As threatened, I've done some thinking about gravity. I apologise for any repetition, and look forward to some lively objections.
One of the things Einstein did in the general theory of relativity was to demonstrate that it was not possible, in the absence of external evidence, for an observer to distinguish between the effects of acceleration, on one hand, and of gravity on the other. With a nod in the direction of Preearth, I acknowledge that Einstein was not the first person to think of this, but he took what had previously been just an observation, and revolutionised scientific thinking with it. The equivalence principle highlights the similarity between the effects of gravity and of acceleration.
Not only did Einstein build on the equivalence principle, he also placed upon it an interpretation that was subtly different from that of Newtonian physics. In Einstein's version, the principle asserts that in free-fall the effect of gravity is totally abolished in all possible experiments and general relativity reduces to special relativity, as in the inertial state. What this means is that if you were free-falling in a windowless, soundproofed (so that the sound of rushing through the atmosphere didn’t give the game away) box, you would have no way in which you could tell that gravity was having any influence on your situation. Thus, any experiment you might try, would tell you nothing about gravity, and could be totally accounted for by the precepts of special relativity, which ignores gravity. So you could ignore gravity, at least until your box made contact with the Earth, after which you might well lose all interest in gravity and GR.
Gravity is something we tend to take for granted. Not until Newton did anyone seriously consider gravity as the driving force that kept the planets orbiting the sun, and the moon orbiting the Earth. It was Newton who not only proposed the theory of universal gravitation, but also worked out the mathematics to prove it. There are, however, a couple of things to be kept in mind. The first of these is that although Newton’s equations described the cosmic situation so well that their use has permitted men to land successfully on the moon, and, more importantly, to return safely, as well as facilitating non-manned visits to far flung bodies in our solar system, these truly groundbreaking observations and calculations do not tell us why or how every body in the Universe attracts every other body, nor, indeed, why any lump of matter should attract any other. What is more, they do not offer any explanation for the provenance of this force; which brings us to the second point that must be kept in mind. Where does the seemingly inexhaustible supply of energy come from? Why, for example, after thousands of millions of years of holding the moon in orbit, and keeping vast quantities of loose objects “stuck” to the Earth’s surface, does gravity show no signs of lessening. Does it not seem that nature might have achieved the “impossible dream” of perpetual motion – an endless supply of renewable energy? If this were the case there would have to be some very serious re-thinking done within the hallowed halls of physical academia. Of course, there has to be another explanation.
The apparent violation of the law of conservation of energy, implied in the preceding paragraph, could not be left unexplained. The concept that came to the rescue was the work function. The work function was originally a tool used by engineers to quantify the amount of work done by a specific process or machine. The equation used to express this function was W = F d, where W is the work done, F is the force used and d is the distance over which the work is carried out. Energy can be related to work done, and fuel requirements can be calculated from energy output. Try moving a very heavy object. If the attempt is successful the work function can be applied to it, but suppose you are unable to move the object. The distance then becomes zero, so F in the second half of the work function equation must be multiplied by 0. Obviously, any quantity multiplied by zero must equal nothing, so the equation becomes W = 0. No work has been done; therefore no energy has been expended. In spite of absurd results like this, the work function has been used in this way to explain that objects are held on the Earth’s surface by gravity without expenditure of energy, because there is no movement. No movement means no force, no work and no energy exchange.
To the original W = F d has been added the term cosθ. Now the equation is W = F d cosθ. Theta can be anything from 0 to 360 degrees. cos converts it into a value between minus one and one. The angle “θ” represents the angle between the direction in which the object is pushed and the direction in which it actually moves. If the object moves in the direction in which it is pushed, then the angle “θ” is zero degrees and, because cos (0) equals one the equation effectively remains in its original form: W = F d. However, should the object not move straight forward the element cos θ will move towards minus one, thus becoming a negative fraction. In the worst case scenario, if the direction of movement lies at 90 degrees to the direction of push, cosθ becomes zero, because cos 90° = 0, thus F d must be multiplied by zero. So the amount of work done becomes W = 0. It might be argued that it is unlikely that a consistent push, or pull, in one direction will result in motion in a direction at an angle of 90 degrees; but this is exactly the situation in the case of one body orbiting another, larger body. If we reason that the moon is attempting to travel in a straight line past the Earth, but because of the gravitational attraction between them it is being constrained to orbit the Earth, then it follows that the line of force holding the moon in its orbit lies perpendicular to the direction in which the moon is travelling.
