## Creation in a Singularity

We have talked about the condition of the universe very soon after the beginning, empty and formless but seething with energy in the form of very strong gamma rays. This is not the way some people envision the very beginning of the universe. Let’s try to understand why.

Physics trips over a stumbling block trying to describe a wave at the very beginning of the universe. The block is the idea of causality. In the physical world a cause must precede its effects. The theory of special relativity refines this criterion when the cause produces an effect at some distance from the cause. A consequence of Einstein’s special relativity is that the fastest possible speed in the universe is the speed of light. An effect cannot be the result of a distant cause if the cause has to travel faster than the speed of light to reach the place where the effect occurs in time to precede the effect.

Gamma rays are electromagnetic waves and therefore occupy a finite volume in space. The volume that contains the photon travels with it at the speed of light. Within that volume the electric and magnetic fields are coordinated with one another. A graph of any field property along a line that passes through the volume will give a picture that looks, unsurprisingly, like a wave. A wave graph is not chaotic, with sudden leaps up and down or gaps with nothing between. It varies smoothly between crest and trough and back again. This smooth variation or coordination between separate parts of a wave occurs physically because the cause of the effects propagates between the different parts at the speed of light. Indeed the whole wave travels at the speed of light.

There is no violation of causality if the source of the wave emits it continuously, because the cause occurs right at the source.

It takes a certain amount of time to establish a wave. The amount of time is the length of the wave train divided by the speed of light. If the wave train repeats all its properties over a certain distance then the wave is periodic, the distance is the wavelength, and the period is the time any feature of the wave takes to propagate one wavelength. The minimum time to establish a periodic wave is one period.

Now we come to the problem of the existence of waves, that is, extended, coordinated structures at the very beginning. What could have coordinated the structures over any appreciable distance? Of course, if the waves appear a little after the beginning, they may be the result of physical causes that began to operate at the beginning. But if a certain amount of time has passed then we are no longer describing conditions at the very beginning.

To describe how things were at the very beginning people must confine their theories to point sources. The distance across a point is zero in any distance units, zero meters, zero inches, zero cubits or zero handbreadths. If the source of a wave is a point, then there is no distance between any cause and its immediate effects.

No one has observed a single point that was the source point at the beginning. We can’t see back to times earlier than 380 000 years after the beginning, when the first atoms formed and left the universe transparent. What is visible, what we have photographed, is a huge hollow shell, not a point.

Some people say that the universe began as a ball of a certain size. However they must deal with very serious problems of causality to make their theories credible. Most people find it simpler to think of the universe as a dimensionless point at the very beginning.

However, the idea that everything was squeezed down to a point at the beginning leads on to another hard problem. We have already discussed the four forces. The strong and weak nuclear forces cannot begin to act until there are protons or neutrons or similar heavy particles. Experiments with cyclotrons and other particle accelerators have shown that protons and neutrons are made up of particles called quarks. By the time protons and neutrons appeared some rather complicated physical causes must have acted to make the protons and neutrons up from quarks. We need not deal with the strong and weak force when considering the very beginning.

The two other forces are gravitation and the electromagnetic forces. When we calculate the effects of gravity or electromagnetic fields, we often find ourselves dividing by the square of the distance between the cause and the effect. This leads immediately to a mathematical snag when we try to apply the laws of physics within a point source at the very beginning. In a point the distance between cause and effect is zero. The square of the distance is also zero, because zero times zero is zero. Then the effect is proportional to the inverse square of zero—and there we stop. Division by zero is an undefined mathematical operation.

Calculus permits mathematicians to approach division by zero as a limit when the distance between two points becomes smaller and smaller. As one divides a numerator by a smaller and smaller denominator the magnitude of the quotient becomes larger and larger. We say that the formula “blows up,” or that we “approach infinity.” But infinity is not a well defined number in mathematics. This means that cosmologists can write formulas that apply a short time after the beginning, and then see what results they get when they make the time since the beginning smaller and smaller. But this procedure doesn’t work in the limit of zero time after the beginning. The formulas for gravitation or electromagnetic fields simply are not valid at the very beginning.

This is what physicists mean when they say that there is a singularity at the beginning. An illustration of a singular point is the North Pole or the South Pole of the Earth. The sense of the word “singular” is isolated or unique. If we stand at any other point on the Earth’s surface, we have a choice of walking north, east, south, west, or any direction in between. We turn and go in any direction we choose. But at the North Pole after turning in any arbitrary direction one can only walk south. From the South Pole one may only walk north. The poles are singular points because the usual rules for all other points do not apply.

