iSoul In the beginning is reality

# Two ways to symmetry

There are two ways in which the length and the time (duration) of a motion are symmetric. The better-known way is the use of a conversion factor, notably the speed of light, which is the same for all inertial observers. All lengths can be turned into time intervals or all time intervals can be turned into lengths. But this does not change the character of the original measurements, which always remain either lengths or time intervals. So this is an optional and artificial symmetry between space and time.

The other way is to note that the extent of a motion can be measured in two ways: as a spatial extent or as a temporal extent (duration). Either of them may be chosen as the independent variable, with the other as the dependent variable, making a symmetry between the two. This is an inevitable and natural symmetry between space and time.

A simple example shows the difference: Consider a running track that is 400 metres long. The winning time on a race is 50 seconds. The winning rate may be expressed as 400 m / 50 s = 8 m/s or as 50 s / 400 m = 0.125 s/m = 125 s/km. In the former case the time is the independent variable and in the latter case the length is the independent variable. People are more familiar with speed, in which the independent variable is time, but with the fixed length of a race, the length is actually given first and so would be the independent variable.

Compare this with converting the length, 400 m, into a time interval, dividing it by c, the speed of light, which is 299 792 458 m/s: The result is 1.334… × 10-6. Or compare the time interval, 50 s, converted to a length, multiplying it by c: The result is 14 989 622 900 m. These extreme numbers may have a place in theoretical calculations (notably, the invariant interval of relativity) but have little meaning beyond that.

The direction of motion can also be measured in two ways: as an angle (spatially) or as a turn (temporally). The spatial angle is measured with something like a protractor, and the temporal angle is measured with a standard rotation such as one of the hands of a circular clock. As with measuring the extent, the two ways are measuring the same motion but they are measuring different aspects of it.