The most successful theory in physics is the combination of quantum mechanics and Einstein’s Special Relativity to give Quantum Field Theory or QFT in short.

QFT treats particles as excited states of an underlying field, so they are called field quanta.

Horizon Radiation
excited fields

In QFT, quantum mechanical interactions among particles are described by interaction among their corresponding underlying quantum fields.

QFT will play an important role in explaining Horizon Radiation.

Both theories of relativity, special and general tells us that many things are observer dependent. Different observers might disagree about speed, lengths or time, but the laws of physics should be the same for everyone. And for 2 observers with very different but constants speed, the vacuum in itself should appear the same.

But relativity can throw some weirder insights into the mix, especially the idea of horizons.

Horizon Radiation

For example the event horizon of a black hole out of which no information can travel, or the cosmological horizon that limits the observable universe, and a strange type of apparent horizon that appears when we accelerate.

Horizon Radiation

Generally speaking, a horizon is a boundary in space-time, from beyond which no influence or information can pass. It limits an observer’s causal connection with the part of the universe.

However there is one eternal fact that must remain intact, that is the laws of physics shouldn’t change if we go near a black hole or start accelerating or encounter any horizon.

But enforcing this isn’t automatic and something has to give. And as it turns out is, what gives is the nature of vacuum. And in fact, the notion of what a particle is, becomes observer dependent.

highly suggested to read the article ,THE NATURE OF VACUUM” once in order to better understand Horizon Radiation.

Both Hawking’s  Radiation and the Unruh effect are a result of this observer dependent vacuum.

Following this article(Horizon Radiation) we will surely delve deeper into Hawking Radiation, but before that, we need a strong foundation on the effects of the horizon on space-time and what actually happens due to the introduction of a horizon.

And trust me after this article, the compelling and very famous explanation of Hawking Radiation will sound inaccurate and incomplete.

To get at this idea of observer-dependent particles and vacua we need to use QFT as a tool.

In QFT we think about each particle type having its own quantum field that exists at all locations in space.

Only when the field vibrates with a single quantum of energy we get to see a particle. That oscillation can be distributed in some region of space, representing the possible positions of the particle.

Horizon Radiation

The property of these particles is encoded in the property of the field itself.

Horizon Radiation

The laws of physics as we know them are the rules defining how particles interact. For the laws to be consistent, the fundamental properties of this field must be the same for all observers.

Horizon Radiation

Imagine I fire a pair of photon which annihilate to produce an electron-positron pair. All observers, be they are floating in empty space or orbiting a black hole, should agree on the basic level of interaction, that two photons upon interaction must create an electron-positron pair.

That means everyone has to agree on the fundamental nature of the quantum fields to describe these particles and the way they interact. There is no conflict for constant speeds and inertial observers.

In fact, QFT is Lorentz invariant, which simply means that special relativity is already built into it, so the equations describing the interaction transforms clearly between inertial reference frames in a way that leaves the laws of physics and the nature of the vacuum to be intact.

But when an observer who sees horizon tries to write down this equation, in order to preserve the laws of physics, they find that they have to redefine the nature of the vacuum itself.

To explain this we need to see how particles are described in QFT.

Imagine, the simplest type of quantum field is comprised of oscillators at all locations a bit like a bed of springs which are all attached to each other, or even better analogy would be an infinite extremely flexible drum skin.

Horizon Radiation

Now imagine an oscillation in any one of these locations. A particle perfectly localized in space would be like a single spring or a single point on the drum skin. That oscillator can increase or decrease in energy in discrete quantum chunks.

Horizon Radiation

We interpret each quantum of energy , representing a single particle.

Now every point in this field, this drum skin rather, is connected to the neighbouring points and the equations we write to describe the oscillations are coupled to each other.

This coupling allows the oscillations or the particles to evolve through space. But this also makes it devilishly difficult to solve these equations, because these individual oscillations in space cannot be solved independently.

Horizon Radiation

Think of it like trying to find all the 2-dimensional wave equations of each and every oscillation on the drum skin with a high degree of accuracy.

However there is a neat way to get around this.

Through Fourier Transforms we can describe any vibration or wave in two ways.

 For example sound waves can be described as variation over time or variation over frequency.

Similarly, quantum wave functions and quantum fields can be described in terms of variation of position or variation of momentum.

Horizon Radiation

So instead of writing the field having value over every possible point in space, we can write it as having a value for every possible momentum, and we can switch between them through Fourier Transform.

So let’s take our simple spatial quantum field, our drum skin with its single localized particle and transform to momentum space.

Horizon Radiation

The momentum field also has infinite oscillators, but now each one represents a different possible momentum for the particle.

Horizon Radiation

Bizarrely in momentum space, that single perfectly localized position oscillation can also be described as an infinite number of un-localized momentum oscillations.

