The Stephen Hawking Legacy

The Stephen Hawking Legacy

He was perhaps the greatest genius of our time.

Stephen Hawking peered behind the curtain of reality and glimpsed the true workings of the universe, he inspired all of us to pursue our curiosity no matter the obstacles.

However, his true legacy is his work. He made profound contributions across physics from quantum theory to cosmology.

Our tribute is to bring you Stephen Hawking’s most famous discovery.

Soon after Einstein revealed his great general theory of relativity in 1915, physicists realized that allowed for the possibility of a catastrophic gravitational collapse in places of extreme density, like the dead core of a massive star.

Space and time could be dragged in words to create a hole in the universe, a boundary in space-time called an event horizon, that could be entered but from beyond nothing could return.

Once formed there was nothing in theory or imagination that could bring material consumed back to the outside universe.

Stephen Hawking

These black holes should exist forever only growing never shrinking, or so we thought, until 1974 when a young physicist named Stephen Hawking published a paper in Nature entitled black hole explosions.

In this and in a follow-up 1975 paper, he attempted a new union of quantum mechanics and general relativity to show that black holes should not be so black after all, they should leak what we now know as Hawking radiation.

There’s a popular description of how Hawking radiation works it goes something like this:
empty space is filled with activity of pairs of virtual particles, matter, and antimatter, that spontaneously appear, and then annihilate each other, briefly borrowing energy from the vacuum itself.

But when this happens near a black hole, one of the pairs will be swallowed by the event horizon leaving the other free to escape and taking its toll and energy.

That energy can’t come from nothing and so the black hole itself pays the debt by slowly leaking away its mass.

This is a nice picture but how accurate is it in fact?

If we follow the narrative of Hawking’s original calculation the story sounds rather different.

We’ve come a long way over the past few months building up the knowledge we’ll need to follow that calculation, if you think you are ready, let’s take a deep dive into the quantum field theory of curved space-time, to glimpse the true nature of Hawking radiation.

Space is filled with quantum fields, they can oscillate with different frequencies much like the many possible vibrational modes on the guitar string. A particle is like a note on the string, and just like a real guitar note, real particles tend to be comprised of many vibrational modes.

Those underlying vibrational modes are still present in the absence of real particles, they fluctuate energy due to quantum uncertainty, and those fluctuations give us what we think of as virtual particles.

Now don’t take the existence of virtual particles too seriously, they’re really just a tool for calculating the infinite ways in which a fluctuating quantum field can behave.

Stephen Hawking

One way the quantum fields are very different to guitar strings is that they can have both positive and negative frequencies, a negative frequency can be thought of as a mode that travels backward in time and can be interpreted as corresponding to antimatter.

Now that’s a whole level of weird or on its own, when a quantum field is in a vacuum state there’s a balance between positive and negative frequency modes, which you can crudely think of as a balance between virtual matter and antimatter particles.

These all virtually annihilate or cancel out so that no real particles exist, this is all fine in flat space but, spatial curvature can mess with the balance of the underlying quantum field modes, by introducing horizons.

Horizons cut off access to certain modes of the quantum fields, disturbing the balance that defines the vacuum.

Stephen Hawking knew that black holes with their insane space-time curvature would wreak havoc on quantum fields, but what would the effect be?

To answer that properly he would need a full union of general relativity and quantum mechanics, a theory of quantum gravity, a theory of everything.

It didn’t exist then and it doesn’t exist yet.

Not to be deterred by the impossible, Hawking came up with an ingenious workaround, the narrative of Hawking’s mathematics goes something like this: he imagined a single space-time path, a Lightspeed trajectory that extends from far in the past to far in the future.

This perilous path passes through the location of a black hole in the instant before it forms, in fact, it is its very last trajectory, it emerges barely ahead of the forming event horizon.

Stephen Hawking

Hawking imagined a simple quantum field tracing this path, a field that is in a perfect vacuum state before the formation of the black hole. , but he found that the close shave with the black hole disturbs the fundamental vibrational modes that define the fluctuations of the vacuum by the time.

This trajectory has found its way back out into flat space again. And those fluctuations look like real particles.

A distant future observer sees radiation coming from the black hole, Hawking’s imaginary path from the distant past to the distant future was brilliant, it allowed him to compare the state of the vacuum in two regions of flat space, far from the black hole regions where the nature of vacuums, quantum fields, and particles are perfectly well understood.

But to understand the effect of the close encounters with the black hole, he required an uneasy marriage of quantum mechanics and general relativity.

