In 1995, physicist Ted Jacobson demonstrated that Einstein’s field equations of general relativity can be derived from thermodynamic principles applied to the quantum vacuum, suggesting that gravity may not be a fundamental force but rather an emergent phenomenon arising from quantum entanglement and thermodynamics.
This result grew out of puzzling features of black hole thermodynamics discovered in the 1970s: Bekenstein showed that black hole horizon area measures entropy, Hawking showed that black holes radiate thermally, and the Unruh effect revealed that an accelerating observer perceives the quantum vacuum as a thermal bath.
The key puzzle was that the second law of thermodynamics requires total entropy to increase, yet a shrinking black hole has decreasing horizon entropy. The outgoing Hawking radiation must carry enough entropy to compensate, and the precise accounting only works if Einstein’s equations hold exactly. This suggested that gravity somehow “already knows” about thermodynamics, motivating Jacobson’s insight that gravity might literally be thermodynamics in disguise.
How Jacobson Derived Einstein’s Equations from Thermodynamics
Jacobson’s derivation proceeds by treating every point in spacetime as if it lies on a horizon, using the concept of a local Rindler horizon.
A Rindler horizon is not a physical object but a causal boundary: in flat spacetime, pick any point and consider the wedge-shaped region bounded by light rays emanating from that point. This wedge has a boost symmetry (a Lorentz boost), and the quantum vacuum, when restricted to this wedge, appears as a thermal state with respect to the boost Hamiltonian, not the ordinary time-translation Hamiltonian.
This thermal character is a consequence of quantum entanglement: the vacuum state of quantum fields is a pure state globally, but when you restrict observations to one side of a boundary, the degrees of freedom on that side are entangled with those on the other side. Tracing over the unobserved side yields a mixed thermal state. The amount of entanglement entropy is proportional to the area of the boundary, not its volume.
Jacobson then imposed three conditions on each local Rindler horizon: (1) the entropy of the region is proportional to the horizon area (the Bekenstein-Hawking relation), (2) energy is conserved (the stress-energy tensor is divergence-free), and (3) the Clausius relation holds (the change in entropy equals the heat flux divided by temperature). Combining these, he found that the area must evolve consistently with Einstein’s equations. In this sense, the Einstein field equations are the equation of state of the quantum vacuum.
The Quantum Vacuum, Entanglement, and the Origin of Thermodynamic Behavior
The quantum vacuum is not empty; it is a highly entangled pure quantum state whose local restrictions appear thermal.
In quantum field theory, fields can be decomposed into modes, each behaving like a quantum harmonic oscillator. The vacuum state has these modes in their ground state globally, but when space is divided into two regions, the modes on one side are entangled with partner modes on the other side. The correlation is in the occupation number of each mode: the excitation level of a mode on one side is correlated with that of its partner on the other side.
If an observer only has access to one side, the entangled partner degrees of freedom are unobserved, and the state appears mixed and thermal. This is not a physical wall or coordinate artifact in the usual sense—it is a consequence of restricting to a subset of observables that are causally disconnected from the other side.
The Unruh effect is a special case: a uniformly accelerating observer follows a boost symmetry orbit and perceives the vacuum at a temperature proportional to their acceleration. But the deeper statement is that the global vacuum state is thermal with respect to the boost Hamiltonian, independent of any particular observer.
Why Entanglement Entropy Is Finite: Gravity as a Natural Cutoff
Naively, the entanglement entropy of the quantum vacuum across a boundary is infinite, because there are infinitely many field modes at arbitrarily short wavelengths near the boundary.
This divergence is cut off by gravity. Fluctuations at very short distances carry very high energy (by the uncertainty principle), and when the energy density is high enough, the gravitational back-reaction becomes strong: the fluctuations create a miniature horizon that engulfs them, making it meaningless to ask which side of the boundary they are on.
This gravitational cutoff occurs at the Planck scale and yields a finite entropy proportional to the boundary area divided by the Planck length squared, matching the Bekenstein-Hawking formula. Thus, gravity is what makes black hole entropy finite and, more generally, what gives spacetime its thermodynamic character.
Local Rindler Horizons and the Meaning of “Heat” in the Derivation
The “heat” in Jacobson’s Clausius relation is not ordinary energy but boost energy—the conserved quantity associated with the boost symmetry of the Rindler wedge.