Let’s notice, however, that the singularity at the poles of the Earth is removable. One can turn through any arbitrary angle and then walk away from a pole just as one can at any other point. The difference is only in the name we customarily give to a direction. At a pole all directions have the same name in our conventional system of directions. But we are free to invent any new system of names to help us maintain our orientation at the poles. Pilots who fly regularly close to the North Pole use a special system of coordinates for navigation when they are near the pole. The system has a conventional grid, not a set of lines that converge at a point.

In mathematics some singularities are called poles. In the theory of complex variables there is a mathematical technique for removing some kinds of poles and evaluating their “strength.” However, the theory also recognizes that some singularities are “essential.” No one can remove an essential pole because it is too complicated.

In Newton’s theory of gravity the force is inversely proportional to the square of the distance between two gravitating objects. The closer the two objects are to each other, the greater is the force. Einstein’s theory of general relativity modified Newton’s theory of gravity and mechanics, but it did not remove the singularity. The singularity at the beginning is an essential singularity.

In electrostatics the force between two charged particles is likewise inversely proportional to the square of the distance between them. However, there is a way around the singularity that appears when the particles are as small as electrons. One still has to solve the equations that govern electromagnetic fields. We have mentioned those equations earlier. They are Maxwell’s equations.

Maxwell put the four equations together in the 19th century, before people knew about the internal structure of atoms. His equations still apply in the subatomic world, but the Heisenberg uncertainty principle does not allow fixing an electron at a definite point. The source of an electric field, either a proton or an electron, must be treated as a kind of cloud that gives the probability of finding the source at any point in a small volume. This allows physicists to calculate the average force between an electron and a proton in a hydrogen atom. They can even take into account the possibility that the electron and proton are at the exact same point, with zero distance between them. In that case the force between them is infinite, incalculable, “blown up,” or whatever one likes to call it. However the infinite force only occurs for the one case of zero distance. The probability of that case is vanishingly small.

Let’s use terms that are not quite mathematically exact to help us understand. One multiplies the infinite force by a zero probability, and gets an undefined quantity, but that quantity makes an insignificant contribution to the calculation of the average force.

The above calculation is valid because of the change from the older, “classical” electromagnetic theory to quantum mechanics. In a classical description one may conceive of charged particles localized at a point. In quantum mechanics all particles are treated as de Broglie waves, all waves are quantized and come in discrete packets called quanta, and the Heisenberg uncertainty principle applies. This is all very complicated, but it does let physicists solve an extremely hard problem, one that has an essential singularity. They do so by noting that the one point with infinite forces has zero probability of occurrence, and so they leave it out.

Physics trips over a stumbling block trying to describe a wave at the very beginning of the universe. The block is the idea of causality. In the physical world a cause must precede its effects. The theory of special relativity refines this criterion when the cause produces an effect at some distance from the cause. A consequence of Einstein’s special relativity is that the fastest possible speed in the universe is the speed of light. An effect cannot be the result of a distant cause if the cause has to travel faster than the speed of light to reach the place where the effect occurs in time to precede the effect.

Gamma rays are electromagnetic waves and therefore occupy a finite volume in space. The volume that contains the photon travels with it at the speed of light. Within that volume the electric and magnetic fields are coordinated with one another. A graph of any field property along a line that passes through the volume will give a picture that looks, unsurprisingly, like a wave. A wave graph is not chaotic, with sudden leaps up and down or gaps with nothing between. It varies smoothly between crest and trough and back again. This smooth variation or coordination between separate parts of a wave occurs physically because the cause of the effects propagates between the different parts at the speed of light. Indeed the whole wave travels at the speed of light.

There is no violation of causality if the source of the wave emits it continuously, because the cause occurs right at the source.

It takes a certain amount of time to establish a wave. The amount of time is the length of the wave train divided by the speed of light. If the wave train repeats all its properties over a certain distance then the wave is periodic, the distance is the wavelength, and the period is the time any feature of the wave takes to propagate one wavelength. The minimum time to establish a periodic wave is one period.

Now we come to the problem of the existence of waves, that is, extended, coordinated structures at the very beginning. What could have coordinated the structures over any appreciable distance? Of course, if the waves appear a little after the beginning, they may be the result of physical causes that began to operate at the beginning. But if a certain amount of time has passed then we are no longer describing conditions at the very beginning.

To describe how things were at the very beginning people must confine their theories to point sources. The distance across a point is zero in any distance units, zero meters, zero inches, zero cubits or zero handbreadths. If the source of a wave is a point, then there is no distance between any cause and its immediate effects.

No one has observed a single point that was the source point at the beginning. We can’t see back to times earlier than 380 000 years after the beginning, when the first atoms formed and left the universe transparent. What is visible, what we have photographed, is a huge hollow shell, not a point.

