I understand that gravity is the curvature of space time. If that’s true, doesn’t it fully explain gravity? Why do we need gravity waves and/or gravitons?
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Gravitational wave *is* a consequence of general relativity, the very thing that gives you gravity as curvature of spacetime.
Gravitational wave is an unavoidable consequence of relativity. If information cannot travel faster than light, then causation cannot either, and hence no objects can instantly change how the way its gravity act on another object instantly; even though gravity works without range limit. Hence there is a need for some amount of time for the effect of this change to propagate, at the speed of light. This propagation is caused by gravitational wave.
Over a mostly flat region of spacetime, there can be small deviation. The deviation from this follows an equation approximately if it’s small enough, which turns out to be the wave equation. Hence the name gravitational *wave*.
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Graviton is a hypothetical particle that extend the idea of quantum mechanics to include general relativity. Currently, general relativity and quantum mechanics are at odd with each other, despite the fact that special relativity is compatible with both. Many other waves had been successfully quantized, for example electromagnetic wave is made into photon. So it’s a reasonable thing to believe that gravitational wave can be quantized as well. Now, this belief had so far ran into several obstacles, so it’s not like it an accepted idea either, it’s just the most well-known.
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