How Newton modeled time
By Newton's time, people were getting pretty good at modelling space using the Euclidean distance function. But what about time? Can we model space and time together using some version of the Euclidean coordinate system and distance function adapted from space to space-time? Isaac Newton thought about this and decided:
Absolute, true and mathematical time, of itself, and from its own nature, flows equably, without relation to anything external.
Let's examine some system in motion and see what Newton meant by this. For example, the three cars to the left move differently. Let's say that the red one is going 60 mph, the blue one is travelling at 30 mph and of course, the green car is going 0 mph.
What do their paths look like if we try to extend space to spacetime?
Using Newton's model for time as flowing exactly the same for all observers, we could draw a coordinate system with time on one axis and space on the other. Then the path of the car in space can be plotted against time, as was done in the figure below:
If we measure time on the vertical axis and space on the horizontal axis, the paths of the cars appear as shown to the left. Notice that the green car's path is just a line parallel to the time axis itself. This means the green car is staying at the same place in space but moving through time.
A car that stayed at the same moment in time but moved through space would follow a path parallel to the horizontal axis.
We know from observing Nature that such paths are not found. Yet in the Newtonian model for spacetime there seems nothing to prevent such a path from existing.
This is where Einstein and Special Relativity come in, to give us a mathematical model for spacetime that reflects the observed behavior of Nature in that nothing can go faster than the speed of light. (At least not anything observed in a laboratory or in outer space as of yet.)
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