What Would the Death of a Black Hole Look Like?

Nov 12, 20131:59 PM

Scientists look at a section of CERN’s Large Hadron Collider during maintenance work on July 19, 2013, near Geneva. The LHC did not create micro black holes, as some had feared.
Photo by Fabrice Coffrini/AFP/Getty Images

This question originally appeared on Quora. Answer by Frank Heile, PhD in physics from Stanford University: Black holes do have a finite lifetime due to the emission of Hawking radiation. However for most known astrophysical black holes the time it would take to completely evaporate and disappear is far longer than the current age of the universe. For example, a black hole with the mass of the Sun would take $2\times10^{67}$ years to evaporate whereas the age of the universe is only $13.8 \times 10^9$ years (thus it will take more than $10^{57}$ times the current age of the universe for that black hole to evaporate!).

Black Hole Evaporation Time

The formula for the evaporation time of a black hole of mass $M$ is:

$t_{ev} = \frac{5120 \pi G^2 M^3}{\hbar c^4}$

Since the time is proportional to the mass cubed, a black hole with 10 times more mass will take 1000 times longer to evaporate and a black hole with 10 times less mass will evaporate in 1/1000th the time. If you get a black hole with a small enough mass, it will evaporate in a short time. For example, a black hole with a mass of $2.2 \times 10^5 kg$ will evaporate in about 1 second! This is approximately the mass of a blue whale - but the diameter of the “blue whale” black hole would only be $6\times10^{-22}$ meters which is about 1 millionth of the diameter of a proton! In contrast, a Solar mass black hole has a diameter of 6000 meters (or 3.7 miles) and an Earth mass black hole has a diameter of 1.7 cm (or 0.7 inches).

Black Hole Temperatures

The reason why black holes evaporate is that they have a temperature given by:

$T=\frac{\hbar c^3}{8 \pi G M k_B}$

and all objects with a temperature will emit radiation. Note that as the mass decreases, the temperature will increase! For a one Solar mass black hole this formula gives a temperature of only 60 nano Kelvin (60 billionths of a Kelvin) - VERY cold, but not quite at absolute zero. At 60 nano Kelvins, the Solar mass black hole will be mostly be emitting radio waves at a frequency of about 1800 cycles per second (far below the AM radio band) which would have a wavelength of about 160 km (or 100 miles).

Now, the cosmic microwave background (CMB) radiation has a temperature of 2.7 Kelvin. A black hole with a mass equal to our Moon would have a temperature of about 2.7 Kelvin. This implies that black holes that have a mass larger than the mass of the Moon will actually be gaining mass in our current universe since their temperature would be less than the CMB temperature - hence they will be gaining more energy from the CMB photons that they are capturing than they would radiate in Hawking radiation. But the temperature of the CMB will continue to fall with time, approaching 0 K, so eventually all black holes will evaporate.

The Last Second of a Black Hole

The black hole mentioned previously that will last for only 1 second (with a mass of a “blue whale”) will start that last second at a temperature of $5.6 \times 10^{17} K$ and in that last second of its life the total energy of all the radiation it emits will be 5 million megatons of TNT! This amount of energy is about 1000 times the total nuclear arsenal of all the nations on Earth and is about 1.4 times the total Solar energy hitting the surface of Earth in a day. Now, if the mass of the black hole were twice as big, it would take 8 seconds to evaporate (since it increases by $M^3$ ) and it would double the total energy emitted. So half of that total energy will be emitted in the last second and the other half will be emitted in the preceding 7 seconds. Thus the energy per unit time emitted by a black hole increases exponentially - but it starts at such an extremely tiny initial value for typical astrophysical black holes that it “almost” takes forever for them to blow up! Clearly, in the last stages of black holes evaporation they act like very hot bodies emitting lots of radiation and they eventually explode at the very end of their lives. Note that all black holes end their lives in the same way with the same kind of explosion. All their initial mass does is change how long it takes to evaporate and explode.

In Hawking’s original publication of his 1974 result titled “Black hole explosions?”, he states:

…Near the end of [a black hole’s] life the rate of emission would be very high and about $10^{30}$ erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs.

What is Emitted in Hawking Radiation? During most of the lifetime of a black hole, it would mostly be emitting very long wavelength (cool) photons (in other words, low frequency radio waves).

Neutrinos have a non-zero rest mass, so as the temperature rose (as the black hole shrunk), at some point, the black hole could start to emit neutrinos in addition to photons. We think the mass of the neutrinos are some fraction of 1 eV so the black hole would start to emit neutrinos when its temperature rose above (that same fraction) of 11,600 K. At this point, the black hole will become “white” since it will be emitting photons in the visible range of the spectrum - like our Sun. This neutrino emission would start when the black hole has a mass equal to about a millionth of the Earth’s mass (or 50 trillion( $10^{12}$ ) “blue whales”). This regime of emitting visible light and neutrinos will start when the black hole is still $10^{23}$ times the current age of the universe away from exploding. A black hole of this mass is only about as big as the thickness of a cell membrane. So it will be tiny and hot for a very long time!