This is where someone will point out that gravity is not a force, and that the moon orbits the Earth, not because it is held in place by a force acting perpendicular to its line of travel, but because it is following the curvature of spacetime caused by the presence of the mass of the Earth. However, the question of how this curvature might be caused and maintained without expenditure of energy becomes significant.
Consider a rock lying on the ground. If you pick up that rock, then let go of it, it will fall back to the ground. Your action in picking it up has involved a transfer of energy: some energy from your muscles has been converted into gravitational potential energy as you raise the rock. If, when you have lifted the rock, you place it on a shelf, and let go of it, it will not fall, but it will still have the energy you transferred to it, still in the form of potential energy. Obviously, if it is pushed off the shelf it will fall back to the ground. The argument here is that the energy you put into the rock as you lift it equals the energy that would be necessary for gravity to bring it back to the Earth’s surface, so there is no net expenditure of energy. Could this explain how gravity seems to work without any apparent energy source? A little thought about this situation must raise some doubts. For example, if the attraction of gravity is directly related to the amount of energy put into the rock you are lifting, why does gravitational attraction not increase with distance, as would be the case if you were stretching a spring? If, having picked up the rock you altered your position so that you were holding your rock over an open well. When you released the rock it would fall to the bottom of the well, in spite of the fact that you transferred to it only enough energy to take it as far as the ground surface. Finally, we would have to wonder why a spacecraft that travelled from Earth to the moon would be attracted by the moon’s gravity. Not only would it not have been lifted from the surface of the moon, but it should have enough potential energy to whisk it straight back to Earth as soon as it stops moving away.
GR postulates that matter "curves" spacetime in its vicinity. Obviously this involves curvature in four dimensions which is difficult for most of us to imagine, so physicists usually suggest we think of spacetime as a rubber sheet stretched out flat. If there are no large masses around, the sheet stays flat, and so any object placed on it and given a push will move around in straight lines. However, a large mass, such as the Earth, makes a dip in the sheet because it actually warps spacetime. Now any other object with smaller mass, like the moon, moving about in spacetime rolls into the dip as it comes past the Earth. It appears to be attracted to the large mass, but is in fact following the most direct route through curved spacetime. This effect of warping spacetime is what gives rise to gravity. It is quite easy to draw a diagram of the two-dimensional sheet depressed by a large mass, but progressing from two dimensions to three dimensions becomes quite complicated, and would involve a quasi-infinite number of depressed two-dimensional sheets, centred on the Earth, facing in every possible direction. We are now faced with the question as to whether or not any work is being done to maintain this situation. The easiest way to think about this is to return to the two-dimensional sheet. If we take the rubber sheet into space and place a massive object on it, it will not depress the sheet. In fact it will work only if the whole experiment is conducted is conducted in a gravitational field. Obviously some force is required in order, not only to distort the sheet, but also to maintain it in its distorted shape. If, having distorted the sheet, we took the whole setup out of the gravitational field the elasticity of the sheet would “lift” even the most massive object so that the sheet could return to its non-distorted shape; and just to prove that there was a transfer of energy involved, the massive object would continue moving in a straight line until acted upon by some other force. Similarly, spacetime that has been distorted by the presence of a massive object will not maintain that distortion if the massive object moves away. It seems that an energy exchange is needed, even in the gravitational model of GR.
