Entropy and Penrose
We need a key to understanding how the high frequencies and short wavelengths of the early universe produce a great degree of order. A measure of order is the number of digits required to specify a physical quantity like frequency.
The musical note A has a frequency of 440 Hz. This means that the sound wave oscillates between high and low pressure and back again 440 times per second. It takes three digits to specify A. All the members of an orchestra must tune their instruments to that precision or their music will sound terrible.
An AM radio station has an assigned broadcasting frequency, say 1 120 kHz. The station operators must tune their transmitter to oscillate 1 120 000 times per second. If they are inaccurate their station will interfere with other stations, and they must readjust or stop broadcasting. It takes seven digits to specify their frequency.
An FM station will have a higher frequency, say 102 700 000 Hz. It takes nine digits to specify their broadcasting frequency. This precision is so great that most people could not tune the radio without help. The help they ordinarily receive is an electronic circuit called Automatic Frequency Control, built in to most receivers. All the listeners have to do is tune their receivers close to the station’s frequency. The AFC circuit homes in the rest of the way.
It takes 14 digits to specify the frequency of a laser’s light wave, and at least 20 digits to specify the frequency of rays that could materialize as a positron and electron. The high-energy gamma rays that collided at the beginning of the universe had frequencies much higher than any of the waves we have named so far. Specifying their frequencies required a huge number of digits.
Besides the information in the frequency we have to consider the information in the direction. A low-frequency wave has little directionality. Throwing a stone into a still pond will make a circular wave that expands in all directions. A circular wave has no directionality at all. However, one may combine waves so they reinforce each other in some directions, and cancel each other in other directions. The higher the frequency of the waves, the greater is their directionality. One sometimes sees sunlight glinting from windows in distant buildings. Usually only a few of the windows are bright. They are the ones whose perpendicular happens to bisect precisely the angle between the Sun and the observer. Other windows, even on the same face of the building, glint in slightly different directions. Sunlight is much more directional than radio waves because its frequency range is much higher.
Ordinary space has three dimensions. The three dimensions correspond to three distances between the observer and the glinting window we just mentioned. We would have to say how far the observer is north or south, east or west, and up or down from the window. The ray from the window to the observer also has three dimensions, but they are not space dimensions. One is its frequency. The other two are the angles the incoming ray makes with the window. These angles have to be precise if the ray is to reach the distant observer. In all, we have six dimensions for each ray.
No one can visualize a six-dimensional space very well. Physicists call this space a “phase space” for a ray. They visualize it two or three dimensions at a time.
Rays require six dimensions or six numbers for complete specification. Objects like planets also require six dimensions—three in ordinary space to tell where they are, and another three in “velocity space” to give their speed and direction. The values of these six dimensions at any given time determine the future position and velocity of the planet, provided we know what gravitational fields the planet will encounter.
The solar system is far from other stars. Distant stars have weak gravitational fields in our neighborhood. Also weak in our neighborhood are the gravitational fields of the moons of other planets, the asteroids, the Kuiper Belt objects, and the comets, because all those bodies are small and usually far away. The gravitational fields that mainly influence the Earth’s movement are those of the Sun, Moon, and the other eight planets. With the Earth that makes a total of 11 objects all interacting with their significant mutual gravitation. To describe their initial state we need 11 individual phase spaces. Altogether we need 66 dimensions for the phase space that describes the solar system. We can determine the significant gravitational fields each object will encounter from the positions of the other objects at the time. Any given starting point in that phase space therefore determines the future of the solar system.
If we were to determine the starting point of the solar system we would have to choose very carefully. For instance, if we chose to have the planets widely separated as they usually are but we negligently left all the initial velocities equal to zero, all the planets would fall into the Sun. Or if the initial velocities were too large, the planets would escape from the Sun and freeze in the dark and cold of deep space. Most choices would be disastrous. Only a relatively small number of choices would lead to a stable, habitable solar system.
We simplified the above scenario by leaving out the asteroids and comets. Choosing their initial positions or velocities badly would eventually cause disastrous collisions with the Earth. The phase space must be larger to include them, too, and the choice of the starting point becomes more stringent.
Even so, we are oversimplifying by taking the Sun, Moon, and planets as preformed objects. We know that they formed from dust particles. The phase space must be big enough to include the starting position and velocity of every dust particle in our region of the Milky Way. Such a phase space is completely unmanageable. There is no computer large enough to hold all the required data. Does that boggle the mind? If it doesn’t, consider that such a phase space is really too tiny to predict the tranquil future of the Earth. The dust of the Milky Way came from exploded stars. We would have to consider the origin and history of the first stars. That leads us back to the phase space of the universe soon after creation. One needs six dimensions for every gamma ray in the early universe.
Thermodynamics and quantum mechanics permit us to divide up phase space into cells. The dimensions of position or velocity are not continuously variable if the universe has a limit to its size. Each ray has a limited but very large number of starting possibilities. Multiplying this large number by the number of rays gives the total number of choices.
Penrose calculates the accuracy the Creator would need in pinpointing the beginning point in phase space for the universe.[i]
[i] Penrose, Roger, “Cosmology and the Arrow of Time”, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford: Oxford University Press, 1989), pp. 302–347.
The musical note A has a frequency of 440 Hz. This means that the sound wave oscillates between high and low pressure and back again 440 times per second. It takes three digits to specify A. All the members of an orchestra must tune their instruments to that precision or their music will sound terrible.