Ordinary energy is the conserved quantity from time translation symmetry. Boost energy is the conserved quantity from Lorentz boost symmetry (hyperbolic rotations in spacetime). The vacuum is thermal with respect to this boost Hamiltonian, so the relevant energy flux in the thermodynamic derivation is the flux of boost energy across the horizon.
This is why the derivation works: the temperature in the Clausius relation is the Unruh temperature associated with the boost symmetry, and the heat flux is the boost energy flux. The area-entropy relation then ties the geometry (area) to the thermodynamics (entropy and energy flux), yielding Einstein’s equations.
Diffeomorphism Invariance, Boundary Observables, and the Black Hole Information Paradox
In general relativity, coordinates are arbitrary—only diffeomorphism-invariant quantities are physically meaningful. This has profound consequences for what counts as an observable.
Don Marolf showed that in a diffeomorphism-invariant theory, the Hamiltonian that generates time evolution is a surface integral at the boundary (a gravitational flux integral). This means all observables can be expressed in terms of boundary observables, and their time evolution is governed entirely by the boundary algebra.
If the boundary algebra is closed (commutators of boundary observables yield other boundary observables), then information at the boundary is never lost—it evolves unitarily within the boundary algebra. This is Marolf’s “boundary unitarity” argument.
Applied to the black hole information paradox: the apparent loss of information when a black hole forms and evaporates arises from using local quantum field theory concepts (like entanglement across the horizon) that are not well-defined diffeomorphism-invariant observables. The paradox dissolves when one restricts to properly defined boundary observables, which evolve unitarily. The information is not lost; it is encoded in complicated correlations in the gravitational field at the boundary, though exactly how to extract it remains an open question (work by Suvrat Raju and collaborators is exploring this).
Corvino Gluing and the Non-Uniqueness of Interior Realities
Corvino gluing is a mathematical technique in general relativity showing that different interior spacetime geometries can be glued to the same exterior geometry while satisfying Einstein’s equations.
This means that identical exterior measurements (what an outside observer can see) can correspond to radically different interior realities. It challenges a naive form of determinism: the exterior does not uniquely determine the interior.
This is related to the broader theme that in general relativity, the physical content is in diffeomorphism-invariant relationships, not in the specific geometric details that can be changed by coordinate transformations or gluing constructions.
Holographic Duality (AdS/CFT) and the Emergence of Spacetime
The AdS/CFT correspondence, discovered by Maldacena, is a concrete realization of holography: a gravitational theory in a (d+1)-dimensional bulk spacetime is exactly equivalent to a non-gravitational conformal field theory (CFT) on the d-dimensional boundary.
In this framework, spacetime and gravity in the bulk emerge from the boundary CFT. The boundary theory is sharply defined and well-understood; the bulk description is fuzzy and approximate, involving strings, D-branes, and other objects depending on the state of the CFT.
Jacobson notes that this duality is not symmetric: the boundary CFT is the more fundamental description, while the bulk spacetime is emergent. However, the duality as currently formulated has a fixed rigid conformal geometry on the boundary, which is likely a stepping stone rather than a fundamental feature—just as Newtonian absolute space was a stepping stone to general relativity’s dynamical spacetime.
The Ryu-Takayanagi formula connects entanglement entropy in the boundary CFT to minimal surface areas in the bulk, providing a direct link between quantum entanglement and spacetime geometry. This inspired Jacobson’s 2015 work on the maximal vacuum entanglement hypothesis.
ER = EPR: Entanglement as the Fabric of Spacetime Connectivity
The ER = EPR conjecture (Maldacena and Susskind) proposes that Einstein-Rosen bridges (wormholes) and Einstein-Podolsky-Rosen entanglement are fundamentally the same phenomenon.
In AdS/CFT, two entangled CFTs on separate boundaries are dual to a single connected spacetime with an Einstein-Rosen bridge (a wormhole) between them. More generally, the entanglement of the vacuum fluctuations across any imaginary boundary is what makes space connected across that boundary.
If you could somehow disentangle the vacuum fluctuations on either side of a boundary, the energy required would be enormous, and the gravitational back-reaction would likely “cleave” space—making it impossible to pass from one side to the other. This is related to the firewall proposal in black hole physics.
