

with reference to ourselves. I mean, even the units of measurement that we're familiar with

So, you know, the foot or the meter or those things or the inch or the centimeter

What are those things? Ultimately, historically, they're based on properties of the human body

So they're based on biology, really. So a meter might be, you know, the length of this kind of length and a foot is kind of that kind of length

And that's what we did historically, because why would you do anything else? But of course, the history of physics tells us, as we go into the 17th century, the 18th century

the 19th century, 20th century, we then begin to understand that there are things that are much

bigger than us and much smaller than us. And so is the meter, for example, which is based on

a property of the length of my arm or the length of my foot or whatever it is

is that really, is that fundamental? Is that something that tells us something deep about

the structure of nature well the answer is no it tells us about something it tells us about biology

on earth and how we've evolved on this planet and and and how how big cells are ultimately i suppose

and how many cells you need to make an intelligent multicellular being like a human based on the fact

that we live on this planet with this particular gravitational force and all those things so it

tells you about all that stuff, but it doesn't tell you anything profound or deep about the deep

structure of the universe. And so Max Planck, so of Planck's constant fame and quantum mechanics

fame, came up with a system of units, right? So a way of saying, well, what are the fundamental

quantities, as far as we can tell, that really tell us something about the structure of nature

We're looking, I suppose, for units of measurement that we could, you could imagine if we met some

aliens from some different civilization, that they might be, they might not even have arms

right, or feet, but they might be very different in size and scale from us

So what would the common language be? Is there some units of measurement that we could all agree on

And so Planck thought about that. So what are the fundamental constants of nature as we understand them

So things that really tell us about the underlying structure of the universe

So one would be the speed of light. So the speed of light, that would be, although we can talk about what it is in metres per second or miles per hour or whatever it is

it is a property of the universe. Everything that is massless travels at the speed of light

at this speed, whatever it is. And if you have any mass at all, you cannot travel, you cannot

accelerate to this speed, a universal speed limit. So it's something deep about the property

of the universe. It's to do with perhaps the geometry of space-time, or we don't know where

it comes from, the particular number, but there it is, the speed of light. Another one would be the

strength of the gravitational force. So what is the force between two objects of a particular mass

or in Einstein's theory, a deeper description, how does a particular amount of matter or energy

distort the fabric of the universe? The number that tells you about that is Newton's gravitational

constant, which was first measured back in the 1780s, 1790s. So it's the strength of gravity

that another one And then there Planck constant itself So you can say this is the associated quantum theory what is it So you can read many different textbooks you find many different ways in

You could say, for example, that there's a fundamental limit on how accurately we can

know the position of a particle and the momentum of a particle. You can't know them both with

absolute precision. There's a fundamental limit and it's a roundabout Planck's constant. So the uncertainty on the measurement of the position of something

multiplied by the uncertainty on the measurement of the momentum of something

is always has to be greater than Planck's constant. So it's a fundamental

property of the universe and there are different ways of thinking about it

Planck first introduced it in the context of the frequency or the wavelength of light emitted from objects

Photons. What's the energy of a photon? A packet of light. It's Planck's constant multiplied by the frequency

So it's a deep property of nature. Those three numbers, those three things, the speed of light, strength of gravity and Planck's constant

allow you to define some distances, a particular distance called the Planck length

I want to do it by reading it off something because I don't carry all these numbers around in my head

It would be pointless to do that because you can look them up. So I could take Planck's constant, multiply it by the strength of gravity

and divide it by the cube of the speed of light and then take the square root of the whole thing

So it's HG divided by C cubed square root. You'll find that that has

if you put those things in in terms of metres and kilograms and things

you'll find that it's a length, it has the dimensions of metres

and it's a tiny length. it's about 10 to the minus 35 meters, 0.000000, 35 knots, one of a meter. But that is a length

that we've calculated by measuring the strength of gravity, the speed of light and Planck's constant

