student34
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Thanks a lot for this. It is a lot to take in all at once, but I am sure I will refer to at to help guide my questions and thought process.PeterDonis said:This question as you ask it is not well posed, for the reasons I have already given. However, there are some valid statements that can be made in this context that might be helpful, although they will most likely just illustrate how much more work you have to do to discard your current intuitions. All of these statements apply to any diagram of Minkowski spacetime that is drawn the way your diagrams in the OP are drawn, i.e., that is a diagram from the viewpoint of some inertial frame.
(1) Any straight line that has a slope (relative to horizontal) of less than 45 degrees is called a "spacelike" line. Any such line represents a spacelike 3-space that has Euclidean geometry. However, there will only be one inertial frame (which will only be the frame the diagram is drawn in if the line is exactly horizontal) in which the Euclidean geometry of this spacelike 3-space is obvious from the metric (by setting ##dt = 0##).
(2) Any straight line that has a slope of exactly 45 degrees is called a "null" or "lightlike" line. Any such line represents a portion of a light cone, which is the set of all possible light rays to or from a given event. The "geometry" of a light cone has no simple analogue in ordinary geometry.
(3) Any straight line that has a slope of more than 45 degrees is called a "timelike" line. Any such line represents the worldline of a timelike observer who is always inertial, i.e., always moving in free fall with zero proper acceleration. Every such observer is at rest in some inertial frame; lines that are exactly vertical represent the worldlines of observers who are at rest in the specific inertial frame in which the diagram is drawn. The "geometry" of any worldline is simply a straight line, but this is not very helpful since it just means the points on the line each represent individual events at which the observer's clock reads a particular time, and those times can be treated as real numbers ordered in the usual way from past to future.
(4) To have any region of spacetime that has Minkowski geometry, you must have a region that is represented by an area on the diagram, not a line. No single line, no matter what its slope, will represent any region of spacetime that has Minkowski geometry.