Spacetime
Spacetime Mechanics
3D Diagrams
Zero Intervals
Proper Time
The Two Towers
Spacetime & Reality
Zero Intervals
As noted above, the surface of the two cones in the 3D diagram represent the spacetime points where the spatial and time intervals are exactly the same. This has the strange effect that, the spacetime interval at each point on the surface of the cones is zero (with respect to event A).
It is very easy to misinterpret the meaning of a zero spacetime interval. What it signifies is that light travelling towards event A would move radially along the surface of the back cone, until it reached event A. Similarly, light travelling away from event A would move radially along the surface of the front cone until it reached its target (event B). In both cases, the resulting spacetime interval is zero. So, to light, there is no time passage, nor spatial distance, between it's emission and reception. I.e. They happen at the same time and the same place, in the rest frame of the light. For this reason, the cones are often referred to as light cones.
We have shown the light cones at 45 degrees to the 2D surface representing the present, and they are quite distinct from the present, and from each other. However, for light, the cones are flattened onto each other by the lack of any time lapse. This also superimposes them on the present. That may have led to many misunderstandings about the present in spacetime, as it is only true for light (and other electromagnetic phenomena). Macro objects experience the universe very differently from this, as is displayed in the following example...
A Simple Flashlight Example
Two observers, Alfie and Betty, have a flashlight each and are mutually at rest one light-second apart. Alfie flashes his torch, and this is event A. A second later Betty flashes her torch, this is event B. To light passing from event A to event B, these events are simultaneous and co-located.
But Alfie and Betty do not experience them in that manner. A second after Betty flashes her torch Alfie will see the flash. As Betty is a light-second distant from him, Alfie knows that the light took a second to reach him. Hence Alfie concludes that event B happened a second after event A. So Alfie experiences the flashes of light as happening a second apart in time, and a light-second apart in distance.
This means that zero spacetime intervals do not necessarily mean that:
- The events happen at the same place (i.e. are co-located).
- The events happen at the same time (i.e. are simultanous).
To macro objects, a zero spacetime interval just means that the spatial and temporal "distances" beween two events are the same magnitude. So two events could be co-located and simultaneous, or be a thousand light-years distant and separated by a thousand years in time.
Furthermore, if this is true for zero intervals, it is also true for non-zero intervals. E.g. If a second after Betty flashes her torch, she flashes it again (event C):
-
To the light passing from event A to event B, event A and event C are separated by a timelike spacetime interval of 1 second (the world line B to C).
- To Alfie, event A and event C are separated by a spatial distance of 1 light second, and a time interval of 2 seconds.
Hence, it would appear that the property of spacetime whereby time intervals and spatial distances cancel does not describe how macro objects experience space and time.
This problem with the interpretation of spacetime intervals may lie at the root of popular myths about spacetime, such as the Andromeda Paradox. In this the fact that events far distant in time may have a zero spacetime interval with respect to an event on the Earth, if they are similarly distant in space (such as from the Earth to Andromeda), is taken to mean that they are simultaneous. As we have noted, this is only true for light and other electromagnetic phenomena. It does not accord with the way that macro objects experience spacetime.
