Abstract
Electro-vacuum black holes are scale-invariant; their energy-momentum tensor is traceless. Quantum corrections of various sorts, however, can often produce a trace anomaly and a breakdown of scale-invariance. The (quantum-corrected) black hole solutions of the corresponding gravitational effective field theory (EFT) have a non-vanishing Ricci scalar. Then, the presence of a scalar field with the standard non-minimal coupling $\xi \phi^2 R$ naturally triggers a spontaneous scalarisation of the corresponding black holes. This scalarisation phenomenon occurs for an (infinite) discrete set of $\xi$. We illustrate the occurrence of this phenomenon for two examples of static, spherically symmetric, asymptotically flat black hole solution of EFTs. In one example the trace anomaly comes from the matter sector -- a novel, closed form, generalisation of the Reissner-Nordstr\"om solution with an $F^4$ correction -- whereas in the other example it comes from the geometry sector -- a noncommutative geometry generalization of the Schwarzschild black hole. For comparison, we also consider the scalarisation of a black hole surrounded by (non-conformally invariant) classical matter (Einstein-Maxwell-dilaton black holes). We find that the scalarised solutions are, generically, entropically favoured.
Highlights
AND MOTIVATIONThe Schwarzschild black hole (BH) is scale invariant
In one example the trace anomaly comes from the matter sector—a novel, closed form generalization of the Reissner-Nordström solution with an F4 correction—whereas in the other example it comes from the geometry sector—a noncommutative geometry generalization of the Schwarzschild black hole
At the level of some effective field theory that takes into account the leading quantum effects (e.g., Euler-Heisenberg nonlinear electrodynamics [8]), this anomaly is materialized in the appearance of an energymomentum tensor trace, which generically implies, via the semiclassical Einstein equations, a nonvanishing Ricci scalar
Summary
The Schwarzschild black hole (BH) is scale invariant. That is, even though vacuum general relativity introduces a scale, the Planck length, lP, via Newton’s constant, a classical Schwarzschild BH with a Schwarzschild radius RS of the order of the Planck length RS ∼ lP and another with RS ∼ 109 M⊙ (like the one at the center of M87 [1]) are identical, up to a scale transformation. As argued below, both quantum-corrected BHs emerging within some effective field theory and classical solutions beyond electrovacuum can have a nonvanishing Ricci scalar, here we study the possibility that spontaneous scalarization exists due to this coupling. The latter are entropically preferred over the scalar-free ones in the model where entropy in unambiguous These observations support the suggestion that spontaneous scalarization occurs dynamically, even for BHs that would be classically scale invariant, once quantum corrections are taken into account.
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