The Nature of Space - Continued

twistedIn the last section we defined an inertial reference frame, or more precisely an infinite number of reference frames which were all moving with uniform velocity relative to each other. (Remember 0*ihat + 0*jhat + 0*khat is a perfectly valid relative velocity.) In every member of this set of drifting frames space is both homogeneous and isotropic so they are all inertial frames. There is no experiment which might be conducted in one of those frames to distinguish it from any of the others.

Also in the last section I made the bold assertion that the space in the vicinity of the Earth must not be homogeneous, otherwise an object dropped in a laboratory would just stay where it was released and not fall to the floor. Suppose for the moment that we believe that it is something about space itself which causes the velocity of objects in the space to change. What property of space is it, that is different at different locations, making the space inhomogeneous. Evidently it is not a property that is directly observable to us. Empty space appears to us to be totally featureless.

Let's go back to a one dimensional space for a minute. So far we have made the assumption that a one dimensional space must occupy only one dimension. A sharper definition of a one dimensional space is that it is a space in which every point may be located using only one coordinate. I said somewhere along the way that the real number line was a one dimensional space and so it is but it is not necessary that all one dimensional spaces be straight lines. If we had a curved one dimensional space, one coordinate would still be enough to identify any point in the space. We would just measure along the curve so much distance one way or the other from whatever point we chose to be the origin of our one dimensional reference frame.

curved 1d spaceTo inhabitants of a one dimensional space, the curvature of that space would be undetectable. Every point in their universe can be uniquely specified by a single number. The concept that a second dimension might exist would be completely outside their everyday experience. Some residents of this dimensionally challenged space might speculate that such an extra dimension existed but others would vigorously deny it, citing the evidence of their senses. A straight line in such a curved space would be a line lying along the curve. The only 'real' distance between two points would be the 'shortest' distance. Run the Curved 1D Space display for an illustration.

Notice that when we introduce curvature to the one dimensional space, although it still looks one dimensional from the inside, from outside the space, where we are, the space requires at least two dimensions to display it. Why 'at least'? What about the case where the one dimensional space is shaped like a slinky, you know, that floppy, spiral spring looking thing we played with as kids. Such a space would be one dimensional but occupy three dimensions. For now let's take the simplest view, that the space is curved in only one non-distance dimension.

Once we expand our thinking to include bent spaces, we have a property of space which might be different at different points in the universe. The extent into this non-distance dimension might be different at different locations. Also the amount of curvature or the departure from flat could be more at some places and less at others. Another way to think of the departure from flat is to consider it the rate of change of the non-distance dimension with respect to position. The rate of change of distance with respect to distance is the same everywhere. The rate of change of this other dimension with respect to distance can vary. This allows the possibility that our space might be inhomogeneous and still appear from inside the space to be all the same.

effect of matter on 1D spaceWe already suspect that matter interacts with space such that other matter in the neighborhood is affected, otherwise our satellites would just fly off in whatever direction their initial velocity vector pointed. Suppose now that it is the presence of matter in our space which causes it to be curved. The more matter the more curvature. Run the Matter Effect,1D Space display for an illustration of what this might look like in a one dimensional space.

If space is stretched into some non-length dimension by the presence of matter, what sort of measure might apply to this odd dimension. Suppose for the moment that the metric applicable to this dimension was minus Joules per KG. Then the dimples shown in the Matter Effect on Space display would be potential energy holes whose depth depended on the amount of mass at the bottom of the hole.

effect of curved 1D space on matterNow let's look at the other side of the interaction between space and matter. Suppose we place a test mass into a space which is contorted as you might have seen in the last display. What effect would the bent space have on the test mass. A test mass by the way is one which is small enough that its own effect on the space can be neglected. We will assume in the 1D Space Effect, Matter display that some fixed masses distorted the space and that they are not going to move. Then what we observe will be the reaction of the test mass to that fixed spatial configuration.

Now let's extend our idea of bending space into an additional dimension to a two dimensional space. Imagine that mass in this space stretches the fabric of space itself into a third dimension. In the next display you are presented with a view down into the mouth of a trumpet bell shaped piece of space. A large mass is at the origin of the (x,y) plane and has stretched the (x,y) plane into the third dimension so that a potential energy well is created, down the negative z axis.

space-matter interaction in 2DThe circles (or ellipses)in gray represent contour lines of equal potential energy. The radial lines are added to help convey the notion that this is a transparent surface we are looking at. The large central mass is displayed in blue at the bottom of the potential well so it does not interfer with viewing the orbits. Before you click the action button, you might want to use the pitch angle control to look at this surface from different perspectives. When you do click on Action you will see two views of the satellite orbit. One from the perspective of the 2D space dwellers, the heavy green orbit. They of course are unaware of the existence of a third dimension so all motion seems to them to be in the (x,y) plane. The other view is the actual orbit of the satellite in 3 space. It is displayed by the thin green line. As you will see, it not only goes around the z axis. It also dips down into the well of the stretched space. Again conservation of energy demands that the satellite go faster over those parts of the orbit where the potential energy is less. Run the Space - Matter Interaction, 2D display.

