Space-Time: The Biggest Problem in Physics

This is the Planck length. It’s a trillion trillion times smaller than an atom. It’s also the biggest problem in  physics.

Because when we try to ponder what's happening near 10 to the minus 33 centimeters, the laws of nature break down. The stage of our universe – space-time – seems to dissolve, and we can't make sense of the awful chaos underneath. Decades of investigations have converged on a haunting conclusion: Our best descriptions of nature, quantum mechanics and general relativity, are failing us.

To dig into the deepest layers of reality, we’re going to need new physics. In this video, we’ll build the stage of space-time from the ground up. And then we’ll see how we have no choice but to tear it all down again.

And in the wreckage, we’ll look for clues to paths forward – towards a quantum theory of gravity. During our journey, we’ll be forced to  challenge our basic assumptions. Is space-time real, or simply a large-scale illusion?

At the fundamental level of nature, do questions like “where? ” and “when? ” even have answers?

In 1637, the French mathematician Rene Descartes imagined a hidden  mathematical framework for space. He invented the Cartesian grid, which  labels points in space using x-, y-, and z- coordinates. He thought of this grid  as the backdrop of our stag e of reality.

A few decades later, Isaac Newton described time  in a similar way, as an absolute and rigid part of the stage. But for Descartes and Newton, space  and time existed independently of each other. Then, in 1905, Albert Einstein entered the  scene with his theory of special relativity.

Einstein’s first revelation  was that time is relative: an observer’s measurement of the time between  events depends on their motion in space. This was a clue that we should think of space and time as a single entity: the physical fabric of our universe, with three  dimensions of space and a fourth of time. With this picture of space-time  as a unified continuum, Einstein began to wonder about its shape.

To get a sense for the mathematical model of  space-time, let’s begin with one flat grid. Now, let’s take another one,  and another, and another. When we stitch them together, we create  a mathematical object called a manifold.

Let’s imagine we’re a cat walking on the surface  of this manifold. Locally, the grid looks flat, with straight-line coordinate axes everywhere. But if we zoom out, the manifold turns out  to be made up of curved coordinate axes.

Einstein’s big revelation, in  his general theory of relativity, is that space-time is a manifold that bends  and curves in the presence of matter or energy. The effects of this curvature produce  what we experience as gravity. To measure this curvature,  let’s return to our flat space.

Thanks to Pythagoras, we know that if you have a triangle with sides x and y and  diagonal s, then s^2 = x^2 + y^2. Now, suppose you make the triangle really  small, and call the displacements in the x and y directions along the two sides dx and dy.  Then the diagonal ds satisfies this formula.

But this is also simply the distance  between the endpoints of the diagonal. So we can think of this formula  as giving us a ruler for 2D space. What happens when we put these coordinates on  a rubber sheet and stretch it?

They’re still separated by the same number of grid spacings,  but their physical distances have changed. Now, we need a new ruler that tells us how  much stretching and skewing is happening on our manifold. This is the metric: our  modified ruler for curved 2D space-time.

Einstein expanded on this ruler, creating a metric for how space-time  curves in a 4D universe like ours. To describe the geometry of space-time, you have to calculate where and by how  much matter tells our manifold to curve, producing the effects of gravity. This brings  us to the crown jewel of general relativity: Einstein’s field equation for how the distribution  of matter and energy curves space-time.

This is the Einstein tensor, which describes the curvature of space-time. It’s a function  of the metric – our ruler for 4D space-time. This next part accounts for the cosmological  constant, which is the energy intrinsic in space.

This part is the stress-energy tensor. It  describes the energy density, momentum density, and pressures of matter and energy at  each location in space. In other words, it tells us where stuff is  and how much there is of it.

You can think of this equation as matter  and energy telling space-time how to bend. In a simpler world, this  would be the end of physics: an elegant theory of distances in space and  time, described geometrically by the manifold. But buried in Einstein’s blueprints are places  where the theory describes its own demise.

