Page last updated Sep 27, 2017|
(addition of a link to What Is The Gravitational Constant, G?)
Natural Philosophy and Physics
F = m a
Philosophy is "the rational investigation of the truths and principles of being, knowledge, or conduct" (Webster's Unabridged Dictionary). As such, it covers a wide range of topics, some of which are so esoteric as to be incomprehensible or irrelevant to the average person (e.g., the question of whether or not the observation of things verifies their actual existence and properties, or merely some imaginary construct which leads to their apparent existence and properties). One subset of this range of topics is natural philosophy, which is concerned with the nature of the physical world, and physics, the application of natural philosophy to those parts of the physical world which can be explained by theories which are consistent within themselves, and with each other, and can be tested by experimental means. Note that if a theory cannot be tested by any means, it may be a perfectly acceptable theory philosophically, but it is not a proper theory of physics. It is this sort of distinction that is sometimes used to try to separate religion, which is based on faith, rather than experimental proof, and science, which is presumably based on experimental proof, and not faith. In this context it should be noted that the term "theory" means different things to different people. In the ordinary world, saying that something is "only a theory" implies that it is mere speculation, worthy of no serious interest or consideration. In physics, a theory is an exact, specific statement about the way that things work, which is in some way testable by the application of physical thought and mathematics, and experiment; and most physical theories are so well grounded in experiment and careful comparison with other physical considerations that they are, within the experimental error, as accurate a statement of how things work as any other statement that we can make.
This does not, unfortunately, mean that all theories are correct, and this discussion is centered on three theories of motion in general and gravity in particular that have occupied Western thought during the last few millennia. Each of these theories explain the basic experimental observations, but they look at the nature of motion, gravity, and even the space and time we live in, in very different ways.
What Is Gravity?
Gravity is the easily observable phenomenon that things fall, and have weight. If you pick up something, you perceive the "gravitational" effect on it as the weight that presses downward against your effort to lift it. If you let go of it, you perceive the effect as the nearly constant downward acceleration that it experiences, causing it to fall, faster and faster, until it hits the ground or the air resistance caused by its speed prevents its going any faster. Note that gravity is always downward, and in fact is used to define "down" and the opposite direction, "up". Why things have weight and fall is not known; it is just a fact of nature, which we describe with mathematical statements, and explain with one physical world-view (that is, one theory) or another. At the present time, most people use a theory proposed by Newton, which states (as discussed in more detail below) that gravity is a force exerted by objects on each other according to their mass and distance from each other. In almost every circumstance known, Newton's Law of Universal Gravitation perfectly explains the gravitational effects and motions we observe; but there are rare circumstances in which his law is not correct, and according to Einsteinian Relativity Theory, gravity is not a force at all, but a condition of space-time (a rather odd amalgam of space and time, which varies according to the motion of the observer), which merely mimics a force. The exact mathematics and physics of Einsteinian gravity are beyond the comprehension of most people (not because they are incapable of understanding the theory, but because they don't have the required background in mathematics and physics), but Newton's description is perfectly adequate for all ordinary, every-day purposes. Only certain extreme situations, such as a complete understanding of black holes, or the structure of the Universe, require the use of Einstein's relativity theories, and even there the basic concepts can be described reasonably well by the use of various analogies.
Hellenic (Ancient Greek) Motion and Gravity
Before going into what is right or wrong with Newtonian physics, let's take a look at its predecessor in Western thought, the Hellenic concepts of motion and gravity which were "common sense" knowledge, in the early Renaissance.
The basic idea in Hellenic concepts of motion is that everything has a "natural" place, which it will strive to reach, if not in that place, and that if it has reached that place, or is as close to it as is possible, it has a "natural" motion, which is to be at rest. Thus, we observe that rocks which are lying on the ground "prefer" to remain there, as they obviously belong on the ground, being a part of the Earth, which is made of the same stuff they are, and are lying on. But if you pick up a rock, it will resist your effort to remove it from its natural place, and try to return to the ground. We perceive the rock's effort as its weight, and if we let go of it, we see the successful effort it makes, as it falls to its former position. In other words, in this view, weight is the desire of an object to return to the ground, and falling is the natural result of its being allowed to do so.