Some people say that the universe began as a ball of a certain size. However they must deal with very serious problems of causality to make their theories credible. Most people find it simpler to think of the universe as a dimensionless point at the very beginning.

However, the idea that everything was squeezed down to a point at the beginning leads on to another hard problem. We have already discussed the four forces. The strong and weak nuclear forces cannot begin to act until there are protons or neutrons or similar heavy particles. Experiments with cyclotrons and other particle accelerators have shown that protons and neutrons are made up of particles called quarks. By the time protons and neutrons appeared some rather complicated physical causes must have acted to make the protons and neutrons up from quarks. We need not deal with the strong and weak force when considering the very beginning.

The two other forces are gravitation and the electromagnetic forces. When we calculate the effects of gravity or electromagnetic fields, we often find ourselves dividing by the square of the distance between the cause and the effect. This leads immediately to a mathematical snag when we try to apply the laws of physics within a point source at the very beginning. In a point the distance between cause and effect is zero. The square of the distance is also zero, because zero times zero is zero. Then the effect is proportional to the inverse square of zero—and there we stop. Division by zero is an undefined mathematical operation.

Calculus permits mathematicians to approach division by zero as a limit when the distance between two points becomes smaller and smaller. As one divides a numerator by a smaller and smaller denominator the magnitude of the quotient becomes larger and larger. We say that the formula “blows up,” or that we “approach infinity.” But infinity is not a well defined number in mathematics. This means that cosmologists can write formulas that apply a short time after the beginning, and then see what results they get when they make the time since the beginning smaller and smaller. But this procedure doesn’t work in the limit of zero time after the beginning. The formulas for gravitation or electromagnetic fields simply are not valid at the very beginning.

This is what physicists mean when they say that there is a singularity at the beginning. An illustration of a singular point is the North Pole or the South Pole of the Earth. The sense of the word “singular” is isolated or unique. If we stand at any other point on the Earth’s surface, we have a choice of walking north, east, south, west, or any direction in between. We turn and go in any direction we choose. But at the North Pole after turning in any arbitrary direction one can only walk south. From the South Pole one may only walk north. The poles are singular points because the usual rules for all other points do not apply.

Let’s notice, however, that the singularity at the poles of the Earth is removable. One can turn through any arbitrary angle and then walk away from a pole just as one can at any other point. The difference is only in the name we customarily give to a direction. At a pole all directions have the same name in our conventional system of directions. But we are free to invent any new system of names to help us maintain our orientation at the poles. Pilots who fly regularly close to the North Pole use a special system of coordinates for navigation when they are near the pole. The system has a conventional grid, not a set of lines that converge at a point.

In mathematics some singularities are called poles. In the theory of complex variables there is a mathematical technique for removing some kinds of poles and evaluating their “strength.” However, the theory also recognizes that some singularities are “essential.” No one can remove an essential pole because it is too complicated.

In Newton’s theory of gravity the force is inversely proportional to the square of the distance between two gravitating objects. The closer the two objects are to each other, the greater is the force. Einstein’s theory of general relativity modified Newton’s theory of gravity and mechanics, but it did not remove the singularity. The singularity at the beginning is an essential singularity.

In electrostatics the force between two charged particles is likewise inversely proportional to the square of the distance between them. However, there is a way around the singularity that appears when the particles are as small as electrons. One still has to solve the equations that govern electromagnetic fields. We have mentioned those equations earlier. They are Maxwell’s equations.

Maxwell put the four equations together in the 19th century, before people knew about the internal structure of atoms. His equations still apply in the subatomic world, but the Heisenberg uncertainty principle does not allow fixing an electron at a definite point. The source of an electric field, either a proton or an electron, must be treated as a kind of cloud that gives the probability of finding the source at any point in a small volume. This allows physicists to calculate the average force between an electron and a proton in a hydrogen atom. They can even take into account the possibility that the electron and proton are at the exact same point, with zero distance between them. In that case the force between them is infinite, incalculable, “blown up,” or whatever one likes to call it. However the infinite force only occurs for the one case of zero distance. The probability of that case is vanishingly small.

Let’s use terms that are not quite mathematically exact to help us understand. One multiplies the infinite force by a zero probability, and gets an undefined quantity, but that quantity makes an insignificant contribution to the calculation of the average force.

The above calculation is valid because of the change from the older, “classical” electromagnetic theory to quantum mechanics. In a classical description one may conceive of charged particles localized at a point. In quantum mechanics all particles are treated as de Broglie waves, all waves are quantized and come in discrete packets called quanta, and the Heisenberg uncertainty principle applies. This is all very complicated, but it does let physicists solve an extremely hard problem, one that has an essential singularity. They do so by noting that the one point with infinite forces has zero probability of occurrence, and so they leave it out.