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MATT O'DOWD: Between them, general relativity and quantum
mechanics seem to describe all of observable reality.
And yet, they can't be simultaneously true.
They must be united in a deeper, yet undiscovered, theory.
After a century of work by the greatest
minds in all of physics, why does this union still elude us?
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The first few decades of the 20th century
was a time of miracles for physics.
First, Einstein's relativity utterly
changed the way we think about space, time, motion,
and gravity.
Then the quantum revolution of the '20s and '30s
overturned all of our intuitions about the subatomic world.
Together, general relativity and quantum mechanics
have allowed us to explain nearly
every fundamental phenomenon observed.
And they've predicted many unexpected phenomena
that have since been verified.
And yet these two theories contradict each other
in fundamental ways.
In the century since that golden era of physics,
we've been trying to reconcile the two without success.
But today, on "Space Time," I'm going
to begin our discussion of the great quest
for this union, the quest for a theory of quantum gravity
and for a theory of everything.
This is a big topic.
So in this episode, I want to give you the motivation.
What exactly are the conflicts between general relativity,
or GR, and quantum mechanics?
I'll save the solutions for future episodes.
Let's start with summaries.
General relativity, GR, is Einstein's great theory
of gravity.
In it, the presence of mass and energy
warp the fabric of space and time.
And the motion of objects is, thereby, altered.
This results in the effect we perceive as gravity.
General relativity incorporates the earlier special relativity,
which describes how our perceptions of space and time
also depend on motion.
Unlike the earlier ideas of Isaac Newton,
in which space and time are treated
as separate and universal, special and general relativity
blend them together into a combined and mutable
space-time.
Where general relativity describes the universe
of the large and the massive, quantum mechanics
talks about the subatomic world.
It describes particles as waves of infinite possibility
whose observed properties are intrinsically uncertain.
Our experience of the universe appears
to be plucked from this landscape of possibilities
in strange, but mathematically predictable, ways.
That math started with the Schrodinger equation,
which tracks these probability waves through space and time.
But the Schrodinger equation treats space and time
as fundamentally separate in the old-fashioned Newtonian way.
So clearly, there's a problem.
We already talked about how Paul Dirac fixed part of the problem
with a relativistic wave equation for the electron.
Nowadays, modern quantum field theories
fully incorporate the melding of space and time predicted
by special relativity.
And yet they still don't directly
incorporate the warping of space and time predicted
by general relativity.
This causes issues-- some mild and fixable, others
catastrophic.
Starting with the mild, we have the black hole information
paradox.
We've gone on about that at length.
The black holes of pure general relativity
swallow information in a way that can remove it completely
from the universe, especially when those black holes
evaporate via Hawking radiation.
That's a big conflict with quantum theory
right there, which tells us that quantum information should
never be destroyed.
But that same Hawking radiation offers part of the solution
to the information paradox.
Following the work of Hawking, Jacob Bekenstein, and Gerard
't Hooft and others, it has become clear
that information swallowed by black holes
can be radiated back out into the universe via their Hawking
radiation.
In a sense, both the source and the solution
to the information paradox came from the discover
of Hawking radiation.
Hawking, actually, derived the latter
by finding a way to unite general relativity
and-- in quantum field theory.
But that union was approximate and incomplete.
Really, it was a brilliant hack.
And you can check our previous episode for the gory details.
In fact, it's very possible to shoehorn the curved geometry
of general relativity into the way quantum field theory
deals with space and time.
But that approach completely fails
when you have strong gravitational effects
on the smaller scales of space and time,
like the central singularity of the black hole
or at the instant of the Big Bang.
For that, you need a true quantum theory of gravity.
But even thinking about the structure
of curved space on the smaller scales
leads to craziness and catastrophic conflicts.
I want to talk about these in two ways--
first, very conceptually, then a bit more technically.
Let's start by thinking about what
it means to define a location in a gravitational field
with perfect precision or, in other words, what
it means to talk about very, very tiny chunks in the fabric
of space.
In order to measure a location in space--
say, the location of a particle--
you need to interact with it.
You would typically do that by bouncing a photon
or other particle off the object.
The more precisely you want to measure position,
the higher the energy of that interaction.
That's why we use electron microscopes or X-rays or even
gamma rays to take images of extremely small things.
