Jack SarfattiLessons from Classical Gravity about the Quantum Structure of Spacetime∗ T. Padmanabhan IUCAA, Pune University Campus, Ganeshkhind, Pune 411007, INDIA. email:paddy@iucaa.ernet.in
Jack Sarfatti"Lesson 2: The guiding principle to use for understanding the quantum microstructure of the spacetime should be the thermodynamics of horizons. Combining the principles of GR and quantum theory is not a technical problem that could be solved just by using sufficiently powerful mathematics. It is more of a conceptual issue and decades of failure of sophisticated mathematics in delivering quantum gravity indicates that we should try a different approach. This is very much in tune with item (iii) mentioned in Sec. 1. Einstein did not create a sophisticated mathematical model for mi and mg and try to interpret mi = mg. He used thought experiments to arrive at a conceptual basis in which mi = mg can be interpreted in a natural manner so that mi = mg will cease to be an algebraic accident. Once this is done, physics itself led him to the maths that was needed. Of course, the key issue is what could play the role of a guiding principle similar to principle of equivalence in the present context. For this, my bet will be on the thermodynamics of horizons.[1, 10] A successful model will have the connection between horizon thermodynamics and gravitational dynamics at its foundation rather than this feature appearing as a result derived in the context of certain specific solutions to the field equations. We will see evidence for its importance throughout the discussion in what follows." Yes, I have independently come to similar conclusions, but our future cosmic horizon is primary.
Jack SarfattiYes! THINK OFF-SHELL! Both UV dark matter and IR dark energy are OFF-SHELL virtual particle quantum vacuum effects in my opinion. "Lesson 3: Think beyond Einstein gravity, black hole thermodynamics and think off-shell. There are four technical points closely related to the above conjecture (viz., thermodynamics of horizons should play a foundational role) which needs to be recognized if this approach has yield dividends: • One must concentrate on the general context of observer dependent, local, thermodynamics associated with the local horizons, going beyond the black hole thermodynamics. Black hole horizons in the classical theory are far too special, on-shell, global constructs to provide a sufficiently general back-drop to understand the quantum structure of spacetime. The preoccupation with the black hole horizons loses sight of the conceptual fact that all horizons are endowed with temperature as perceived by the appropriate class of observers. Observer dependence [11] of thermal phenomena is a feature and not a bug!" EXACTLY! BY JOVE I THINK HE'S GOT IT. ;-)
Jack Sarfatti"The quantum features of a theory are off-shell features. But, fortunately, action principle provides a window to quantum theory because of the path integral formalism. Therefore any peculiar feature of a classical action functional could give us insights into the underlying quantum theory much more than the structure of field equations. This suggests that we need to look at the off-shell structure of the theory using the form of action principles rather than tie ourselves down to field equations." Agreed
Jack Sarfatti"Lesson 4: Temperature of horizons does not depend on the field equations of the theory and is just an indication that spacetimes, like matter, can be hot, but in a observer-dependent manner." YES! "One can associate a temperature with any null surface that can act as horizon for a class of observers, in any spacetime (including flat spacetime)." EXACTLY! "This temperature is determined by the behaviour of the metric close to the horizon and has nothing to do with the field equations (if any) which are obeyed by the metric." OK
Jack Sarfatti"The simplest situation is that of Rindler observers in flat spacetime with acceleration κ who will attribute a temperature kBT = (h/c)(κ/2π) to the Rindler horizon — which is just a X = t surface in the flat spacetime having no special significance to the inertial observers." YES. "While this result is usually proved for an eternally accelerating observer, they also hold in the (appropriately) approximate sense for an observer with variable acceleration [15]. In general, this result can be used to show that the vacuum state in a freely falling frame will appear to be a thermal state in the locally accelerated frame for high frequency modes if κ−1 is smaller than the local radius of (spacetime) curvature." CORRECT
Jack Sarfatti"Lesson 5: All thermodynamic variables are observer dependent. An immediate consequence, not often emphasized, is that all thermodynamic variables must become observer dependent if vacuum acquires an observer dependent temperature. A “normal” gaseous system with “normal” thermodynamic variables (T, S, F etc.....) must be considered as a highly excited state of the inertial vacuum. It is obvious that a Rindler observer will attribute to this highly excited state different thermodynamic variables compared to what an inertial observer will attribute. Thus thermal effects in the accelerated frame brings in [11, 17] a new level of observer dependence even to normal thermodynamics. One need not panic if variables like entropy now acquire an observer dependence and loses their absolute nature." RIGHT
Jack Sarfatti"Lesson 6: In sharp contrast to temperature, the entropy of horizons depends on the field equations of gravity and cannot be determined by using just QFT in a background metric."
Jack Sarfatti"One would have expected that if integrating out certain field modes leads to a thermal density matrix ρ, then the entropy of the system should be related to lack of information about the same field modes and should be given by S = −Tr ρ ln ρ. This entropy, called entanglement entropy, (i) is proportional to area of the horizon and (ii) is divergent without a cut-off [18]. Such a divergence makes the result meaningless and thus we cannot attribute a unique entropy to horizon using just QFT in a background metric.2 That is, while the temperature of the horizon can be obtained through the study of test-QFT in an external geometry, one cannot understand the entropy of the horizon by the same procedure. This is because, unlike the temperature, the entropy associated with a horizon in the theory depends on the field equations of the theory, which we will briefly review. Given the principle of equivalence (interpreted as gravity being spacetime geometry) and principle of general covariance, one could still construct a wide class of theories of gravity."
