4.1.4 magnetism from radar-distance

Historically, Maxwell's theory of electromagnetism in the 19th century was informed to relativity before we learned (in the early 20th century) about relativity and the connection that it implies between space and time. Magnetism, and today's myriad applications of electromagnetic induction (e.g. to motors) and of electromagnetic waves (e.g. to communications), like gravity certainly qualifies as an everyday phenomenon for most readers of this note.

We earlier solved this problem for time-dilation, which involves but two events in spacetime, since the metric-equation automatically describes the proper-time elapsed on a moving clock thus eliminating the need for a second map-frame. Length-contraction, in spite of its frequent introduction as complementary to time-dilation, in fact involves three instead of two events in spacetime and requires a definition of distance from the vantage point of a traveling observer.

The good news is that radar-time approaches to extended-simultaneity do just this, and allow us to derive the traditional Lorentz-transform (or tangent free-float-frame) definition of extended-simultaneity (along with traditional expressions for length contraction) as a limited special case, specifically for fixed-velocity travelers in flat-spacetime. This is illustrated with our radar-time solution to Rachel E. Scherr's "Mount Rainier versus Mount Hood" simultaneity-puzzle. Since the spaceship is at Mount Rainier when it erupts at cτ=0, the last chance to influence and the first chance to detect the Rainier eruption are concurrent. This happens only about 6 spaceship-ticks before the spaceship can first detect Mount Hood's eruption, but something like 16 spaceship-ticks after the ship's last chance to influence that eruption. From the spaceship's perspective, Mount Hood erupts halfway between those opportunities and therefore well before the eruption at Rainier.

Since distances may be measured using times-elapsed on only one clock associated with any given world-line, this radar-time approach allows measurement of apparent-length differences (not to mention event-timing) in curved-spacetimes with no extended-arrays of either yardsticks or synchronized-clocks! This is good news since e.g. on planetary surfaces (like earth) the differential passage of time with altitude makes coordinate-systems with truly synchronized-clocks difficult to arrange.

One bottom-line relative to everyday life is this: If the speed of light were to increase, magnetism would decrease. If you wake one morning having imagined that the lightspeed constant in our universe suddenly went up, you need only check the magnets still sticking to your refrigerator to be assured that it was only a dream.

Related references:

• • R. E. Scherr (2007) "Modeling student thinking: An example from special relativity", Amer. J. Phys. 75, 272 abstract.

  • Carl E. Dolby and Stephen F. Gull (2001) "On radar time and the twin paradox", Amer. J. Phys. 69 (12) 1257-1261 abstract.

  • P. Fraundorf (2012) "A traveler-centered intro to kinematics", arxiv:1206.2877 [physics.pop-ph].