It seems that there is this concept that one cannot know that he is existing in a volume (plane) of simultaneity, because the only things experienced (observed) directly in the mind are thoughts based on information arriving from the past. There is no external existence within ones instantaneous Lorentz space that can be directly observed until sometime later—then it is too late—it is already in your past by then. This idea is certainly compatible with the Vienna logical positivist school and gave strength to the idealists like Berkeley who maintained that reality existed within the mind (ultimately, the mind of God). The idea is favored by solipsists as well (Einstein warned about falling into the trap from which there would be “…no escape from solipsism”).
Here, we will appeal to Hawking’s more flexible approach he refers to as “model-dependent realism: the idea that a physical theory or world picture is a model (generally of a mathematical nature) and a set of rules that connect the elements of the model to observations. This provides a framework with which to interpret modern science.” This is from “The Grand Design” by Stephen Hawking and Leonard Mlodinow. Actually, PeterDonis, I believe the ideas you are advancing here are generally much more in line with Hawking. So, I should not be accusing you of advocating logical positivism or solipsism. (Perhaps I owe you and apology on that count.)
So, here is a space-time diagram with a sequence of events. The diagram represents a sort of sequence of thought experiments in which observers in a Lorentz space send messages back and forth as they move through 4-dimensional space. At the end of the sequence of experiments, they get back together and compare notes to see if there is a basis for knowing that each other existed in any of the Lorentz space simultaneous external world volumes.
In the sketch below the brown, light brown, and blue observers are together. They are initially together in the brown rest frame, synchronizing space-time markers, displays of actual distance traveled in 4-dimensional space, referenced from a point assigned as zero distance. The distances displayed (and photographed at selected points) will correspond to distances traveled along the respective world lines.
All three are together at event 1 (events in the space-time diagram are brown, light brown, or blue circles) where they synchronize their distance markers). It is planned that the light brown guy and the blue guy will move to a new position that puts them in brown’s instantaneous plane (3-D volume) of simultaneity at event 8. Light brown and blue have used Lorentz transformations to assure that their distance markers display the same values at event 8 as the brown guy’s markers display at event 2. Light brown and blue both transmit pictures of their displayed values so that brown can validate their numbers when he (brown) arrives at event 3 in the space-time diagram (brown calculates how far he has traveled along the 4th dimension since leaving event 2.
At event 9 the light brown guy transmits a photo of his distance display, which is received by blue at event 14 and received by brown at event 4. The blue and brown guys do calculations that demonstrate that events are still occurring in agreement with theoretical physics. At this point the brown guy is able to confirm that the light brown guy existed in his plane of simultaneity back when he (the brown guy) was at event 3). From the data received from the blue guy at event 4 he is also able to determine that the blue guy was also in his (brown’s) plane of simultaneity back at event 3.
Just one experiment doesn’t seem enough, so they continue acquiring data.
The blue guy arrives at the brown guy’s position at event 5. So, here the blue guy and the brown guy simultaneously occupy the same position at the intersection of their X4 axes. Special relativity tells them that if the light brown guy really still exists, then the light brown guy must exist at event 11 in blue’s instantaneous 3-D space volume, while simultaneously existing at event 12 in brown’s simultaneous space. However, as PeterDonis points out, they can’t really be sure, because they have no way of getting information from those events instantly while they are at event 5. They must wait until later for confirmation from the light brown guy.
Brown gets his confirmation when he arrives at event 6, receiving the picture of light brown’s event 12 photo of his distance traveled along the 4th dimension, which is exactly the same distance that the brown guy recorded for his own trip when at event 5. Thus, brown concludes that the light brown guy must have been in his simultaneous space which included both event 5 and event 12 simultaneously.
The blue guy has to wait until his event 16 for confirmation that light brown was at event 12 simultaneously with event 5 (when both blue and brown guys were simultaneously at event 5). Of course the blue guy saved a copy of brown’s distance position along brown’s X4 dimension when they were together at event 5. So, now he had confirmation that light brown was in brown’s simultaneous space, i.e., both brown and light brown were in the simultaneous space of events 5 and 12.
But, now, blue asks whether the light brown guy was in his (blue’s) simultaneous space when blue was at event 5. Fortunately, light brown included a photo of his X4 position corresponding to event 11. Light brown and blue both used Lorentz transformations to figure out what each other’s positions should be along their respective X4 axes when blue arrived at brown’s position, event 5. Light brown transmitted his computations that he had made about what blue’s X4 position should be when light brown was at event 11. And blue computed the X4 reading that light brown should have when he (blue) was at event 5.
Light brown and blue wanted to be sure science was working right, so they took photos of their respective X4 distances corresponding to blue’s simultaneous space at blue’s event 16. Light brown did calculations (by prearranged agreements) at event 13. At event 17 blue found that light brown was in his (blue’s) simultaneous space when blue was at event 16 and light brown was at event 13.
Without including it in the space-time diagram we have all three observers get together at the end of the experiments and review all of their data. They conclude that sure enough, when the brown guy and blue guy were at event 5, the light brown guy simultaneously existed at event 12 (in brown’s simultaneous space) and event 11 (in blue’s simultaneous space). They then conclude that the light brown guy is actually a 4-dimensional object, and-- by extension--they all are. Thus, we have a model in which objects are 4-dimensional extending into a 4th spatial dimension.
Further, once it is recognized that we have a 4-dimensional spatial universe populated by 4-dimensional objects, it is obvious that the objects do not move. There is a perception of motion related to 3-dimensional cross-sections of the 4-dimensional objects, and this is compatible with the notion of consciousness moving along the 4-dimensional world line of an observer, the consciousness having a deep connection to the perception of the "flow of time." Finally, it implies two possible models of consciousness: 1) A 3-D consciousness that moves at light speed along the observer's world line, or 2) A 4-D consciousness that is coupled to the 4-dimensional material structure of the observer over the full extent of the observer world line.
These experiments (albeit thought experiments) would satisfy the criteria for a good model fit, at least in agreement with Hawking’s criteria:
1. Is elegant
2. Contains few arbitrary or adjustable elements
3. Agrees with and explains all existing observations
4. Makes detailed predictions about future observations that can disprove or
falsify the model if they are not borne out.
To satisfy number 4 in a more convincing manner, The three observers could have sat down together and planned the entire set of experiments in advance, predicting all of the readings that would be observed at each of the events in the scenario.
