2015 05 23 Mike Polioudakis Angular Momentum: Spinning a Box around on a String I have never been happy with the idea of angular momentum as it is presented in books and taught in courses. I don’t deny angular momentum. I don’t want to over-stress how different it might be from regular straight line translational momentum; hereinafter called “regular” momentum. I do not deny the Relativistic insistence on equivalence of frames. I just sense some oddness in how we think about it, and I think we over-stress how similar angular momentum is to regular momentum. The idea that regular momentum and angular momentum are exactly equivalent and exactly the same is like the idea that the Sun moves around the Earth. Seeing that they might be different is like seeing that it makes more sense to say the Earth moves around the Sun. Yes, it is possible to construct and defend scenarios in which we see the Earth at the center of the universe and all motion of all kinds as taking place relative to a stationary Earth. But that is ridiculous. It is a lot of trouble, and not very productive. It is easier, and far more productive, to accept that the Sun is relatively still compared to the Earth and the other planets, and to describe their orbits in terms of that. In the same way, a determined physicist we can take an object that looks as if it has angular movement to comparatively stationary and so take angular momentum as not absolute, and construct explanations for all movements in relation to a (formerly) spinning body as if the (formerly) spinning body were not spinning – but this way is ridiculous. It is easier to assume the body is spinning and to account for all the motion in relation to the spinning body on the premise that it is spinning. It seems some accounts of spinning motion wish to deny this simple common sense. I don’t like it. Newton’s bucket spinning on a rope was really spinning, not the rest of the universe. Mach can shut up. I find this account sometimes in texts: imagine a steel door. I find it easier to imagine a “spinner” gun target oriented with its axis vertical instead of horizontal. Somebody throws a ball at the door. When the ball hits the door, the door spins on its hinges (axis). By usual normal accounts, translational regular momentum turned into angular momentum. Now imagine that some puts a ball into the path of the rotating door. The door hits the ball, and, by luck, comes to a complete stop (heavy ball), putting all its momentum into the ball. The ball goes off in a straight line. The incident turned the angular momentum of the door into the translational regular momentum of the second ball. The door served as a momentum repository between the incoming ball and the outgoing ball by converting regular momentum into angular momentum and then converting angular momentum into regular momentum. Momentum is conserved. I like this. It is clear, straightforward, and conserves total momentum. But it does not conserve angular momentum. So here is what texts do with it. Think of the axis of the door as the axis by which we measure all angular momentum. As the first ball approaches the door, the angle between it and the door axis changes; it gets smaller. In effect the ball has angular momentum in reference to the door axis. When the first ball hits the door, it transfers its angular momentum to the door. When the door hits the second ball, the door transfers its angular momentum to the ball. As the second ball goes off, the angle between it and the door axis changes continually; it gets bigger. So all we ever had was angular momentum, and it is conserved. Not really. There is a way to preserve this scenario using projective geometry but I don’t want to do that here. It flies in the face of my physical intuition. Return to the scenario that tries to save angular momentum as the only momentum involved. If the ball had true angular momentum around the door hinge axis, the way the angle changed would follow the pattern of a circle, ellipse, hyperbola, or some similar allowable closed path. Although it is possible to make the way that the angle changes follow a hyperbola, to do so stretches the situation, as with using projective geometry. To force the way the angle changes to represent an allowable path turns a lot of normal trajectories into odd hyperbolas. For what we usually think of as angular momentum, the path has to follow a circle or ellipse, some simple closed path, and the ball does not. Its change of angle does not follow any change sequence that conforms to a closed path. It is just a lot easier not to think of the trajectory of either the incoming or outgoing ball as having angular momentum in relation to the hinge axis. It is easier to think of it as having regular momentum. Now to another standard image from physics, the empty box swung around in a circle (or ellipse or other similar closed path). It is easiest to imagine an almost empty box at the end of a string. People inside the box need not feel that the box has angular momentum or any other kind of momentum. They feel that there is a gravitational field below their feet. They feel that the gravitation field bends even the path of a light ray. All of this scenario is common and so I don’t go into details. Now imagine that the box is made of glass; the string is invisible; and all around the box, but not directly in the path of the box, is a thin evenly spaced dust that reflects light. A denizen of the box turns on a flashlight (aimed in the direction that we see the box travelling). As the light beam goes through the box, another denizen measures the light beam as travelling in a curved path. Both inhabitants interpret the data to mean that the box is in a gravitational field. Fine. Now the intense light beam goes through the walls of the box, where it steadily encounters little bits of dust as it goes onward. Some of the light reflects back to the box, where the inhabitants record the data. The inhabitants can get a steady stream of data, and so record the apparent path of the light beam. According to us, this is what they see: the light curves as it goes away from the box. It curves fairly quickly. Even if there is no obvious object out there to cause the curve, the inhabitants have to interpret the light as travelling in a gravity field. Assume the box is travelling in a circle (another closed path would work as well but a circle is easiest). Call where the box first released the light beam the “bottom” of the arc. As the box spins around the arc, the speed and direction of light will appear to change for no obvious reason. I do not go through the details here; you can work it out if you start at the bottom of the arc. For light to speed up, slow down, and change directly slightly, for no apparent reasons, would seem to violate a few laws. It is just easier to assume the box spins around at the end of a string. I don’t know what this implies for thought experiments that went into constructing General Relativity.