Two Roads to the Null Energy Condition Maulik Parikh Department of Physics Arizona State University M. P., “Two Roads to the Null Energy Condition,” arXiv: 1512.03448. M. P. and A. Svesko, “Thermodynamic Origin of the Null Energy Condition,” arXiv: 1511.06460. M. P. and J.-P. van der Schaar, “Derivation of the Null Energy Condition,” arXiv: 1406.5163; Phys. Rev. D. Science or Science Fiction? Gödel universe Gott time machine Morris-Thorne wormhole Bouncing cosmologies Big Rip phantom cosmology Negative mass black holes Super-extremal black holes Einstein on the Gödel Universe “... cosmological solutions of the gravitation equations have been found by Mr Gödel. It will be interesting to weigh whether these are not to be excluded on physical grounds.” “Physical Grounds” or Ad Hoc Criteria? No closed causal curves Stable causality Global hyperbolicity Cosmic censorship/strong asymptotic predictability Geodesic completeness Generalized second law of thermodynamics Energy conditions The Null Energy Condition The most basic of the energy conditions is the null energy condition (NEC): T kµkν 0 µν ≥ where k is any light-like vector. The weak and strong energy conditions both imply the null energy condition. All Metrics are Solutions of Einstein’s Equations If there are no conditions on the energy-momentum tensor, any metric becomes an exact solution to Einstein’s equations: Simply define the energy-momentum tensor to equal the Einstein tensor. Importance of Energy Conditions Singularity theorems. Positive energy theorems. Black hole no-hair theorem. Laws of black hole mechanics. Exclusion of traversable wormholes, construction of time machines, creation of a universe in the laboratory, and cosmological bounces. H˙ = (⇢ + p) 0 − Energy Conditions from Matter? The energy conditions are conventionally viewed as conditions on the energy- momentum tensor for matter. Matter We have a fantastic framework for understanding matter: quantum field theory. Violation of the Null Energy Condition in QFT Consider a well-behaved NEC-obeying theory of matter, theory A. Let theory B have the same action as theory A but with an overall minus sign. Theory B then violates the NEC. But theory B is otherwise just as well-behaved as theory A! Hence the NEC does not follow from QFT. How the Energy Conditions are Used in Practice In gravitational theorems, the energy conditions are always used in conjunction with the equations of motion. 1 T kµkν 0 (R Rg )kµkν 0 R kµkν 0 µν ≥ ⇒ µν − 2 µν ≥ ⇒ µν ≥ Why not just start with a condition on the Ricci tensor? Perhaps the real null “energy condition” is R kµkν 0 µν ≥ We will derive this condition. Goal of the Rest of the Talk Derive R kµkν 0 µν ≥ We will continue to refer to this as the “null energy condition”. But note that this is now a condition on geometry. Where Could the NEC Come From? The geometric form of the NEC does not come from general relativity. We have argued that the matter form does not come from QFT either. Hence we should look for the origin of the NEC in theories that contain both matter and gravity. That is, the NEC is a clue to the unification of matter and gravity. Unification Options Option 1: The origin of matter is gravity (Kaluza-Klein theory). Option 2: The origin of gravity is matter (emergent gravity paradigm). Option 3: Both matter and gravity have a common origin (string theory). Two Possibilities 1. The null energy condition comes from string theory. M. P. and J.-P. van der Schaar, “Derivation of the Null Energy Condition,” arXiv: 1406.5163; Phys. Rev. D. 2. The null energy condition comes from thermodynamics in emergent gravity. M. P. and A. Svesko, “Thermodynamic Origin of the Null Energy Condition,” arXiv: 1511.06460. Stringy Origin of the Null Energy Condition Worldsheet String Theory Polyakov action µ 1 2 ab µ ν (2) S[X ,hab]= d ⇤√ h gµν (X)h ⌅aX ⌅bX + αΦ(X)R −4⇥ − Xµ(σ, ⇥) spacetime coordinates of the string hab(σ, ⇥) worldsheet metric R(2) two-dimensional Ricci scalar of worldsheet gµν (X) spacetime metric Φ(X) dilaton Worldsheet string theory = two-dimensional conformal field theory of D fields + two-dimensional gravity Strings in Minkowski Space For a string propagating in flat space, the worldsheet CFT is free. µ 1 2 ab µ ν (2) S[X ,hab]= d ⇧√ h ⇥µν h ⌃aX ⌃bX + ⇤R −4⌅ − The equation of motion for the worldsheet metric is the 2D Einstein’s equation: λ 1 R(2) R(2)h = T −2⇡ ab − 2 ab ab ✓ ◆ As the Einstein tensor vanishes in 2D, the worldsheet stress tensor must be 0. Spacetime Null Vectors from the String Worldsheet Consider the Virasoro constraint: 1 T =0 η ⇥ Xµ⇥ Xν h (⇥X)2 =0 ab ⇒ µν a b − 2 ab In light