I am a fan of a cartoon strip drawn by Johnny Hart called B.C. It is populated with little cave man characters, one of whom is always trying to catch clams moving from one place to another. He knows they move but they are always lying still when he sees them. His hypothesis is that clams have legs. His chosen mission is to catch one in motion so he can tell the world that 'Clams got legs!!!'.

 

This program deals with relationships among quantities which define a system and its state, where the state of a dynamical system is the position and velocity of all its moving parts. These relationships we have called laws of nature. It was to our benefit to have those laws of nature take the simplest possible form. And the form they take will depend on the reference frame in which those laws are formulated.

The last sentence in the preceding paragraph is one of those which in some textbooks is assumed to be either so obvious or so obscure that farther explanation is either unnecessary or hopeless. I will take a crack at further explanation.

So far the reference frame has just been a convenient bit of space into which to place a coordinate axes system so we could determine the location of things with some precision. We suggested that depending on the degree of freedom of the particles in our system we might choose a space of 1, 2 or 3 dimensions. What we need to do next is to expand on this notion of a reference frame a bit.

Let's look at a reference frame from the point of view of an observer whose universe is the reference frame itself. What must the world look like to inhabitants of a 2 dimensional reference frame. They would be conscious for instance of left and right, and forward and back, but not of up and down. Some things which seem to us 3D dwellers to be pretty ordinary might appear quite mysterious to a person limited to 2 dimensions.

What sort of law of nature might a 2D scientist concoct based on his observation that something appears from nothing, changes size and position and disappears into nothing again, repeating the whole performance periodically. In the Two Dimensional Phenomena display we will see such an object from the perspective of the inhabitant of a 2D reference frame. A dark object appears in our green 2D space, changes size and later disappears, only to repeat the process again and again.

The 3D version of the phenomena is no more mysterious than a swinging clock pendulum. I wonder if some of the mysterious phenomena we observe would appear simpler if we had the benefit of an additional dimension or so. Perhaps we should keep an open mind on this.

OK, if we accept that observers in a reference frame can not observe anything outside their frame of reference, then it is likely that the laws of nature derived by such observers will depend on the frame itself. The example I gave was a fairly trivial and obvious one involving reference frames of different dimension. How about a reference frame that is rotating at a constant angular velocity. Would the laws of nature take the same form in that frame as in a frame that was not rotating. Our instinct says, 'probably not'.

For our purposes it will be helpful to visualize a reference frame as a coordinate system in an otherwise empty space. Recognize that truly empty space does not exist in nature. Even in the region of the universe farthest removed from concentrated matter, the space has in it radiant energy, the occasional stray atom and possibly other stuff which we do not know about yet. We can imagine an empty space however, and for now that is all that is required. Next we may imagine particles, observers or whatever we wish to be in our frame. Furthermore, we may designate some objects to be in more than one frame simultaneously. In fact since we are creating this universe, we can do just about anything we please.

Now let's imagine two reference frames somewhat displaced from one another, and a particle which is observable from both. Observers in the two reference frames would not agree on where the particle was because each would be measuring from her own origin. So physical quantities, like distance are different when measured from different frames. Run the Fixed Reference Frames display to experiment with some examples. We will work in two dimensions because the cursor works best there. Adding a third dimension changes nothing.
 

Now suppose the frames are moving at a uniform velocity relative to one another and there is a particle observable from both frames. The adjective "uniform" applied to velocity means unchanging in either speed or direction. In this situation one observer would claim the velocity was one thing and the other a different value. Here again a physical quantity, the velocity, is different in different frames. Use the Moving Reference Frames display to illustrate.

Let's think now about the nature of the space itself, which fills our reference frames. We have said it is empty except for things which we put into it. How can empty space have any properties? One obvious example is that our reference frame space has the property of dimension. We explicitly choose the dimensions of the space we wish to use. There are two other properties of space that I want to introduce at this point. These are homogeneous and isotropic.

"Homogeneous" means the same throughout. In a homogeneous substance for example any small sample is the same regardless of where the sample is taken. For space to be homogeneous, there must be nothing that physically distinguishes one point from another. On the face of it this seems to be the case for empty space. How about the situation where I insert coordinate axes in the frame, does that make the space nonhomogeneous because now every point is at a different location relative to the origin? No, remember that we are talking here about a property of the space itself.

The presence or absence of the coordinate axes is undetectable because no experiment which we can imagine within a neighborhood around a point in our space which excludes the origin, can detect the location of the origin.

I used the term "neighborhood" above, which will be a useful concept to define. Let's take a neighborhood to be a region of space around a point which we may shrink to as small as necessary to exclude any other point we may choose. This concept is a valid one as long as our space is continuous, meaning that there is no point that does not have another point next to it in all directions. So we have arrived by way of the back door at another property of our reference frame space, that of continuity. Implicit in this discussion of neighborhood is the notion that a "point" has no extent. It is dimension less.

"Isotropic" means the same in all directions. An example of a non-isotropic medium is a flowing stream. Anchor your canoe in mid-stream and notice that it will swing around to point upstream every time. All directions then can not be equivalent, otherwise it would remain pointing in whatever direction you left it. The canoe seems to prefer one direction over all others. Allow me from time to time to attribute preferences and other mental activities to inanimate objects like canoes and particles. I suspect that they do not think anything but it is sometimes useful to imagine that they do, to make it easy to express some ideas.