Of course, not all objects are rocks, or behave like them. Liquids such as water behave in a fairly similar manner, running downhill until they are at as low a place as possible; but once there, they may jiggle back and forth (e.g., wave motions). And air and fire float around, and rise upward. In Hellenic times, these different behaviors were explained by presuming that different kinds of matter had different properties -- and in particular, different natural places and motions -- according to the mixture of elements that they were composed of. The elements, as described in this view, are not the same as the elements that we are familiar with today -- oxygen, carbon, nitrogen, gold and so on -- which differ in terms of their atomic structure; but are based strictly on their natural places and motion, things that can be observed on a large scale, without the aid of modern optical and physical equipment.
The best-known grouping of materials into elements, partly by accident in terms of what ancient texts have survived to this day, and partly by general agreement in the days following its proposal, was set forth by Aristotle, and is the basis of the four materials mentioned above -- air, earth, fire and water. Other elemental groupings were proposed by different philosophers, but Aristotle's views are the ones that survived into, and were generally accepted during, the Renaissance.
It might be noted that this description of motion and gravity does not apply to the heavens, whose motions are totally different. All heavenly bodies move across the sky in some combination of uniform circular motions: three for the stars (the daily westward motion around the axis of the sky's rotation, a long-term precession of the Equinoxes which turns out to be caused by a change in the direction of that axis, and a shorter-term nutation, or "nodding" of the axis, relative to its average long-term change), and additional motions for the Sun, moon, and planets. Since these objects repeat their motions, over and over, they can't be getting any closer to their natural places, so to speak, and if they were like the materials described by Aristotle, they would cease moving, since they aren't accomplishing anything by moving. So we presume that they are already in their natural place (the heavens), but unlike the four elemental materials of our experience, their natural motion is not to be at rest, but continually in motion, in some combination of uniform circular motions. This implies that the heavens are made of some material different from the four elements of earthly existence -- a fifth element, or "quintessence" (Latin for fifth element), sometimes referred to as the "ether" (ethereal hence coming to mean heavenly, among other things).
Summary of Hellenic theories of motion and gravity: All things have a natural place. Things made of some combination of the four elements which make up earthly materials seek their natural place when moved from it, but remain at rest if already in it. Things in the heavens, made on a fifth element, are already in their natural place, but have a natural motion consisting of some combination of uniform circular motions. Gravity consists of the natural effort made by things which belong near the Earth, to return to it. If they are not prevented from doing so, they will fall down; but if they are prevented from doing so, they will exert a downward force on whoever or whatever is preventing them from falling, which we perceive as their weight.
Newtonian Motion and Gravity
Newton's description of motion and gravity is completely different from the Hellenic concepts. As a result of Kepler's supposition that the Earth, like the other planets, moves around the Sun, Galileo's observations of the heavens, and Galileo's experiments (real and thought experiments) involving motion, certain concepts had become common sense by Newton's time, and were summarized by him in his first two laws of motion:
The Law of Inertia (Newton's First Law of Motion): An object which has no force acting on it will maintain a constant motion. If it is at rest, it will remain at rest. If it is in motion, it will continue in motion, with constant direction and speed.
The Force Law (Newton's Second Law of Motion): An object which has a force acting on it will be accelerated in the direction of the force, by an amount which depends upon the amount of the force, and the inertia of the object, according to the formula
where F is the force acting on the object, a is the acceleration or rate of change of motion of the object resulting from that force, and m is the mass, or inertia of the object.
F = G m M / r2
It should be noted that the concept of inertia occurs in Hellenic thought, as well, even though not mentioned above; but in Hellenic thought, it represented a different concept -- namely, the resistance of an object to any effort to change its position. In Newtonian physics, inertia is the resistance of an object to any effort to change its motion, not its position. Of course, if an object is at rest, changing its position also requires changing its motion, so in that case, the concept of inertia is very similar in both views; but in Newton's physics, it is the change of motion, or acceleration, that requires a force, and a very slow change of position might not require much force, whereas in Hellenic physics, any change of position should require a more or less comparable force.