Perhaps there is a way round all this that does not deny the work done by gravity, and might even identify an energy source. We will return to the example of the rock, lifted from the ground and thus endowed with potential energy which allows it to return to the ground with no net expenditure of energy. If we accept that the Earth distorts spacetime, and that it is this distortion that holds the rock on the surface and causes it to return if it is moved away, then we have to explain why increased movement away from the Earth does not make movement of the rock increasingly difficult. One way of tackling this is to accept that the distortion of spacetime caused by the Earth is in fact a distortion caused by the combined mass of the Earth and the rock. Of course, the contribution made by the rock is so small in relation to that of the Earth that it would be practically undetectable. However, as the rock is lifted from the surface, a secondary distortion of spacetime (albeit a minute one) is caused. While this secondary distortion remains part of the major distortion the rock “seeks” to return to the surface in order to restore the single, combined distortion. If the rock breaks free of the distortion caused by the Earth it will be free to travel through space, accompanied by its own mini-distortion of spacetime. If it does not break free, the rock will eventually return to Earth, either by the apparently direct route by which it ascended, or by going into orbit around the Earth and gradually spiralling down, perhaps over many years, until friction with the atmosphere ignites its surface and causes it to plunge as a meteor to Earth. This latter route appears to be very much the less direct, but GR tells us that the rock is actually following a geodesic, which is defined as the shortest distance through curved spacetime between two points.
Here the scientifically minded may be constrained to object that the effects of gravity seem to be unbounded in there range, so nothing actually breaks free of the gravity of a massive body. However, acknowledging that gravity decreases as a square of the distance involved, there must come a point where its effects are negligible.
GR is interpreted as saying that there is no force of gravity? David Deutsch (The Fabric of Reality) argues that there is no gravitational force. “In the nineteenth century, few things would have been regarded more confidently as real than the force of gravity. Not only did it figure in Newton’s then-unrivalled system of laws, but everyone could feel it, all the time, even with their eyes shut – or so they thought. Today we understand gravity through Einstein’s theory rather than Newton’s, and we know that no such force exists. We do not feel it! What we feel is the resistance that prevents us from penetrating the solid ground beneath our feet. Nothing is pulling us downwards. The only reason why we fall downwards when unsupported is the fabric of space and time in which we exist is curved.”
GR tells us that matter and energy, simply by their presence in spacetime cause the distortions that give rise to the phenomenon of gravity. Impressive as this assertion is, it leaves some unanswered questions. What GR tells us is that the presence of a massive body, or indeed a body of any proportions, in spacetime causes spacetime to distort. Should it be possible, somehow, to remove this body, spacetime would resume its original shape. So, not only does the presence of mass distort spacetime, it also holds spacetime in that distorted shape, in spite of the fact that spacetime seems to favour returning, like a rubber sheet, to its original shape. Let us return to the sheet with the bowling ball on it. While the ball sits there in its depression, with the whole system in equilibrium, all is still. The sheet is being held in its distorted shape by the “weight” of the ball. The weight of the ball is simply a measure of its mass in relation to the “gravitational attraction” between it and the mass of the Earth. If the bowling ball is removed from the equation, the sheet resumes its original shape; if the Earth could be removed from the equation, it would have the same effect. We know is that if we lift the bowling ball off the sheet and take it up to a height of a few feet above the sheet, thus increasing its gravitational potential energy, then release it, it will fall onto the sheet causing a temporary depression that will be deeper than the depression it caused when it was at rest. The sheet will then push back hard against the ball, possibly throwing it into the air. What does all this tell us? A few things will be immediately obvious. First, when the ball was sitting at rest in its depression, it was being prevented from falling further towards the centre of the Earth because the distorted sheet was pushing it upwards. Newton tells us that for every force, there is an equal and opposite force, and this is something that relativity has not relegated to the shadows. However, we are invited to believe, that that there is no force acting against the force being exerted by the sheet, that somehow the state of equilibrium is being maintained, in contravention of Newton’s law, by the curvature of spacetime, which behaves as though it were a force, but is not.
