* Institute of Physics, Vietnam Academy of Science and Technology (VAST), Hanoi, Vietnam.

Recent research published in The European Physical Journal C [3] has highlighted an important advancement in the study of modified gravity, providing new insights into the dynamics of black holes, stars, and gravitational wave emission. The researcher constructs a "perturbative f(R) theory" to modify Einstein's general relativity (GR) and makes predictions for new physical phenomena, such as the potential of an ideally spherical star to emit gravitational waves.

2.1 Beyond general relativity: the motivation for f(R) gravity

Einstein's general relativity (GR) has been highly successful at describing gravity at the scale of our solar system. It is not successful, however, in describing large-scale cosmological phenomena like dark energy, dark matter, and the early inflationary era of the Universe. Scientists have thus been driven to look for extended theories of gravity, and one of the simplest modified theories of gravity is the f(R) theory.

The Ricci scalar, L_G = R, is the Lagrangian in general relativity (GR). The f(R) theory generalizes this by allowing the Lagrangian to be an arbitrary function of R, i.e., L_G = f(R). This minimalist modification may lead to fascinating new physics. This paper focuses on one such approach called the perturbative-f(R) theory. The technique considers models that are slight deviations from GR of the type f(R) = R + λh(R), where the requirement ∣λh(R)∣ ≪ ∣R∣ is guaranteed. Most importantly, this is a small adjustment to the underlying Lagrangian of the theory, as opposed to an approximation for weak gravity. This makes the method usable even in regions of extremely high spacetime curvature, e.g., in the region of the event horizon of a black hole.

2.2 Key findings: from clarifying GR to unveiling new physics

The paper is divided into two principal sections: a thorough re-examination of a charged, spherical source of gravity in general GR and calculation within the newly developed perturbative-f(R) theory on the same setup.

2.3 Revisiting the Reissner-Nordström metric in GR

Before changing GR, the paper clarifies some age-old problems with the metric of a charged, spherical body.

• Static outside, dynamic inside: The study corroborates the fact that away from a spherical source, the gravitational field is static (time-independent). Inside the source, however, the metric may be time-dependent, for example, when the star is collapsing or pulsating.

• Gravity is always attractive: The article resolves a long-standing historical debate by correctly defining gravitational acceleration in physical distance and time intervals. It proves that the gravitational field of a charged source is always attractive, contrary to some previous claims. The gravitational force is infinitely strong at the event horizon.

• A single event horizon: A black hole that is charged contains a single physically meaningful event horizon, r_sq. Importantly, it is always smaller than the horizon of an uncharged Schwarzschild black hole with the same mass—not larger, as some earlier claims suggested.

2.4 New phenomena in perturbative-f(R) gravity

As in the perturbative method, it reveals a variety of effects not available within standard GR.

• Gravitational waves from a spherical source: In GR, a fully spherical, pulsating, or collapsing object cannot radiate gravitational waves. However, the perturbative-f(R) theory suggests that if the time-varying radius of the source occurs, then its exterior metric also becomes time-varying and can radiate gravitational waves. This is in conflict with Birkhoff's theorem, a pillar of GR for spherical systems. The finding that gravitational waves could have a spherically symmetric source will be relevant for the study and discussion of experimental results, such as measurement results at LIGO in the near future.

• No electromagnetic waves: Even if the gravitational field is time-dependent, it is still proven in the literature that a spherically symmetric source never radiates electromagnetic waves.

• Einstein's solutions improved using "effective mass": The theory improves GR solutions by replacing total mass M_f with an effective mass M_ff(t) that depends upon time t. The effective mass contains correction terms that depend upon the specific f(R) model and can be time-dependent if the source is non-static.

• Reconfiguring black hole horizons: The effective mass directly impacts the size calculated for the event horizon of a black hole. For Sgr A*, the supermassive black hole at the center of the galaxy, the authors estimate that in the model f(R) = R + λR², the event horizon would be about 0.08% less than GR calculates, a very small but potentially measurable difference.

2.5 New methods and broader implications

Besides these tangible outcomes, the paper introduces novel analytic techniques and addresses fundamental questions in modified gravity.

2.6 A new embedding method

The paper presents a new and simpler method for embedding the local spacetime of a black hole or star into the expanding universe background (Friedmann–Lemaître–Robertson–Walker, FLRW cosmology). The existing methods were typically convoluted and entailed putting some special conditions in place. The new method is simpler and less numerically demanding, providing a more plausible model of astrophysical objects in our evolving cosmos.

2.7 The uniqueness of solutions

One of the typical pathologies of f(R) gravity is that when solving for equations in the absence of a matter source, it may happen that one can have an infinite number of solutions, and thus one cannot define the physically correct solution. In this work, it is argued that this issue is an artifact of the matter source being ignored. By solving the field equations completely, when the right-hand side of the equations has the energy–momentum tensor (T_μν) into the field equations, they are constrained and yield one consistent solution. This vindicates the physical hypothesis that a specific gravitational source will warp spacetime in just one way.

2.8 The TOV equation

The Tolman-Oppenheimer-Volkoff (TOV) equation is the equation of the internal pressure and density structure of a star. Earlier attempts to form a TOV equation in general f(R) gravity yielded expressions that were vastly complex and not explicit. Within this work, a new, explicit, simplified TOV equation is derived with the perturbative method and, for the first time, is applicable to charged gravitational sources.

2.9 Conclusion and outlook

This research presents a concise and elegant framework to investigate gravity beyond Einstein. The perturbative-f(R) theory not only corrects GR predictions for phenomena like planetary precession and black hole masses but also introduces fully new effects, for instance, gravitational waves from spherical sources.

With the development of simpler tools for cosmological embedding and stellar structure, and with the solution of the fundamental problem of uniqueness of solutions, this work lays a firm foundation for additional research. It is a significant step toward finding an ideal f(R) model that will solve the grand cosmological puzzles and comply with precise observations of stars and black holes.

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