One example that uses Special Relativity is the Twin Paradox
Former UCSC Physics professor Peter Scott was very generous to provide his input on the study of SRT and its applications.
1. What would need to happen in order for the SRT (special
relativity theory) to become a part of the practical sciences?
SRT is by now very well established, and is, at least in some
sense, already always taken into account when discussing the
behavior of physical systems (which could be assumed to be
"practical sciences").
SRT describes how classical Newtonian mechanics must be modified
whenever the velocities of the particles or other parts of any
physical system become high, that is, approaching the speed of
light. Normally, since there are always limits to the accuracy
with which we can measure things (three decimal places is typical),
the low speed behavior is well-described by classical Newtonian
mechanics, identical to such behavior even if SRT is taken into
account. However if the velocity of some particle or other body
(a rocket ship?) is high, we need to take SRT into account when
describing the behavior of the particle. For example, a mu-meson
arriving at the Earth's atmosphere will typically be traveling at a
high speed, and therefore will have a longer lifetime (before
decaying into its constituent particles) than if that same particle
were at rest. This behavior is by now very well-tested, not only
with particles moving through the space beyond our Earth, but also
in every particle accelerator.
2. Do you believe the rate of development of SRT is accelerating or
slowing down?
Since it is well established, and has been for decades (see above),
it is neither.
3. If a new discovery comes to disprove SRT, how would the study of
physics and the understanding of our universe change?
Any change in our understanding of our universe would surely depend
on the nature of such a "new discovery". While I don't expect any
new discoveries that might disprove SRT, there are currently
behaviors of our universe that are NOT understood, such as the
"Standard Model" of particle physics, and the apparent presence of
"dark energy" and "dark matter". These are the main puzzles that
are now being worked upon, by folks much more familiar with them
than I am.
Former UCSC Physics Professor Peter Scott