We should look more closely at the concept of the “equal and opposite force”. When I pick up a stone from the surface of the Earth, then let go of it, it falls back to the surface. Is it the “equal and opposite force” that causes it to fall? It might be tempting to think so, especially in view of the idea that it is the energy I put into the stone by lifting it that caused it to fall back to Earth, but, in fact the answer has to be “no”. When I pick up the stone, my weight increases; in other words my feet press more firmly against the ground. This must be the “equal and opposite force”, but wait, it does not end here. As my feet exert more force against the Earth, the Earth pushes back more firmly than it did before, so where does it end? It seems that it is not the “equal and opposite force” that is involved in gravitational attraction.
One problem with the rubber sheet and bowling ball analogy is that it is simple to visualise it in two dimensions – which, of course, is why it is used as an illustration – but it is much more difficult to visualise the whole thing in three dimensions, and that is without adding the fourth dimension of time. It is, therefore, easy to think in terms of the sun, for example, as sitting in the bottom of a depression in spacetime, with the various objects comprising the solar system circling the sides of the depression at various heights, and lateral distances, just waiting their time to make the final descent. Of course, this is a serious over-simplification, but it is a trap that is easy to fall into, especially when the alternative is trying to come to grips with the idea of a “pit”, the bottom of which is in the centre, and the sides of which slope outward in every direction. It is a lot easier to think of this as a sphere of attractive “force”, at the centre of which is a lump of matter, which is doing the attracting. However, General relativity tells us that we must think in terms of a sphere of spacetime that is distorted in such a way that the shortest distance between any two points is not a straight line, it is a curved line (unless the two points lie on a radial line), and that all these curved lines trend towards the centre of the sphere. Put another way, this is saying that within the sphere the “downward” direction is towards the centre from every point on, or inside, the sphere. Furthermore, it says that, unless any downward path follows a radial line, that path will be curved
Perhaps the use of the word force when writing about gravity is only a sort of “shorthand” which authors employ in order to avoid convoluted, if strictly more correct, terminology. It is also possible that there is a duality associated with gravity, similar to the wave/particle duality of quantum objects. If so, it must mean that that an essentially quantum feature is manifest in the realms of GR.
The apparent force of gravity is the manifestation of the fact that objects move, naturally, through spacetime along the straightest possible paths, but, because of the curvature of spacetime, these straightest paths do not represent Euclidean straight lines. These lines resemble the straightest lines between any pair of points on the two-dimensional surface of a sphere. These curved straightest lines are geodesics, and it appears that we must now add to our laws of nature the law of geodesic motion. Wolfson (Simply Einstein) is emphatic about this. “This law of geodesic motion ultimately covers everything from falling apples to planets and space shuttles and on to the overall behaviour of the Universe as a whole.”
It appears that the degree of curvature of spacetime is directly related to both the mass and density of the body causing the curvature. For example, a body of the mass and density of the sun will cause relatively gentle curvature over a large area. If this mass were compressed to the size of the Earth, the curvature of spacetime around it would be much more severe. Given a situation in which an enormous mass, such as the total mass of the Universe, is compressed into an unthinkably small “speck”, the curvature would be extreme. This approximates to the state of the Universe at the instant of the Big Bang. Therefore, it follows that every particle of matter and energy in the Universe, at the start of its life – or of this cycle of its life – occupied the same point in spacetime. The energy, whatever its source, that caused this infinitesimal, primordial speck to expand, transforming itself into billions of light years of spacetime, would also have caused the curvature of spacetime to expand and to “soften”, but, it would always remain curved, thus it would always tend to return to its original condition, like the rock falling back to Earth once the restraining force has been removed. This would mean that the energy which drives gravitational attraction is the potential energy imparted to every particle in the Universe by the Big Bang. Thus, there is sufficient potential energy within the Universe to bring every particle back to an infinitesimally small speck, unless some external force intervenes. Every particle distorts spacetime around it to a minute degree. As particles clump together, not only are their masses added together, but so is their power to distort spacetime. What is more, without a continued expenditure of energy to prevent this clumping from taking place, it must continue until all the matter and energy in the Universe has returned to its starting point. This would imply that the real mystery is not where the energy of gravity comes from, or why it seems to be inexhaustible, but rather where the energy comes from that is causing the expansion of the Universe not just to continue, but to accelerate, as modern observations assure us that it is.
