Black holes in scalar-tensor gravity (1109.6324v2)

Abstract: Hawking has proven that black holes which are stationary as the endpoint of gravitational collapse in Brans--Dicke theory (without a potential) are no different than in general relativity. We extend this proof to the much more general class of scalar-tensor and f(R) gravity theories, without assuming any symmetries apart from stationarity.

• The paper generalizes Hawking’s work by demonstrating that stationary black holes in scalar-tensor gravity obey GR solutions when the scalar potential meets specific conditions.
• It employs both Jordan and Einstein frames to analyze scalar dynamics, highlighting the essential role of conformal transformations in preserving GR symmetries.
• The study underscores that stable, non-divergent scalar fields are crucial to ensuring that black hole solutions remain compatible with general relativity.

Overview of Black Holes in Scalar-Tensor Gravity

This paper by Thomas P. Sotiriou and Valerio Faraoni extends the results of Stephen Hawking's 1972 work on black holes in Brans-Dicke theory within the broader context of scalar-tensor and $f(R)$ theories of gravity. The authors seek to determine whether stationary black holes, as the endpoints of gravitational collapse, manifest differently in these alternative theories compared to general relativity (GR). They conduct their analysis without imposing any additional symmetry constraints beyond stationarity, thus adding generality to their investigation.

Scalar-tensor theories, which include Brans-Dicke theory as a particular case, introduce a scalar field alongside the metric tensor of general relativity. These models are often seen as effective low-energy approximations for quantum gravity theories, encouraging exploration beyond classical GR descriptions. The action for a general scalar-tensor theory is characterized by two arbitrary functions $\omega(\phi)$ and the potential $V(\phi)$.

The analysis moves beyond the special case of spherically symmetric spacetimes, which had already been shown to yield black holes indistinguishable from those predicted by GR, to address more general setups. By utilizing both the traditional Jordan frame and the conformally transformed Einstein frame, the paper examines scalar dynamics and the conditions required for scalar-tensor theories to produce GR-consistent black hole solutions.

Key Results

The paper makes several assertions that hold significance in theoretical physics:

1. Stationarity and Asymptotic Conditions: Hawking's conclusions regarding stationary black holes in Brans-Dicke theory are generalized. In these extended theories, black holes that are outcomes of gravitational collapse remain solutions of GR, provided certain conditions on the scalar potential hold. Specifically, the requirement is that there be no effective cosmological constant (i.e., $V(\phi_0) = 0$).
2. Scalar Stability: The authors emphasize the stability of the scalar field. They proclaim that, barring instabilities (indicated by conditions on $U(\phi)$'s second derivative in the Einstein frame), only GR solutions are scalable to gravitating black holes under these theories.
3. Role of Conformal Transformations: The scalar is typically non-minimally coupled to gravity in the Jordan frame. However, in the Einstein frame, the action's vacuity of second scalar derivatives allows the imposition of the Weak Energy Condition. This transformation is crucial as it maintains essential symmetries, preserving the solution characteristics aligned with GR predictions.
4. Implications of Divergence: The scalar field is expected to remain finite throughout spacetime. Any divergence could lead to discrepancies with GR-approved black holes, possibly indicating exotic physics beyond the capability of regular scalar-tensor frames.

Implications and Future Research

This exploration offers concrete conditions and limitations within scalar-tensor theories that resonate closely with GR's predictions for black holes. The proofs ensure a wider class of theoretical models align with the classical dogma of GR, thus restricting spurious modifications to quintessentially GR-compatible configurations unless specific breakdowns occur. This supports GR's robustness in describing black holes unless significant deviations imposed by unusual scalar-tensor dynamics, such as instability or potential divergence, are encountered.

The paper paves avenues for further investigations, particularly in quantum gravity frameworks where these alternative theories could have more profound implications. Moreover, the boundary conditions rejected here could guide searches for novel solutions or approximate models representing quantum aspects, dark energy, or early universe phenomena. Understanding the nuances of conformal transformations and the constraints posed by non-trivial potentials will remain crucial as researchers continue to interrogate the foundational principles of gravity and its cosmic applications.