How will this universe end?
This may be the one prediction of general relativity
that we never directly verify until, well, the very end.
Yet the fact that Einstein's theory
allows us to learn the answer to this question is incredible.
Let's see how this is even possible.
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Soon after Albert Einstein proposed his general theory
of relativity, industrious physicists--
and, in particular, a brilliant Russian
named Alexander Friedmann-- applied the new theory
to the whole universe.
Combined with observations of the redshifts of galaxies,
this showed us that the fabric of space itself is expanding.
Recent astronomical observations have revealed something
even stranger-- the future expansion of the universe
will be dominated by a mysterious influence
that physicists have come to call "dark energy."
In order to get a truly meaningful sense
of the forces governing the fate of our universe,
and especially the nature of dark energy,
we're going to have to go pretty deep.
We'll need to peer into the mathematics
of general relativity.
Trust me, it'll be worth it for the amazing insights
we'll gain.
For today's episode, we're going to start
by describing the universe without dark energy.
Because, well-- baby steps.
And the answer we get for the fate of our own universe
will still be right in one very important aspect.
At the heart of general relativity
are the Einstein field equations,
which look like this.
"Equations," plural.
See, that G and T are what we call "tenses"--
multicomponent, multidimensional quantities.
G is the Einstein tensor.
And it defines the shape, the curvature, of spacetime.
T is the stress-energy tensor.
And it describes all of the energy, the pressure,
the momentum, and more-- all of the stuff
within that spacetime.
Both have 10 independent components, giving 10
independent field equations to describe
the response of the fabric of the universe
to everything it contains.
John Archibald Wheeler put it more
simply-- "Spacetime tells matter how to move,
while matter tells spacetime how to curve."
Now, before we do any general relativity,
let's develop intuition a bit by thinking about gravity
from a Newtonian perspective-- as a force,
rather than as an Einsteinian spacetime curvature.
If I throw an apple in the air, then gravity
pulls it back down.
The faster I release it, the further
it travels before falling back.
If I throw it at 11 kilometers per second, which is the escape
velocity on Earth's surface, then,
by the time it almost stops moving,
it will be so far away that Earth's gravitational pull will
be close to zero.
Now this escape velocity comes right out
of Newton's universal law of gravitation,
which itself can be derived from the Einstein equations.
The mathematical step from Newton's gravity
to escape velocity comes from thinking about energy.
As the apple rises, its kinetic energy, its energy of motion,
is sapped by the gravitational field
and converted into potential energy.
Remember, energy is always conserved.
There's a minimum kinetic energy that the apple needs in order
to escape the energy-sucking gravitational-potential well
of the Earth.
That minimum kinetic energy tells you the escape velocity.
In this Newtonian analogy-- and, remember,
ignoring dark energy-- the universe
also has an escape velocity that lets
distant galaxies escape each other's gravitational
influence.
Based on the amount of stuff in the universe,
there's some current expansion speed
that would allow the future expansion rate of the universe
to slowly grind to a halt over infinite time.
How do we calculate the escape velocity
for the whole universe?
By solving the Einstein field equations
for the whole universe, of course.
It's mind-blowing that we can actually
do this and only possible because, on its largest scales,
billions of light years, all of the galaxies and galaxy
clusters are very evenly dusted across all of space.
When we ignore the bumps and wiggles caused
by individual galaxies, the resulting smooth universe
lets us reduce those 10 Einstein equations to only two
relationships, called the Friedmann equations,
devised by our boy Alex.
The first one can be written like this.
That may not look simple, but trust me,
we can read this in simple English.
That letter a is called the "scale factor,"
and it represents the size of the universe.
Although it's more accurate to think of it
as the average distance between galaxies.
Now the Friedmann equation tells us how a evolves over time.
That's what that a with a dot on top
represents-- the speed of the expansion of the universe.
Now, if you guys have taken some calculus,
that's the time derivative of a.
Let's stick to the left-hand side
of the equation for the moment.
It reflects the same balance between kinetic
and gravitational potential energy
that we saw in our rising apple.
In fact, this Friedmann equation is an energy equation.
That first piece, the a-dot over a squared,
is analogous to the kinetic energy of expansion--
how much outflowing oomph the universe has.
But that oomph is resisted by the gravitational effect
of all the master and energy in the universe.
That's this rho thing-- the density-- how packed with stuff
the universe is.
So this second piece represents the capacity of the universe
to slow itself down and is analogous to the gravitational
potential energy.
The balance between these two energy like terms
tells us the fate of the universe.
So, what are the possibilities?
If the kinetic energy of expansion
and the potential energy of collapse
are perfectly balanced, then the universe
will expand to a ginormous size and grind to a near halt.
Essentially, that's if the expansion is exactly
equal to the escape velocity.
In that case, the left-hand side of the equation adds to zero.
