1. The case for emergent gravity
Josh Erlich
William & Mary

2. what is emergent gravity?
Any scenario in which gravitation is not part of the fundamental description, but instead emerges from the dynamics of the theory.

3. What isn’t emergent gravity?
Einstein-Hilbert
Action, perhaps
with matter
Quantized
Gravity
Quantization procedure

4. A. Sakharov (1967): - Induced Gravity
“Here we consider the hypothesis which identifies the [Einstein- Hilbert action] with the change in the action of quantum fluctuations of the vacuum if space is curved.”
- Induced Gravity

5. Michael Faraday’s laboratory notes, Vol. 5
“I have been arranging certain experiments in reference to the notion that gravity itself may be practically or directly related by experiment to the other powers of matter, and this morning proceeded to make them.”

6. Induced (Semiclassical) Gravity
A.D. Sakharov (1967); Visser (2002)
Assuming a generally covariant regulator, the expectation value of the energy-momentum tensor satisfies Einstein’s equations, plus corrections.
To calculate , couple a theory covariantly to a background spacetime metric (or vielbein).
... + + +
... +

7. Entropic Gravity (E. Verlinde, 2009)
Macroscopic systems experience forces if they are not at a minimum of their free energy.
Gravitation might be such an entropic force.
strengthperformance.com

8. Spacetime from Entanglement
(Van Raamsdonk; J. Maldacena, L. Susskind; C. Cao, S. Carroll)
Entangled particles (or entangled black holes) are connected in some sense.
Perhaps the connections are via wormholes which collectively give rise to an effective spacetime geometry.
Perhaps networks of entangled states describe the Hilbert space of theories with curved spacetime. 40

9. AdS/CFT Correspondence (J. Maldacena 1997)
The AdS/CFT correspondence relates nongravitational theories to gravitational theories with an extra noncompact dimension.
Both the extra dimension and gravitation may be considered emergent features of the field theory.

10. Interactions from Constraints
Amati, Testa (1974)
“Confinement”

11. Gravity as a Cutoff Artifact
C. Carone, JE, D. Vaman; + Claringbold; + S. Chaurasia, Y. Zhou
D=Number of
spacetime dimensions
Matter fields
Clock and Ruler fields
This action is invariant under diffeomorphisms, and in the quantum theory, spacetime geometry emerges as an artifact of short-distance physics.

12. Gravity as a Cutoff Artifact
The clock and ruler fields are integrated over configurations without critical points - a diffeomorphism-invariant constraint.
Gauge fix the clock and ruler fields:

13. ... ... 2-particle scattering = + + + = + massless graviton
Einstein’s gravity emerges from quantum short-distance physics. There is a composite graviton in the spectrum.
There is no physical meaning to spacetime here except for the emergent gravitational interaction.

14. Some Technical Details
c.f. M. Suzuki (2016) for emergent E&M
The gauge-fixed action takes the form
where
V0 determines
perturbative scale
To O(1/V0),
Flat space energy-momentum tensor
where

15. Some Technical Details
Example:
External lines factor out:

16. Take regulator (e.g. dim reg) as physical.
Example:
If we choose
then the amplitude has the massless spin-2 pole of Einstein gravity with no cosmological constant: =

17. The Cosmological Constant
The tuning
sets the cosmological constant to zero. Other choices introduce a cosmological constant source for

18. Same idea, with fermions
(Carone,JE,Vaman, to appear)
Motivated by the supersymmetric D-brane action
(Aganagic, Popescu, Schwartz 1997)
𝑆=−∫ 𝑑 𝐷 𝑥 𝐷/2−1 |𝑉(𝜓)| 𝐷/2−1 𝑑𝑒𝑡( ∂ 𝜇 𝑋 𝐼 + 𝑐 0 𝜓 𝛾 (𝐽 ∂ 𝜇) 𝜓 𝜂 𝐼𝐽 )

19. …and maybe gauge fields, too

20. Another perspective - Geometry from Vanishing Energy
Consider a gauge theory minimally coupled to an unspecified
spacetime metric:
𝑇 𝜇𝜈 =0.
For each field configuration, fix the metric by the constraint
Then,
This leads to the same theory as before (c.f. Nambu-Goto vs Polyakov)

21. But it’s not a theory without a UV completion!
We have seen that it is easy to generate quantum models of
gravity.
But we want a quantum theory of gravity.
The barrier is a physical UV regulator that defines physics
in the regime where the model breaks down.

22. Possible Physical Cutoffs
1. Nonlocality? (Ask John Moffat.)
2. Discreteness of spacetime - borrow ideas from dynamical triangulations or simple latticization.
3. Horava-Lifshitz - violate Lorentz invariance
4. Modification of quantum theory?

23. Stochastic Mechanics/Field Theory
Stochastic Mechanics (Edward Nelson, 1966,1985)
Stochastic Relativistic Field Theory (Guerra and Ruggiero, 1973)
Langevin Equation - describes
individual particle trajectories with
Brownian motion and drift b. 𝑑𝜉
Fokker-Planck Equation - describes evolution of probability density
with trajectories satisfying Langevin equation. 𝜌

24. Stochastic Mechanics/Field Theory
Given a quantum mechanical wavefunction
𝜓= 𝜌 𝑒 𝑖𝑆/ℏ If
𝐛=𝑢+𝑣
with
𝑢=𝐷 𝛻𝜌 𝜌
and
𝑣= 2𝐷 ℏ 𝛻𝑆
then
The discrete time step can act as a regulator.
In that case, very short distance physics is classical(!) 𝑑𝑇

25. Is there evidence for emergent gravity?
1) The appearance of Einstein gravity at long distances depends primarily on diffeomorphism invariance. It would be wasteful for Nature to add unnecessary fundamental degrees of freedom.

26. Is there evidence for emergent gravity?
2) The Second Law of Thermodynamics and the success of inflationary models suggest that the universe began in an extremely uniform state, which would have extremely low entropy in the presence of gravity. (Penrose)
If gravitation emerges only after the universe cools sufficiently, then the uniform initial state is instead one of high entropy, i.e. a typical state.

27. Is there evidence for emergent gravity?
3) Where does the signature of the vacuum spacetime metric come from? In emergent gravity a natural background spacetime geometry is also emergent.

28. Is there evidence for emergent gravity?
4) Speculation: A second-order gravitational phase transition might generate a nearly scale invariant spectrum of fluctuations in the early universe. (Is inflation necessary?)

29. Is there evidence for emergent gravity?
Summary: Perhaps.