So let's say we shoot a particle with a beam from a particle
accelerator to measure its location
with extreme precision.
The Heisenberg uncertainty principle
tells us the minimum energy of our beam for a given precision.
It turns out that to measure a position to an accuracy better
than a Planck length around 10 to the power of negative 35
of a meter, the amount of energy you would need
to put into that region of space would make a tiny black hole
with an event horizon one Planck length in diameter.
Try to measure more precisely, and you need more energy.
That means you make an even larger black hole.
So general relativity plus Heisenberg
say it's impossible to measure a length smaller than the Planck
length.
Steady viewers will remember that the uncertainty principle
talks about the trade-off between position and momentum.
But large momentum also means large energy.
The uncertainty principle also defines the precision trade-off
between time and energy.
So this same argument can be used
to suggest a fragmentation of time.
Try to measure any time period shorter than 10
to the power of negative 43 seconds, the Planck
time, and boom--
black hole.
For those of you who already watched our episode
on the Heisenberg uncertainty principle,
here's another way to think about this.
We know that for a particle to have a highly defined location,
its position wave function needs to be constructed
from a wide range of momentum wave functions
that include extremely high frequencies or extremely
high momenta, i.e., the more certain
its position, the less certain its momentum.
And so large momenta are possible.
So position can be defined to within a Planck length.
And then momentum becomes extremely uncertain
and includes the possibility of ridiculously high values.
That means ridiculously high kinetic energies.
Particles whose positions are defined within a Planck length
can spontaneously become black holes.
Of course, those black holes don't really happen.
Rather, they're an absurdity that
tells us that something is missing
in our description of either quantum theory
or general relativity, or both, at the smaller scales.
Let's look at the real conflict.
Standard quantum theories treat the fabric of space-time
as the underlying arena on which all the weird quantum
stuff happens.
Given that sensible underlying structure,
it's relatively routine to apply quantum principles,
or quantize, most of the forces of nature.
For example, classical electromagnetism
becomes quantum electrodynamics when
you quantize the electron field and the electromagnetic field.
But in the resulting math, the new quantum fields
still lie on top of a smooth, continuous grid
of space and time.
So what if you want to quantize gravity?
The gravitational field doesn't lie on top of space-time.
It is space-time.
To quantize gravity, you have to quantize space-time itself.
That leaves no clean coordinate system
on which to ground your theory.
This sounds annoying.
In fact, it's a disaster.
It leads to several problems.
But I'll focus on the one that wrongly
predicts these crazy fluctuations on the Planck
scale.
In general relativity, the presence of mass or energy
warps the gravitational field.
There can be no exceptions.
Any energy must cause space-time curvature.
If not, you could build perpetual motion machines,
for example, using the Casimir effect.
In quantum gravity, gravity itself
becomes an excitation in our quantized space-time.
The energy of those excitations should themselves
precipitate more space-time curvature, represented
as further excitations.
In other words, gravity should produce
more gravity, ad infinitum.
This type of self-interaction or self-energy
is seen in other quantum field theories
and is hard to deal with, even there.
For example, in quantum electrodynamics,
the electron has a self-interaction
due to its electric charge messing with the surrounding
electromagnetic field.
In QED, the mess is fixed with something
called perturbation theory.
It's a scheme to calculate a complex interaction,
like the buzzing electromagnetic field around an electron,
with a series of corrections to a simple, well-understood
interaction, which might be the electron
in a quiet electromagnetic field.
We talk about this more in our episode on the g factor.
So perturbation theory is applied
throughout quantum field theories of the standard model.
And it works because, one, these corrections
are small and/or, two, even in the case where
the corrections appear large or even infinite,
they can be constrained.
They can be brought back to reality
by actual physical measurements of a few simple numbers
in a process called renormalization.
For example, measurement of the mass and charge of an electron
renormalizes quantum electrodynamics
to allow incredibly precise calculation of the electron's
self-energy.
None of this works when you try to quantize general relativity.
When you have strong gravitational effects
on the quantum scale, the self-energy corrections
blow up to infinity.
But unlike other quantum field theories,
there are no simple measurements you
can do to renormalize those corrections.
In fact, you would need infinite measurements to do so.
We say that a quantized space-time
of general relativity is non-renormalizable.
The non-renormalizability of quantized general relativity
is connected to the idea that precisely localized particles
produce black holes.