In the absence of a theory of quantum gravity, Hawking needed a hack, and that hack was the Bogoliubov transformations.

These can be used to approximate the effect of curved space-time on quantum fields, by smoothly connecting regions of flat space, they describe a sort of mixing of the positive and negative frequency vibrational modes that are caused by those curves.

The physical interpretation of this mixing via the above transformations is tricky, in fact, there isn’t just one valid interpretation.

Hawking’s calculation talks about scattering certain modes of the quantum field. Scattered or deflected by the gravitational field of the forming black hole, they have nudged off their narrow escape path and lost behind the forming event horizon.

Meanwhile, other modes avoid scattering and continue unscathed, with the loss of certain fundamental modes, the vacuum state must be constructed from the remaining modes that distorted vacuum.

Stephen Hawking

Looks like it’s full of particles the nature of the lost modes, it tells us what Hawking radiation should look. Like black holes tend to scatter modes with wavelengths similar to their own sizes, the quantum field that emerges is distorted in the same wavelength range, and so it produces wave packets.

It produces particles that also have wavelengths about as large as the event horizon so, the more massive the black hole the longer the wavelength of its radiation Hawking. Calculating the frequency distribution of this radiation found something incredible, it should look exactly like thermal radiation.

Black holes would have a heat glow with an apparent temperature that depends on their mass, more directly, it’s proportional to the surface area of the event horizon. Large black holes should appear cold, radiating excruciatingly slowly, but small black holes should appear hot, and the smallest should radiate explosively.

What about the whole picture of particle-antiparticle pairs being pulled apart by the event horizon?

Hawking’s math describes splitting or mixing of these pure positive and negative frequency modes, it’s fair to interpret this mixing as the promotion of what were once virtual particles into reality, and for the escaping modes there exists a corresponding set of modes linked by quantum entanglement, that are trapped behind the event horizon.

We can interpret those as corresponding to the swallowed antiparticle partner, so the split matter-antimatter part of the picture is reasonable, but there are reasons to dismiss aspects of this picture.

Stephen Hawking

Firstly, this radiation is not localized. Remember the Hawking radiation has wavelength the size of the event horizon, the size of the entire black hole?

Well, these debris wavelengths have created particles, and they tell us that there is an enormous quantum uncertainty in the location of these particles. Hawking radiation must appear to come from the global black hole, not from specific points on the event horizon. In fact, an observer in free fall through the horizon sees nothing to them, space is locally flat, the vacuum should look like a vacuum.

This radiation is visible only to distant observers, there is one exception when you turn on your jetpack and hover a fixed distance above the horizon, then you do see particles. You see Unruh radiation.

Hawking radiation is mostly going to be photons and other massless particles, to produce particles with mass, the energy of the radiation has to be high enough to cover the rest mass of the particle, so it’s okay to interpret the narrative of Hawking’s calculation as the splitting of entangled matter and antimatter pairs.
Even if it really is just a heuristic interpretation, it’s the cause of the splitting that’s hard to pin down.

We can think about positive and negative frequency modes being mixed due to scattering, perhaps by the as yet undiscovered graviton. Other physicists have derived Hawking’s result with very different-seeming narratives, for example, in 2001, scientists got the same thermal spectrum for Hawking radiation by thinking about particles escaping from beneath the event horizon, through quantum tunneling.

The common thread is quantum uncertainty, for example, uncertainty in position or momentum can lead to particle pairs too, or once in the same location, or merge through once on the same world line. Becoming separated by the event horizon alternatively, uncertainty in energy can lead to particle creation.

Whichever way you interpret it, it’s hard to avoid the conclusion that black holes emit particles, the fact that different derivations lead to exactly the same result, or that the radiation looks thermal, can’t be by chance.

it’s hard to make Hawking radiation go away, and believe me, Stephen Hawking himself tried.

However, these calculations are all hacks albeit, brilliant ones, without a full quantum theory of gravity the origin of Hawking radiation will remain mysterious, and there are other mysteries that we haven’t touched on for example what happens to the particles or modes trapped by the black hole? How do they end up reducing the black hole’s mass instead of increasing it?

And then there’s the famous information paradox in which Hawking radiation appears to destroy.

For now, we must conclude that black holes radiate and in doing so evaporate.

The scariest monsters of general relativity are ultimately unraveled by the brilliant mind of Stephen Hawking and a mysterious quirk of quantum space-time.

It is his legacy for us.

Stephen Hawking

Stephen Hawking – 1942 – 2018