Theory – III. Gravity
Is general relativity (GR) the fundamental theory of gravity? Should it be modified? This is the central topic under investigations. It is also related to the most fundamental problem about gravity: its conflict with quantum nature.
From the conflict between gravity and quantum nature, one expects the gravity law be different from GR at high energies or small length scales where quantum fluctuations of gravity (if exist) cannot be ignored. This refers to the UV modification of gravity. From the need of anti-gravity and extra gravity in cosmology, one may smell a hint of a different behavior of gravity at cosmological scales, which refers to the IR modification of gravity. In addition, practically speaking, because the Einstein equations are nonlinear and the stress energy invoked therein is always averaged over space-time at some scale, in principle the form of the field equations can change with the averaging length scale. To sum up, there are three kinds of modification of gravity: the UV modification which serves quantum gravity, the IR modification as an alternative to dark energy (hardly dark matter), and the modification due to the averaging in space-time.
Modified theories of gravity in general possess more degrees of freedom than GR. These addition degrees of freedom may cause problems and make the theory fail. It is essential to examine the possible problems such as instability, the ghost problem, acausality, etc. Concerning these problems, our investigations show that GR is the best theory of gravity so far.
For the interplay between gravity and quantum physics, black holes provide a wonderful stage. On this stage several plays about quantum gravity have been well performed, such as the black hole radiation, entropy and temperature, the information loss problem, and entanglement. They provide the threads of the reconciliation between gravity and quantum nature.
The conflict between gravity and quantum nature is about the formalism of the theories: the former in the classical framework and the latter in the quantum. To reconcile them, naively their formalism needs to be unified. Since the quantum framework is more fundamental than the classical, people usually resort to the quantization of gravity. One may quantize metric (Canonical Quantum Gravity) or some other quantities about gravity (Loop Quantum Gravity). One may quantize something else which eventually gives quantum gravity magically (String Theory).
We expect, as a criterion, that quantum gravity can solve the cosmological constant problem, in particular, giving a low-energy solution. However, the above quantum gravity theories deal with the quantum nature of gravity which manifests at high energies, but do not give gravity nature distinct from GR at low energies (e.g. microns); yet they do not pass the criterion. That suggests a more thorough change of thinking may be needed.
A fundamental change concerns the formalism, i.e. the framework to formulate gravity. An intriguing example for classical gravity is ENTROPIC GRAVITY, which puts GR on the level of thermodynamics. In this framework the gravitational field equations are obtained via extremizing a thermodynamic free energy (e.g. entropy) whose relation to metric is a prioi given. It suggests that GR be a macroscopic description of gravity, and the geometrical quantities in GR such as metric be macroscopic quantities. If GR is macroscopic, there should be a microscopic fundamental theory of gravity yet to be discovered. As Statistical Mechanics gives the microscopic framework for the underlying theory of (macroscopic) Thermodynamics, it is possibly the statistical framework that one should invoke to formulate (microscopic) fundamental gravity. This suggests a new formalism for gravity: the STATISTICAL FORMULATION OF GRAVITY.
The above line of though gives two new directions to the reconciliation between gravity and quantum nature: One remains in quantization and the other goes beyond. For the former, one needs to find the microscopic theory of gravity to quantize. For the latter, quantum nature needs to be recast in a framework compatible with the statistical formalism. Both are novel and worth exploration.