In the Book "The Geometry of Minkowski spacetime" by Naber, he says we may assume the following:

Any two admissible observers agree on the temporal order of any two events on the worldline of a photon, i.e., if two such events have coordinates $(x^1, x^2, x^3, x^4)$ and $(x^1_0, x^2_0, x^3_0, x^4_0)$ in $S$ and $(\hat{x}^1, \hat{x}^2, \hat{x}^3, \hat{x}^4)$ and $(\hat{x}^1_0, \hat{x}^2_0, \hat{x}^3_0, \hat{x}^4_0)$ in $\hat{S}$ then $x^4 - x^4_0$ and $\hat{x}^4 - \hat{x}^4_0$ have the same sign.

i.e. that two observers agree upon the ordering of events in spacetime. Why is he making this assumption? Isn't the whole point of SR to say that simultaneity is relative?

  • 1
    See: physics.stackexchange.com/questions/169631/… Two events on the worldline of a photon are lightlike separated. – BowlOfRed Dec 24 '17 at 3:45
  • Please don't post images of text: transcribe the text into the post as a bloack quote (use > for markup). This has two advantages: first the quote is machine readable and therefore searchable and second it is possible for user to cut-n-paste from the block quote to highlight important points. – dmckee Dec 24 '17 at 5:37
  • Here's an important point that should be highlighted: the author does not make this claim for arbitrary pairs of points, he makes it for "any two events on the worldline of a photon". That is a very restricted case, and you have to analyze the situation with that in mind. – dmckee Dec 24 '17 at 5:38

For completeness [especially for those without easy access to the reference], it might be good to include the other assumptions.

  1. Each admissible observer presides over a 3-dimensional, right-handed, Cartesian spatial coordinate system based on an agreed unit of length and relative to which photons propagate rectilinearly in any direction.
  2. Each admissible observer is provided with an ideal standard clock based on an agreed unit of time with which to provide a quantitative temporal order to the events on his worldline.
  3. For each admissible observer the speed of light in vacuo as determined by the Fizeau procedure is independent of when the experiment is performed, the arrangement of the apparatus (i.e., the choice of P), the frequency (energy) of the signal and, moreover, has the same numerical value c (approximately $3.0 \times 10^8$ meters per second) for all such observers.
  4. "causality assumption": Any two admissible observers agree on the temporal order of any two events on the worldline of a photon, i.e., if two such events have coordinates $(x^1, x^2, x^3, x^4)$ and $(x^1_0, x^2_0, x^3_0, x^4_0)$ in S and $(\hat x^1, \hat x^2, \hat x^3, \hat x^4 )$ and $(\hat x^1_0, \hat x^2_0, \hat x^3_0, \hat x^4_0 )$ in $\hat S$, then $x^4 − x^4_0$ and $\hat x^4 − \hat x^4_0$ have the same sign.
  5. The Relativity Principle: All admissible frames of reference are completely equivalent for the formulation of the laws of physics.

The causality assumption seems to be a requirement that observers agree on the same sense of "future" [for lightlike related events]. (Temporal-inversions are excluded.) Furthermore, he uses this to define "causal automorphism" as used in the Zeeman theorem, which Naber discusses later on that page and in the rest of the book.

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