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## Main Question or Discussion Point

Here's something that puzzled me for some time: why are space-proper-time (SPT) diagrams not used more frequently? I did search this forum for the term, but without success. It may be known under some other technical term that I don't know about.

I'm using the term with the meaning as per Lewis Carroll Epstein's "Relativity Visualized", where the orthogonal axes of two inertial coordinate systems in relative motion are simply rotated by an angle

I originally thought that SPT diagrams represent a weird combination (as Kip Thorne wrote) of "my space and your time" but the more I toyed with SPTs, the more useful they seemed. I think one problem is that SPT diagrams cannot show space-like intervals, but for time-like intervals, they seem to be much easier to picture and use than Minkowski space-time diagrams.

For one thing, the calibrating marks on the two sets of orthogonal axes have identical lengths. Other benefits that I can think of include:

i) Time dilation and Lorentz contraction are both read off directly by projecting one coordinate system's proper-time intervals or space intervals onto the other system.

ii) It shows the freedom of choice of reference system very clearly - no preferred frame is remotely suggested - just rotate the chart!

iii) A straight vector summation of a space interval and a proper-time interval gives a "space-proper-time interval" that is invariant under change of inertial frames. There may be an official term for this interval, but I don't know it.

iv) Maybe more…

Apart from space-like intervals, I still have to find a situation that cannot be represented on the SPT diagram. I know that here velocity additions cannot be done by adding angles as in the rapidity case, but still…

So where's the catch? I fully expect to be hand-slapped for suggesting that SPT diagrams could be useful, but I simply must know "if not, why not?"

Jorrie

I'm using the term with the meaning as per Lewis Carroll Epstein's "Relativity Visualized", where the orthogonal axes of two inertial coordinate systems in relative motion are simply rotated by an angle

**phi = asin(-v/c)**relative to each other, as shown in the attached figure for a relative velocity (red coordinates relative to blue coordinates) of**v = 0.8c**.I originally thought that SPT diagrams represent a weird combination (as Kip Thorne wrote) of "my space and your time" but the more I toyed with SPTs, the more useful they seemed. I think one problem is that SPT diagrams cannot show space-like intervals, but for time-like intervals, they seem to be much easier to picture and use than Minkowski space-time diagrams.

For one thing, the calibrating marks on the two sets of orthogonal axes have identical lengths. Other benefits that I can think of include:

i) Time dilation and Lorentz contraction are both read off directly by projecting one coordinate system's proper-time intervals or space intervals onto the other system.

ii) It shows the freedom of choice of reference system very clearly - no preferred frame is remotely suggested - just rotate the chart!

iii) A straight vector summation of a space interval and a proper-time interval gives a "space-proper-time interval" that is invariant under change of inertial frames. There may be an official term for this interval, but I don't know it.

iv) Maybe more…

Apart from space-like intervals, I still have to find a situation that cannot be represented on the SPT diagram. I know that here velocity additions cannot be done by adding angles as in the rapidity case, but still…

So where's the catch? I fully expect to be hand-slapped for suggesting that SPT diagrams could be useful, but I simply must know "if not, why not?"

Jorrie

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