2. The Entanglement Space and Time
The Relativity of Simultaneity…
As Einstein grappled with the problems in electrodynamics that gave us the special theory of relativity, the discovery of one key misapprehension about space and time allowed him to reconcile two apparently incompatible notions, the principle of relativity, demanded by experiment, and the constancy of the speed of light, demanded by Maxwell’s electrodynamics. He could assert both if he was willing to suppose something utterly at odds with classical theory, that observers in relative motion may disagree on which spatially separated events are simultaneous. Two events judged to occur at the same time by one observer might be judged to be sequential by another in relative motion. This result is the relativity of simultaneity. It is described in careful detail in the first section of Einstein’s (1905) celebrated “On the Electrodynamics of Moving Bodies,” for all that follows in Einstein’s paper depends upon it. It expresses a profound entanglement of space and time, a moral worthy of inclusion in our catalog.
This notion lay behind Minkowski’s (1908, p. 75) immortal declaration when he introduced the concept of spacetime:
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
Prior to relativity theory, space and time were treated separately. One considered space at one instant of time; and then successive spaces as the instants passed. Minkowski combined these into a single four dimensional spacetime manifold of events. That much was not incompatible with classical theory. In it, just as in relativity theory, the set of all events in space and time form a four dimensional manifold. On pain of violation of Novelty we cannot claim it as a moral of relativity theory. It was just a classical possibility that was not exploited. The novelty lies in the way the new spacetime can be decomposed to spaces that persist through time. It reflects a new entanglement of space and time. In classical theory, the decomposition is unique. There is one way to do it. In relativity theory, as shown in Figure 1, each inertially moving observer finds a different way to slice spacetime into spaces. There is a different decomposition associated with each inertial frame of reference.
Figure 1. Simultaneity in classical theory and special relativity
Each such space consisted of simultaneous events. Since these observers could not agree on which events are simultaneous, they cannot agree on how to form the spaces. We cannot select one slicing as the correct slicing. Each is geometrically identical and any criterion that would elevate one would do the same to all the rest. It is just like seeking diameters of a perfect circle. There is no one correct diameter that bisects the circle. There are infinitely many and they are all identical in their geometric properties
…Is not Robust
Important as it is, these traditional presentations of the relativity of simultaneity cannot stand as a moral since they fail the requirement of Robustness. With the advent of the general theory of relativity, spacetime took on a more varied geometrical structure. It could still be sliced in many ways into spaces that persist with time, but, in important cases, just one slicing is preferred geometrically. To visit some familiar examples, consider the Robertson-Walker spacetimes used in standard big bang cosmology. Just one slicing gives spaces filled with a homogeneous matter distribution. Any other slicing mixes events from different epochs with differing densities of matter. Or consider a Schwarzschild spacetime, the idealized spacetime of our sun. Just one natural0 slicing gives us spaces whose geometric properties remain constant with time. Analogously many chords might bisect the area of an ellipse, but bisection along the principle axis is geometrically distinct from all the others.
Infinitesimal Neighborhoods of Events in Classical and Relativity Theory
What of the celebrated entanglement of space and time brought by the special theory. Has the general theory parted what the special theory had joined together? It has not. To find a robust entanglement, we must seek it in a more subtle way. We will find it by exploiting a fundamental fact about the spacetimes of both the special and general theory. They differ in domains of any finite extent. However, if we select just one event and consider the events infinitesimally close to it, then we have found a mini-spacetime that is the same in both special and general relativity. This mini-spacetime mimics the bigger spacetime of special relativity. The entanglement of space and time of the relativity of simultaneity can be found within it. We can use it to formulate this entanglement in a way that is robust under the transition from special to general relativity.
Let us proceed to these mini-spacetimes. In order not to violate Novelty, I will first construct as much of them as I can in a way that is compatible with both classical and relativistic theories. In both, the set of all events forms a four dimensional manifold. That means that we can label events with four real numbers, the spacetime coordinates, and we can then use those numbers to decide which events are near which. This notion of nearness lets us extract the mini-spacetime of events infinitesimally neighboring some arbitrary event O, as shown in Figure 2.
Figure 2. Extracting an infinitesimal neighborhood in both classical and relativistic spacetimes.
Within the mini-spacetime, we can identify events, such as T, which come temporally later than event O. A definite amount of time will elapse between event O and T as we pass along the spacetime trajectory OT. That time is physically measurable, for example, by counting the ticks of a clock that moves along the trajectory OT. Similarly, we can find an event S that occurs simultaneously with O (for at least one observer). The distance along the interval OS is physically measurable. For example, we might contrive a measuring rod to pass through events O and S so that opposite ends occupy events O and S simultaneously (at least for one observer). In the mini-spacetime, the trajectory OT represents an inertially moving body and the interval OS a straight line.
We may also sum intervals represented by OT and OS using the familiar parallelogram rule for vectors. If we add OT and OS, as shown in figure 3, we arrive at OT’. If OT is the trajectory of some body, then OT’ will represent the trajectory of a body moving in the direction of OS with respect to the first body.0
Figure 3. Measuring time elapsed and spatial distance between infinitesimally close events; the addition of displacements between infinitesimally close events.
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