# Choice of Origin of Coordinate Systems

• Fantasist
Consistent in the sense that the observers don't arrive...at the same conclusions about what is happening?If the observers are using different coordinate systems then they would not arrive at the same conclusions about what is happening. If the observers are using different coordinate systems then they would not arrive at the same conclusions about what is happening.

#### Fantasist

I am having a personal discussion with somebody elsewhere (not on Physics Forums) and we are stuck at the moment because of a disagreement that I narrowed down to the question whether, in the context of SR, two observers in different reference frames can choose the origin of their coordinate system independently of each other, or whether one observer is bound by the choice of the other. Let me explain in more detail:

assume you have two rulers with distance markings on them (assumed to be identical for the sake of the argument) but without any numerical (or other) labeling defining the origin. When the two rulers are moving past each other, all that the observer in each reference frame sees are the bare markings on his and the moving ruler. Analogously, assume the same applies to the clock ticks that each observers registers for his and the other system's clocks. One observer can now obviously arbitrarily assign the origins to a certain markings/ticks both on his as well as the moving ruler/clock and thus be able to numerically define the coordinate systems. The question is, can the other observer independently choose his own origins to describe the scenario, or is he bound by the choice of the first observer? If the latter, from which postulate is this constraint derived?

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It doesn't seem right to me. Because the choice of origin is completely arbitrary and affects nothing. In fact the choice of origin only affects the names that the two give to different ticks\markings. By choosing one origin, a particular tick\marking will be called 5 and by choosing another one, the same tick\marking will be called 94231. Changing the name of ticks\markings affects nothing. And also the names given to ticks\markings by different observers, don't have any relationship with each other and so there is no constraint. Its just that both should know the other one's "naming convention" and how the two conventions are related so that they can translate each other's statements to a form which is in terms of their own naming convention.

The usual form of the Lorentz transformations $$x^\prime = \gamma (x - vt) \\ t^\prime = \gamma (t - vx/c^2)$$ assumes that the origins of both frames coincide. You can generalize it to non-coincident origins: $$(x^\prime - x_0^\prime) = \gamma [(x - x_0) - v(t - t_0)] \\ (t^\prime - t_0^\prime) = \gamma [(t - t_0) - v(x - x_0)/c^2]$$ where ##(x_0, t_0)## and ##(x_0^\prime, t_0^\prime)## are the coordinates of the same "reference event" in the two frames. Letting ##(x_0, t_0) = (0,0)## and ##(x_0^\prime, t_0^\prime) = (0,0)## gives you the usual form.

Oh...yeah...sorry!

jtbell said:
The usual form of the Lorentz transformations $$x^\prime = \gamma (x - vt) \\ t^\prime = \gamma (t - vx/c^2)$$ assumes that the origins of both frames coincide. You can generalize it to non-coincident origins: $$(x^\prime - x_0^\prime) = \gamma [(x - x_0) - v(t - t_0)] \\ (t^\prime - t_0^\prime) = \gamma [(t - t_0) - v(x - x_0)/c^2]$$ where ##(x_0, t_0)## and ##(x_0^\prime, t_0^\prime)## are the coordinates of the same "reference event" in the two frames. Letting ##(x_0, t_0) = (0,0)## and ##(x_0^\prime, t_0^\prime) = (0,0)## gives you the usual form.

Yes, OK, this is how one of the observers defines the origins of the two reference frames. What about the other observer? Can he independently choose different origins i.e. different values ##(x_0, t_0)## and ##(x_0^\prime, t_0^\prime)## to describe the scenario?

If you want to use the Lorentz transform then the two coordinate systems must share a common origin event. However, the Lorentz group is a sub group of the full Poincare group which also includes space and time translations.

DaleSpam said:
If you want to use the Lorentz transform then the two coordinate systems must share a common origin event.

So the moving-frame observer could not give a consistent account of the same event(s) unless he knows the choice of origin made by the rest frame observer?

