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Original by Michael Weiss.

Too Many Explanations: a Meta-Objection

An old lawyer joke:

"Your Honor, I will show first, that my client never borrowed the Ming vase from the plaintiff; second, that he returned the vase in perfect condition; and third, that the crack was already present when he borrowed it."

Or to quote Shakespeare: "Methinks the lady doth protest too much."

Why so many different explanations? Are the relativists just trying to bamboozle their opponents? To prevail, a defense attorney just has to stir up doubt about the plaintiff's case; she's not required to give her own theory of what happened. But a physical theory should tell a single coherent story.

Relativity here pays the price of permissiveness. It says to us, "Pick whichever frame you like to describe your results, or use spacetime diagrams and don't choose a reference frame at all. They're all equivalent, I don't mind." No wonder that one explanation ends up looking like three or four.

Most physicists feel that the Spacetime Diagram Explanation is the most fundamental. It does amount to a sort of "Universal Interlingua", enabling one to see how superficially different explanations are really at heart the same.

Figure 1 is the basic spacetime diagram for our hero and heroine. By adding lines one way or another, we will get all the various explanations. (Oh yes: choose units so that c=1 throughout. So light rays graph as 45 degree diagonal lines in all our diagrams.)

Figure 2 is the diagram for the Doppler Shift Explanation. The red lines at 45 degrees are the pulses of light one twin sends to the other. (To reduce clutter, I've made two copies of the diagram. The left one shows Stella's pulses, the right one Terence's.)

The time dilation factor in the diagram is two: Terence ages twice as much as Stella. (Notice that Stella has time to send off a mere 16 pulses, while Terence fires off 32.) The emissions are spaced evenly from the viewpoint of the respective senders; not so the receptions, which are redshifted or blueshifted according to the relative motion of sender and receiver. All pulses are properly accounted for; check out the Doppler Shift Explanation for full details.

Figure 3, the diagram for the "GR" Explanation, adds lines of simultaneity (in blue) instead of light pulses.

These lines represent collections of events that all happen simultaneously, according to Stella. You can see how the lines are closely bunched near Stella, and spread apart near Terence. This is a graphical representation of "gravitational" time dilation. If we pretend that Stella's notion of simultaneity is the Truth (just for a moment!), we'd have to say that Stella's clock is running much faster than Terence's during the Turnaround.

Modify Figure 3 slightly, and we have a portrayal of the Time Gap Objection (Figure 4).

Here we have let the Turnaround become instantaneous. On the Outbound Leg Stella uses one frame of reference, and one notion of simultaneity. On the Inbound Leg she switches to another. The "gap" (the section of Terence's worldline devoid of blue lines) is a consequence of this abrupt switch.

These are just a few of the ways we can decorate our simple diagram with extra lines. In the laissez-faire spirit of General Relativity, we could cover the diagram with almost any network of grid lines, and base a description on the resulting coordinate system. (I hasten to add that there are some pitfalls for the unwary: see Section 6.3 of Misner, Thorne, and Wheeler for the fine points.)

One territory, many charts.

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