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1.Modern gravitational physics has fundamental problems, including
1)the Planck-scale divergence problem,
2)the black hole singularity problem,
3)the inflation problem, and
4)the dark energy problem.
The continuous appearance of these anomalous phenomena suggests that an important element is missing from our present understanding of gravity.
To address these problems, we first need to examine whether they share any common features.
2.Although these major unresolved problems in gravitational physics appear to differ in scale and physical properties, they all arise under conditions of high compactness
1)The Planck-scale divergence problem: According to quantum mechanics, probing a spatial scale R requires the wavelength of theprobe to be at least comparable to, or shorter than, that scale. Therefore, decreasing thelocalization scale requires increasingly large momentum and energy. At the Planck scale, the Schwarzschild radius corresponding to the energy required for localization becomes comparable to the spatial scale being probed.
The Planck length is defined as l_P = (ℏG/c^3)^(1/2) = Gm_P/c^2 = 0.5 R_S(m_P). Conceptually, the Planck length is smaller than the Schwarzschild-equivalent radius associated with the Planck mass, satisfying l_P=R_S(m_P)/2. This places Planck-scale localization in a high-compactness regime. Because the Planck scale belongs to the region (R<R_S), the trans-Planckian regime is essentially a problem that occurs in a high-compactness state, (R_S/R>1).
2)The black hole singularity problem: The interior of a black hole is defined by (R<R_S). Therefore, the singularity problem is one of the central pieces of evidence that general relativity is incomplete under high-compactness conditions, (R_S/R>1).
3)The inflation and dark energy problems: Although the average density of the observable universe appears remarkably low, its global mass-energy distribution tells a different story.
Even in an infinite and homogeneous spherical distribution, the acceleration of an expanding shell at radius R is determined by the fluid contained inside that shell and is independent of the matter outside it. Including the relativistic contribution of pressure, the acceleration is
d^2R/dt^2=−(4πG/3)(ρ+3P/c^2)R.
Thus, even when the universe extends beyond the chosen region and matter continues to exist outside it, the radial gravitational dynamics of a spherical domain is described in terms of the energy density and pressure enclosed within that domain. This is the basis of the shell theorem and Birkhoff-type spherical reasoning commonly used in cosmology.
Accordingly, calculating the mass enclosed within the observable universe or the particle horizon region as M(R)=(4π/3)ρ_mR^3 and then evaluating 2GM(R)/Rc^2 does not amount to replacing the FLRW metric with the Schwarzschild metric. It is simply a calculation of the dimensionless gravitational compactness associated with the matter content enclosed within a spherical region of radius R.
If the notation R_S for the Schwarzschild radius unnecessarily suggests a black-hole interpretation, the same physical quantity can simply be written as 2GM/c^2R, or, apart from the numerical factor 2, as GM/c^2R. These are dimensionless gravitational quantities constructed from G, M, R, and c, and their physical meaning does not require a prior black-hole interpretation.
For an observable universe with a radius of R = 46.5 Gly and a critical density of ρ_c ≈ 8.5x10^{-27}[kg/m^3], the corresponding Schwarzschild-equivalent radius is approximately R_S ≈ 477 Gly. This yields a cosmic compactness ratio of R_S/R ≈ 10.3 > 1 (Even when considering only matter (baryons and dark matter), the compactness is still greater than 1, with R_S/R ≈ 3.23>1). Therefore, on cosmological scales, the observable universe has an extremely low average density and weak local spacetime curvature, yet its global gravitational compactness exceeds unity. This demonstrates that low density and weak local curvature do not necessarily imply weak gravitational compactness on large scales. Classical General Relativity is incomplete in describing the observable universe, and additional components, such as an inflaton field and dark energy, are required to explain the expansion history of the universe within classical General Relativity.
Overall, classical general relativity is presumed to be incomplete in high-compactness regimes and, therefore, in physical environments characterized by such high compactness, including black hole interiors, the Planck-scale regime, and the observable universe.
3. The need for an effective repulsive gravitational response at high compactness or critical scales
Inflation and dark energy account for the accelerated expansion of the early and late universe, respectively. Such accelerated expansion corresponds to an effective repulsive gravitational response.
The black hole singularity problem requires a physical mechanism capable of halting unlimited gravitational collapse at a finite radius. An effective repulsive gravitational response could provide such a mechanism.
The Planck-scale divergence problem likewise requires a physical mechanism that limits unlimited spatial localization. If a repulsive gravitational response emerges at sufficiently small radii, a minimum radius may arise beyond which further localization becomes impossible, opening a possible path toward resolving ultraviolet divergences.
Thus, these major problems share a common feature: they may be resolved through an effective repulsive gravitational response in high-compactness regimes (R_S/R ≳ 1).
The difficulties encountered by gravitational theory in these three distinct regimes suggest the possible existence of a physical quantity that is almost negligible in ordinary weak gravitational fields but becomes significant in high-compactness regimes. Furthermore, this single physical quantity may provide a common physical explanation for, and a potential solution to, the problems arising in all three domains.
Therefore, it is worth considering an extension of General Relativity that explicitly incorporates into the gravitational source the physical quantities that become significant in high-compactness regimes.
Accordingly, upon reviewing the derivation process of the field equations of General Relativity... it seems that the problem lies in the XY part.
