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-v ^{2})* 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-.8

^{2}) = 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} + x^{2}*)),
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

**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**

*B***A**'s perspective can be seen from that of

**B**.

.

*figure 2*

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

measures.

*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.

**The Twins**

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 **A _{1}**
and

**B**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

_{1}**A**and

_{2}**B**are the world-lines of the twins in the uniform part of B's return trip, also according to Twin A. As in

_{2}*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.

*figure 3*

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

**A**and

_{1}**B**represent the un-accelerated

_{1}*away*

segment of B's journey, and vectors

**A**and

_{2}**B**represent the

_{2}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-v ^{2})*, which in
the example would be 4*

*SQRT(1-.8*= 2.4

^{2})*ly*, which is Twin B's measure of the space traversed in each direction per 3

*yr*period at a relative .8

*c*. 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*.