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When we restrict our attention to homogeneous Lorentz Transformations, we restrict two observers to have the same space time origin. Yet they could be moving w.r.t each other with a constant velocity.

  • So what does the sentence same "space-time origin" mean physically?

I know that mathematically it means that two observers have the same origin in the Minkowski space. But I want to know what the statement means in physical terms.

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Physically, it usually means that
they meet [possibly only momentarily] at this origin event
and that they set their wristwatches to read zero then
and assign spatial xyz-coordinates (0,0,0) there.

Update:

  • When they "meet" (i.e. their worldlines intersect), they are at the same point in space at the same time. (They can shake hands.)

  • For the purposes of using more-easily comparable coordinate systems, they each agree:

    • to set their wristwatches to read zero at that meeting and
    • to lay out metersticks along their x-, y-, and z- axes, with the zeroes at their common meeting position.

Of course, physics can go on without them meeting and without them agreeing on a set of coordinates to use. It's just that extracting the physics from their numerical measurements is made more difficult by their different choices of standards.

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  • $\begingroup$ This is exactly how Einstein describes it in On The Electrodynamics of Moving Bodies, 1905. Two observers on two different train tracks sync their respective wristwatches and then he derives the Lorentz transformations from that condition. $\endgroup$ – oemb1905 Dec 8 '17 at 1:19
  • $\begingroup$ So they do not start off from the same point in space? Could you elaborate your answer a bit further? $\endgroup$ – Abhikumbale Dec 8 '17 at 8:13
  • $\begingroup$ I made an update to address your comment. $\endgroup$ – robphy Dec 8 '17 at 13:51

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