Notes on Relativity and Cosmology - Physics Department, UCSB

98 CHAPTER 4. MINKOWSKIAN GEOMETRY
4.3 More on Minkowskian Geometry
Now that we’ve ironed out the twin paradox, I think it’s time to talk more
about Minkowskian Geometry (a.k.a. “why you should like relativity”). We
will shortly see that understanding this geometry makes relativity much simpler.
Or, perhaps it is better to say that relativity is in fact simple but that we so far
been viewing it through a confusing “filter” of trying to separate space and time.
Understanding Minkowskian geometry removes this filter, as we realize that
space and time are really part of the same object. (I am sure that, somewhere,
there is a very appropriate Buddhist quote that should go here.)
4.3.1 Drawing proper time and proper distance
Recall that in section 4.1 we introduced the notion of the spacetime interval.
The interval was a quantity built from both time and space, but which had the
interesting property of being the same in all reference frames. We write it as:
(interval) 2 = ∆x 2 − c 2 ∆t 2 . (4.3)
Recall also that this quantity has two different manifestations 3 : proper time,
and proper distance. In essence these are much the same concept. However, it is
convenient to use one term (proper time) when the squared interval is negative
and another (proper distance) when the squared interval is positive.
Let’s draw some pictures to better understand these concepts. Suppose I want
to draw (in an inertial frame) the set of all events that are one second of proper
time (∆τ = 1sec) to the future of some event (x 0 , t 0 ). We have
−(1sec) 2 = −∆τ 2 = ∆t 2 − ∆x 2 /c 2 .
Suppose that we take x 0 = 0, t 0 = 0 for simplicity. Then we have just x 2 /c 2 −
t 2 = −(1sec) 2 .
You may recognize this as the equation of a hyperbola with focus at the origin
and asymptotes x = ±ct. In other words, the hyperbola asymptotes to the light
cone. Since we want the events one second of proper time to the future, we
draw just the top branch of this hyperbola:
3 Insert appropriate Hindu quote here.