Suppose that we are in flat Minkowski spacetime. There are many inertial observers moving at constant speeds relative to eachother and a single non-inertial observer moving with constant acceleration. In all of the inertial frames, the spacetime diagram looks like this: one world-line coincides with the time axis, many straight world-lines represent the many inertial observers moving at constant speed and one curved world-line representing the accelerating observer. We can perform Lorentz transformations to build the spacetime diagram of another inertial observers and in all of them there will be many straight world-lines and the one curved world-line. However, what does the spacetime diagram of the accelerated observer look like? How can that transformation be performed? Is there any good simulation/image of what such transformation and spacetime diagram would look like?
The Penrose Carter diagram for flat Minkowski spacetime looks like:
The Penrose-Carter diagram of the flat Minkowski spacetime f or an inertial observer looks like a diamond, in which coordinate lines connect opposite apexes which are the spac elike and timelike infinities. Borders of this diamond are the lightlike infinities where world lines of light rays end. The proper spacetime of an uniformly accelerated observer is a small diamond included in the first one with one common spa celike infinity. At first sight, the coordinate lines for such an observer seems similar to those for the whole spac etime (compare figure 1 with figure 4) but there are big differences:
•The time coordinate lines for the uniformly accelerated obs erver end on the lightlike infinities of the whole spacetime, while position coordinate lines end on one extre mity at a spacelike infinity and on the other extremity, due to the horizons, at a finite position.
•Considered as functions of ξ or ψ , the slope at extremities of the coordinate lines for the uni formly accelerated observer varies from − 1 to 1, while the slope at extremities is vanishing for the coo rdinate lines of the inertial observer.
•There is not a perfect symmetry between time and position coo rdinate lines for the uniformly accelerated observer as it is the case for the coordinate lines of the iner
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