How do you look at them? And when (by your own clock)?I look at the clocks on the rear bumper and the front bumper and observe they indicate the same time.
How does he look at them? And when (by his clock)?A second observer agrees with me on the rear bumper clock but reads the front bumper clock as a different time.
Note, particularly, that for both observers, what you are thinking of as "the object at some instant of time" is *not* what you actually see. The light arriving at your location at some particular instant of time by your clock was emitted by the object at *different* instants of time by your clock. You do not see a "snapshot" of the object at a single "instant of time" by your clock; you have to *construct* that "snapshot" from observations that you make at different times by your clock.
Sure, and that will be as it was one light-travel time ago by your clock (assuming that your clock and the bumper's clock are synchronized). But note that if you and the second observer are in relative motion (which I assume you meant), it is impossible for *both* of your clocks to be synchronized with the rear bumper's clock; at least one of your clocks must be ticking at a different rate than the rear bumper's clock. This means you have to be really careful interpreting what the clock readings you see "mean".I would think that we saw the rear bumper as it was at the time it's clock indicated.
Here it gets even more important to be precise. Note that for you and the second observer to see different clock readings from the front bumper, you must be spatially separated; if you were spatially co-located at the instant you observed the front bumper, you would have to see the *same* clock reading (because you would both have to be observing the same light beam from the front bumper). Since you are spatially separated, you can see different clock readings from the front bumper even if you both are at rest relative to each other; your spatial separation means the light travel time to you from the front bumper is different than the light travel time to the second observer. So in order to interpret what you are seeing, you have to separate this effect from the effect of your relative motion.I would think that we saw the front bumper as it was at different times, one ahead of the other by the difference between the two different times shown on the clocks.
I strongly advise becoming familiar with them; it's much harder to correctly analyze relativity scenarios if you don't understand them.I've not worked with worldlines
Yes. The "local plane" is often called a surface of simultaneity.I would think that the nature of a particular inertial rest frame defines a kind of local plane through the spacetime (a certain orientation of the plane). The geometric result is intersecting the plane through multiple worldlines (parts of things, or multiple things)
As long as the events in question are spacelike separated, yes. Changing frames can't change the time ordering of events along a single worldline, and it can't change the time ordering of events on different worldlines that are timelike or null separated (i.e., one event is within the other event's past or future light cone, so they can be causally connected).determining time sequence - another orientation would "cut" the worldlines at a different "angle" and present a different sequence.
So before you can even figure out what a change of frame can do to the time ordering of events, you have to figure out how they are causally related to each other (i.e., are they timelike, null, or spacelike separated). But if you know how they are causally related to each other, you can figure out *all* of the physics; you don't even have to go through the extra step of figuring out the time ordering in a particular frame. The main reason we still talk about inertial frames, IMO, is interpretation: it helps to put the physics in terms we can grasp intuitively.
Yes, in the sense that the 3-D geometric shape of the intersection of an object's "world tube" (the set of all worldlines of points within the object) with a particular surface of simultaneity can depend on the orientation of the surface of simultaneity (i.e., on the state of motion of the inertial frame that defines the surface of simultaneity). But note, once again, that because of light travel time delays, this "shape" will *not* in general be the same as the shape that you actually see when you observe the object. Google "Penrose-Terrell rotation" to see an example of how this works.I suppose this would also determine the shape configuration, too.