## Q Lu X Figure 2 Three alternative cases of a stone moving along a straight line in space as it emits three flashes, #7, #2, and #3 The space locations of emissions are the same in all three cases, as are the times of first and last emissions #/ and #3 But emission time for the middle flash #2 is different for the three cases We ask At what time will a free stone following a natural path pass the intermediate point and emit flash #2~> We answer this question by demanding that the total wristwatch time x from first to last flash emissions be an extremum From this requirement comes an expression for the energy of the stone as a constant of the motion

### FIXED Positions

Figure 2 Three alternative cases of a stone moving along a straight line in space as it emits three flashes, #7, #2, and #3 The space locations of emissions are the same in all three cases, as are the times of first and last emissions #/ and #3 But emission time for the middle flash #2 is different for the three cases We ask At what time will a free stone following a natural path pass the intermediate point and emit flash #2~> We answer this question by demanding that the total wristwatch time x from first to last flash emissions be an extremum From this requirement comes an expression for the energy of the stone as a constant of the motion

### Now for the full step-by-step derivation.

1. Let t be the frame time between flash #1 and flash #2 and let s be the frame distance between these two flashes. Then the metric  tells us that the wristwatch time zA along segment A is

To prepare for the derivative that leads to extremal aging, differentiate this expression with respect to the intermediate time t:

dxa dt

2. Next, let T be the fixed time between flashes #1 and #3 and S be the fixed distance between them. Then the frame time between flash #2 and flash #3 is (T-t) and the frame distance between them is (S-s). Therefore the wristwatch time T3 along segment B is

Time

Time Wristwatch time along segment B = Tb

Wristwatch time along segment B = Tb

Wristwatch time along segment A = Ta

Space

Figure 3 Three alternative cases of a stone moving along a straight line in space as it emits three flashes, #1, #2, and #3 These are the same three cases shown in Figure 2, but here we plot the stone's path in space and time Such a spacetime plot is called a worldline. On each of three alternative worldlines, flash emissions #7 and #3 are fixed in space and time Flash emission #2 is fixed in space (horizontal direction in figure) but its time is varied (up and down in the figure) to find an extremum of the total wristwatch time x = xA + xB from #7 to #3 The result is an expression for a quantity that is a constant of the motion the energy of the stone

Again, to prepare for the derivative that leads to extremal aging, differentiate this expression with respect to the intermediate time t: 