The so-called "invariant interval" -- a magnitude ascribed to changes in B's location between two events from any coordinate system -- usually given by s = SQRT(x2 - t2) (again, with x proportional to c and calibrated with t), yields a negative square root for relative velocities less than c -- an imaginary number. But the relationship can be just as well transformed to
s = SQRT(t2 - x2)
and expressed in the example by
s = SQRT(102 - 82) =
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Thereby s, the interval is revealed in the relativistic spacetime diagram not as an abstract imaginary, graphically inexpressible, but as a physical quantity, the proper time, the observed speed of the clock of body B.
A significant implication profiled by figure 2 is that there are actually two invariants involved in a relativistic relationship: 1) the conventionally recognized interval, reinterpreted here as the proper time of B between two events along its world-line, which is invariant when measured from any reference frame, and 2) the equality of spacetime intervals of the world-lines of A and B. In the Minkowski diagram the world-line of an observer is not recognized as being equivalent in length to the world-line of a body being observed; the latter is treated as a function of the observer's time and the observer's measure of distance traveled by the observed, so a ray of light would supposedly terminate at coordinates (10,10) after 10 seconds according to the observer, giving the light a world-line 14.14 in length. But in the relationship shown in figure 2 between an observer and a body in relative motion (now, for the sake of comprehension, substituting s (proper time) for t', letting t and t' represent the lengths of the world-lines of A and B, and setting x = v so that t' = SQRT(s2 + x2)), the spacetime interval of the observer (t) can be shown as necessarily equivalent to any observed world-line:
From the equation for the invariant interval, which was reformulated as
s = SQRT(t2 - x2) we can derive
t = SQRT(s2 + x2)
which equates t with the hypotenuse of the triangle formed of s and x, and therefore also equal to t', which is the hypotenuse. The world-line of B is therefore necessarily equal in length with the world-line of A. We can extrapolate and declare that all world-lines (assuming inertial reference frames3) must be equal in length with all others for any given period, regardless of coordinate system.
It is important to note that both the Lorentz Transformations and the equation for the invariant interval indicate a Euclidean relationship between space and time, and between bodies in relative motion. For although the relationship between clocks in relative motion given by t' = SQRT(t2-x2) is indeed parabolic, as is generally stressed in connection with the Minkowski diagram, the fact that a hypotenuse relates to the sides of a Euclidean triangle by a parabolic function presupposes the right-angle. And as figure 2 shows, the temporal component of any body's relative motion in spacetime is at a right-angle to the observer's space axis, and parallel with the observer's own motion in time.
The spacetime diagram works to represent the relationship determined by the Lorentz transformation only if a body moving uniformly in time is actually moving perpendicular to space. Given that another body in relative motion is also moving perpendicular to space along the time-axis of its own coordinate system, its space axis must be different than that of the body taken to be at rest. Accordingly, figure 3 shows two reference frames at once, with A and B each moving in time perpendicular to space according to their own coordinate system. It depicts, as the Minkowski diagram cannot, the curious phenomenon wherein each observer measures the other's clock as moving more slowly than her own.
Figure 3 is a fully accurate depiction of the relativistic relationship. It expresses the duality that students of relativity often have difficulty comprehending: Each body has its own orientation in spacetime, each moves in time perpendicular to space, and each mirrors the relativistic effects of the other. Both the Lorentz Transformations and the (modified) equation for the invariant interval indicate a perpendicular relationship between space and time for any body (except light) at-rest or moving uniformly. The motion in time of a body A is perpendicular to its orientation in space, and the relative motion of a body B, although proceeding in a relative orientation that is partly temporal, partly spatial according to A, is moving uniformly in time perpendicular to space in its own coordinate system.
The relative motion of light as represented in these terms is especially noteworthy. Whereas the speed of light is commonly expressed as ~300,000 km per second, to fully describe its observed motion relativistically is to report that it travels 1 ls in space relative to an observer's spatial reference, and zero seconds in time relative to the observer's temporal reference of 1 sec, as is given both by the Lorentz transformationsand the equation for the invariant interval. A world-line representing a ray of light in figure 4 therefore has a spacetime interval of 10 but a proper time of zero, and lies directly along the x-axis of observer A. (The interval in this case is s = SQRT(102 - 102).)
The world-line of light doesn't move in time relative to the observer, therefore it is a misrepresentation to place in on a diagonal, which in effect treats the observer's clock as absolute.
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