But if the left-hand side comes out just a little bit positive,
that represents a tiny bit of extra oomph.
There will be some expansion energy remaining
after gravity is diluted to nothing,
and the universe will expand forever, never stopping.
And if the left-hand side is negative, in that case
there was never a high-enough expansion rate
to reach that extreme size.
The universe will eventually fall back inwards,
and we'll see many of those distant galaxies
up very close and personal as the universe undergoes
the Big Crunch.
So, what's the answer?
Will the universe expand forever, or collapse?
Well, we know the expansion speed.
We measure it from the redshifts of galaxies as well
as other independent methods.
The expansion rate now is what we call the "Hubble constant."
It's around 70 kilometers per second per megaparsec.
For a long time-- until the late '90s--
it was believed that the answer to whether the universe would
recollapse lay in measuring rho--
the density of the universe.
Astronomers worked for decades to weigh up the galaxies
across vast swaths of the universe,
including their dark matter.
But the density of the universe turns out
to be too low-- only about a quarter of what
is needed to reverse the expansion.
The density term, the recollapsing term,
falls short of the expansion term.
The left side of the Friedmann equation is positive.
There's no way around it-- the universe will expand forever.
In a way, we're lucky to be living in an era
where we can still even see the distant galaxies.
Over billions of years, they will
slip beyond our cosmic horizon, leaving only darkness
beyond the local region of the Milky Way.
But wait-- we never got to the right side of the equation.
We know that it should also be positive,
because, well-- equals sign.
But that right side describes something completely
different to the left.
It describes the shape of the universe--
its spatial curvature.
According to the first Friedmann equation,
the fate of the universe-- as determined
by its expansion and density-- should be intrinsically tied
to its shape.
After all, matter tells spacetime how to curve.
We'll see in the next episode that there's
a surprising mismatch between the left and the right sides
of the first Friedmann equation that tells us that we're
missing something crucial.
That something is dark energy.
I'll show you why the Friedmann equations tell us
that dark energy must exist in this universe,
and what these equations can tell us about its true nature.
Now, don't get your hopes up.
Dark energy won't save us from infinite expansion.
Quite the opposite, in fact.
Dark energy accelerates the expansion.
And to understand how does so will take us
far beyond the capacity for Newtonian analogy.
It'll shatter our intuitions about energy conservation
and gravity on the largest scales of spacetime.
Hey, guys-- before we get to comments, a quick shout-out.
It's webby time.
PBS Digital Studios has been nominated for Best Science
and Education Video/Channel.
If you guys could head over there and vote,
that will be really helpful.
Link in the description.
OK.
So last week we talked about nuclear synthesis.
Let's see what you had to say.
Enrique Rigitano points out that there
seems to be this main sequence of elements produced
in the cause of dying high-mass stars
and asks how elements out of this sequence get produced.
OK, that's a great question.
In fact, there are many different fusion reactions
occurring in the cores of those dying high-mass stars.
Some of the most common involve either adding helium-- that's
helium capture-- or losing helium
after another fusion reaction.
So, if you start with an even-atomic-number element--
like carbon, with its six protons--
this means that you end up with a much higher abundance
in even-numbered elements than odd-numbered elements.
Of the 10 most abundant elements in the universe,
only two have an odd number of protons-- hydrogen
and nitrogen. However, there are many ways to build elements out
of this sequence, in fusion reactions that just
aren't quite as productive.
As for fluorine, that's actually one of the more confusing ones,
because it's not produced efficiently
in any known reaction in high-mass stars.
But recent work has suggested that it might actually
be formed in lower-mass stars like our sun
after they enter the red giant phase.
There's this brief period when it's
possible for material near the core
to be carried to the surface by convection
and then blown into space as the outer layers are blown away
into a planetary nebula.
A lot of you were interested to know
where the actual elements in our bodies came from
and where the supernovae that produce those elements
and their remnants are now.
So the disk of the Milky Way-- it rotates approximately once
every 230 million years or so.
However, stars don't all orbit at exactly the same rate.
Orbits aren't perfectly circular,
and stars drift apart as they move in and out
of the spiral arms and above and below the galactic disc.
The Sun has orbited the Milky Way around 18 times
since it formed.
And so, by now, it's wandered very far
from the giant cloud of gas from which it formed.
That cloud is long gone, anyway.
It collapsed into the Sun's sibling stars,
which are also scattered across the galaxy by now.
And it was blasted away by supernovae
from the most massive stars it produced.
The elements that form the Earth and that
form you may, in part, be from those supernovae,
but more will be from previous generations of stars
that enriched that giant cloud before it
formed stars of its own.
And, last of all, thanks to you, Rod Lendaeta,
for sharing this beautiful Serbian proverb.
"Be humble, for you are made of Earth.
Be noble, for you are made of stars."
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