In places called singularities,  such as at the moment of the Big Bang and at the center of black  holes, matter becomes so dense that it squeezes infinitely – forcing  space-time to curve infinitely, too. In physics, when you encounter infinities in your  equations, it’s a signal they’ve broken down. So we have to conclude that general relativity  fails to describe physics at these singularities.

Einstein’s theory is also ignorant  of physics at the subatomic level, where particles are too light to noticeably curve  space-time and physics is quantum mechanical. To investigate space-time at the fundamental  level – to really start to ask questions about what it’s made of – we need to look  at it from a quantum point of view. This is the quantum stage.

In this probabilistic  universe, subatomic particles don’t have fixed positions in space. Instead, they  have “amplitudes. " Amplitudes are like probabilities.

But unlike  probabilities, they can be complex numbers, which means that amplitudes can cancel each other  out. Quantum mechanics says that  to go from here to there, a particle can take many different  paths, each with an amplitude. These amplitudes must be summed up to find  the total amplitude of that transition.

So, on our quantum stage, physics  is not described by saying where things are and how they move, but how  amplitudes for different possibilities change over time. This evolution is  described by the famous Schrodinger equation. And then there’s quantum entanglement.

When two particles are entangled,  their amplitudes become contingent. If you measure either particle,  the outcome is uncertain, but, if you measure one, it appears to  collapse the state of the other, so that the outcome of the second measurement  is suddenly completely determined. Ever since physicists realized that quantum  mechanics is the underlying language of our universe, we’ve been trying to fit everything  we know about nature into its strange laws.

The result of these efforts is Quantum  Field Theory, a more sophisticated version of quantum mechanics. It’s the framework  that allows us to apply quantum principles, or “quantize,” all of the matter and force fields  that fill the universe – except one – gravity. That’s because in all quantum field theories, the  matter and force fields are described as lying on top of a smooth, fixed, continuous grid of  space-time – the special relativity stage.

But to describe gravity, we have  to quantize the space-time stage itself. How are we supposed to do  this, without a stage to stand on? This is the problem of quantum  gravity.

It brings us to the edge of our understanding of physical reality  – to the end of space-time as we know it. Let’s see what happens when we apply  quantum mechanics directly to our manifold. Just like particles, the manifold should  behave quantumly.

This means it can’t be fixed. It must have a probability  of being in many different states. There is even some probability for space-time  to start shredding itself, resulting in an awful soup of quantum uncertainty, where bits  of spacetime pop in and out of existence.

All of this happens at the Planck  length, because here, gravity is quantum. At these short distances, the notions  of “here” and “there” become murky. If we try to place coordinates on  our manifold, they quickly lose meaning.

Our metric dissolves in  the chaos of quantum uncertainty. In the quest to understand what  happens at very small distances, experimental physicists have built  powerful particle accelerators. Today, the Large Hadron Collider at  CERN can probe physics at 10-17 cm.

But to see what’s happening at the Planck  length, you'd need a particle collider 1,000-trillion-times more powerful – one  about as big as our entire Milky Way galaxy. And even if we could build one, the collisions  it would produce would put so much energy into such a tiny region of space that the  region would collapse into a black hole. There's simply no operational way of  probing space-time below this length.

That suggests that space-time below  the Planck length doesn't have meaning. Something deeper is happening  here. We might be tempted to ask: What if space-time isn’t the base  layer of reality?

What if there is a more fundamental description of physics  that produces what looks like space-time? In the 1970s, Jacob Bekenstein  and Stephen Hawking stumbled on a compelling clue about the  nature of quantum gravity. To appreciate the profound  implications of what they discovered, we need to first review the laws of thermodynamics  and their origin in statistical physics.

Suppose you have a system  with large numbers of atoms or molecules and macroscopic  properties, like temperature. The entropy of such a system measures the  number of different possible molecular arrangements, or microstates, that produce  a macroscopic system at this temperature. While using the equations of general  relativity to explore black holes, Bekenstein and Hawking discovered that black holes  act as if they have an entropy, and that quantum mechanics causes them to radiate particles in  a particular way that gives them a temperature.

