Space and Time in Classical Physics

©Fernando Caracena

Descriptions of space, time and motion in physics

As babies we learned many of the concepts of classical physics in a very natural way: through our actions. The knowledge that we acquired then, becames the intuitive basis for our ideas relating to space, time and motion. These concepts can be framed in an elegant mathematical form that relies on several millenia of human thought and debate, to which Galileo Galilei was an important contributer.

We do not know what time and space really are. At birth, we have no innate knowledge of them. We become habituated to them by learning to operate in this world. Jean Piaget (1896 – 1980), the Swiss psychologist famous for his child development studies, noted that infants first operate in the “out of sight, out of mind” mode, but later develop the notion of object permanence—the ability to recognize that when an object disappears behind something else, it is still back there somewhere. Even then, infants have to learn how to connect the dots. For example, when mommy disappears behind a screen, the infant may be expecting to see her reemerge from where she first disappeared, only to be jolted by seeing mommy come out the other side of the screen. For the infant, our ordinary experience with time and space is full of surprises. This is what make peek-a-boo games so much fun.

We learn to connect the dots for a moving body as occupying a succession of positions corresponding to a sequence of times. After much trial and error, we learn to do this intuitively. Baseball players learn to do this well enough to sometimes connect the bat with a ball approaching home plate at about a hundred miles per hour, occasionally hitting a home run. A lot of this intuitive knowledge is acquired by programming our nervous system. We learn it through conscious use of our muscles, and successful results are automatically written for us as subroutines in our cerebellum. Note that we did not invent the first computer, we had one all along in our brains.

From experience, we have learned that objects in motion have smaller and smaller displacements over progressively smaller time intervals. We see this when a moving body is photographed using a time-lapse camera. Newton, imagined (mathematically) that the motion of material objects was continuous; that is, as the time interval between observations approaches zero without reaching it, the corresponding displacement also approaches zero without reaching it. But are space and time continuous, or do they become discontinuous at small scales? Is motion on a very small scale really a complex of discontinuous jumps in position and time? Examine the picture on a television screen and you will see that a TV image is made up of many, discrete dots (pixels) of many colors. The dance of colored pixels fools our eyes. The TV image is nothing but colored dots. Does the universe itself fool our senses in a similar way? Modern theories of physics are toying with the idea of abandoning the idea that time and space are continuous.

Classical physics uses notions learned by trial and error during our infant years and elevates them to mathematical theory. You can think of physics as human experience rendered into mathematical theory and the search for new experience not already so rendered. Before we can delve deeper into the properties of space and time, it is a good idea to see how these concepts enter into physics.

Suppose we do a series of experiments, making a time exposed photograph of of the trajectory of a white ball against a black background, which is illuminated with a stobe lamp. Each flash registers the image of the ball. In the multiply exposed photograph, the images of the ball at different positions outline the trajectory of the ball. Among the various photographs made under identical conditions but with a different strobe frequencies, where the strobe lamp has the highest frequency of flashes, the images of balls are closer together.

Ball photo at low strobe rate

Suppose that the film used is so fast that an image of the ball registers at each flash no mater how high the strobe frequency. The photograph will capture the moving ball's motion as a series of white dots that trace out the path of the ball. If the strobe frequency has been low, the dots are spaced far apart. At higher strobe frequencies, the dots will be spaced closer. Now suppose that we turn up the strobe frequency. As we let the strobe frequency go higher and higher, the dots will eventually blend together into a continuous image that looks like toothpaste squeezed out of the tube. That toothpaste limit is the classical concept of the motion of an object moving continuously through space and time: at every time, the object has a position; and time flows continually; therefore, position changes continuously. Common sense you say; but to the Greek philosopher, Zeno, it was a paradox.

Ball photo at high strobe rate

There are a number of images generated by strobe-photograhy on the Internet, see for example, here.