Jacobson extends this idea: the metric of spacetime may be entirely encoded in the pattern of vacuum fluctuations. If you knew the quantum state of the vacuum everywhere, you could extract the metric from the correlations in that state, making the metric redundant as a fundamental ingredient.
In 2015, Jacobson reformulated his approach using ball-shaped regions of space rather than local Rindler horizons, proposing that the vacuum state maximizes entanglement entropy for a fixed spatial volume.
This is a statistical rather than thermodynamic approach: instead of imposing the Clausius relation, he assumed that the vacuum has the maximum possible entanglement entropy across the boundary of a ball, holding the ball’s volume fixed. From this assumption, he was able to derive Einstein’s equations.
The requirement to hold volume fixed (rather than radius or area) is striking and unexplained. It is necessary to get the correct numerical relationship between the gravitational constant and the Bekenstein-Hawking entropy. Jacobson finds this deeply puzzling and considers it a clue to a deeper principle.
The Metric as Redundant: Spacetime from Quantum Fields Alone
Jacobson suspects that the metric of spacetime is not a fundamental degree of freedom but is entirely determined by the quantum state of the vacuum fluctuations.
If you could write down quantum field theory without a pre-existing metric—replacing the metric everywhere with the one extracted from the quantum field state itself—you might get a self-consistent scheme where spacetime is strictly emergent. The metric would be “superfluous and redundant” if the vacuum state is known.
This is still speculative, but it aligns with the broader theme that spacetime and gravity are not fundamental but arise from more basic quantum informational or thermodynamic principles.
Historical Analogies: Thermodynamics as a Guide to New Physics
Jacobson draws an analogy between his work and the historical role of thermodynamics in guiding physics to new discoveries.
Carnot correctly deduced the maximum efficiency of heat engines using the (wrong) concept of caloric. The effort to understand why Carnot was partially right led to the correct microscopic theory of heat and statistical mechanics.
Planck correctly derived the blackbody radiation formula but could not explain it without inventing quantum mechanics. The thermodynamic result was correct even though the underlying physics was unknown.
Similarly, Jacobson’s thermodynamic derivation of Einstein’s equations may be “fault-tolerant”—correct at a macroscopic level even if the microscopic theory of quantum gravity is unknown. The approach is likely to be partially right and partially wrong (like Carnot’s caloric), but it is asking the right questions and pointing toward the correct problems.
Discreteness of Spacetime
Jacobson initially thought that the finiteness of black hole entropy might imply discrete spacetime, but his later work suggests gravity itself provides a natural cutoff without requiring discreteness.
He now suspects that spacetime is ultimately discrete at the fundamental level, but not in a naive lattice-like way (which would break Lorentz symmetry and other important symmetries). Any discretization must be non-local in some sense.
Don Marolf’s constraint supports this: gravity cannot emerge from a system with locally defined observables that commute at spacelike separation. The underlying structure from which spacetime emerges must be fundamentally non-local.
Comparison with Other Approaches
Jacobson’s work is conceptually distinct from Erik Verlinde’s entropic gravity proposals, which he finds unclear in detail.
It is closer in spirit to the work of Mark van Raamsdonk and others in AdS/CFT who derived Einstein’s equations from entanglement entropy considerations. Van Raamsdonk’s work directly inspired Jacobson’s 2015 paper.
Jacobson’s approach does not rule out specific quantum gravity programs (string theory, loop quantum gravity, etc.) because it operates at a “fault-tolerant” level that is blind to the microscopic substructure, much as thermodynamics is blind to the atomic composition of a gas.
Open Questions and Current Thinking
Jacobson is currently re-examining the principles behind his earlier work, particularly the puzzling role of volume in the maximal entanglement hypothesis.
He suspects gravitons exist as effective excitations (like phonons in a Bose-Einstein condensate) but are not fundamental—at sufficiently short wavelengths, the concept of a graviton dissolves into whatever the underlying degrees of freedom are.
The question of what spacetime “emerges from” remains open. Jacobson does not think it is a question to be answered naively right now, but insights from AdS/CFT and entanglement are stepping stones. The ultimate theory is likely non-local and discrete in some subtle way, beyond current conceptual horizons.