1)the Planck-scale divergence problem,
2)the black hole singularity problem,
3)the inflation problem, and
4)the dark energy problem.
The continuous appearance of these anomalous phenomena suggests that an important element is missing from our present understanding of gravity.
To address these problems, we first need to examine whether they share any common features.
2.Although these major unresolved problems in gravitational physics appear to differ in scale and physical properties, they all arise under conditions of high compactness
1)The Planck-scale divergence problem: According to quantum mechanics, probing a spatial scale R requires the wavelength of theprobe to be at least comparable to, or shorter than, that scale. Therefore, decreasing thelocalization scale requires increasingly large momentum and energy. At the Planck scale, the Schwarzschild radius corresponding to the energy required for localization becomes comparable to the spatial scale being probed.
The Planck length is defined as l_P = (ℏG/c^3)^(1/2) = Gm_P/c^2 = 0.5 R_S(m_P). Conceptually, the Planck length is smaller than the Schwarzschild-equivalent radius associated with the Planck mass, satisfying l_P=R_S(m_P)/2. This places Planck-scale localization in a high-compactness regime. Because the Planck scale belongs to the region (R<R_S), the trans-Planckian regime is essentially a problem that occurs in a high-compactness state, (R_S/R>1).
2)The black hole singularity problem: The interior of a black hole is defined by (R<R_S). Therefore, the singularity problem is one of the central pieces of evidence that general relativity is incomplete under high-compactness conditions, (R_S/R>1).
3)The inflation and dark energy problems: Although the average density of the observable universe appears remarkably low, its global mass-energy distribution tells a different story.
Even in an infinite and homogeneous spherical distribution, the acceleration of an expanding shell at radius R is determined by the fluid contained inside that shell and is independent of the matter outside it. Including the relativistic contribution of pressure, the acceleration is
d^2R/dt^2=−(4πG/3)(ρ+3P/c^2)R.
Thus, even when the universe extends beyond the chosen region and matter continues to exist outside it, the radial gravitational dynamics of a spherical domain is described in terms of the energy density and pressure enclosed within that domain. This is the basis of the shell theorem and Birkhoff-type spherical reasoning commonly used in cosmology.
Accordingly, calculating the mass enclosed within the observable universe or the particle horizon region as M(R)=(4π/3)ρ_mR^3 and then evaluating 2GM(R)/Rc^2 does not amount to replacing the FLRW metric with the Schwarzschild metric. It is simply a calculation of the dimensionless gravitational compactness associated with the matter content enclosed within a spherical region of radius R.
If the notation R_S for the Schwarzschild radius unnecessarily suggests a black-hole interpretation, the same physical quantity can simply be written as 2GM/c^2R, or, apart from the numerical factor 2, as GM/c^2R. These are dimensionless gravitational quantities constructed from G, M, R, and c, and their physical meaning does not require a prior black-hole interpretation.
For an observable universe with a radius of R = 46.5 Gly and a critical density of ρ_c ≈ 8.5x10^{-27}[kg/m^3], the corresponding Schwarzschild-equivalent radius is approximately R_S ≈ 477 Gly. This yields a cosmic compactness ratio of R_S/R ≈ 10.3 > 1 (Even when considering only matter (baryons and dark matter), the compactness is still greater than 1, with R_S/R ≈ 3.23>1). Therefore, on cosmological scales, the observable universe has an extremely low average density and weak local spacetime curvature, yet its global gravitational compactness exceeds unity. This demonstrates that low density and weak local curvature do not necessarily imply weak gravitational compactness on large scales. Classical General Relativity is incomplete in describing the observable universe, and additional components, such as an inflaton field and dark energy, are required to explain the expansion history of the universe within classical General Relativity.
Overall, classical general relativity is presumed to be incomplete in high-compactness regimes and, therefore, in physical environments characterized by such high compactness, including black hole interiors, the Planck-scale regime, and the observable universe.
3. The need for an effective repulsive gravitational response at high compactness or critical scales
Inflation and dark energy account for the accelerated expansion of the early and late universe, respectively. Such accelerated expansion corresponds to an effective repulsive gravitational response.
The black hole singularity problem requires a physical mechanism capable of halting unlimited gravitational collapse at a finite radius. An effective repulsive gravitational response could provide such a mechanism.
The Planck-scale divergence problem likewise requires a physical mechanism that limits unlimited spatial localization. If a repulsive gravitational response emerges at sufficiently small radii, a minimum radius may arise beyond which further localization becomes impossible, opening a possible path toward resolving ultraviolet divergences.
Thus, these major problems share a common feature: they may be resolved through an effective repulsive gravitational response in high-compactness regimes (R_S/R ≳ 1).
The difficulties encountered by gravitational theory in these three distinct regimes suggest the possible existence of a physical quantity that is almost negligible in ordinary weak gravitational fields but becomes significant in high-compactness regimes. Furthermore, this single physical quantity may provide a common physical explanation for, and a potential solution to, the problems arising in all three domains.
Therefore, it is worth considering an extension of General Relativity that explicitly incorporates into the gravitational source the physical quantities that become significant in high-compactness regimes.
Accordingly, upon reviewing the derivation process of the field equations of General Relativity... it seems that the problem lies in the XY part.