O.K. You can see where this is going, now how about a three dimensional space, one with three distance dimensions. Nothing we have said so far prevents us from imagining that our space may be stretched into an additional, non-distance dimension. The main difficulty we have with that idea is that we can not visualize the situation. We have the same problem that I suggested might bother the inhabitants of lesser dimensional spaces. You will hear vigorous arguments that the whole notion is absurd, and those arguments may be right. But the idea does give us a way of organizing our thinking. As a by product of this idea, gravity now is no longer a mysterious force. It is just the normal F = minus the rate of change of potential energy with respect to position.

bent space
Now run the Space - Matter Interaction, 2D display again and watch the satellite go around for a while. From our vantage point we would say that the space in which the satellite is moving is definitely not homogeneous. Clearly there is a potential energy "gradient" at work here, bending the trajectory of the satellite into a closed orbit. A gradient is the rate of change of a quantity with respect to position. Remember that the rate of change of potential energy with respect to position was the force on an object.

Now imagine yourself in a laboratory inside the satellite. What would you conclude about the nature of the space in the lab based on any experiments which were carried out entirely without reference to anything outside? You would discover that in the absence of external forces, a body at rest in the lab remains at rest. And a body in uniform motion continues in that motion until something interacts with it. This is the weightless condition about which you have heard.

So our internal experiments convince us that the space in the lab is homogeneous and any reference frame in uniform motion relative to the lab is an inertial frame. Here we have a predicament. The same space is both homogeneous and not homogeneous, depending on where you stand to look at it. This is the sort of ambiguity up with which we can not put. To paraphrase a once prominent politician.

Let's think about it this way. When we observe the satellite from outside, from the surface of the planet for example, we are using its motion to infer the presence of a potential energy gradient. It goes something like this. I measure the position of the satellite at several times, and take the rate of change of that position to be its velocity. Then take the rate of change of velocity to be its acceleration. Acceleration implies a force and a force implies a potential energy gradient. Therefore the space is not homogeneous. The whole argument hinges on a measurement of position.

Measurement only works relative to something else so how do know that our measurement of satellite position was made relative to a proper reference. If we take measurements of position from non-inertial reference frames, might we not expect to get some bogus results. In fact if the reference frame from which we take measurements was non-inertial in exactly the right way the error introduced might be just enough to make curved space look flat.

microgravityWhat would the conditions have to be in the observer's reference frame in order that the curvature of space caused by a nearby planet be exactly hidden by the effect of bad measurement? If the reference frame was accelerated precisely the same as the acceleration due to the curvature of space, the measurement of acceleration would be in error by exactly enough to give a null result. That is what we get when we take measurements from a frame attached to a body in free fall around a planet. To the inhabitants of this space ship, the space they find themselves in is homogeneous and isotropic. Every point appears to be the same as every other, hence there is no tendency to move from one point to another.

Now suppose we have a reference frame which is accelerated uniformly with respect to an inertial reference frame. Let's make this a space ship far from any big pieces of matter which might bend the nearby space. The space would be homogeneous in the absence of any such matter. What would we conclude from any experiments we carried out on the ship to detect the presence of a potential energy gradient? The same bogus measurements from an accelerated reference frame which allowed us to conclude that space was flat where it was curved will now allow us to conclude that space is curved where it is indeed flat.

The force experienced by objects in an accelerated reference frame can not be distinguished from the force experienced by objects in an unaccelerated reference frame in the presence of a potential energy gradient. It was this line of reasoning which led Einstein to his general relativity theory. The classic illustration is the ride in an elevator. When it accelerates upward, no experiment could tell for certain that the effect was not due to a sudden increase in the mass of the Earth. Of course so far it has always been an upward acceleration so we have a strong suspicion which case applies.

The force on an object due to the potential energy gradient divided by the resulting acceleration is constant for any given object. Accelerating an object in the absence of a potential energy gradient also requires a force. The ratio of that force to the resulting acceleration is also constant for a given object. And for the same object these ratios are precisely equal. The first ratio is the gravitational mass of the object, the second is its inertial mass. The fact that these are equal was dismissed by Newton as a lucky coincidence. Einstein took this as evidence of a profound simplicity in nature.

Newtons second law, F = m*A, is derivable from the energy point of view we have been using. The details are beyond our scope here but at least we have lifted the hood and looked at the engine which makes it go, the interaction of space and matter.

moonscapeSo far we have concentrated on the curvature of space caused by the presence of matter. This is a very weak effect, so much so that it takes an object the size of the Moon to make a pound of butter weigh three ounces. There are other things which may bend space, possibly into yet other dimensions. Among these things are electrical charges. All matter is made up of charged particles. For the most part the positive and negative charges balance each other out so the effect on space of these positive and negative charges is negligible until you get very close to an object. At short range, the tiny distances between the positive and negative charges becomes significant and the space in the neighborhood becomes bent.

Every object then has this shield of twisted space surrounding it at very short distances so that every interaction between objects occurs through their mutual distortion of each other's space. The collision between billiard balls which we observed is in fact the result of a very steep potential energy gradient at the ball surfaces. Since all interactions occur through this space link, then for every action there must be an equal and opposite reaction because the rules by which every object distorts space are the same for all.

The Earth 'pulls' on the Moon due to the distortion in space at the Moon's location caused by the Earth. The Moon also causes a distortion in space at the location of the Earth according to the same rules so the 'pull' of the Moon on the Earth must be equal and opposite. The same proposition holds for the distortions in space caused by electrical charges.

Well, with all this we have filled in some of the underlying thoughts supporting Newton's laws. If you are intrigued by this way of looking at the universe, I recommend you look into some books on non-Newtonian physics and relativity. In fact I intend to write one covering both topics as soon as I find the time.

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