Would Gravitons have Gravity in it of themselves?

Question

If you held a handful of gravitons, would you be holding a handful of gravity, drawing things towards it, or do gravitons have to be exchanged or transmitted for gravity to take effect, in which case, a handful of stationary gravitons would not draw things towards itself?

Answer 1

At the present time, the graviton is a hypothetical particle, and we won't know its properties until we have a complete theory of quantum gravity. However, in most models the graviton is a massless particle like the photon. So, like the photon, it always travels at the speed of light. If these models are correct there is no such thing as a "stationary graviton". However, gravitons - again, like photons - carry energy and so contribute to the energy density of spacetime which causes the curvature of spacetime in general relativity. And it is this curvature of spacetime that we call gravity.

Answer 2

Talking about "gravitons" is inherently somewhat speculative, as there is no good quantum theory of gravity.

that said, John Archibald Wheeler worked out "geon" solutions to the classical Einstein equation, where a spacetime containing nothing but gravitational waves at early time ends in a collapse to a black hole at a late time, which indicates that gravitational waves (which, in a quantum theory, would be made up of many "gravitons") do, in fact, gravitate.

Note that this is a natural thing to expect from the Einstein equation, which is very nonlinear, which means that gravitational waves do couple with and interact with each other.