Einstein's Equation of Mass and Energy

In the previous discussion we noted the grave change introduced by relativity theory in our understanding of the space-time framework within which we seem to see a universe of objective events involving changes in matter and energy — involving changes in the states of rest or motion of material bodies and transformations in their energies. But the change introduced by relativity theory in our understanding of that mass and that energy is equally grave and, as we shall see in the subsequent discussions, it has taken U9 far in our search for the utterly simple which constitutes the goal of the scientific quest and which underlies the apparent complexity of this universe of gravity, electricity and inertia. This is the second great change that was required in our physics to make possible the joining of the cosmological map of European science with that of Advaita Vedanta. The physics of the last century took for granted that the universe was objective in three dimensions and that it consisted of real particles with real mass and real energy moving through real space in real time. Mass was considered to be one thing and energy, another. Space was considered to be one thing and time, another. It had not yet been noticed that space and time are opposites and that they must, therefore, in some sense, be identical. By opposites, in physics, we mean two quantities, like plus and minus electrical charges, which are related in such a way that, in some sense, if we have the same amount of both it is like having none of either. If in an atom the positive charges on the nucleus are balanced by the negative charges of the electron cloud around it, the total charge on that atom is zero. we also understand that in order for two things to be opposites they must, in some sense, be identical. Plus and minus electrical charges are opposites only by being identical in that they are both electrical charges. It's like money into the bank and money out of the bank. Rupees into the bank and rupees out of the bank are opposites because they are both rupees. Dollars into the bank and dollars out of the bank are opposites because they are both dollars. There is no such thing as the opposite of a dog. Space and time are opposites in the sense that if, between two events, the space separation is equal to the time separation then the total separation between those two events is zero. Between the event of perception and the event perceived the space separation exists only by contrast to the time separation, and the time separation exists only by contrast to the space separation. They are identical in that together they form the two halves of the framework within which we seem to see a universe of mass and energy. We turn now to a discussion of that mass and that energy. Long before the end of the last century we had understood that the inertia, or mass, of a physical body was measured by the resistance which that body offered to changes in its state of rest or motion. Newton, in 1666, put it quite beautifully: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare. (Bodies all persevere in their states of quiescence or of motion, uniform in direction, unless, by forces impressed upon them, they are compelled to change their states.) Now it is quite singular that matter should behave in this way — that it should resist changes in its states of rest or motion — and this behavior, like the existence of the electrical and gravitational fields, cannot be explained by classical physics. Gravity, electricity, inertia and the conservation laws which govern transformational causation are completely inexplicable by that causation. They are completely inexplicable to classical physics. But matter does behave in this way. If, in a moving vehicle, you collide with another vehicle, which has pulled to a stop on the highway in front of you, the consequences to your vehicle depend largely on the mass of the other vehicle. The greater its mass, the greater the resistance which it will offer to changes in its motion on impact. Often we measure the mass of an object by the resistance which it offers to being shaken. A man in a blindfold can easily distinguish a baseball from a tennis ball simply by shaking it. It requires energy to throw a baseball only because the mass of the ball offers resistance to every change in its state of rest or motion. Likewise, energy is required to catch it or to change the direction of its motion with a baseball bat. Mostly we study physics in the playground, and we understand it in our muscles. But here on Earth, where the gravitational field is about the same From playground to playground, we sometimes think that the mass of an object may be measured by the resistance which that object offers to being picked up. Rut that is wrong — or, rather, only partly true. weight, or the resistance of an object to being picked up, varies with the strength of the gravitational field against which it is being picked up. On the moon, where the gravitational field is much less than in our playgrounds on Earth, a baseball would offer much less resistance to being picked up. On the moon it would weigh only one-sixth as much as it weighs on Earth, but it would be just as hard to shake — every bit as hard. Energy, in classical physics, was understood as the capacity to do work — the capacity to affect changes in the states of rest or motion of matter. A force is measured by the rate of change which it produces in the state of motion of a given mass. Energy is measured by the total change. When we wind up a clock, we wind it up against the force of the spring, but it is how far we wind against that force which determines the energy of the wound spring and, therefore, how long the clock will run. When we pick up a weight from the floor, we pick it up against the force of gravity, but it is how far we pick it up against that force which determines the energy of the raised weight and what it might do on falling. The energy of the raised weight is determined by the product of the weight (the force exerted by gravity at that particular place on that mass) and the distance through which work has been done