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We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field.
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Some details of the gedanken construction: 1. Place the N N copies of the system S S at distances r r apart, where r r is much larger than the size of S S . We can stabilize the copies against their mutual gravitational attraction by assuming a repulsive force mediated by a boson with mass ∼ 1 / r \sim 1/r . This finite range interaction is negligible at asymptotically large distances. 2. Consider the graviton field at distances R R much larger than N 1 / 3 r N^{1/3}~r (i.e., far from all of the matter sources). 3. Take the limits R , N , r → ∞ R,N,r~\rightarrow~\infty such that the leading contribution to the Newtonian potential at large R R is given by the total energy E = N E n E=N\,E_{n} , up to corrections that can be made as small as desired
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R. Casadio, [arXiv:2103.00183 [gr-qc]]
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Subleading corrections such as the cubic corrections to the action can also generate 1 / r 5 1/r^{5} corrections. However, these will be suppressed by a higher power of the Planck scale, i.e., the leading cubic corrections are proportional to l p 4 / r 4 ( G M / r ) l_{p}^{4}/r^{4}(GM/r) . It is important to notice that our correction is renormalization group invariant as it does not depend on μ \mu , which is a nontrivial consistency check. The presence of the corrections (and thus the hair) is gauge independent, but the coordinate dependent form of these corrections will depend on the choice of reference frame, see [ 12 ]
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