[MUSIC PLAYING] 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? [MUSIC PLAYING] 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. [MUSIC PLAYING]
I want to take the occasion to discuss the common claim that a theory of quantum gravity should be able to describe black hole singularities, or even that that is the whole point of a quantum description of black holes (not a criticism of the video, just want to talk about this). It's such a weird and dated statement.
Black holes generically do not have singularities at the semiclassical level; apart from the edge case of Schwarzschild, singularities only ever appear behind Cauchy horizons where one expects an instability. If you really want there to be a Kerr ringularity, you need to keep the time travel and traversable wormhole that come with it, which is absurd. Much simpler to accept the semiclassical suggestion that there is a natural cutoff before the craziness at the Cauchy horizon.
They're also not needed, and do not enter the question of the information paradox. "Explaining" or "resolving" a BH singularity does not solve the info paradox which has much more to do with the event horizon.
Hey, it's me in the final comment! I thought he was referring to 't Hooft's recent papers, but he was evidently talking about the early proposals on holography.