against that weight (against that force). Now energy, like mass, is a conserved quantity. What is meant by that is that in any transformation of energy, the total amount of energy which we have at the end is exactly the same as the total amount which we had at the beginning, even though the form of that energy may have changed. If a baseball is thrown straight up with a certain kinetic energy, it will reach a certain height where its kinetic energy will have disappeared completely. Working against gravity on the way up, the kinetic energy has been transformed to gravitational energy which, on the way down, is re-transformed to kinetic energy so that the ball will reach the ground with the same amount of kinetic energy which it had when it began its rise. We all understand this in our muscles, and it is important to remember that we do. The energy of a physical system was defined as the capacity of that system to produce changes in the state of rest or motion of a body of a given mass. The inertia, or mass, of a physical body was defined as the capacity of that body to resist changes in its state of rest or motion. Energy is the capacity to produce such changes, and mass is the capacity to resist them. In this sense mass and energy were understood, in classical physics, to be opposites. What we had not noticed is that in order to be opposites they must, in some sense, be identical. Their identity followed from Einstein's equations. Einstein's equations led to a revolution in our physics. Our entire system of physics was wrong and required to be corrected for this new understanding. We had misunderstood the relationship of space to time and with it the relationship of mass to energy. From the equations of relativity it followed that, for a particle at rest, its energy is its mass. It was like a mountain having two names because it had been seen from two different points of view without the realization that both views were of the same mountain. That which we had been measuring all along as mass turned out to be some kind of energyf and, for a particle at rest, Einstein's equation of mass and energy states that E = m This equation is often written as E = mc 2, but the c2 is necessary only if we refuse to measure space and time in commensurate units. If space is to be measured in centimeters then time should be measured in "jiffies." A jiffy is the length of time required for light to travel one centimeter. The speed of light, c, is then one centimeter per jiffy and c2 goes to unity and we have Einstein's original equation, E = mc2. The energy of a particle varies with its motion but the energy which a particle has when not in motion, that is, the energy which a particle is when seen at rest is called its rest energy or its rest mass. But what kind of energy is it? As mentioned earlier, all the matter of the observable universe can be traced back to hydrogen, dispersed through space and falling together by gravity to galaxies and stars. The other chemical elements are fashioned from hydrogen at extreme temperatures in the bellies of the stars and in the brilliant stellar explosions which scatter those heavy elements through interstellar space. The problem is to understand the hydrogen, because the mass of these other elements is just the mass of the original hydrogen out of which these other elements have been built. Here, at last, through relativity theory, we are able to understand that the mass of that hydrogen is simply some kind of energy. But what kind of energy? It is a tremendous simplification to understand that, for particles their mass is their energy. But what kind of energy is the mass of the primordial hydrogen from which the entire universe arises by transformation? Suppose we have two identical clocks. One we wind up; the other we leave unwound. Which one is more massive? Which one will be harder to shake? The wound one will be harder to shake because it is wound up against the resistance of the spring and will therefore have a greater energy and a greater mass. Now against what are the protons and the electrons of the primordial hydrogen wound up to give them their mass? The problem is simplified by the fact that we have only a few kinds of energy from which to choose — gravity, kinetic energy, radiation, electricity, magnetism and nuclear energy. It is further simplified by the fact that kinetic energy, radiation and magnetism cannot exist at rest and cannot, therefore, give rise to the rest energy of the primordial hydrogen. We are left, then, to choose between gravity, electricity and nuclear energy. Since we are living in what is called the nuclear age of physics, many would think that the choice should go to nuclear energy, but that would be wrong. Even if the matter of the universe began as hydrogen, which has the maximum available nuclear energy, and ended as iron, which has the least, even then the nuclear energy released to other forms could account for only about one percent of the rest mass of that matter. Therefore, the choice must go to gravity, electricity, or both. Embodied beings such as ourselves have a genetic response to being wound up against gravity. We have a fear of heights. We have muscles to handle the problem if we jump down from a chair because the wind-up of climbing to the seat of a chair is not very great. But very few of us have muscles capable of handling the problem if we jump down from the edge of the roof, because the gravitational energy represented by our bodies on the roof is more than our muscles can handle on landing. None of us has muscles to handle the problem if we fall from an airplane or from a high cliff because the gravitational energy represented by the physical separation of our bodies from the ground is simply too great. The farther away from the earth our bodies are, the more wound up they are and, therefore the more massive they are. However, the gravitational wind-up in picking our bodies up from the surface of the Earth represents only a minute fraction of their total gravitational wind-up because the surface of the Earth in no way represents the gravitational floor of the universe. An object falling to the surface of the Earth, even