Lightcone Structure on the Set of Events
If light has a universally constant speed to all inertial observers and if light is the stuff with the largest speed, then light must constitute the ideal material with which to communicate between spatially and temporally separate events.
Hermann Minkowski realized that the special character of light implies a new kind of causal structure on the set of all spatio-temporal events associated with the class of inertial observers.
Given the 'absolute universality' of light's properties and its observer-independent properties, we can see that the set of events is partitioned by the events which trace out light paths into four distinct subsets:
- the set of events tracing out light's path to and
from a given event
- the set of events which occur in the past of a
given event
- the set of events which occur to the future of
a given event
- those not in the above sets-- called the 'elsewhere'
Given an event e, we can illustrate these sets of events as follows:
The special theory of relativity has these causal relations valid for each event in the set of all spatio-temporal events. Thus there is a tight weaving together of all events in the past, present, and future of any inertial observer. We can summarize this via the following diagram:
The Interval: a Quantity Invariant in Form
for All Inertial Observers
Consider two inertial observers A and B with B traveling with respect to A at speed v in a given direction. We let them have their coordinate systems origins cross and synchronize each observer's clock at that event. Each observer then records the time of occurrence of the same event off their worldlines. The spacetime diagram of these happenings is shown below:
The two observers can agree on a quantity which is the same for
each observer. To see this, we analyze the times of sending and receiving
of the probe signals as follows.
So, it follows that the product of tB1 and tB2 is:
We can easily re-express this in terms of the time of occurrence and the spatial distance functions as follows:
so that
Then we find:
Thus, the difference between the square of the time of occurrence and the spatial distance to an event of interest is what is invariant for the two inertial observers. This fact flows directly from the postulates of the special theory and was one of the factors which inspired Hermann Minkowski to suggest that the special theory demands a revolution in our conceptions of the very structure of space and time.
We see that while neither the temporal interval nor the spatial interval is the same for the two observers, the interval I is. The revolutionary character of this discovery is evidenced by the fact that Einstein took this perspective from about 1909 on in his searches for a generalization of the special theory to encompass gravitation.
To connect the interval with our discussion of the four causally distinct regions of events, we remark here that each type of causal region has a unique characteristic in terms of I:
- Spacelike related events have I < 0
- Timelike related events have I > 0
- Lightlike related events have I = 0
Note that if we focus on -I, then spacelike events will have I > 0 and timelike events I < 0. This is the usual convention.
Minkowski suggested that the appropriate 'distance' function for a world in which the special theory of relativity is valid is given as follows. Let ds be the infinitessimal interval between events at point (t,x,y,z) and (t+dt,x+dx,y+dy,z+dz). Then the numerical value is: