Maybe another stupid question, but what's the energy of a graviton? Is it $\hbar \omega$?
Does it emit gravitons when an apple falls onto the ground, like photons be emitted when an electron transits from a higher energy level to a lower one?
Answer
It is assumed, but not measured that e =hw. See this reference about the difficulty of measuring a single graviton. http://arxiv.org/abs/gr-qc/0601043
It is an interesting exercise to compute the amount and wavelength of gravitons from a falling apple. As a first approximation, one can compute the gravitational waves emitted by an apple orbiting the earth and use the formulas developed for binary black holes and neutron stars by the LIGO VIRGO etc experiments. In low earth orbit, the apple will radiate at a wavelength of approximately 90 light minutes, give or take a factor of two. This is very roughly 10^12 meters. Each graviton will then carry about 10^-30 ergs, a very small amount. According to the Wikipedia gravitational wave article the sun earth system emits 200 watts of gravitational radiation, but this would typically emit 10^-34 erg gravitons. 200 watts is 2 10^9 erg seconds, so the sun earth system is emitting 10^43 gravitons per second. Using the formula from the Wikipedia article, the earth-apple system with a one tenth kilogram apple would emit 10^46 times less gravitational wave power, or 10^-42 watts or 10^-35 ergs/second. This implies an average of one graviton every 10^5 seconds, or about once every twenty 5400 second orbits. If your falling apple falls for about one second, it should emit one graviton once out of every one hundred thousand tries. To be more analagous to the orbital picture, your apple should be thrown horizotally like a baseball, rather than falling vertically
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