Newton also enunciated a third law, the Law of Action and Reaction, which states that any action (which in his day meant a force) results in an equal and opposite reaction. In other words, if you push on something, the force that you use on the object automatically (the action) generates a resisting force (the reaction) which is equal in magnitude, but opposite in direction, and in terms of the object acted upon. Your action pushes another object one way, and its reaction pushes you equally hard, in the opposite direction (hence the action and reaction act in opposite directions, on different objects). Note that the effect of the two forces is not necessarily the same, but depends upon the masses, or inertias, of the object. The Earth and Moon are always pulling on each other, with exactly equal and opposite forces (the Third Law doesn't say anything about the nature of the action and reaction, so it is irrelevant to this discussion just what the forces are, but it happens that in this case, they are the gravitational force that is the main topic of this discussion). But although the forces are equal, their effects are not, because the Earth resists a change in motion 81.6 times more than the Moon (that is, it has 81.6 times more inertia, or mass, than the Moon). Hence, looking at the Second Law of Motion, with equal forces, but different masses, the Earth will be accelerated (its motion will change) 81.6 times less than the Moon, causing any motion produced by that acceleration to be 81.6 times smaller than the corresponding motion produced by the acceleration of the Moon (see Gravitational Interactions of the Earth and Moon).
Now, given Newton's way of looking at things, how would he explain gravity? You pick something up, and feel a downward force on you (a reaction to your action), equal to the force you had to exert to pick it up. If you let go of it, it falls to the Earth. Why should it fall? The Law of Inertia says that when you let go of it, it should remain in constant motion, if there is no force on it; and you certainly aren't exerting a force on it, anymore. So why does it fall? Because the Earth, in some way, creates a force which pulls the object downward, which we called the force of gravity, and express mathematically by Newton's Law of Universal Gravitation as
where F and m are as before, M is the mass of the Earth, r is the average distance between the two masses, and G (the gravitational constant, is the force of gravity exerted on a mass of one unit by another mass of one unit at a distance of one unit (units typically being either in Newtons, kilograms and meters, or in dynes, grams and centimeters). This force, acting on the object, gives it a downward acceleration, called the acceleration of gravity, which is equal to
g = F / m = (G m M / r2) / m = G M / r2 = a constant, at the surface of the Earth
Summary of Newtonian theories of motion and gravity: Objects do not have a natural place, but a natural motion, which is to continue moving as they are, unless some force acts on them. If a force acts on them, they are accelerated in the direction of the force, at a rate proportional to the force, and inversely proportional to their mass, or inertia. The weight that we experience when we pick something up is a downward force, exerted by the Earth on the object. As long as we oppose that force, the object can remain suspended and motionless; but if we remove our opposition to the force, it will fall downward, with an acceleration equal to the acceleration of gravity.
Contrast between Hellenic and Newtonian theories of gravity: In Hellenic natural philosophy, the weight of an object is its desire to return to the ground; in Newton's physics, the weight of an object is a force which the Earth exerts on the object, compelling it toward the ground. In Hellenic natural philosophy, things fall because they want to return to their natural place, the ground; in Newton's physics, things fall because the force of gravity compels them to do so.
Einsteinian Motion (Relativity Theory) and Gravity
As noted above, Newton's physics is the common-sensical approach that almost everyone takes in viewing gravity, even today. Things have weight, caused by the Earth's gravitational force on them, and thus they fall, when that force is not opposed. But quite remarkably, nearly a century ago, Albert Einstein showed that this is completely wrong; that gravity is not a force, but a curvature of space-time (admittedly caused by the Earth, in our case) which makes the amount of space between another object and the Earth less in the future, than it is in the present, unless some force opposes the natural tendency of things to move through space along a geodesic -- the straightest path possible in that curved space-time, given the velocity of the object -- and the apparent weight of an object is a reaction to the force (or action) which prevents it from doing that. In other words, when you see something fall, it does so not because the Earth exerts a downward force on it, but because the Earth warps the fabric of space-time in such a way that in the future, the object is closer to the Earth. And when you pick up something, and feel its "weight", you are not feeling the Earth pulling it down, but a reaction to the upward force you are exerting, which is keeping it from falling, the way it is supposed to.
Needless to say, this is a very strange way of looking at things, and it wouldn't hurt to explain how Einstein came to this conclusion, and why we believe him, strange though the results may appear to be.
