Hawking Radiation — How does radiation come from black holes if nothing can come out of it?

Hawking Radiation is Stephen Hawking’s most cited paper. Hawking radiation is the radiation from black holes, which will lead the black holes to eventually evaporate!

But didn’t we all hear that nothing comes out of black hole? If so, how is radiation coming from it? The short answer is that — Hawking Radiation is not coming from inside the black hole. It is coming from the surface just outside the black hole.

The physics of Hawking radiation goes like this.
  1. As per quantum physics, space everywhere is filled with fluctuating fields. Even the space just outside the black hole’s event horizon (surface beyond which there is no return) has tiny fluctuating fields.
  2. The quantum fluctuations can lead to creation of particles from the quantum fields (more on it below). It’s happening all around us but such particles exist only for a very short period of time and get annihilated.
  3. When such particles are created just at the surface of black holes, it may happen that one particle falls into the blackhole, while the other falls outside the black hole. The one that falls into the black hole doesn’t return, while the one that falls outside the blackhole is emitted as “Hawking radiation”
  4. In case of Hawking radiation, there is a net creation of particles (that don’t annihilate). The energy for creation of such particles must come from somewhere. It comes from the blackholes itself (technical version of this phenomenon in FAQs below). It means that as the Hawking radiation is emitted, blackholes lose a part of their energy. If you wait long enough, black holes lose all their energy and evaporate!
The Hawking radiation and subsequent evaporation of blackholes is a significant breakthrough in physics. Two direct consequences of the formulation of Hawking radiation are:
  1. Negligible threat of blackhole catastrophe in CERN’s Large Hadron Collider (LHC): There is a theoretical possibility of creation of tiny black holes in LHC. But we don’t need to worry about them because they evaporate in a short time, due to Hawking radiation

  2. Holographic principle: Holographic principle says that our 3D world may just be a projection of the information on the 2D surface at the outer edge of the cosmos, just like holograms where we see a 3D object on a 2D surface. The physics of holographic principle will be a topic of another post but just note that Hawking’s radiation led to the formulation of holographic principle. Holographic principle was also part of the famous Big Bang Theory TV series.

FAQ

1. How are particles created from quantum fluctuations/fields?

We all know the famous Einstein’s equation E = mc², which essentially says that mass is a form of energy. If you can get enough energy in a concentrated form, you can create a particle.

Heisenberg’s uncertainty principle gives a chance for such energy concentration to happen. In school, we have studied that position and momentum of the object cannot be measured simultaneously.

Δx * Δ p≥ h
It turns out that such relation is applicable not just to position and momentum. Other pair of attributes also follow this principle. Energy and time are one of them.

Δ(E) * Δ (t) ≥ h

where Δ(E) and Δ(t) are uncertainty in Energy and Time respectively.

We can think of it this way. Δ(E) is the energy borrowed to create particles but such borrowed energy must be returned within Δ(t) so that the above inequality is not violated. The inequality above says that the more you borrow the shorter is your payback time. But as long as you are paying back in stipulated time period, you can borrow enough.

If Δ(t) is small enough to create a sufficient concentration of borrowed Δ(E), particle formation will happen (mass is concentrated energy). Such formation happens in pairs of particle and anti particle to conserve laws of physics.

2. How does falling of a particle (created at its surface) into blackhole decrease blackhole’s energy?

In short, it’s because of the negative magnitude of the gravitational potential energy. Let’s see this in detail.

Gravitational potential energy of a body of mass m in presence of a gravitational field of mass M is given by (-) GMm/r. G is the gravitational constant and r is the distance between m and M. The negative sign is because of the “attractive” nature of gravitation.

Total energy of a particle can now be written as rest-mass energy (mc²) + kinetic energy + gravitational potential energy

E = mc² + (1/2)mv² + (-GMm/r)

c is the speed of light and its value is 3 x 10⁸ m/sec.

The mc² value is usually large and compensates for the negative gravitational potential energy, making the total energy positive.

The situation near blackholes changes because the (negative) gravitational potential energy is high due to the strong gravity of black hole. But, we are not yet under the threat of total energy not turning negative because, a body freely falling into blackhole will acquire enough kinetic energy that, together with mc², it can still balance the negative value of potential energy. However, if we somehow ensure that bodies that fall into blackhole don’t have high velocities, there is a threat of total energy turning negative. It’s what happens in Hawking radiation.

In Hawking radiation, the particle pair is produced at the surface. Hence, they don’t fall with high velocity. It means that their total energy has a threat of turning negative.

E = mc² + (~0) — GMm/r

Since gravitational potential energy (-GMm/r) is large and negative, it overrides mc² making the total energy negative, as the particle falls into the black hole.

In essence, when a particle created at the surface of blackhole is falling into the blackhole, it is adding negative energy to the blackhole. As a result, the total energy of blackhole would decrease. If enough particles fall into blackhole, meaning enough negative energy is added, the blackhole would eventually lose all its energy and vanish. We call this as the evaporation of blackhole.

3. Is there a sense of the magnitude of the time required for the blackholes to totally evaporate?
We can get a sense of this from the evaporation procedure discussed in FAQ 2. The process of loss of energy is due to addition of small chunks of particles with negative energy. Black holes are usually massive objects but the negative energy added is only in small magnitude relatively. We should hence a large, large time for massive black holes, even greater than the current age of the universe. Ethan says that
black hole the mass of our Sun would take 10⁶⁷ years to evaporate; the one at the Milky Way’s center would require 10⁸⁷ years, and the most massive ones in the Universe could take up to 10¹⁰⁰ years!
However, if the black holes are of small mass, they evaporate almost instantaneously, like the case of those that are formed in LHC. Those black holes evaporate approximately in the order of 1/10⁸⁰ seconds — extremely tiny tiny tiny tiny tiny tiny time.