A research group including Masato Minamitsuji has developed a class of transformations of the spacetime metric involving higher-order derivatives, which has long been thought to be impossible, and employed it to drastically extend the known framework of gravitational theories. This novel framework provides the largest class of gravitational theories and hence allows us to handle a variety of theories in a unified manner. The framework could address mysteries in cosmology, e.g., inflation in the early Universe, dark energy that drives the late-time cosmic acceleration, black holes and gravitational wave emission. It could resolve tensions between the observational data and the theoretical prediction of general relativity, and also exhibit peculiar phenomena that can be tested by future observations.
Einstein's general relativity has passed various gravitational experiments as well as cosmological observations and is now commonly accepted as the standard model of gravitational theory. On the other hand, various extensions of general relativity have been proposed so far for several reasons, e.g., because general relativity is a low-energy effective theory and should be modified at high energies and/or because extended gravitational theories serve as a good candidate that can be tested against general relativity.
General relativity describes gravitation in terms of the metric tensor that governs the spacetime geometry. Mathematically, it can be characterized as "the most general theory written in terms of the metric that yields second-order equations of motion." In fact, a system with higher-order equations of motion in general possesses unstable degrees of freedom called Ostrogradsky ghosts. In this sense, it is reasonable to require the second-order nature of the equations of motion. Indeed, the equation of motion in Newtonian dynamics and the Maxwell equation in electrodynamics are both second-order differential equations.
In contrast to general relativity, extended gravitational theories in general involve additional degrees of freedom on top of the metric, most of which can be effectively described by scalar-tensor theories (i.e., those involving a scalar field on top of the metric). The most general scalar-tensor theory having second-order equations of motion is already known, which is now called the Horndeski theory. The Horndeski theory itself can be regarded as a general class of theories that encompasses many scalar-tensor theories.
A useful strategy for exploring the theory space is to study how scalar-tensor theories are related to each other via a transformation of the metric. A commonly used transformation is the so-called disformal transformation, which is an invertible transformation of the metric that contains the derivative of the scalar field up to the first order. Here, a transformation is said to be invertible when one can freely go back and forth between two theories that are related to each other by the transformation. The disformal transformation enables us to extend the Horndeski theory without the aforementioned problem of Ostrogradsky ghosts. On the other hand, it has long been thought to be impossible to construct an invertible transformation with higher-order derivatives of the scalar field.
Despite the common belief, the research group has developed a class of invertible disformal transformations involving higher-order derivatives and employed it to drastically extend the existing framework of scalar-tensor theories. The extended framework has been named "the generalized disformal Horndeski theory." This framework serves as the most general framework of extended gravitational theories and is expected to accelerate inclusive research of gravitational theories both from theoretical and observational points of view.
The results of this study have been published in Progress of Theoretical and Experimental Physics. The article：Progress of Theoretical and Experimental Physics 2023, 013E01 (2023),https://doi.org/10.1093/ptep/ptac161
(Figure Caption) Schematic picture of the extended gravitational theories. Performing the disformal transformation on the Horndeski theory ("H" in the figure), one can obtain a broader class of disformal Horndeski theory ("DH" in the figure). In this study, using a generalization of the disformal transformation involving higher-order derivatives, the generalized disformal Horndeski theory ("GDH" in the figure) has been constructed, which greatly extends the existing frameworks. It has also been shown that this strategy can be systematically generalized to transformations involving arbitrary higher-order derivatives.