The critical thing to realize, in thinking about gravity, is that it is a very strange force. Most forces, operating according to Newton's Second Law, produce an acceleration a which is equal to F / m, or the ratio of the force to the object's inertia. Now suppose that you were on a movie lot, in a rocky hilly area, and a scene was to be shot in which the hero was tumbling down a hillside, with large boulders tumbling past him. Obviously, if real boulders were used, our hero might be seriously squashed; so fake boulders are used, which are made to look like the real thing, but have a much lower mass. At the top of the hill, there is a large collection of boulders, real and fake, set up in preparation for the "shoot". Our hero leans against a real boulder, which has a large mass, strongly resists the force he exerts against it, and remains resolutely in place. He is then pushed off the hillside by the villain, who pushes a number of (hopefully fake) boulders after him. It is easy for the villain to do this, because the fake boulders have hardly any mass, and are easy to move (mass being a resistance to a change in motion); but let's suppose that a couple of real boulders are accidentally dislodged, as well (off to the side, so that our hero isn't all mussed up, anyway). How does gravity know, as it pulls on the fake boulders, which take very little effort to accelerate, that it shouldn't pull on them very hard, and as it pulls on the real boulders, which are hard to accelerate, that it must pull on them very hard, so that even though it is much harder to move the real boulders, they fall in exactly the same way as the fake ones. In other words, how does gravity manage to always make everything fall at exactly the same rate, regardless of how massive (or not) it is, and regardless of what it is made of? For not only fake and real boulders, but also wood, oil, water, gold, light beams and (if they existed) fairy dust and horsefeathers all fall, under the influence of gravity, in exactly the same way (presuming that we can ignore the effects of air resistance). Of course, the examples of fairy dust and horsefeathers are inserted for humor, such things presumably not existing; but light beams, even though they have no inertia (resistance to a change in motion) at all -- that is, even though they have no mass -- fall under the influence of gravity exactly the same as any other object. This may be hard to believe, but if you shine a beam of light across a room (in the case of my classes, fifty feet from one side of the room to the other), during the time the beam would take to cross the room (in the case cited, about five thousandths of a millionth of a second) it would fall exactly the same amount as any other object would, during that short time.
So, accepting this rather hard to believe fact -- that everything falls the same, in a given place, as a result of gravity -- why does it work out that way? No other force in nature works in this way. All other forces act more strongly on some things, and less strongly on others. Is it really possible that gravity doesn't work this way? And if it does, how can we explain it?
The first possibility, that different things are acted on differently by gravity, has been carefully tested, over the century or so starting some time before Einstein's theories, and since then. And within the limit of experimental error (which the last I heard, involved about 24 digit accuracy), all things do indeed react to gravity in exactly the same way. But if that's the case, then how can we explain this strange result?
What Einstein proposed, is that the presence of gravity is like being in a rocket ship, which is accelerating upward at one "gee" (the acceleration of gravity), and the absence of gravity is like being in an elevator whose cables have been cut. In the first case, anything not attached to the ship will appear to fall, as the ship rises, with a motion which is a mirror image of the upward acceleration of the ship. Since the ship has the same motion, regardless of which object we are looking at, the mirror image of its motion is the same for every object. In other words, if the Earth's surface were accelerating upwards, everywhere, at one "gee", we could observe what we do. That isn't possible, of course, as it would make the Earth bigger and bigger; but that doesn't mean that we can't get the same effect, in a different way.
The other example, being in an elevator which is falling, because its cables have been cut (or jumping off a roof, and observing your resultant motion), is exactly analogous to what happens to an astronaut in orbit. Anything in the falling elevator, you (jumping off the roof), or the astronaut (in orbit around the Earth) would appear to be "weightless". That is, you wouldn't feel any gravitational force acting on you. And in fact, according to Einstein, that would be absolutely true, because there isn't any force acting on you. You are falling toward the Earth because it is natural to do so, according to a revised Law of Inertia: Things which have no force acting on them have constant motion along their space-time geodesic. In the presence of a large mass, like the Earth, that geodesic is curved so as to make things closer together in the future; so things get closer together, or fall, but without any force to make them do so. A good analogy to this would be to imagine people moving northward along parallel lines, such as the meridians of longitude which cross any given parallel of latitude. Each of them is moving exactly parallel to the others (perpendicular to their common parallel of latitude), so they should never get any closer together; and on a flat Earth, where parallel lines never meet, they wouldn't get any closer together; but on the real Earth, where all the meridians of longitude come together at the Poles, the individuals would find that they are gradually getting closer together, even though they never move toward each other, at all. Of course, on the real world, we can measure the shape of the surface, and "see" that the reason for this odd situation is the curvature of the surface; but in space-time, you can't see or measure empty space or time, so your only perception is that you get closer and closer together; and in the absence of a true understanding of the situation, you ascribe your movement together as being due to a gravitational force.