But black holes, like everything else, need  to comply with the laws of thermodynamics. So if black holes have an entropy, they  must be made of more primitive parts: quantum micro-states that can be  thought of as the “atoms” of space-time. Usually, the entropy of a  system is related to its volume, because it depends on how many ways you  can arrange all the microstates inside.

But Bekenstein and Hawking  showed that with black holes, something surprising happens:a black  hole’s entropy is proportional to its area – more specifically, the area  of its event horizon, or boundary. It’s as if we can know all of  the possible microstates of a black hole’s interior structure just by  counting the ways of arranging things on its surface. But how could a surface know  everything about an interior volume?

This strange finding is the best clue we have  about the quantum nature of space-time. If a 3D object like a black hole is best  understood using only two dimensions, could the same be true for the entire universe? In the 1990s, these questions coalesced into a  compelling idea called the holographic principle.

The holographic principle says that space-time is  like a hologram “projected” from the information available on some lower-dimensional  surface like the boundary of the universe. On the holographic stage, the  fabric of our reality – space-time and gravity – actually emerges from a  quantum description in a lower dimension. If the holographic principle is true, the math  describing gravity and the geometry of spacetime should be equivalent to the math of quantum  physics in a space of one fewer dimension.

In the last few decades, physicists  have searched for these mathematical equivalences with the goal of creating  a dictionary of physics that bridges the dual descriptions. The best example so far  is something called the AdS/CFT duality. This (CFT) is a well-understood quantum  theory known as a conformal field theory, or CFT.

It has no gravity in it. Let’s consider a  CFT in two spatial dimensions, and of course time. This is a kind of space-time permitted in  general relativity known as anti–de Sitter space, or AdS.

It has gravity. Let’s consider an AdS  space with three spatial dimensions, and time. The AdS/CFT duality is a dictionary that  relates the math of the two theories.

You can use it to calculate anything in  one theory in the language of the other. We might describe this situation by saying that  the AdS spacetime is “emergent” from the CFT, because some of its dimensions appear out of  the dynamics of the lower-dimensional CFT. But if space-time is emergent, what are  the quantum processes that produce it?

Here, AdS/CFT gives us a compelling  hint. The AdS space-time geometry, along with Einstein’s equations of general  relativity, emerge from entanglement – a sort of quantum inseparability between  the states of the particles of the CFT. The idea is that if two things  are entangled in one description, they become physically connected in the other.

So, at the deepest level of reality, it’s possible  that quantum entanglement is knitting space-time together in this way, giving rise to the geometry  of the space-time manifold – and our universe. If space-time is emergent in this way, entanglement doesn't happen in space-time-- entanglement CREATES space-time. Below the Planck length, it’s entirely  possible that the quantum entanglement knitting spacetime fluctuates wildly.

This  would mean that things are connecting and disconnecting all the time in such a  way that distance as we understand it may cease to exist. Maybe this is why we can’t  measure distances below the Planck length. The holographic principle is one place where we’ve  made lots of progress toward a theory of quantum gravity.

There’s a catch, though. The AdS universe  has a different space-time geometry than our own. So while the AdS/CFT duality  is remarkable in its own right, it’s not proof that we live on  a holographic stage.

Instead, you can think of it as a toy model of how  space-time can emerge from entanglement. To make further progress, we have to find a  way to extend these ideas to a model describing our universe with its particular geometry,  particles, and peculiarities, like dark energy. Ideally, we should also find a way  to test the holographic principle experimentally.

That’s going  to require creative new ideas. To me, space and time are the two of  the most fundamental concepts in all of science. The very language we use to  describe nature depends on them.

So if space and time  are emergent -- in some sense "not really there," we have to  figure out what replaces these concepts. This is the exciting quest for  the next generation. There is hardly anything deeper and more  inspiring to work on than that.