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By Root 880 0

A is x = 10 meters and the distance in time between the two events is t = 1 hour, where x and t are measured by me. We haven’t decided what c is yet, but when we do we will know ct and we can then go ahead and use the distance equation to calculate s, the distance in spacetime between events O and A. Our hypothesis is that while x and t can and will be different if they are measured by someone flying past at close to the speed of light, the distance s will stay the same. In other words, x and t can and will change but they must change in such a way that s never changes. To risk overemphasizing the point, we want to remind you that our goal is always to build the laws of physics using invariant objects in spacetime and the distance s is just such an object. If that sounds too abstract, then we can say it again but this time using less mathematically fancy language: Nature’s rules must express relationships between real things, and those things live in spacetime. A thing living in spacetime is akin to an object sitting in a room. Spacetime (or the room) is the arena in which the thing lives. The nature of real things is not a matter of opinion and in that sense we say they are invariant. A three-dimensional example of something that is not an invariant might be the flickering shadow of an object sitting in a room illuminated by a warming fire. Clearly the shadow varies depending on how the fire is burning and where the fire is but we are never in any doubt that a real, unvarying object is responsible for it. Using spacetime, our plan is to lift physics out of the shadows and hunt down relationships between real objects.

FIGURE 5

The fact that two different observers can disagree on the values of x and t, provided s is the same, has a very important consequence, which can be visualized quite simply. Figure 5 shows a circle centered on O, the waking-up event, with a radius s. Because we are, for the moment, using the Pythagorean form of the distance equation, every point on the circumference of the circle is the same distance s away from O. This is a pretty obvious statement: The distance s is the radius of the circle. Points outside the circle are farther away from O while points inside are closer to O. But our hypothesis is that s is the distance in spacetime between events O and A. In other words, the event A could lie anywhere on the circumference of the circle and still be a distance s in spacetime from O. At what point on the circle should event A lie? That depends on who is measuring x and t. For me in the house, we know exactly where it should be since x = 10 meters and t = 1 hour. This is what we have drawn on the diagram and labeled A. For a person flying past in a high-speed rocket, the distance x in space and the distance t in time will change, but if s is to remain the same, then the event must still lie somewhere on the circle. So different observers record different positions in space and time separately for the same event, but subject to the constraint that we only slide the point around on the circle. We’ve labeled two possible positions A’ and A”. For position A’, nothing particularly interesting has happened, but look carefully at position A”. Something very dramatic indeed has happened. A” has a negative distance in time from O. In other words, A” happened before O. It is now in the O’s past. This is a world where you finish your breakfast before you wake up! Such a circumstance is a clear violation of our cherished axiom of causality.

As an aside, pictures like the ones shown in Figures 4 and 5 are called “spacetime diagrams” and they often help us work out what is going on. They really are simple things. Crosses on a spacetime diagram denote events and we can drop a line down onto the line marked “space” (the space axis) from the event to work out how far apart in space the event lies from the event O. Likewise, a horizontal line drawn to the line marked “time” (the time axis) tells us the time difference between the event and the event O. We can interpret the area above the space axis as the