The final flash
Throw any object up in the air. It raises up, slows down, momentarily pauses and falls back to Earth. The harder you throw it, the higher it rises before coming down. When thrown hard enough, at roughly 11 kilometers per second, the object leaves earth entirely and never returns. This threshold is called the escape velocity.
The alternative to this improbable superman feat with bare hands is that we build rockets to send satellites to reach outer space and beyond. The fuel in the rockets deliver continuous thrust to reach the escape velocity and escape Earth’s gravity. The more massive a planet is, the stronger its gravity. Therefore, it requires an even higher escape velocity.
Now, imagine packing matter into a small region of space. As we pack more matter, the density grows and the region’s gravity intensifies. Eventually, you reach a threshold, where the escape velocity within that region exceeds the speed of light. Since nothing can travel faster than light, everything including light cannot escape! The boundary around the region that marks the point of no return is called the event horizon.
Congratulations, you have created a black hole!
In nature, black holes are not built this way. They form naturally when a massive star runs out of nuclear fuel and collapses under the weight of its own gravity compressing enormous mass into a small volume. There are other means by which black holes can form. But all of them involve a large amount of mass being squeezed into a small region.
Now, this leads us to an intriguing puzzle. Since nothing can escape from the black hole isn’t everything inside it lost forever? Nature seems to always come up with novel ideas to keep its tab on the stuff and energy that goes into any physical system. It turns out that black holes have a subtle way of returning energy to the universe through a phenomenon called the Hawking radiation named after the famous physicist Stephen Hawking who predicted it.
According to him, a black hole gives off a type of radiation that is the same as the radiation given off by any hot object, such as a hot pan, oven or even our warm bodies. This radiation is called black body radiation which has an associated temperature. Not only that, but the black hole also slowly loses its mass and energy through this radiation! How is that possible? Isn’t a black hole supposed to prevent anything from escaping it? The answer, as Stephen Hawking discovered, lies in the interplay between gravity, time and our notion of empty space or vacuum near a black hole. Let’s first explore what we mean by empty space.
Something rather than nothing
A thermos flask keeps your drinks hot by having a vacuum in the gap between its inner and outer walls. With almost no air, heat conduction can hardly happen. The pressure within the insulating space of a thermos flask can range from a hundredth to a millionth of atmospheric pressure for some of the highest-grade industrial vacuum. This means that almost 99% of the air in the region between the inner and outer walls of the insulating layer is removed. But even the best thermos flask insulation is about a few trillion times denser than interstellar space! Interstellar space, in turn, is not completely empty and is filled with around one atom per cubic centimeter of space.
Suppose we remove even that single atom to create the region of perfect ‘nothingness’. Would that space be truly empty? Not really, according to Quantum field theory or QFT.
QFT is the unification of quantum mechanics, which describes the physics of the very small and special relativity, which describes the physics of objects moving at high speeds. In QFT, the fundamental entities are not particles, but quantum fields that pervade space. For every fundamental particle like the electron, quark, photon and so on, there is an associated field. Particles arise when their associated field is excited, just like how sound is created when a violin string is plucked. The electron, for instance, is an excitation of the electron field.
Interestingly, even in empty space or perfect vacuum these fields due to their quantum nature are never completely at rest. Imagine a pond with the water surface never at rest, always rippling randomly due to wind. The quantum fields in vacuum are similar¹. The random rippling or vibrations of the fields corresponds to the lowest energy state, the vacuum state. The vacuum state does not contain any real particles as there is no net excitation of the fields.
Now imagine a star collapsing under its own gravity to form a black hole. The quantum fields that were in the vacuum state before the collapse evolve in this gravitationally changing region of space. Could a distant observer still describe those fields as empty or in vacuum state? The answer lies in something strange about the nature of time itself near the star as it collapses into a black hole.
When time stops… almost
Our head ages faster than our legs! No, I am not talking about some biological phenomenon, but rather an intriguing consequence of Gravity. Time itself runs at different rates depending on how close you are to a massive object like our Earth.
This effect is called gravitational time dilation.
According to Einstein’s general theory of relativity, clocks closer to a massive object tick more slowly than clocks far away, when their rates are compared. Imagine standing on the surface of the earth. Your head is slightly farther from Earth’s center than your feet. As a result, your head ages slightly faster than your feet. How much is this difference? For a height difference of two meters, the head ages more than our legs by about 10^-11 seconds in a day! That’s a tiny difference.
No, you cannot keep your mind younger by walking upside down on your hands! The effect is far too small to notice. Even over an entire lifetime, your head would age by roughly half a microsecond more than your feet! Hardly something consequential.
Or is it?
Gravitational time dilation has a very practical consequence for a technology that we use every day, the Global positioning system or GPS. GPS satellites orbit far above the Earth, where gravity is weaker than it is on the ground. Clocks in GPS satellites therefore tick faster than those on Earth. This effect amounts to about 45 microseconds per day. The satellites high orbital speed introduces an opposing effect due to special relativity which we will not discuss here. That results in their clocks ticking slower by about 7 microseconds per day. The net effect is that the clocks in the satellites run about 38 microseconds faster than clocks on the ground in a day.
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That might sound like a small difference. But GPS rely on accurately measuring travel times of radio signals between the satellites and your smartphones. Without corrections for that small difference, the timing errors would translate to errors of your location by about 10 kilometers!
What all this tells us is that time is not absolute. Clocks can tick at different rates depending on gravitational fields in which they operate. On Earth, the differences are small. But near a black hole where gravity is extreme, the consequences become much more dramatic. That brings us to the last rays of light before a black hole forms.