So it would seem that that should be, it should have something to do with the deep

structure of the universe and indeed it does how important is it um well let me give you some

examples of the Planck length so it turns out that if you ask a question how much information

in bits is stored inside a black hole right that would seem to have no what's that got to do with

all this, right, how much? It turns out, in a calculation that was done by Jacob Bekenstein in

the 1970s, it turns out that the entropy of a black hole, which is the amount of information

hidden within it, is equal to the surface area of the event horizon of the black hole in square

plank lengths. That's an astonishing result. But just think about that for a moment. It's telling

us that the amounts of information stored in a region of space, in this case a black

hole, is equal to the surface area surrounding that region in square plank lengths. That's

a bizarre result. So it does seem that the plank length is fundamentally important. And

here's another property of the plank length. So let's say we want to make an observation

of some very small thing. So how do you observe something that's very small? Well, one way to do

it is shine a light on it to see where it is, right? Together, that's the way that we observe

things with our eyes. So you might say, well, it's tiny. So I have to shine light with a very

small wavelength onto this thing to see the tiny thing. The wavelength can't be bigger than the

tiny thing, otherwise you won't see it. So the smaller this thing is, the smaller the wavelength

of the light that I've got to shine on it. But remember quantum mechanics tells us

that the smaller the wavelength of the light the higher the energy of the

photons. So I have to start bombarding this thing with higher energy photons to

see it. What happens if you try to approach something that's the Planck

length? You get so much energy in there that what you do is you form a black

hole and then you put more energy in, you try to see what's going on, and the

black hole grows and so you get to a point which is around the plank length in size where you can in principle try to resolve the structure of this thing It starts to go bigger again because you make

a bigger and bigger black hole and it grows. So the Planck length does seem to be, I would

say is, a fundamental property of the universe. And now the nature of that thing, so if we

talk in terms of black holes, then you have these kind of plank-sized pixels, in some sense

tiling the event horizon. Does that mean that these are building blocks of space that are that size

It would seem so, but this is where we get to, we're at the edge of our current understanding

I think it is legitimate to make the argument that given what we know about the universe

given the measurement we make of the strength of gravity, the measurement we make of Planck's constant

and the measurement we make of the speed of light, then there is something fundamental about this very tiny length

10 to the minus 35 metres. Just one caveat, though, one caveat

Is it important, that number, 10 to the minus 35 metres? There are theories, and we test these theories

at places like the Large Hadron Collider, for example. There are theories where, as an example, there are extra dimensions in the universe

So not only the four-dimensional space-time of Einstein's theory, but more dimensions

And the dimensions can be curled up at little places in points, or they can be big extended sheets, and all different configurations

If there are extra dimensions in the universe, then you find that you can, let's say that you could see those extra dimensions, energies just around the energies that we collide particles at Large Hadron Collider

Then the explanation for why gravity is so weak, so our measurement of Newton's gravitational constant would change at those higher energies

It might mean that this Planck scale would drop, and so the Planck length would expand

and we'd see this kind of physics, much lower energies than we might otherwise have anticipated

So I think it's worth a caveat that the Planck length itself does seem to be a fundamental property of the universe

But actually what it is, is a measurement. and at the moment our measurement is given what we measure then it's 10 to the minus 35 meters

but you could imagine configurations of the universe where it's rather bigger than that

Now the Planck length, this unimaginably small number in meters, you might say well how does that

how does that affect our everyday lives right? Does it come into any calculation of a thing that

we can conceive of. And there is a beautiful calculation. It's one of my favorite calculations

in all of physics that was initially done by the great mathematical physicist Chandrasekhar

back in the 1930s. And it's a stunning calculation. It's about the mass of white dwarf stars

So let's think about a white dwarf star. What is that? So what is a star, first of all

so a star is is a some material it's mainly hydrogen and helium collapsing under its own

gravity so it's so gravity is trying to squash this thing down what holds it up well as the star

contracts as it forms um our sun formed around four and a half billion years ago through this

process a gravitational collapse that means the core heats up and it heats up which means that

The hydrogen and helium is a triggering around very fast. And ultimately, you switch on nuclear fusion reactions in the core