Now, it is beyond the scope of an introductory astronomy course, even one which considers, however briefly, such topics, to discuss how Einstein arrived at his conclusions in any more detail than this; and as already mentioned, in most cases, it is not necessary to use Einstein's rather odd ideas of space-time and gravity, at all. Newton's concepts of motion and force are perfectly adequate, under all normal circumstances. But there are places where this is not true. As we move closer to the Sun and its gravitational influence, the curvature of space-time produces slightly different motions than Newtonian physics predicts; such differences were noted in the 1800's, in the motion of the perihelion of Mercury, and in 1918, in the bending of light rays which pass very close to the Sun, and the differences are exactly as predicted by Einstein's physics, even taking into account the much more accurate measurements of these phenomena possible with current technology. Also, when we discuss places where gravity is extremely strong, such as the "surfaces" of dead stars, such as neutron stars and "black holes", Newton's physics does not give as accurate results as Einstein's theories of special and general relativity. And when we discuss the physics of the Universe as a whole, we run into an even stranger situation, for it turns out that Einstein's physics predicts not one, but three different ways in which space-time can bend, and only one of them corresponds to gravity, as we know it.
When there is a large amount of mass in a given region, Einstein's theory works as just discussed: the mass curves space-time in a way that makes things closer together in the future than you would expect, if they followed forceless Newtonian paths. But just as mass (or energy, which is equivalent in Einstein's physics), or "being", curves space-time in one way, empty space, or "not-being", curves space-time in the opposite way. Why empty space should have any effect on the curvature of space-time is beyond any normal description, but it does; and when there are large amounts of empty space between the masses in a region, the tendency of the empty space to curve space-time in the opposite way from the way that the mass wants to curve it can produce a cancellation of the two curvatures, causing the space-time to be "flat", instead of curved. In such a space, things moving in parallel paths would move (unless they got very close to masses) in more or less straight lines, never getting any closer or further apart, more or less as we would have expected in the first place. And in fact, the two effects -- an apparent gravitational effect, making things curve toward each other in the vicinity of large masses, and no apparent gravitational effect, allowing things to move in straight-line paths far from large masses, is exactly what we would expect in Newton's physics, given the inverse-square nature of the Law of Gravity. Namely, close to a massive object, where r is small, the forces are large, and paths curve noticeably; but far from any massive object, things would have essentially zero force acting on them, and move in essentially straight-line paths.
But as mentioned, there is a third possibility in Einstein's physics, which does not occur in Newton's physics (or in Hellenic thought); namely, that if you have enough empty space (tens or hundreds of millions of light years of essentially empty space), all that empty space will cause a curvature of space-time which causes things to become further apart, in the future -- in a sense, the empty space creates more and more of itself, so that the distances between things grow and grow, without any end. If some object created this effect, we would be tempted to call it anti-gravity; but it is not produced by any object, but by the absence of any significant mass, in a vast region of otherwise empty space; and it's hard to call it anti-gravity, if nothing is creating the effect (or perhaps we should say, the nothingness of empty space is creating the effect).