from outer space, comes in with kinetic energy only a few times greater than the energy needed to vaporize that object. An object falling to the surface of the sun would develop considerably more kinetic energy because the gravitational field at the surface of the sun is nearly thirty times as strong as the gravitational field at the surface of the Earth. The surface of the sun represents a lower floor, and the farther down we fall, the more gravitational energy is converted to kinetic energy, and, therefore, the more dangerous it is when we land. But there are collapsed stars, such as black dwarf stars, with a density of a hundred thousand pounds per pint where the consequences of falling would be quite severe. And there are neutron stars, collapsed to a density of perhaps a hundred thousand battleships per pint, where falling would develop kinetic energies equivalent to about ten percent of the rest mass of the falling objects. The splash o£ a ten-gram marshmallow falling to the surface of such a star would have enough energy to evaporate a town. It would have the energy of one atom bomb. But the total energy of a ten-gram marshmallow is equal to the energy of ten atom bombs. Is there a still lower floor to the universe to which we might drop the marshmallow to convert its entire rest energy to kinetic energy? Yes, theoretically there is. If all the matter of the observable universe were in one place, that would be the floor. The mass of the universe, as we see it now, is equal to the energy required to pick it up, or space it out, from that floor. The universe is wound up against gravity — it is massive — just by being dispersed. We can understand the mass of the protons as due to the fact that they are not all in one place. It is that simple. The curious thing is that we can understand the rest mass of the hydrogen from an entirely different consideration. The hydrogen atom is made of electrical particles — one proton and one electron — and they have opposite charges. Now since opposite charges attract each other, there should be a certain amount of energy represented by the fact that the charges are separated from each other. That, however, would be only a small part of the energy. A great deal more energy is represented by the smallness of the particles themselves. Like charges repel each other. We may put this another way: Like charge repels itself. We can see, then, that the smaller the geometrical size of a given charge, the greater the electrical energy represented by its smallness because its smallness represents pushing like charge toward itself. If a single, electrical charge were allowed to become infinitely big, its rest mass would go to zero. If a single, electrical charge were forced to become infinitely small, its rest mass would go to infinity. It is only because this charge is seen squeezed down against itself to a finite size that it has a finite rest mass. Seen from this point of view, we can understand the rest energy of the primordial hydrogen as due entirely to the smallness of the electrical charges. Thus, we have two apparently different and seemingly independent ways of accounting for the rest energy of the primordial hydrogen. It is wound up against its gravitational field because the atoms are seen spaced out, and it is wound up against its electrical field because the atoms are seen to be composed of minute electrical particles. But spacing out against the gravitational field and spacing in against the electrical field are themselves opposites. The primordial hydrogen is massive because it is seen as spaced out against the condensational gravitational field and spaced in against the dispersional electrical field. Between two protons, gravity pulls and electricity pushes. In this sense, they are clearly opposites. Yet both operate on the inverse square law. Between two objects, the gravitational field, like the electrical field, falls off inversely as the square of the distance between them. If the distance is doubled, the pull or push falls off to one-quarter. If the distance is increased threefold, it falls off to one-ninth. The remarkable thing is that no other fields fall off in this way, and that the strengths of the fields are such that the rest energy of the primordial hydrogen turns out to be the same when looked at from either side. Rest energy is simply a geometrical wind-up. Both the gravitational and the electrical rest energies of the primordial hydrogen are what are called energies of position in space. Although one is wound up against space by dispersion and one by condensation, they are both wound up against space, and without space they are inconceivable. They are opposite in the direction of wind-up, but identical in that they are both wound up against space on the inverse square law. (It might be helpful to our understanding if we simply dropped the words 'gravity' and 'electricity' and thought of the one as the result of dispersion and the other as the result of smallness or condensation.) When we break a cookie, we require space in which to break it. There must be space between the parts. Likewise, if we break it to crumbs, there must be space around the crumbs. Further, the more divided it is, the smaller the parts .To say that it is divided is to say that it is small. You can't break a cookie into larger and larger parts. Dividedness and smallness are but one idea. In this sense, electricity and gravity are identical. In the previous paragraphs no mention has been made of time. nut space cannot exist except by contrast to time. We live in a four-dimensional universe. We see momentum in the x direction, momentum in the y direction, momentum in the z direction and momentum in the time direction. The time component of the momentum is the energy. It is only through relativity theory, through an understanding that space and time are identical as well as opposite, and that neither can exist without the other, that we have come, at last, to understand that energy and mass are also identical as well as opposite and that neither can exist without the other. For a particle at rest, its energy is its mass. E = m. It is energy itself which is hard to shake.