One of the obvious signs that the basic Newtonian system wasn't sufficient was the orbit of Mercury. Rather than being a pretty stable orbit, it moves around kind of like a spirograph. Einstein's theory of general relativity actually accounts for that, but for it to work,
there MUST be such a thing as "gravitational waves" - the source of the "force" known as gravity. Basically, as the theory goes (cause it's still a theory), when two LARGE masses spin around each other, it generates "gravitational waves," which can be thought of as longitudinal waves that compress space itself.
Now, in the case of electromagnetic waves, it was fairly easy to build a generator and a receiver to prove that they exist and are measurable. Unfortunately, to build a gravitational wave generator, they've calculated that the two large masses need to be about the mass of the sun, and they need to spin around each other at almost the speed of light.
However, there is INDIRECT proof, in the form of data collected over 30 or so years on two black holes or neutron stars (not sure which) orbiting around each other. The theory that assumes there are gravitational waves predicts with uncanny accuracy the behaviour of these two large masses (which the basic newtonian theory does not).
There are currently several observatories in the world built for the purpose of measuring these waves. Unfortunately, the waves are ridiculously weak and very hard to measure. The one facility I visited (LIGO in eastern washington) has a probability of maybe 1/30 each year of finding something. However, with the upgrades and new instruments they're building in over the next 5 years or so, they expect to be able to measure about 50 events in a year.
Here's more info on how one would attempt to measure something as subtle as ripples in time and space, see here:
LIGO Overview
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