Horizon Radiation

Each one of these momentum modes exists at all spatial points of the universe.

If we add all the momentum oscillations together, with the right the right weightings, they cancel out everywhere except the spatial location of the original particle.

Horizon Radiation

The superposition of the infinite universe sized momentum oscillators, or the “momentum particles” can represent a single special oscillator, i.e one particle at one point in the universe.

Now all this is really abstract and hard to relate with right?

So lets get back to the more tangible and slightly less abstract, the infinite drum skin.

Now it starts out with no oscillations which means the vacuum state of the quantum field. If we were to make an excitation at a particular spot, that is, make a particle, we would need to hit that spot with a drumstick and set the oscillation going.

Horizon Radiation

Most of the amplitude will be where we hit it directly, but really the vibration will extend everywhere in the drum.

Horizon Radiation

We could in principle make same oscillation by simultaneously hitting the drum everywhere with infinite number of drumsticks, adding global vibrations, but with different frequencies which will ge cancelled out everywhere except that one spot.

But why exchange a single spatial equation for infinite equations in momentum space?

Well because they make the mathematics much easier. The momentum oscillators have an important quality. They behave like simple harmonic oscillators.

Horizon Radiation

So their value over time is a simple sine wave.

Horizon Radiation

But even more importantly, they are uncoupled, the field at each momentum spot oscillates independently from its neighbouring momenta.

Dealing with uncoupled equations allows it to add and subtract oscillations without affecting the neighbouring momenta.

However, changing the momentum modes does affect the superposition – the sum of all the oscillation modes.

 A single particle can be described as many oscillators in the momentum space. Those oscillations can be reconfigured to add new particles or remove old ones.

For example- to describe a particle-antiparticle interaction, like a two-photon annihilating into an electron-positron pair, there is a nice mechanism in quantum field theory, and that is the field operator.

Horizon Radiation

It is comprised of the creation and annihilation operator that can raise or lower the number of particles one at a time by changing the number of particles or oscillations in each momentum modes.

Here the field operator also represents the field properties and for the laws of physics, this is the thing that needs to be the same for everyone.

But there is no such restriction for the creation and annihilator operators. Also, these operators actually define the vacuum.

As we already saw that the vacuum is actually a sea of virtual particles.

In momentum space, we think of it as a superposition of infinitely many momentum modes. Infinite spatially undefined particles with defined momenta, which just happens to cancel out each other leaving zero particles or a “vacuum”.

So what happens when we add a horizon to our infinite quantum field.

What if there is an edge to the drum skin? Or we cut a hole in the middle of the drum skin ? will our drum skin behave in the same way as it used to?

Horizon Radiation

The answer as we can clearly imagine is NO.

Now our drum skin will respond very differently.

We no longer have the access to the part of the drum skin and also oscillations can reflect from the boundary.

If we want to make same particles, same excitation like before we would need to strike the surface in various parts of it with many different ways.

Well the same is true with the universe as well.

If we introduce an event horizon, then we lose access to some momentum modes.

Horizon Radiation

That means we have to reconfigure our old field operator in order to create and annihilate the same particles as we had in the infinite horizon of the universe.

We need to recheck the field operator for the laws of physics to stay consistent. The new annihilation operator needs to be a mix of old annihilation operator and the old creation operator.

Horizon Radiation

Now when we use the new field operator to describe the vacuum some momentum modes that once cancelled out no longer cancels out now.

Horizon Radiation

What was once a vacuum now has particles!!!

These additional oscillations are nothing but heat. The vacuum suddenly acquires a non zero temperature and appears to be bathed in thermal particles that don’t exist otherwise.

Horizon Radiation

So by now we can start to get a good idea how the hawking radiation comes into picture. However there are still few more concepts we need to clear.

So till then let’s keep our eyes sharp at the horizon and observe what it does to the quantum contents of the space-time!

More articles will be coming soon on Horizon Radiation.

please leave a comment below or ask any question you want to regarding this article, “Horizon Radiation” .

I would highly recommend a few books that would really help you to know  in depth regarding the cosmos:

  4. A Brief History of the Universe: From Ancient Babylon to the Big Bang (Brief Histories)
  5. The Physics Book: From the Big Bang to Quantum Resurrection, 250 Milestones in the History of Physics (Sterling Milestones)
  7. Cosmos
  8. Relativity: The Special and the General Theory (Routledge Classics)
  9. Black Holes: The Reith Lectures
  10. The Oxford Companion to Cosmology (Oxford Quick Reference)

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nature of vacuum


I am Mayukh Bagchi. 3rd year Undergraduate Engineering student at SRM University , physics and in general Astrophysics is my passion and I want to pursue the same. This blog is dedicated to serve the community with honest and clear insights to the cosmos.

1 Comment

VAIBHAV PATIL · August 19, 2018 at 5:46 pm

That’s awesome work bro!!!!

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