of the observed body B is projected as moving in space as measured
by the observer A and in time both according to B's own clock and
the clock of A. According to Special Relativity, a body moving at 80%
the speed of light will go 4 light-years (ly) by A's reckoning and in
5 years (yr) on A's clock, but B's clock will be observed by A to have
elapsed only 3 yr.
The relativistic relationship can be expressed by t' = t*SQRT(1-v2) per a Lorentz transformation (with t being the observer's time, t' the observed time, and v the relative velocity proportional to c), which in the above example yields 5*SQRT(1-.82) = 3. Strictly speaking therefore, body B travels a relative 4 ly in 5 yr according to A, with an elapse of 3 yr according to B's proper time (its clock is observed by A to elapse 3 years).
(It will be relevant in later discussion to note that world-lines as in figure 1 are necessarily equal in length: This is because the world-line of an observer relates to the observed body's time and velocity as the hypotenuse to the sides of a right triangle (t = SQRT(t'2 + x2)), with t' as the observed clock and x as the distance traveled by the observed body according to the observer), and the world-line of the observed body is the hypotenuse.)
The Utility of a Relativistic Spacetime Diagram
The spacetime diagram works to represent the relationship determined by the Lorentz transformations only if a body moving uniformly ("at rest") in space is actually moving, time-wise, perpendicular to space in its own coordinate system, thus providing the observer with a basal frame of reference in the diagram. Given that a body observed to be in relative motion is also moving along its time-axis perpendicular to space in its own coordinate system, its space axis must be different than that of a body taken to be the observer at rest. Accordingly, figure 2 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 strange phenomenon wherein each observer measures the other's clock as moving more slowly than her own. By rotating the diagram, the mirror image of A's perspective can be seen from that of B.
Two bodies in two different coordinate systems, x-y and x'-y', are
shown to mirror their relativistic effects. By rotating the diagram
either system can be represented as at rest in space and the other
in relative motion with a slower clock, and each by the same
Figure 2 is a fully accurate depiction of the relativistic relationship. It expresses the duality that students of relativity often have difficulty comprehending: It is because each body has its own orientation in the spacetime continuum that each mirrors the relativistic effects of the other.
Granting the absolute effects of acceleration on clocks, as has been satisfactorily confirmed by experiments, and given the relative effects of un-accelerated (uniform) motion, any uniform motion that might be included in a test of the so-called Twin Paradox can be considered in abstraction from the accelerations. By doing so the problem of mutual time dilation during periods of uniform relative motion and its reconciliation at the twins' reunion can be isolated and more easily resolved.
Figure 3 takes the perspective of a stay-at-home
Twin A's coordinate system. World-lines A1
and B1 represent the spacetime
paths of the twins in the uniform part of Twin B's journey to a distant
star-system according to Twin A; vectors A2
and B2 are the
world-lines of the twins in the uniform part of B's return trip, also according
to Twin A. As in figure 1, the Pythagorean 3-4-5 relationship that obtains from a
relative velocity of .8c is used for the sake of simplicity and clarity.
Uniform portions of the journey of Twin B to-and-from a distant
star-system are shown from the perspective of the stay-at-home
Twin A. Vectors A1 and B1 represent the un-accelerated away
segment of B's journey, and vectors A2 and B2 represent the
period of B's un-accelerated return segment. At the 5 year mark
on Twin A's clock corresponding to her observation of the
uniform part of Twin B's journey away, the clock of the latter
indicates that she has begun her deceleration near the
destination after 3 years of moving uniformly. Following her
acceleration for the return home, she coasts for 3 years before
beginning her deceleration to earth, where both twins agree
that 6 years of uniform motion have elapsed.
Figure 3 illustrates how Twin A can agree that Twin B's clock ends up with an identical recording of 6 years moving uniformly: Because Twin B had already began her return, according to A's clock, two years prior to A's corresponding time of 5 years, they both agree that Twin B has spent 6 years moving uniformly while A was waiting at home for 6 years.
(Note that Twin B's vectors are drawn to intersect the spacetime points of destination and reunion for the sake of clarity, but it would be the decelerations subsequent to the uniform segments that would actually mark those arrivals.)
There are several indications that figure 3 is an imperfect representation of two-way periods of uniform motion in spacetime: The diagram isn't rotate-able, as was figure 2; to accurately treat two reference frames that separate and re-converge it would be necessary to somehow balance their perspectives, otherwise the time dilation observed by one is treated as more "real" than the other. But Twin A is portrayed as having been absolutely at rest and Twin B as in absolute motion in the un-accelerated segments spanned by the diagram, with no way to reverse or balance their roles, because one twin reverses directions while the other maintains her continuous direction in spacetime -- one world-line thus forms an angle, and the other does not. Another problem is that the sum of the lengths of the twins' world-lines are not equal like they are in figures 1 and 2, as Twin B's vectors add to 10 units compared to 6 units for A's; this is because Twin B is moving in A's coordinate system according to A's measure of space, which requires the representation of B to conform as-if to an absolute space metric. The Lorentz transformation for the relative measure of space corresponding to the transformation related to time is x' = x SQRT(1-v2), which in the example would be 4*SQRT(1-.82) = 2.4 ly, which is Twin B's measure of the space traversed in each direction per 3 yr period at a relative .8c. Unlike Twin B's proper time (3 yr each way in the example), her own measure of the distance involved is not directly observed by Twin A, and isn't therefore represented (or representable) in a spacetime graph depicting A's observations; this accounts for the over-extension of Twin B's world-lines in figure 3.