Similarly, when the temperature of the black hole reaches 10 billion Kelvin ( $10^{10} K$ ), electrons and positrons can be emitted. At this point, the black hole will also be emitting very high energy gamma rays along with the electrons, positrons, and neutrinos. This all starts when the black hole has shrunk to about 56 million “blue whales,” and it will still be 400,000 times the current age of the universe away from its final explosion.

As the temperature gets higher and higher, more kinds of particles and antiparticles will be emitted by the black hole - basically, if the temperature of the black hole is high enough to create these particles it will. The details of the last few microseconds or nanoseconds of the lifetime of a black hole will depend on the details of quantum gravity - and we have no quantum gravity theory at this point that could calculate those details.

Primordial Black Holes?

However, we do not have any evidence for black holes of less than about a Solar mass in our universe right now. Theoretically, there could have been smaller (primordial) black holes created at the time of the big bang. If these were in the right mass range, they could possibly be exploding about now, 13.8 billion years later. Astronomers have searched for the kind of explosion with no success. In order for a black hole to be exploding now, its mass at the time of the big bang would have had to have been approximately 760,000 “blue whales.” In contrast, a Solar mass is almost $10^{25}$ “blue whales.”

There is no known astrophysical process that could create black holes with a mass much smaller than a few Solar masses in our universe today. The smallest black holes that can be created today are from the remnants of supernova explosions of stars. The problem is that for smaller stars, the remnant of the supernova explosion would end being a white dwarf star or a neutron star. It is only the supernova explosion of larger stars that will result in a remnant that is heavier that a few Solar masses. Remnants that are more than a few Solar masses will not stop in the white dwarf or neutron star stage and will instead continue to collapse to become a black hole.
Time Evolution and Power Output of Black Holes Evaporating The table below shows

the time it will take for a black hole to finish evaporating and explode,
the instantaneous power that will be output at the beginning of that time period,
all as a function of the mass of the black hole.

This table uses our convenient “blue whale” mass unit $( 2.2 \times 10^5 kg)$ for the mass of the black hole. The power unit is in terms of the total energy per second that the Sun delivers to the surface of the Earth:

So, a black hole with a mass of 100 blue whales will explode in about 12 days and at the beginning of those 12 days it will be about as “bright” as the Earth would be if it were completely white and reflected all of the Sun’s light. Note that this is not very bright when compared to a typical star! In fact, the total energy emitted during the last second of the black hole’s life will only be about 1/20,000th of the Sun’s total energy output in that same second. However, the Sun mostly emits its energy in visible light photons, whereas most of the black hole’s energy will be in very, very high energy gamma rays and particles of various types. In fact, during the entire last “current age of the universe’s” part of any black hole’s life, it will mostly be emitting high energy gamma rays since its temperature is greater than $10^{11} K$ for the last 13.8 billion years of it’s life. So if a primordial black hole happens to evaporate somewhere nearby while the Fermi Gamma-ray Space Telescope is pointing at it, the satellite may be able to detect the particular gamma ray signature of the explosion. That was one of the design goals for Fermi - however, no black hole gamma ray explosions have been seen so far.
LHC Micro Black Holes? There was some concern that the LHC might have been able to create microscopic black holes. Unfortunately, to accurately calculate anything about black holes of that size, we would require a quantum gravity theory. In “normal” string theory, the smallest black hole would be at about a Planck mass (which would last for a few Planck times) but the LHC energy is 14 orders of magnitude too small to reach that energy so “normal” string theory would say the LHC could never create a black hole. [Note that a Planck mass is about 1% of the mass of a typical mosquito and that the size of a Planck mass black hole will be about a Planck length wide which is also about the length of the strings in String Theory. Finally a Planck time is how long it takes light to travel a Planck length. The Planck length and Planck time are VERY small length and time units; however a Planck mass is “only” microscopic - i.e. it is visible in a microscope!]

However, there are some speculative string theories with large extra dimensions where the LHC might have been able to create a black hole. Those black holes also would have almost immediately evaporated (within a very very small fraction of a second) and would have exploded with exactly the same energy that was used to create them, so to the LHC detectors it would have looked similar to any other LHC collision. The most significant signature of a black hole “explosion” would have been that the particles produced would have been produced in a spherically symmetric distribution. No such events have been seen so far at the LHC.

The concern that the microscopic black hole that the LHC might have created could have gobbled up the Earth was totally spurious since they would have evaporated so quickly. There is actually experimental evidence that these kinds of black holes could not have gobbled up the Earth - the fact that the Earth has not yet been gobbled up! The reason is that the energy of cosmic rays can be many orders of magnitude higher than the LHC energy, so these kinds of black holes would have been created often during the past 4.5 billion years - and yet we are still here!
Further Reading For more information about the fate of the universe see my answer to: What would we see if we could watch as the universe is dying? to learn about what would happen to the entire universe as all the black holes evaporate.

The process of black hole evaporation is described in the Wikipedia article: Hawking radiation and this answer used information and the formulas of that article to make these calculations (mostly using Wolfram|Alpha).