Let’s imagine that we want to drop a single bit of information into a black hole. How can we think of achieving this? One answer is to drop a single photon into the black hole. A photon is a massless particle of light, and one photon can store one bit of information. We can think of a photon as spinning clockwise (0) or anticlockwise (1), which means it can represent one bit. Each photon also carries a fixed amount of energy inversely proportional to its wavelength. This relationship between energy and wavelength was first proposed by Einstein in 1905 and is a key feature of quantum theory. Long-wavelength photons have smaller energy and short-wavelength photons have higher energy. This is the reason UV light from the Sun can be dangerous but the light from a candle can’t: UV photons have short wavelengths and carry enough energy to damage your cells, whereas candlelight photons have longer wavelengths and don’t carry enough energy to cause damage. As a general rule, the position of a photon cannot be resolved to distances smaller than its wavelength. This means we would like to drop a photon with a wavelength roughly equal to or smaller than the Schwarzschild radius into the black hole, since longer wavelength photons would typically reside outside the hole. Now we can calculate the largest possible number of such photons we can fit inside the black hole. This should give us a crude estimate of the maximum number of bits it can store, which is the entropy.** We go through the calculation in Box 9.1. The answer for the number of bits hidden in a Schwarzschild black hole is:
where A is the area of the event horizon.†† One very interesting thing about this equation is the collection of numbers in front of the horizon area, A. This combination of the speed of light, c, Newton’s gravitational constant, G, and Planck’s constant, h, a number that lies at the heart of quantum theory, is well known to physicists. It is the square of the so-called Planck length. We’ve described the significance of the Planck length in more detail in Box 9.2. The short version is that it is the fundamental length scale in our Universe and the smallest distance we can speak of as a distance. The suggestion is that the entropy of a black hole, which is to say the number of bits of information it hides, can be found by tiling the event horizon in Planck-length-sized pixels and assuming that the black hole stores one bit per pixel. This is illustrated in Figure 9.3.
Figure 9.3. The horizon of a black hole, with an imaginary tiling of Planck-area-sized cells. Remarkably, the total number of cells is equal to the entropy of the black hole.
It’s hard to overstate what an intriguing result this is. What is the nature of these Planckian pixels and why are they tiling the horizon when, according to general relativity, the horizon is just empty space? Recall, an astronaut freely falling through the horizon should experience nothing out of the ordinary according to the Equivalence Principle. And yet Bekenstein’s result suggests that they encounter a dense collection of bits. Furthermore, why should the information capacity of a black hole be proportional to the area of its horizon rather than its volume? How much information can be stored in a library? Surely the answer must depend on the number of books that fit inside. For a black hole library, however, it seems as if we are only allowed to paper the outside walls with the pages of the books. It is as if the interior doesn’t exist.
One might wonder whether the black hole might be missing a trick from an information storage perspective, but a simple argument suggests that a black hole of a given mass has the maximum possible information storage capacity (entropy) for that mass. Imagine dropping an object into a Schwarzschild black hole. To obey the Second Law, the black hole entropy must increase by at least the entropy of the object it swallows. The area of its event horizon will grow accordingly, but the area increase depends only on the mass of the object because the area of the horizon is proportional only to its mass. Now imagine instead dropping a super-high entropy object with the same mass as before into the black hole. The horizon area will increase by precisely the same amount, proportional only to the mass of the object. This means that as we add mass to the black hole, it must increase its entropy by the largest possible amount. It is as if objects thrown in get completely scrambled up, to guarantee that our ignorance is maximised.
A black hole therefore has the largest possible entropy. It can store the maximum possible amount of information in a given region of space, and the amount of information measured in bits is given by the surface area of the region in Planck units. This hints at something deeply hidden; everything that exists in a volume of space can be completely described by information on a surface surrounding the region. This is our first encounter with the holographic principle.
BOX 9.1. Black hole entropy
Roughly speaking, only photons whose wavelengths (as measured by a distant observer) are less than the Schwarzschild radius can fit inside the hole. According to quantum physics, a photon has an energy E = hc/l, where l is the wavelength and h is Planck’s constant. Therefore, the smallest possible photon energy is E = hc/R, where R is the Schwarzschild radius. The total energy of a black hole of mass M is, according to the famous Einstein relation, Mc2. The maximum number of photons we can fit inside the hole is:
Now, the Schwarzschild radius R = 2GM/c2, which means we can write:
The horizon area A = 4πR2, and therefore:
You may wonder if more information can be stored using other types of particle (electrons also spin and can be used to encode bits). Unlike photons, other particles carry mass which means fewer of them can fit inside the hole.
BOX 9.2. The Planck length
The Planck length is a combination of three fundamental physical constants; Planck’s constant, Newton’s gravitational constant and the speed of light. The Planck length is a very tiny length. The diameter of a proton is 100,000,000,000,000,000,000 Planck lengths. Max Planck first introduced his eponymous unit in 1899 as a system of measurement that depends only on fundamental physical constants. This is preferable to using things like metres or seconds, which reflect the vagaries of history and have more to do with the size of humans and the orbit of our planet than the underlying laws of Nature. The strength of gravity, the behaviour of atoms and the universal speed limit, however, are independent of humans. If we encountered an alien civilisation and asked them to tell us the area of the horizon of the M87 black hole in Planck units, they would come up with the same number that we do. In a formula, the Planck length is given by:
It is believed to be the smallest distance that makes any sense: smaller than this, it is likely that the idea of a continuous space breaks down.