The thermodynamic interpretation of gravity, proposed by Jacobson in 1995, derives Einstein's equations from the Clausius relation on local horizons. This line of research sits alongside work on the Universe's horizon as a heat engine, on quantum ripples inflating the Universe, and on the entropic symphony of inflation. Most such studies remain at the level of cosmological dynamics, that is, they modify the Friedmann equations but do not rise to the level of field equations. This work goes further: it embeds generalized entropy into the very derivation of the gravitational equations, offering a fundamental path to modified gravity.
The author generalizes the Clausius relation by replacing the Bekenstein-Hawking entropy with an arbitrary function of it. Within Jacobson's formalism this leads to an effective gravitational constant inversely proportional to the derivative of entropy with respect to area. Standard logarithmic corrections are then analyzed, and their inability to eliminate the singularity is demonstrated. To solve the problem, a new entropy is introduced: the horizon is modeled as a set of two-dimensional quantum harmonic oscillators, and the microscopic entropy is computed through the number of microstates. A minimum horizon area A₀ is introduced, playing the role of the ground state. The resulting entropy is substituted into the modified equations, and the cosmological evolution of a flat Friedmann-Robertson-Walker universe is studied.
The generalized Friedmann equation has the form H² = 8πG(A−A₀)ρ / (A−A₀ + 4αG), where A = 4π/H² is the area of the apparent horizon. In the high-density limit the Hubble parameter tends to a finite constant H²_early ≈ 4π/A₀, which corresponds to a de Sitter phase with exponential expansion. This means there is no initial singularity: the scale factor never goes to zero. Entropy and temperature in the early Universe are finite: $S_{earl}$y = A₀/(4G), $T_{earl}$y = 1/√(πA₀). In the late-time limit the equation reduces to H² ≈ (8πG/3)ρ(1 − ρ/$ρ_c$), where $ρ_c$ = 3/(8αG²), which coincides with the effective equation of loop quantum cosmology at leading order. However, the full theory contains higher-order corrections and differs in the ultraviolet regime: instead of a bounce it gives an inflationary phase.
The work shows that modifications of horizon entropy translate universally into changes of the gravitational coupling, which provides a thermodynamically grounded path to modified gravity theories. Unlike neighboring works, which either study the cosmological consequences of specific entropies or discuss quantum aspects of gravity, here a systematic derivation of the field equations from a generalized Clausius relation is proposed. This raises the thermodynamic approach from the level of phenomenology to the level of a fundamental theory. The new entropy with a minimum area introduces a natural ultraviolet cutoff and demonstrates how quantum properties of the horizon can resolve singularities.
The proposed formalism opens the way to studying a broad class of entropic deformations and the corresponding gravity theories. One can study the consequences for black hole physics, in particular evaporation and the information paradox, where the minimum area may play the role of a Planckian remnant. It is also promising to apply the model to perturbation theory and structure formation in order to test predictions against observational data. A generalization to non-metric theories and the inclusion of additional fields are possible.
The results affect early-Universe cosmology by offering an alternative to inflation with scalar fields, as well as quantum gravity by linking entropic corrections to the elimination of singularities. The model may influence the interpretation of cosmic microwave background and large-scale structure observations.
To test the stability of the resulting inflationary solution and compute the spectrum of primordial perturbations. To study the behavior of the model in the presence of matter with an equation of state different from dust, and to consider spherically symmetric black hole solutions.
The work is directly related to the problem of singularities in general relativity and the search for quantum-gravitational mechanisms to eliminate them. It also touches on the nature of dark energy and inflation, offering a geometric explanation without introducing new fields. The question of the microscopic origin of horizon entropy remains open, as does the question of which particular entropic deformation is realized in nature.
The minimum horizon area A₀ plays a role analogous to the ground-state energy of a quantum harmonic oscillator: even in an "empty" horizon there remains an irreducible quantum activity. In the model, allowed areas are quantized as A = nA₀, where n ≥ 2, meaning the n=1 state is forbidden due to logarithmic divergence.
The idea that gravity arises from horizon thermodynamics echoes Liu Cixin's novel The Three-Body Problem, where an alien civilization manipulates fundamental laws through curled-up dimensions. Here, horizon entropy acts as code that rewrites gravity.