The fusion reactions, in the case of our sun, hydrogen is fused into helium that releases energy, which creates a pressure which holds the star up

So the stars are balancing act. Gravity is trying to squash it, heats the core up

Fusion reactions release energy, creates a pressure, holds it up. But of course, that doesn't exist

It can't carry on forever. because the star doesn't have an infinite amount of fuel in its core

And so ultimately we run out of fuel and it will begin to collapse again

And then stars will start to fuse heavy elements and so on. But ultimately you could ask the question

well, when there's no more nuclear fusion can occur in the core

what happens to the star? Does it just collapse without limit? Which would be a black hole as we now understand it

Or is there something else that can hold it up? some other property of matter

This wonderful calculation is a calculation about what happens to electrons in a collapsing star So it a genuine quantum mechanical calculation

So what happens? One of the properties of the universe, one of the fundamental ideas in quantum mechanics, is called the uncertainty principle, Heisenberg's uncertainty principle

So it says that if you try to confine an electron into a smaller and smaller box, which means you're trying to measure its position, you're trying to confine it and squash it down, then the product of the size of the box, the uncertainty in the position, multiplied by the uncertainty in the momentum of this thing, has to be greater than Planck's constant

Fundamental property of the universe. So as you try to, this star's collapsing and all the electrons are getting pushed together and closer and closer together

They try to avoid each other, which is another fundamental property of quantum mechanics called the exclusion principle

which roughly speaking says they don't like to be in the same place, roughly

And so you start to squash them. They try to get away from each other. They go into little boxes in the star and the boxes are shrinking

And so the electrons are getting confined. And so they jiggle faster. The uncertainty on their momentum is faster and faster

And so they start to jiggle around because they're getting squashed into each other and trying to stay away from each other

So this is pure quantum mechanics. The uncertainty principle, the Pauli exclusion principle, jiggling around

That jiggling is like a temperature in a sense. It creates a pressure which can hold the star up

So you could ask, you could do the calculation. But what happens as I squash this down and the electrons start to jiggle more

They make the pressure, they hold the star up. What happens if that jiggling starts to bump into relativity

What happens if it starts to go towards the speed of light? They can't jiggle any faster

And so ultimately, there must be a limit on the pressure that these electrons in this quantum mechanical process can exert to hold this thing up

So you can do the calculation and it's done. I did it in a book called The Quantum Universe

There's a gratuitous bit of advertising. It's in the appendix. So you can do it. It's not a hard calculation

Some details are difficult, and Chandrasekhar did this magnificent calculation. If you do that calculation, you find that the maximum mass of a star

which is just a lump of matter held up by this process, is 1.4 times the mass of our Sun

So it's an astonishing calculation. How does that relate to the fundamental properties of the universe

us because all we've used there, we've used quantum mechanics and we've used the strength

of gravity. Those are the things. These are the things. Planck's constant, the speed of light

Newton's gravitational constant. Those are the things we used when we looked at the Planck units

like the Planck length. So you can use those things, strength of gravity, speed of light

Planck's constant to construct a mass it's called the Planck mass and it's rather big actually so

whereas the Planck length is very very tiny the Planck mass is about the mass of a grain of dust

a mote of dust it's quite a large thing but you can calculate it fundamental property of the

universe using Newton's gravitational constant speed of light Planck's constant comes into that

calculation. Turns out when you do the calculation that roughly speaking that number, the Chandrasekhar

limit, is the Planck mass cubed divided by the proton mass squared. That's what it is. You do

that calculation, put the numbers in, it's about 1.4 times the mass of the Sun. But I find it

profoundly important. It's a beautiful, beautiful result because what we're saying is that you can

calculate the maximum mass of a load of stuff that can hold itself up through this quantum

mechanical process. And it just depends on these fundamental properties of the universe

Strength of gravity, plant's constant, speed of light, that's it. So it's a very, very beautiful

calculation. But it's the best example I know of the relationship between these rather abstract

quantities perhaps and something that you can look at in a telescope it's it's it's a beautiful

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