If one uses the original Minkowski coordinates (x,y,z,w; where w = ict), one can maintain the orthogonality of x vs w and x' vs w' axis pairs in spacetime diagrams.
On May 21, 2009, Michael Edmiston <edmiston@bluffton.edu> wrote:
I agree with John Denker that drawing a spacetime diagram is a really good
way help gain an understanding of various situations involving relativity
and two or more observers. When I first learned how to draw and interpret
spacetime diagrams I was amazed at how much they helped me. My students who
eventually understand spacetime diagrams tell me the same thing.
If you are already familiar with spacetime graphs, then you probably already
understand what I am about to try to explain. If you are unfamiliar with
spacetime graphs and are trying to learn about them, there is a problem that
trips up my students over and over again. This pitfall might also catch
you. Students misinterpret the diagrams because they have been taught an
improper way to find the projections of points onto the graph's axes.
In a conventional "x-y" graph with orthogonal axes, students have been
taught to find the "x-value" by constructing a line that is perpendicular to
the x-axis and that also passes through the point of interested. Likewise,
they find the "y-value" by constructing a line perpendicular to the y-axis
that also passes through the point of interest. Although this method works
when the axes are orthogonal, this method does not work when the axes are
not orthogonal. Indeed, I believe it is not the proper way to interpret the
graph even for orthogonal axes, and I have been asking math and science
teachers to stop teaching it that way. I have not been having very much
success.
The proper way to find the projection of a point onto the x-axis to draw a
line through the point such that the line is also parallel to the y-axis
(not perpendicular to the x-axis.). The fact that the perpendicular method
works with the conventional graph is an artifact of the axes being
orthogonal. The perpendicular method is not the general method for finding
the projection of a point onto a particular axis of that graph.
Some math folks have pointed that my viewpoint is at odds with the common
definition that the "projection" of one line segment (A) onto another line
segment (B) is typically defined as Acos(theta_AB). Yeah, that's a problem,
and it also stems from the fact that typical view of the trig functions
involve a right triangle, and that is the way "projection" is often defined.
However, when dealing with a two-dimensional plot for which the axes may or
may not be perpendicular, it is not the correct way to find the x and y
values.
Perhaps some of you have figured out a better way of saying or explaining
what I am trying to explain. Maybe I shouldn't be using the word
"projection." But regardless of how we say it, if you try to identify the
position and time values of a point on a spacetime diagram by using
perpendicular construction rather than parallel construction, you will
misinterpret the diagram. Hopefully I've stated this clearly enough that if
you understand what I am talking about, then you are probably okay, and if
you don't understand what I am talking about then you need to think a bit
more about how to use spacetime diagrams.
Hopefully John Denker's repeated suggestion that you draw a spacetime
diagram will prod some of you unfamiliar with this tool to start using this
tool. I am just trying to warn you about a pitfall that seems to trap my
students more often than not.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
1 University Drive
Bluffton, OH 45817
419.358.3270
edmiston@bluffton.edu