One "remarkable" feature of spacetime is that by using the interval ds, you can mark off regions of spacetime in terms of some physical ideas. For example, lets start with an Event A (origin) and ask what the magnitudes of the various worldline intervals will be using this event as a basis coordinate. The figure below shows that connecting Event A with Event B, we get an interval in which the time-like coordinate difference is greater than the spacelike coordinate difference. This means that all events similar to B are in the future of Event A, and can be connected to A by the passage of time. It also means that to physically travel to B would require a speed less than that of light, and so in principle, B could be a future event on the worldline of A. This corresponds to ds > 0.
If we connect Event A with Event C, we have a ds that has a larger space-like value than a time-like value. This means that to get from A to C you would have to travel faster than light, and so there is no worldline that can connect A and C. Instead, you could find a reference frame where these represented the endpoints of a meter stick. This also corresponds to ds < 0. Finally, if we connect Event A with any Event on the edge of the two cones formed by the crossed lines, we get intervals for which ds = 0. These events, leading from Event A, represent light rays sent from A traveling forward in time, and outwards in space.
Future and Past Light Cones
Every point in spacetime has exactly this same structure, with two "light cones" joined at a vertex at the location of the spacetime event. The one at positive times is called the future light cone. The one at negative times is called the past light cone. All physical worldlines that include Event A occur within these past and future light cones, and all other events outside these light cones are identified as existing in "Elsewhere" and cannot be directly connected to Event A without traveling faster than light.
This partitioning reveals a very important concept in physics. There may indeed be things going on elsewhere in the vicinity of Event A, but these other things have no cause-effect impact on Event A because no information can reach Event A to register this activity until a later time when the information catches up to the worldline that includes Event A in the future.
In cosmology, this has the very important consequence that it isn't the location of a star or galaxy Now that determines how it will gravitationally interact with us Today that counts. It is where that galaxy seems to be Now based on its visual location. Because gravity also travels at the speed of light, a galaxy located 1 billion light years away is influencing us today by virtue of where it was 1 billion years ago when we are seeing it today, not where it has actually moved to in the intervening 1 billion years.
Related EoC Articles
- Spacetime: Foundations
- Spacetime: Geometry
- Spacetime: Gravity
- Spacetime: Special Relativity
An artist's concept of twisted space-time around Earth. (Source: Spacetime Vortex - NASA.)