s2 = 102 - 82
s2 = 62
Thus s, the interval as it is generally called, is represented in the diagram as the proper time of body B.
A significant implication of the diagram is that there are actually two invariants involved in a relativistic relationship: the conventionally recognized interval, the proper time of B, and in addition, the identical spacetime intervals of the world-lines of A and B. The world line of an observer is not commonly recognized as being equivalent in length to the interval of the world-line of a body being observed; but in the relationship shown in figure 1a between an observer and a body in relative motion (where t2 = s2 + x2), the spacetime interval of the observer is necessarily equivalent to any world-line in relative motion, as the latter forms the hypotenuse of the Euclidean triangle described by the observer's measure of another body's distance traveled in space and the time elapsed on the moving body's clock.
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. Although the relationship between clocks in relative motion given by t' = (t2-x2).5 is indeed parabolic, as is generally recognized, the fact that a hypotenuse relates to the sides of a Euclidean triangle by a parabolic function presupposes the right-angle. And as figure 1a shows, the temporal component of any body's uniform motion in spacetime is at a right-angle to the observer's space axis.
Figure 1b shows both reference frames at once, with A and B each moving in time, and perpendicular to space according to its own frame. The perpendicular relationship of a body's motion in time to its own reference in space will be significant in later considerations.
The relative motion of light as it would be represented in these terms is especially noteworthy. And it should be kept in mind that whereas the speed of light is commonly expressed as (approximately) 300,000 km per second, or 1 ls, 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, as is given both by the Lorentz transformations and the equation for the spacetime interval. A world-line representing a ray of light in figure 1c, depicted below, 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 s2 = 102 - 102.)
Next Page 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
(Note: You can view every article as one long page if you sign up as an Advocate Member, or higher).
