When geologists and biologists showed conclusively that the Earth had to be millions of years old for evolution to have taken place, astronomers were baffled because the Sun's energy output (assuming combustion of carbon fuel in pure oxygen) could only be maintained for a few thousand years.

There had to be some other source of energy for the Sun for it to be able to shine for millions of years. Astronomers already knew the sun was in equilibrium, held together by its gravity and kept from collapsing by internally generated heat, but where was all this heat coming from? One astronomer (sorry, I can't remember
which one) even remarked "it's as if at 20 million degrees energy issues from matter as does steam from boiling water".

Other than combustion, the only other source for such energy was radioactivity, which had just been discovered.

“A star is drawing on some vast reservoir of energy by means unknown to us. This reservoir can scarcely be other than the sub-atomic energy which, it is known, exists abundantly in all matter; we sometimes dream that man will one day learn how to release it and use it for his service.” F.W.Aston’s experiments seem to leave no room for doubt that all the elements are constituted out of hydrogen atoms bound together with negative electrons. The nucleus of the helium atom, for example, consists of 4 hydrogen atoms bound with 2 electrons. But Aston has further shown conclusively that the mass of the helium atom is less than the sum of the masses of the 4 hydrogen atoms which enter into it – and in this, at any rate, the chemists agree with him. (…) Now mass cannot be annihilated, and the deficit can only represent the mass of the electrical energy set free in the transmutation. We can therefore at once calculate the quantity of energy liberated when helium is made out of hydrogen. If 5 per cent of a star’s mass consists initially of hydrogen atoms, which are gradually being combined to form more complex elements, the total heat liberated will more than suffice for our demands, and we need look no further for the source of a star’s energy.”

From “The Internal Constitution of the Stars”, Presidential Address of Professor A.S. Eddington to Section A of the British Association at Cardiff, on 24th August 1920, published in The Observatory Vol. XLIII No. 557, October 1920

In his landmark lecture from 1920, Arthur S. Eddington for the first time ever realised that fusion powers our Sun as well as all other stars, thus solving a major mystery of contemporary science. He could do so only thanks to his exceptionally prompt support to the Einstein’s Theory of relativity, including the relationship between energy and mass E=mc2. The above excerpt from Eddington’s lecture shows his bold and ingenious application of the formula on the brand new results of precise measurements of atomic weights made by Francis William Aston (Nobel Prize in Chemistry, 1922) in the Cambridge Cavendish laboratory – a birthplace of many other important results, including the discovery of D-D fusion. Notice that the knowledge of subatomic structure was very poor in 1920 so the lecture had to have a very vague frameset.

http://www.efda.org/2005/08/discovery-of-the-energy-source-in-stars/

So a few rock hounds and critter gitters, and some navel-gazing stargazers, plus some very boring and meticulous chemistry in the lab, a Polish lady chemist, (Madame Curie) and one crazy-ass German Jew led directly to the Manhattan Project in less than a generation. Dr Eddington died in 1944, not having seen the ultimate consequences of his work.

Nothing in basic science research can be considered a waste of time. We have no idea where it will eventually lead, and its where all the really big breakthroughs come from. Maybe this gravity wave stuff will be what leads us to an inertialess drive, or FTL propulsion, or artificial gravity.