Again considering empty space, it is obvious that the orientation of objects in it does not matter to the objects. The space in the stationary reference frames we have been discussing is isotropic. What about a reference frame moving with a uniform velocity? Might a frame in motion exhibit a preferred direction relative to the direction of motion? This question forces us to think about what we mean by motion.

Now get a firm grip on something. This is going to be a turbulent ride. There is no uniform motion of an isolated object, where isolated means that there is no other object in the neighborhood. What we call motion is really a change in position as time passes. But position is only defined relative to something, like an origin or another object. So to say a reference frame, which is an object, is in uniform motion is nonsense. It can only be in motion relative to something else. Since there is no way for a point in a reference frame to know then whether or not the frame is in uniform motion, the space in the neighborhood of that point must be as homogeneous and isotropic as it was when we said the frame was at rest.

The logic of the preceding paragraph implies that there is no absolute reference frame to which all others may be compared. This concept is known as Galileo's relativity principle. It got him into a lot of trouble since the leading theologians of his time had not thought of it first. We got into this discussion in order to support our instinct that the laws of nature might be different in some reference frames than in others. What we have done is to discover a whole set of reference frames in which the laws of nature must be the same, namely frames in which space is homogeneous and isotropic. Any reference frame for which this condition holds true is called an "inertial" frame of reference. In another course we may begin wondering if time is homogeneous. For now let us assume that it is since we are not getting into Einsteinian relativity.

So why exactly must the laws of nature be the same in all inertial reference frames? If the laws of nature were different in any two inertial frames, then an experiment whose outcome depends on those laws would yield different results in the two frames, making them distinct. Since the only element which is integral to the frame is the space which fills it, any distinction between inertial frames must depend on differences in the space itself. From the fact that the space in inertial frames is homogeneous and isotropic, we know that it is essentially featureless such that one inertial frame must be internally indistinguishable from another. Therefore the laws of nature in inertial reference frames are all the same.

For us to analyze dynamical systems, we need to pick a standard reference frame, so as to get consistent results. The inertial frame is the logical choice since homogeneous and isotropic space is clearly the least complicated spatial structure we will encounter. If we consider an arbitrary reference frame in which space is non-homogeneous and non-isotropic, even if an object interacted with no other objects, its various positions and orientations in space would not be dynamically equivalent. Such properties of space would obviously complicate the description of dynamical phenomena.

So the laws of nature in inertial reference frames take their simplest form. Without saying so we have been using inertial reference frames throughout this program so far.

I told you earlier to keep a firm grip as we went through this material. There is some deep stuff in here. Let me repeat the arguments with a slightly different twist to help you solidify your understanding.

Let us examine some of the consequences of space being homogeneous and isotropic. Consider an object in an inertial frame of reference, not moving relative to the origin. Since every point in the space is the same, the object in a sense can't tell whether it is in one place or another so it just stays where it is. Now suppose that the object initially was moving at a constant speed in some direction, again since all points are the same, the object could not detect its motion through space so it just continues doing what it was doing.

What about the situation where the reference frame is attached to a laboratory on the surface of the Earth, is such a reference frame inertial? Let's do a little thought experiment. Hold a ball 1 meter off the floor and drop it. Does the ball remain at rest in the laboratory frame of reference? I don't think so. So evidently the laboratory frame of reference is not inertial. Does this mean then that the space in the laboratory is not homogeneous? Right you are.

If we had done the ball drop experiment in a drifting starship in deep space far from any matter would the ball have dropped? We know from our reading of science fiction that it would not. So we have discovered that space in the vicinity of a large mass like the Earth is not homogeneous. Somehow the space in reference frames fixed in the presence of matter is distorted so that the points in that space are not all the same, and that lack of sameness causes objects in that space to seek new positions. When an object changes positions or moves, our intuition tells us that it has been subject to some force so we invent a force and give it a name, gravity.

Getting back to the inertial frame of reference. In that case we have concluded that a body at rest remains at rest, a body in uniform motion remains in that motion and any reference frame which moves with constant velocity relative to an inertial frame is also an inertial reference frame. "Whoa! What was that last point?", you say. OK, that was a little fast and loose but you will have to forgive me. The only exercise I get is jumping to conclusions. Let's go back and see where that conclusion came from.

Since space in an inertial frame of reference is homogeneous and isotropic, then any object moving at a constant velocity, that is with fixed speed and direction, will pass through a succession of points that are all identical. If the points are all identical then no experiment we can imagine would allow us to detect the difference between where we are and where we were. Therefore any fixed velocity is equivalent to zero velocity. So if a second reference frame spans the same space as the inertial frame, even if it has a non-zero constant velocity relative to the inertial frame, space as observed from the second frame will be homogeneous and isotropic as well. Otherwise the motion of the second frame would be detectable from within that frame.

In thinking about inertial reference frames, we seem to have arrived at Newton's first law. A body at rest remains at rest, one in uniform motion continues in that motion, in the absence some outside force.

At this point I recommend that you shut down this program temporarily and take a break. I don't know about you but this stuff makes my head spin after a while. It is worth the effort though because a solid understanding of reference frames and the nature of space make the laws of nature somewhat less mysterious. If you insist, you may go on by clicking on Next below.

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