Jack Sarfatti"Lesson 9: Holographic structure of gravitational action functionals finds a natural explanation in the thermodynamic interpretation of the field equations. If the gravitational dynamics and horizon thermodynamics are so closely related, with field equations becoming thermodynamic identities on the horizon, then the action functionals of the theory (from which we obtain the field equations) must contain information about this connection. This clue comes in the form of another unexplained algebraic accident related to the structure of the action functional and tells us something significant about the off-shell structure of the theory. Gravity is the only theory known to us for which the natural action functional preserving symmetries of the theory contain second derivatives of the dynamical variables but still leads to second order differential equations. Usually, this is achieved by separating out the terms involving the second derivatives of the metric into a surface term which is either ignored or its variation is cancelled by a suitable counter-term. However, this leads to a serious conceptual mystery in the conventional approach when we recall the following two facts together: (a) The field equations can be obtained by varying the bulk term after ignoring (or by canceling with a counter-term) the surface term. (b) But if we evaluate the surface term on the horizon of any solution to the field equations of the theory, one obtains the entropy of the horizon! How does the surface term, which was discarded before the field equations were obtained, know about the entropy associated with a solution to those field equations?! In the conventional approach we need to accept it as another ‘algebraic accident’ without any explanation and, in fact, no explanation is possible within the standard framework. The explanation lies in the fact that the surface and bulk term of the Lagrangian are related in a specific manner thereby duplicating the information about the horizon entropy."
Jack Sarfatti"The duplication of information between surface and bulk term in Eq. (12) also allows one to obtain the full action [10] from the surface term alone using the entropic interpretation. In fact, in the the Riemann normal coordinates around any event P the gravitational action reduces to a pure surface term, again showing that the dynamical content is actually stored on the boundary rather than in the bulk.
Jack Sarfattione can construct a variational principle to obtain the field equations, purely from the surface term [29]." "Surface" is like the 2D hologram plate. "Bulk" is like the 3D hologram image.
Jack Sarfatti"Lesson 10: Gravitational actions have a surface and bulk terms because they give the entropy and energy of a static spacetimes with horizons, adding up to make the action the free energy of the spacetime."
Jack Sarfatti"The Avogadro number of the spacetime The results described in the previous sections suggest that there is a deep connection between horizon thermodynamics and the gravitational dynamics. Because the spacetime can be heated up just like a body of gas, the Boltzmann paradigm (“If you can heat it, it has microstructure”) motivates the study of the microscopic degrees of freedom of the spacetime exactly the way people studied gas dynamics before they understood the atomic structure of matter. There exists, fortunately, an acid test of this paradigm which it passes with flying colours."
Jack Sarfatti"Lesson 11: Gravitational field equations imply the law of equipartition E = (1/2)kBTN in any static spacetime, allowing the determination of density of microscopic degrees of freedom. The result again displays holographic scaling." The equipartition theorem is strictly classical breaking down in quantum statistical mechanics where hf/kBT >> 1.
Jack SarfattiThat is Bose-Einstein & Fermi-Dirac quantum statistics in the 3D bulk vs. classical Maxwell-Boltzmann statistics. Anyon fractional statistics on the 2D horizons.
Jack Sarfatti"It is worthwhile to list explicitly the questions which have natural answers in the emergent paradigm while have to be treated as algebraic accidents in the conventional approach: 1. While the temperature of the horizon can be obtained using QFT in curved spacetime, the corresponding entanglement entropy is divergent and meaningless. Why? 2. The temperature of horizon is independent of the field equations of gravity but the entropy of the horizon depends explicitly on the field equations. What does this difference signify? 3. The horizon entropy can be expressed in terms of the Noether current which is conserved due to diffeomorphism invariance. Why should an infinitesimal coordinate transformation xa → xa + qa have anything to do with a thermodynamic variable like entropy? 4. Why do the gravitational field equations (which do not look very “thermodynamical”!) reduce to T dS = dEg+PdV on the horizon, picking up the correct expression for S for a wide class of theories? 5. How come all gravitational action principle have a surface and bulk term which are related in a specific manner (see Eq. (12))? Why do the surface and the bulk terms allow the interpretation as entropy and energy in static spacetimes? 6. The field equations for gravity can be obtained from the bulk part of the action after discarding the surface term. But the surface term evaluated on the horizon of a solution gives the entropy of the horizon! How does the surface term — which was discarded before the field equations were obtained — know about the entropy of a solution? 7. Why does the gravitational field equations reduce to the equipartition form, expressible as E = (1/2)(kBT )n allowing us to determine the analog of Avogadro’s number for the spacetime? And, why does the relevant microscopic degrees of freedom for a region reside on the boundary of the region? 8. Finally, why is it possible to derive the field equations of any diffeomorphism invariant theory of gravity by extremizing an entropy functional associated with the null surfaces in the spacetime, without treating the metric as a dynamical variable? Obviously, any alternative perspective, including the conventional approach, need to provide the answers for the above questions if they have to be considered a viable alternative to emergent paradigm."
Quantum Structure of Spacetime∗
T. Padmanabhan
IUCAA, Pune University Campus, Ganeshkhind,
Pune 411007, INDIA.
email:paddy@iucaa.ernet.in