The last rays
Imagine a massive star collapsing under its own gravity. As the collapse proceeds, outgoing rays of light escape from the star’s surroundings. Some escape early and others later. There is a critical ray that marks the boundary between light that can escape and that which gets trapped. This critical ray corresponds to the event horizon.
Now consider two rays of light that eventually reaches a far observer, one arriving later than the other. The later ray must have escaped from very close to the critical ray. The later a ray when it arrives at the observer, the closer its trajectory must have been to the boundary or event horizon. This separation between the later rays and the critical rays shrinks exponentially as we trace them back in time. This leads to the event horizon creating extreme stretching of time between the early universe and the time measured by a distant observer. A very tiny interval of time in the early history before the black hole formed can correspond to a very large interval in the distant future.
Let’s say a light wave arrives at a distant observer’s detector. How rapidly it oscillates determines its frequency. More oscillations per second means a higher frequency. As we have seen, the relationship between the early history of an escaping ray and time at which it reaches a distant observer becomes extremely stretched near a black hole. Therefore, a wave travelling along that ray will be stretched too. This leads to a decrease in frequency or longer wavelength. This is called a redshift as light is shifted to the red end or longer wavelength of the spectrum. Let’s see what happens to quantum fields near a black hole under similar stretching of waves.
Recall that particles are excitations of the quantum fields. Consider a distant observer detecting a photon of Hawking radiation from the black hole. The photon corresponds to a particular frequency of oscillation of the electromagnetic field. In a simple picture, this means that the peaks and troughs of the wave arrive at regular intervals.
But when traced back through the history of the collapsing star, because of the extreme gravitational effects near the forming event horizon, the relationship between the waves late time regularity and early time history gets exponentially distorted. Equal times in the distant observer’s time correspond to increasingly smaller intervals in the early history. Consequently, a wave that has a single frequency in the late time description no longer corresponds to a single frequency when traced back to the early region. When the mathematical description is worked out (which we will not do here), the quantum field becomes a mixture of many positive and negative frequencies².
In QFT, this mixing means a field mode considered to contain no particles in the early time vacuum before the collapse, can contain particles according to the late time description after the black hole forms. In this way, the formation of the black hole and exponential distortion of the waves leads to an initially empty vacuum to be observed as a stream of real particles and radiation by a distant observer. The key is that the exponential relationship between early and late time waves also determines how much of each of positive and negative frequencies are mixed. When this quantum mixing proportion is worked out mathematically, the result is a thermal spectrum same as that of black body radiation! This is the crux of Hawking radiation.
Evaporation
If the black hole gives off radiation, where does this energy come from? To conserve energy, the black hole must lose an equivalent amount of energy, and therefore mass. According to Einstein’s famous equation E=mc² energy and mass are equivalent. As the black hole radiates, it gradually loses mass. In effect, it slowly evaporates.
Fascinatingly, although Hawking radiation has the characteristics of a blackbody radiation, a black hole behaves very differently from an ordinary hot object. A hot pan cools as it loses heat. A black hole, on the other hand, gets hotter as it loses mass. The smaller it becomes, the hotter it gets and faster it radiates. But how long would it take to evaporate completely?
The temperature and rate of radiation depends on the mass of the black hole. This can be understood as follows. A lighter black hole compared to a heavier one is confined to a much smaller region and has a much smaller event horizon. The gravitational effects on the quantum fields we discussed earlier is much more extreme and therefore results in a higher Hawking temperature. The relationship between the black hole mass and the quantum radiation is such that its temperature rises as its mass decreases. Therefore, as it radiates and loses mass, the temperature rises, increasing the rate of radiation and leading to a runaway process.
For instance, a black hole of the mass of a sun has a temperature of about 6x10^-8 kelvin about 60 billionth of a degree of empty space! It takes around 10⁵⁷ years for such a black hole to evaporate completely, which is unimaginably many times the current age of the universe. But as the black hole evaporates and when it mass reaches say that of the largest animal, the blue whale, its temperature rises to several billion times that of the sun and evaporates in a fraction of a second. The last final moments would be a spectacular burst of radiation exceeding millions of megatons of TNT or far higher than any nuclear device.
The final spectacular flash seems like a contradiction of nature for an object that was destined to be lost forever into the dark oblivion.
Remarks
The hawking radiation has never been observed directly from an astrophysical black hole. For a typical black hole, the radiation is too weak to be detected. For example, a black hole around the mass of the sun has a hawking temperature of about 6x10^-8 kelvin, far below the 2.7 kelvin temperature of the cosmic microwave background (CMB), which fills the universe. Such black holes absorb more energy from the CMB than they radiate away. Furthermore, radiation from hot matter swirling around black holes completely buries the faint Hawking radiation. There were analogue experiments conducted that captures the physics of Hawking radiation. In one such experiment using ultra-cold atoms called Bose-Einstein condensates, researchers created a sonic black hole, where sound waves cannot escape a boundary similar to a event horizon. They observed radiation consistent with the Hawking radiation.
Footnotes
- Note that this is only an analogy. The quantum fields are not substances like water in motion just like how a magnetic field around a magnet is not a substance.
- For the mathematically inclined: A wave that oscillates at a single frequency according to the distant observer’s clock becomes a distorted multi frequency wave when traced back through the black hole’s formation. When this distorted wave is Fourier decomposed, we get both positive and negative frequency modes. In QFT, the positive frequency modes are associated with the annihilation operators and negative frequency modes are associated with creation operators. Consequently, late-time annihilation operator becomes a mixture of early-time creation and annihilation operators. The creation operator contributes to the late-time observer detecting particles even when the field was initially in its vacuum or empty state.