cone coordinates, this becomes µ ν ηµν ⇥+X ⇥+X =0 In other words, k µ ∂ X µ is a spacetime null vector field: ≡ + µ ν ηµν k k =0 Strings in Curved Spacetime For a string propagating in curved space, the one-loop effective action is µ 1 2 ab µ ⌫ 1 2 (2) S[X ,h ]= d σp hh @ X @ X (⌘ +C ↵0R (X )) d σp hC Φ(X )R 0 ab −4⇡↵ − a 0 b 0 µ⌫ ✏ µ⌫ 0 −4⇡ − ✏ 0 0 Z Z Virasoro constraint in light-cone coordinates: µ ⌫ 0=@ X @ X (⌘ + C ↵0R +2C ↵0 Φ) + 0 + 0 µ⌫ ✏ µ⌫ ✏ rµr⌫ At zeroeth order in α we recover our original equation: µ ν ηµν k k =0 But at first order in α we miraculously discover that kµkν (R +2 Φ)=0 µν ∇µ∇ν Einstein-Frame Metric The spacetime metric appearing in the string action is the string-frame metric. It is related to the usual (“Einstein-frame”) metric by scaling: 4Φ + D 2 E gµν = e − gµν Transforming the Virasoro condition to the Einstein-frame metric, we find 4 0=kµk⌫ (R +2 Φ)=kµk⌫ RE EΦ EΦ µ⌫ rµr⌫ µ⌫ − D 2rµ r⌫ ✓ − ◆ That is 4 RE kµk⌫ =+ (kµ EΦ)2 0 µ⌫ D 2 rµ ≥ − This is precisely the desired geometric form of the null energy condition! Summary of Worldsheet Derivation The null energy condition is not about matter or energy at all. The geometric form of the “null energy condition” comes from string theory. The origin of the null energy condition lies in gravity -- 2D gravity! The ``physical ground” for the null energy condition is two-dimensional diffeomorphism invariance. M. P. and J.-P. van der Schaar, “Derivation of the Null Energy Condition,” arXiv: 1406.5163; Phys. Rev. D. Thermodynamic Origin of the Null Energy Condition The Null Energy Condition from Thermodynamics? Let us try to show the universality of the null energy condition by starting from the ultimate universal theory: thermodynamics. Emergent Gravity Suppose there exists some microscopic theory that, when coarse-grained, describes gravity in some infinitesimal region of spacetime. Thermodynamics of the Microscopic Theory This microscopic theory must have a finite coarse-grained entropy: Smax. We also assume that the theory obeys the second law of thermodynamics. Null Energy Condition from Local Holography Amazingly, from these very minimal assumptions -- finiteness of entropy, the second law of thermodynamics -- and one non-trivial assumption (local holography) we can derive the null energy condition. Approach to Equilibrium Now, at late times, as the theory approaches its maximum entropy, we have kt S(t) Smax 1 e− ⇡ − S 1 where k − is the relaxation time, assumed to be Planckian. Then, approaching equilibrium, we have t S¨ 0 In fact, there exists a general statistical- mechanical proof that near-equilibrium systems approaching equilibrium obey this inequality. G. Falkovich and A. Fouxon, New J. Phys. 6 (2004) 50. Entropy and Area To make the connection to geometry, let the coarse-grained entropy of the microscopic theory correspond to the surface area of the infinitesimal region of spacetime. Hence A S = 4 A S˙ = ✓ 4 A S¨ = ✓˙ + ✓2 4 ⇣ ⌘ Null Energy Condition from Thermodynamics The lightsheets of this area obey the Raychaudhuri equation: 1 ✓˙ = ✓2 R kµk⌫ −2 − µ⌫ Rewriting that equation, and using our earlier relations, we find 1 R kµk⌫ = ✓˙ ✓2 µ⌫ − − 2 1 = ✓˙ + ✓2 + ✓2 − 2 ⇣1 ⌘1 = S¨ + S˙ 2 −S 2S2 0 ≥ Once again, we have found precisely the geometric form of the NEC. M. P. and A. Svesko, “Thermodynamic Origin of the Null Energy Condition,” arXiv: 1511.06460. Quantum Violation of the Null Energy Condition The validity of the NEC implies that black holes can only grow. Conversely, if a black hole shrinks, the NEC must be violated. Thus quantum effects -- Hawking radiation -- must violate the NEC. Thermal Violation of the Second Law We also know that the second law of thermodynamics can be violated by thermal fluctuations. What is the probability of a violation of the second law in the underlying, microscopic theory? Hawking Radiation in Emergent Gravity The probability per time of a second-law-violating fluctuation to a macrostate with entropy S f is S S ∆S Γ e f − max e ⇠ ⇠ But this is precisely the probability per time of the emission of Hawking radiation in a tunneling framework in which both matter and gravity are treated quantum-mechanically! M. P. and F. Wilczek, “Hawking Radiation as Tunneling,” hep-th/9707001; Phys. Rev. Letts. The Dual Origins of Gravity We have seen that the null energy condition, when viewed as a property of geometry, can be derived in two different ways, each based on a principle. But why are there two derivations? Dual Derivations of Einstein’s Equations In string theory, Einstein’s equations famously follow from Weyl invariance on the closed string worldsheet.