| |
| T2 minus
X2 equals (spacetime distance)2? I
don't get it. |
| |
|
Although the Minkowski Diagram is
a powerful tool for visualizing the differences in the way
spacetime is split into the appearance of space and time
by different observers, it has some drawbacks:
-
It is not apparent from looking at this
kind of diagram why the invariant distance of Einstein's
(flat) spacetime is obtained by subtracting X2
from T2 rather than by adding. In fact,
from the appearance of the diagram, it would seem that
the distance could be obtained by normal vector addition.
- We cannot read the amount of time and space dilation
directly off the axes without use of a conversion factor. *
|
|
Minkowski diagram |
| |
| |
|
So show
me. |
| |
|
Constant
Velocity Diagram View 1 |
|
In the diagrams on the following pages, we sacrifice the
appearance of symmetry between space and time axes to which
we have become accustomed in the Minkowski diagrams. In return,
we obtain diagrams that allow easy visual comparison between
the quantities X, T, and τ.
In the first view of the new diagram,
we also sacrifice a fixed position for Δτ on the diagram.
This allows us to keep the practice of positioning all observers
at the origin O.
|
| |
| But
where is the hyperbola
in hyperbolic spacetime?
|
| |
|
The second view of this diagram sets the stage for us to
reveal the hyperbola implied by
X2 - T2 = (invariant).
This version restores τ to a fixed position.
All that we give up is the ability to place every observer
at (0,0). The origin is now reserved solely for observers
in the rest frame of the event. (Beyond the small, but possibly
finite, risk of hurting the egos of observers in motion, the
sacrifice seems justified.)
|
|
Constant
Velocity Diagram View 2
|
| |
|
Hyperbolic View |
|
By superimposing many such graphs and then animating them, we
can provide a fairly dramatic revelation of the hyperbola. |
| |
*
This conversion factor can be shown geometrically--and by
using hyperbolas, at that--but it is not particularly obvious how
this should be done. But see, for example, G. Naber's The Geometry
of Minkowski Spacetime: An Introduction to the Mathematics of the
Special Theory of Relativity, 1992, Springer-Verlag, New York,
pp. 32-6.
|