- #51

- 831

- 5

I'm not understanding what these calibration curves are for or even how you derived them. On the bottom one, you show it going through the 5 year Proper Time for the traveling twin and through the 5 year Proper Time of the stationary twin.

Later you said:

Are you saying that you have a calibration scheme that allows you to determine that the twins have synchronized clocks or that it provides some means by which to "calibrate" them? It seems rather trivial to me if all you want to do is draw a line from a particular Proper Time for one twin to the same Proper Time for the other twin but then I don't understand why you would need a calibration curve.

I'll add some more commentary a little later. For now, let me provide these sketches with a little bit of commentary.

Equation 1) is derived from the upper left sketch. We have sketched the space-time diagram for a red guy moving to the left and a blue guy moving to the right. Both red and blue are moving at the same speed with respect to the rest black system. This is necessary in order that line lengths on the screen for red and blue will have the same scaling (one inch along a red coordinate has the same physical value as one inch on the corresponding blue coordinate). If you don’t use symmetric coordinate systems in this way, you must use hyperbolic calibration curves to compare physical distances among coordinate systems. This is what we wish to make clear with the hyperbolic curve derivation accompanying the sketches.

These slanted coordinate systems arise from special relativity theory. These unusual looking coordinates are selected as the only coordinates that always yield the same speed for light: c. That’s because the world line of a photon of light always bisects the angle between the X4 coordinate and the X1 axis.

Note that the X4 axis for a moving observer is rotated with respect to the rest system X4 axis (the slope is proportional to the speed). Then, the moving observer’s X1 axis is rotated so as to always maintain symmetric rotation with respect to a photon world line (which is always rotated to a 45-degree angle in the rest system).

Now, we see that the blue X1 axis is perpendicular to the red X4 axis. You will find this is the situation for any pair of symmetric coordinate systems. Further you can always find a rest system for which observers moving relative to each other will move in opposite directions at the same speed. So, contrary to some objections, this derivation is not a special case—it has completely general application. This allows us to write the Pythagorean Theorem equation involving the red and blue coordinates. The time dilation Lorentz transformation equation can be derived directly from the Pythagorean Theorem.

Here, we just want to derive the Proper Time hyperbola equation, i.e., equation 2) above. This equation may be modified for time scaling, using X4 = ct (we use units of years for time and use the compatible units of light-years distance along the X1 axis as shown when plotting the graphs for equation 3).

Equation 4) is shown for a constant value of 10 for the Proper Time. See the corresponding plot. This plot shows the points along a hyperbolic curve in the black rest system that correspond to a fixed Proper Time value of 10 years. The red slanted lines terminating on the hyperbola represent example world lines (time axes) associated with possible observers moving at various speeds.

So, even though the line lengths on the computer screen are different in the rest black rectangular coordinate system, the Proper Times are all the same.

Last edited: