Jacobson's entropic gravity (JEG) is based on Maximum Entropy Principle (MaxEnt) applied to the metric field equations. In JEG, the entropy functional S[γ] is extremized under the constraint of a conserved energy-momentum tensor Tμν. The resulting equations exhibit similarity to Einstein's field equations but include an additional term proportional to the square root of the determinant of the metric tensor, √|γ|.
MaxEnt is a method for inferring probability distributions that make the fewest assumptions about the underlying system, given available information. In the context of JEG, MaxEnt is used to find the most probable distribution for the metric tensor γμν, given the constraint of a conserved energy-momentum tensor Tμν.
The entropy functional S[γ] is defined as
S[γ] = -∫d4x√|γ|C(R)log(Q[γ]/Q0),
where d4x represents the spacetime volume element, √|γ| denotes the determinant of the metric tensor, C(R) is a function that encodes the constraints from quantum theory, and Q[γ] is the configuration probability density. The normalization constant Q0 ensures that the total probability integrates to unity.
By extremizing S[γ] with respect to variations in the metric tensor δγμν, Jacobson derives the following entropic gravity field equations:
Gμν + Λγμν = 8πT(Tμν + <Tμν>q),
where Gμν represents the Einstein curvature tensor, Λγμν is a cosmological constant term, and <Tμν>q denotes the quantum expectation value of the stress-energy tensor. The term proportional to the square root of the determinant, √|γ|, appears implicitly through the normalization condition for Q[γ].
JEG shares some similarities with GR, such as yielding metric field equations that describe the curvature of spacetime in response to matter and energy distributions. However, it also introduces new terms and concepts, including:
- Quantum corrections: JEG predicts quantum corrections to the classical gravitational field equations due to the presence of the quantum expectation value <Tμν>q in the entropic gravity field equations. These corrections could potentially explain deviations from GR observed in experiments or astrophysical phenomena.
- Temperature dependence: Since entropy plays a central role in JEG, temperature effects are expected to influence gravitational interactions. This could lead to intriguing implications for black hole thermodynamics and the evolution of the universe.
- Cosmology: JEG offers novel insights into cosmological phenomena, such as structure formation and inflation, by treating matter as an emergent property arising from the statistical organization of microscopic degrees of freedom.
Verlinde's emergent gravity (VEG) proposes that gravity arises from the organization of matter into coherent structures, which in turn create a gravitational potential through their entropic forces. The metric field equations are obtained by balancing these entropic forces against the usual partial derivatives of the entropy density.
In VEG, the gravitational force between two objects is derived from their entropic properties rather than geometric curvature. The entropic force Fent is given by
Fent ∝ ∇S,
where S represents the entropy of the system and ∇ denotes the gradient operator. The force law exhibits a power-law dependence on mass and distance, in contrast to the inverse-square law of Newtonian gravity.
The metric field equations in VEG are derived by requiring that the sum of the second partial derivatives of the entropy density σ equals the Einstein tensor multiplied by a constant G:
∇μ(√−gσγμν) = G(Tμν + <Tμν>q),
where g is the metric tensor, Tμν is the classical stress-energy tensor, and <Tμν>q denotes the quantum expectation value of the stress-energy tensor. This equation implies that the metric structure emerges from the distribution of entropy in the system.
VEG offers intriguing predictions distinct from both GR and JEG, including:
- Deviations from GR in strong-field regimes: VEG predicts deviations from GR in strong-field regimes, such as black holes and cosmological expansion, due to the nonlinear dependence of entropic forces on mass and distance. These deviations could be tested through observations of gravitational lensing, gravitational waves, or other astrophysical phenomena.
- Black hole thermodynamics: In VEG, black holes are viewed as regions of maximal entropy rather than singularities. This perspective has implications for information preservation and the holographic principle, as well as potential connections to quantum gravity theories like loop quantum gravity.
- Dark matter and dark energy: VEG provides an alternative explanation for dark matter and dark energy based on the organization of matter into coherent structures and the resulting entropic forces. This approach could offer new insights into these enigmatic phenomena and their relationship to gravity.
Both Jacobson's entropic gravity and Verlinde's emergent gravity represent ambitious attempts to reconcile quantum mechanics with gravitation by treating gravity as an emergent phenomenon arising from the statistical organization of matter and energy at large scales. While sharing some similarities with GR, they also introduce novel concepts and predictions that challenge conventional wisdom and open up new avenues for research. Ongoing efforts aim to address open questions, refine the mathematical formulations, and explore experimental possibilities, ultimately striving for a more comprehensive understanding of the physical world.