Space and time simply cannot behave in the familiar way
below the Planck scale.
And so the simplest approach to quantizing gravity
and space-time must be wrong.
Generations of physicists, starting with Einstein himself,
spent their lives trying to fix this
to unite quantum mechanics and general relativity.
They are still trying.
Even though we still lack an accepted resolution,
the struggle has not been without progress.
There are two main approaches.
One is that you search for a way to quantize general relativity
in a way that avoids the infinities
and non-renormalizability.
The leading example of this is loop quantum gravity,
or you just assume that GR and, indeed,
the mutable fabric of space-time itself
are emergent phenomena from a quantum theory
deeper than our currently accepted theories.
That's exactly what string theory seeks to do.
In upcoming episodes, we'll explore
these and other ingenious approaches
to crack the greatest problem in modern physics,
the quest for a theory of quantum space-time.
Hey, everyone.
So we haven't done a Patreon shout-out in a while.
Why-- because the Patreon crew already
hangs out with us on Google Hangouts
and on the content selection team and, in general,
on the Patreon site.
Want to hang out with us?
Please, we would love to have you.
But I digress.
Thank you so much, Patreon supporters.
You make all of this much easier for us.
And today, a special huge thank you
to Justin Lloyd, who's contributing at the Quasar
level--
Justin, as a special thanks, we're
sending you a box of chocolate-covered Planck-scale
black holes.
If our modern understanding of Hawking radiation is right,
they will evaporate catastrophically
long before they reach you.
Let us know if you get them.
It'll really help us constrain some black hole theory.
OK.
So I want to comment responses.
Today, we're covering both the black hole entropy
enigma and the challenge question episode.
How much information does the universe contain?
A few of you asked why it is that the surface
area of a black hole's event horizon must always increase
and how mass and radius can actually decrease.
Well, let's talk about the latter.
When two black holes merge, a lot of energy
is pumped into gravitational waves.
There's only one place for that energy
to come from, the mass of the black holes.
As a result, the mass of the final merged black hole
is smaller than the sum of the masses of the two originals.
Event horizon radius is proportional to mass.
And so the radius of the final black hole
is smaller than the sum of the radii of the originals.
Note that the final black hole is both more massive and larger
than either of the original black holes taken separately.
Then there's the Penrose process.
It's possible to extract energy from a rotating
black hole by throwing in objects
on near-miss trajectories.
The rotating black hole drags space around with it.
And the incoming object absorbs some of that rotational energy
and get flung out at a higher speed.
The loss of rotational energy by the black hole
also means a loss of mass.
But rotating black holes are slightly squished.
As they lose spin, angular momentum,
they become more spherical.
In that process, the event horizon only changes shape.
It doesn't lose surface area.
dabeste points out that it's important to emphasize
that you're talking about the observable universe, not
the entire universe.
And I totally agree.
For those of you who had doubts, that challenge question
is asking for the size of the storage device needed
to store all of the information in the observable universe.
I said "observable" near the start of the episode.
But I dropped that "observable" part
a couple of times later on.
That's my bad.
VoodooD0g points out that the vacuum isn't really empty,
what, with all the virtual particles
popping into and out of existence.
So what about their information?
Well, actually, those don't really contain information
because they aren't real in the sense
that we think of normal particles.
The phantom virtual particles represent
both the absence of particles and every possibility
of particles.
But in both cases, there's no specific defined state
to keep track of.
No real quantum states means no information except, perhaps,
whatever information you need to track the bulk properties,
like vacuum energy.
youteub akount asks whether the universe has ever
been in a state of too much information in too
little space, particularly during the Big Bang.
That is a really great question.
In fact, it's a great extra-extra-credit question.
I'll make sure one of our winners
is selected from those who answer this question
in their submission.
Clue-- you'll need to go beyond the formula for the Bekenstein
bound in terms of event horizon surface area.
The more fundamental formula is in terms
of radius and contained energy.
Have at it.
Rubbergnome is skeptical about the 't Hooft solution
to the black hole information paradox
and cautions that we don't neglect
other interesting ideas, like complementarity,
the membrane paradigm, fuzzballs, and holography.
Well, to be fair, we did mention complementarity
in the information paradox episode.
And 't Hooft's ideas are the gateway to holography.
We all get to fuzzballs, or better known as the black hole
triple hypothesis.
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