Fantasist said:
So the moving-frame observer could not give a consistent account of the same event(s) unless he knows the choice of origin made by the rest frame observer?

This would be absurd. This is like saying if two people witness the same crime, neither can give the name of the criminal without knowing the other's phone number.

Fantasist said:
So the moving-frame observer could not give a consistent account of the same event(s) unless he knows the choice of origin made by the rest frame observer?
I am not sure what you mean by "consistent account". Personally, I would say that each observer's account is consistent with the facts regardless of the existence or knowledge about any other observer.

DaleSpam said:
I am not sure what you mean by "consistent account". Personally, I would say that each observer's account is consistent with the facts regardless of the existence or knowledge about any other observer.
Consistent in the sense that the observers don't arrive at contradictory conclusions if they use different origins to define the coordinate axes involved in the transformation.

What counts as a "contradictory conclusion"? If I conclude that the light flashed at x=5 and you conclude that it flashed at x'=3, is that a "contradictory conclusion"?

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DaleSpam said:
What counts as a "contradictory conclusion"? If I conclude that the light flashed at x=5 and you conclude that it flashed at x'=3, is that a "contradictory conclusion"?

No, the conclusions would be contradictory for instance if I conclude that you measure a shorter time period between two events than I, but you, with your different choice of origin, conclude the reverse.(for the same two events).

Fantasist said:
No, the conclusions would be contradictory for instance if I conclude that you measure a shorter time period between two events than I, but you, with your different choice of origin, conclude the reverse.(for the same two events).
A period is the difference between two times. Changing to a different origin can change the values of the two times but not their difference.

The guys can each mark off the coordinates on their axes in any way they want (as long as the distances on their yardsticks between tick marks are the same), and their origins don't have to coincide when the t and t' are zero. But you would have to correct for this in the equations you use for the Lorentz transformation. They would have to calibrate to one common event. You couldn't directly use the equations in post #3.

Chet

Fantasist said:
No, the conclusions would be contradictory for instance if I conclude that you measure a shorter time period between two events than I, but you, with your different choice of origin, conclude the reverse.(for the same two events).
The outcome of any measurement is frame invariant. It doesn't even need to be an inertial frame.

What changes from frame to frame is not the measurement but the interpretation of the measurement.

Fantasist said:
So the moving-frame observer could not give a consistent account of the same event(s) unless he knows the choice of origin made by the rest frame observer?

Where did this come from?

1) The stationary observer has an account of events. The account is self-consistent.
2) The moving observer has an account of events. The account is self-consistent.

If one knows the full details of how the account in 1) and the account in 2) were created, it is easy to compare them via the Lorentz transform (if standard accounting practices are used) or a modified version of the Lorentz transform (if nonstandard accounting practices were used).

IF one does not know the full details of how the account in 1) and the account in 2) were created, it may still be possible to compare them by reverse-engineering the details of the accounts to determine the unspecified information . I suppose you could call this "forensic accounting". There will be a minimum amount of information one must have to successfully carry out the reverse engineering process though.

I don't understand the idea that the accounts are "not consistent". An example of what is considered an "inconsistent account" might be helpful. If you are concerned about the time-ordering of space-like separated events, i.e. the issue of simultaneity, I could make some more exact statements, but I'm not sure that that's what the issue actually is, I am just guessing at the moment.

[add]It's been awhile since I posted this, without a clarification by the OP, so I'll give my best guess as to what's going on - though I wouldn't rule out the possibility that I've misunderstood the question.

But I *think* the question boils down to whether or not the simultaneity convention in an inertial frame depends on the choice of origin- if that is the question, then the answer to that is no. Given a specific inertial frame, the choice of frame origin does not affect the set of events that are regarded by that frame as simultaneous I believe that the difference in the defintion of simultaneity depnedning on the choice of inertial frames is being referred to as some sort of "inconsistency", not words I would use to describe the situation. It is true that the set of events regarded as simultaneous depends on the choice of inertial frame (and it is also true that this set does not depend on the choice of the frame origin).

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