An AM radio station has an assigned broadcasting frequency, say 1 120 kHz. The station operators must tune their transmitter to oscillate 1 120 000 times per second. If they are inaccurate their station will interfere with other stations, and they must readjust or stop broadcasting. It takes seven digits to specify their frequency.
An FM station will have a higher frequency, say 102 700 000 Hz. It takes nine digits to specify their broadcasting frequency. This precision is so great that most people could not tune the radio without help. The help they ordinarily receive is an electronic circuit called Automatic Frequency Control, built in to most receivers. All the listeners have to do is tune their receivers close to the station’s frequency. The AFC circuit homes in the rest of the way.
It takes 14 digits to specify the frequency of a laser’s light wave, and at least 20 digits to specify the frequency of rays that could materialize as a positron and electron. The high-energy gamma rays that collided at the beginning of the universe had frequencies much higher than any of the waves we have named so far. Specifying their frequencies required a huge number of digits.
Besides the information in the frequency we have to consider the information in the direction. A low-frequency wave has little directionality. Throwing a stone into a still pond will make a circular wave that expands in all directions. A circular wave has no directionality at all. However, one may combine waves so they reinforce each other in some directions, and cancel each other in other directions. The higher the frequency of the waves, the greater is their directionality. One sometimes sees sunlight glinting from windows in distant buildings. Usually only a few of the windows are bright. They are the ones whose perpendicular happens to bisect precisely the angle between the Sun and the observer. Other windows, even on the same face of the building, glint in slightly different directions. Sunlight is much more directional than radio waves because its frequency range is much higher.
Ordinary space has three dimensions. The three dimensions correspond to three distances between the observer and the glinting window we just mentioned. We would have to say how far the observer is north or south, east or west, and up or down from the window. The ray from the window to the observer also has three dimensions, but they are not space dimensions. One is its frequency. The other two are the angles the incoming ray makes with the window. These angles have to be precise if the ray is to reach the distant observer. In all, we have six dimensions for each ray.
No one can visualize a six-dimensional space very well. Physicists call this space a “phase space” for a ray. They visualize it two or three dimensions at a time.
Rays require six dimensions or six numbers for complete specification. Objects like planets also require six dimensions—three in ordinary space to tell where they are, and another three in “velocity space” to give their speed and direction. The values of these six dimensions at any given time determine the future position and velocity of the planet, provided we know what gravitational fields the planet will encounter.
The solar system is far from other stars. Distant stars have weak gravitational fields in our neighborhood. Also weak in our neighborhood are the gravitational fields of the moons of other planets, the asteroids, the Kuiper Belt objects, and the comets, because all those bodies are small and usually far away. The gravitational fields that mainly influence the Earth’s movement are those of the Sun, Moon, and the other eight planets. With the Earth that makes a total of 11 objects all interacting with their significant mutual gravitation. To describe their initial state we need 11 individual phase spaces. Altogether we need 66 dimensions for the phase space that describes the solar system. We can determine the significant gravitational fields each object will encounter from the positions of the other objects at the time. Any given starting point in that phase space therefore determines the future of the solar system.
If we were to determine the starting point of the solar system we would have to choose very carefully. For instance, if we chose to have the planets widely separated as they usually are but we negligently left all the initial velocities equal to zero, all the planets would fall into the Sun. Or if the initial velocities were too large, the planets would escape from the Sun and freeze in the dark and cold of deep space. Most choices would be disastrous. Only a relatively small number of choices would lead to a stable, habitable solar system.
We simplified the above scenario by leaving out the asteroids and comets. Choosing their initial positions or velocities badly would eventually cause disastrous collisions with the Earth. The phase space must be larger to include them, too, and the choice of the starting point becomes more stringent.
Even so, we are oversimplifying by taking the Sun, Moon, and planets as preformed objects. We know that they formed from dust particles. The phase space must be big enough to include the starting position and velocity of every dust particle in our region of the Milky Way. Such a phase space is completely unmanageable. There is no computer large enough to hold all the required data. Does that boggle the mind? If it doesn’t, consider that such a phase space is really too tiny to predict the tranquil future of the Earth. The dust of the Milky Way came from exploded stars. We would have to consider the origin and history of the first stars. That leads us back to the phase space of the universe soon after creation. One needs six dimensions for every gamma ray in the early universe.
Thermodynamics and quantum mechanics permit us to divide up phase space into cells. The dimensions of position or velocity are not continuously variable if the universe has a limit to its size. Each ray has a limited but very large number of starting possibilities. Multiplying this large number by the number of rays gives the total number of choices.
Penrose calculates the accuracy the Creator would need in pinpointing the beginning point in phase space for the universe.[i]
[i] Penrose, Roger, “Cosmology and the Arrow of Time”, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford: Oxford University Press, 1989), pp. 302–347.
He calculates an inconceivably large number. Penrose says: This is an extraordinary figure. One could not possibly even write the number down in full…it would be ‘1’ followed by 10123 successive ‘0’s!
Even if we were to write a ‘0’ on each separate proton and on each separate neutron in the entire universe—and we could throw in all the other particles as well for good measure—we should fall far short of writing down the figure needed.[i]
[i] Ibid, p. 344.
If a pre-existing intelligence did not select the initial state of the universe, then science cannot explain the existence of a universe capable of supporting life. Getting the starting point right by one chance in Penrose’s large number is not mere good luck. It is a miracle of the highest order. Luck may act like a lady once in a while and favor somebody, but she doesn’t do miracles. The God of the Bible does miracles, but why should He do them for people who will give Luck the credit?