2

I'm not sure exactly how to categorize Robb's treatment of the spacetime interval. But it seems like a gem of simplicity and insight.

The following illustrations are based on MTW Box 1.3. As drawn, we can think of two unaccelerated observers crossing paths at event $\mathcal{A}.$ One moves along the world line to event $\mathcal{B}$ at which a light pulse originating at $\mathcal{P}$ is reflected back to the worldline $\mathcal{AZ}$ at event $\mathcal{Q}$. With $\mathrm{c}=1$ it is easily seen that the distance $x$ is equal to the proper time from $\mathcal{P}$ to the event labeled $t$ which is equal to the proper time from $t$ to $\mathcal{Q}$. From that we easily find the surprising and simple equation:

$$s_{\mathcal{A}\mathcal{B}}^2=-\tau _{\mathcal{A}\mathcal{B}}^2= -\tau _{\mathcal{A}\mathcal{Q}}\tau _{\mathcal{A}\mathcal{P}}$$

enter image description here

Notice that if we drop my requirement that $\mathcal{A}\mathcal{B}$ be time-like, Robb's equation will still work.

The reason this example is so compelling to me is that I found it illuminating when analyzing Einstein's thunderbolt thought experiment. The first of the following graphics show how events are labeled. The left Minkowski diagram shows the worldline of railbed system vertical. The right diagram shows the onboard origin's worldline vertical. The distance $x$ in the above discussion appears as the magenta line segment originating at event $\mathcal{E}_b.$ It is also the proper time separation between the events terminating the tops of the dashed magenta lines, as well as other significant separations in the diagrams.

See Bergmann's Introduction to the Theory of Relativity, Chapter IV for a discussion of this well-known thought experiment.

enter image description here enter image description here

Is there any standard terminology established for Robb's treatment of the spacetime interval?

user12262
  • 4,258
  • 17
  • 40
Steven Thomas Hatton
  • 1,965

3 Answers3

2

I don't know about "standard terminology", but I refer to it as the "product of times formula" in my answer to Invariance of the relativistic interval . ("Product of radar-times" is probably more descriptive.)

It's known, but not well-known.

In addition to Robb's Optical Geometry of Motion (1911) and Misner-Thorne-Wheeler's Gravitation (1973), the formula appears in Bondi Relativity and Common Sense (1962), Markze-Wheeler Gravitation as Geometry (1964), and Geroch's General Relativity from A to B (1978). I'm sure it's in other works (e.g. Synge's Relativity the Special Theory (1956), Born (1922) ).

Ultimately, it is an expression of the square-interval between a pair of events in terms of the "light-cone coordinates" for the area of a causal diamond with those events at opposite corners. In other words, the displacement vector is along the diagonal of its associated causal diamond. This is used in my "rotated graph paper" approach. See the reference in my earlier answer, linked above.

robphy
  • 11,748
1

[...] how to categorize Robb's treatment of the spacetime interval.

It's (also) a special instance of Stewart's theorem, in the setting of flat spacetime given in terms of spacetime interval values $s^2$ for each pair of events (i.e. as a metric space which is suitable generalized to include the inverse triangle equality of timelike intervals).

Consider also generalizations.

user12262
  • 4,258
  • 17
  • 40
1
I'm not sure exactly how to categorize Robb's treatment of the spacetime 
interval. But it seems like a gem of simplicity and insight.

Professor Hermann Bondi developed this beautiful mathematics to teach the theory of relativity at Oxford University. In literature, it is known as Bondi algebra or Bondi $k$-calculus. This particular calculus is predominantly found in the relativity books written by British authors, for example this one.

Mass
  • 2,020