The only thing we demand is that their motion include a noticeable component that makes them drift further and further upwards. As far as angular motion is concerned, we do not care whether they move around the axis clockwise, or counterclockwise, or not at all. They can move towards the axis or away from it, or simply stay at a constant distance. The rule is this: Upwards axial motion is compulsory.įollowing this rule, objects still have great freedom. (A mathematician would say that we have introduced “cylindrical coordinates” to describe such motions.) Now, let us introduce a restriction – a traffic rule for moving objects, if you will. So far, we have done nothing but to introduce a specific way of viewing motion in ordinary, three-dimensional space. Traffic rules for our three-dimensional space with axis Radial motion is motion that changes only the radial coordinate of an object, angular motion changes only its angle α. In a polar coordinate system such as is shown in the following image, the location of every point P is defined by drawing the connecting line to the origin and then noting the length r of that line and the angle α between the connecting line and some pre-defined axis:Īll points with the same value for r lie on a circle with radius r – hence r is called the radial coordinate. In fact, with this view, it is only a small step to see why we have called those directions radial and angular in the first place. We’ll leave out all motion parallel to the axis and only look at the same assortment of radial and angular arrows as in the previous picture:įrom this perspective, it is perfectly obvious that the arrows representing radial motion (red) always points straight at or directly away from the axis, while motion in the angular direction (cyan) traces out part of some circle around the axis. Now the blue axis is no more than a blue dot. It shows the view of an observer looking straight down at the gray plane. If you’re still somewhat unsure what radial and angular motion are all about, the picture below will hopefully make matters clear. This time, we look from a somewhat higher vantage point down onto the gray plane: The next illustration shows a few examples – movement parallel to the axis in blue, movement in the radial direction in red, and movement in the angular direction in cyan. We can represent the three kinds of motion by arrows tracing the path of an object following only that kind of motion – moving only radially, or only along the axis, or in pure angular motion. Conversely, if we know all three components of the object’s motion, we can reconstruct the motion as a whole. If an object is moving in a general way, we can identify the different kinds of motion involved: By measuring how far that object progresses in parallel to the axis, we determine its axis motion by measuring how its distance from the axis changes, its radial motion by tracking its motion sideways around the axis, its angular motion. An object in pure angular motion will trace out a circle around the central blue axis. The third is motion perpendicular to the other two, which we shall call angular motion.
The second is radial motion – either directly towards the blue axis, or directly away from it. The first has objects moving parallel to the blue axis – either directly upwards or downwards -, so let’s call this axis motion. We can define three distinct types of motion. Let’s look at objects moving around in this space. In the middle of this space, we draw an axis – in the following image the vertical blue line: Let’s start by looking at three-dimensional space in a specific way. Visualizing the geometry of familiar three-dimensional space, though admittedly with some additional assumptions that will seem contrived, it is possible to understand an important aspect of the way that a black hole isolates its interior from the rest of the universe. How, in one sense, space and time switch their roles inside a black hole – and why this leads to a black hole’s most characteristic property, namely that nothing can get out An article by Markus Pössel
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