Einstein himself was puzzled by this result, when he was developing his theory of general relativity, because it seemed to imply that (a) if the Universe had a lot more mass than empty space, space-time should curve so that everything would be closer together in the future, than otherwise expected, or (b) if the Universe had a lot more empty space than mass, space-time should curve so that everything would be further apart in the future, than otherwise expected. In the former case, depending upon how extreme the curvature was, the Universe could collapse to a point of zero size (analogous to everyone reaching the North Pole, after traveling all the way along their meridians of longitude, at the same time). In the latter case, the Universe might expand into infinity, leaving any given place infinitely far from every other place. And at the time he came to this conclusion, neither of these results made any sense, because the Universe as we currently conceive it -- a nearly infinite space filled with tens or hundreds of billions of galaxies, each containing hundreds of billions or trillions of stars -- was not known to exist. All that we knew about was inside a single system, which we now call our Galaxy, or the Milky Way Galaxy, which happens to be a perfectly stable system of stars, orbiting in one way or another under their mutual gravitational attraction, without any reason to expect any significant change in size at any time in the past, present or future. Of course, there is the third possibility, not mentioned in this paragraph, that the Universe happens to have a nearly equal mixture of matter and space, so that things can have stable, "straight-line" paths; but the chances of that happening are so close to zero, that Einstein didn't consider it significant. Instead, he presumed that he was wrong, and the third kind of space-time curvature couldn't occur. When, only a decade or so later, Edwin Hubble showed that the Universe is indeed expanding, Einstein declared his failure to make the prediction that empty space could expand the greatest mistake of his life; and is it turns out, not only does empty space expand, but as it does so, there is more and more empty space, so it expands faster and faster (any given part of the empty space expands at the same rate as before and afterward, but with more "pieces" of empty space, the overall expansion is faster).
But for now, we ignore such considerations, as they belong not in a discussion of the basic nature of motion and gravity, but in a discussion of the Expansion of the Universe. Instead, we summarize the three ways of looking at things, one last time, in case this (undoubtedly overlong) discussion has led you to forget them:
Summary of the three theories of gravity presented here: In Hellenic theory, things fall because they want to; in Newtonian physics, they fall because the Earth creates a force of gravity which pulls them downward; and in Einstein's relativity theory, they fall because the Earth curves the fabric of space-time so that in the future, things are closer together than now. Similarly, in Hellenic theory, the weight of an object is a measure of its desire to return to the Earth; in Newtonian physics, the weight is the downward force of the Earth's gravity; and in Einstein's relativity theory, the weight is a reaction force created by our action, in preventing the object from falling, the way it was supposed to, in the first place.
So when we have three theories of gravity, like this, how are we to choose between them? Why do we prefer one over the others? Because careful experimentation says we have to. The Hellenic ideas do not work, if we realize that the Earth is not stationary, as once thought, but moves around the Sun, just like the other planets, according to the Laws of Planetary Motion discovered by Kepler. Under these circumstances, the basic concepts of motion involved in Hellenic physics are blown to bits, and theories of gravity based on them must be discarded. Similarly, Newton's physics works perfectly well under most circumstances, but in some circumstances, it makes incorrect predictions about the nature of motion, while Einstein's theory of relativity provides a consistent, logical (albeit very strange) way of looking at things which makes predictions about the motions of things that are absolutely accurate, to the limit of current experimental technology. Fortunately, the circumstances in which Einstein's predictions are better than Newton's are so far removed from ordinary experience that we can normally ignore the truth, and pretend that things fall because the Earth exerts a force on them; but that is not true. Things fall because the Earth curves the fabric of space-time, and moving without any force acting on them, they follow what we perceive as curved paths, even though, following the Law of Inertia, they are moving in the straightest possible paths through that space-time fabric.
And of course, there is always the possibility that some day, we will find deviations between the predictions of Einstein's theories, and the observed motions of things. There are, in fact, a number of experiments that have been proposed which may yield such deviations, and if so, and if we cannot find a way to explain them using Einstein's physics, we may be forced to abandon that, and accept some still stranger theory of gravity, space and time; but to date, every experiment ever carried out has confirmed the accuracy of Einstein's physics, and it is very unlikely that any reason will be found to abandon it, any time soon.
Still, a century ago, physicists were quite confident that very little remained to be discovered in physics, and since then, relativity theory and quantum mechanics have completely changed our concepts of reality at the very largest and very smallest scales; so who can say that any day now, new experiments won't overturn our current theories of gravity and motion?
Related topics, to be added sometime soon...
Perturbations and Orbital Resonance -- Neptune and Pluto
Tidal Friction and Rotational Resonance -- Mercury and the Moon