Rafael D. Sorkin

Causal Sets: Discrete Gravity

These are some notes in lieu of the lectures I was scheduled to give, but had to cancel at the last moment. In some places, they are more complete, in others much less so, regrettably. I hope they at least give a feel for the subject and convey some of the excitement felt at the moment by those of us working on it.

Forks in the Road, on the Way to Quantum Gravity

In seeking to arrive at a theory of “quantum gravity,” one faces several choices among alternative approaches. I list some of these “forks in the road” and offer reasons for taking one alternative over the other. In particular, I advocate the following: the sum-over-histories framework for quantum dynamics over the “observable and state-vector” framework; relative probabilities over absolute ones; spacetime over space as the gravitational “substance” (4 over 3+1); a Lorentzian metric over a Riemannian (“Euclidean”) one; a dynamical topology over an absolute one; degenerate metrics over closed timelike curves to mediate topology change; “unimodular gravity” over the unrestricted functional integral; and taking a discrete underlying structure (the causal set) rather than the differentiable manifold as the basis of the theory. In connection with these choices, I also mention some results from unimodular quantum cosmology, sketch an account of the origin of black hole entropy, summarize an argument that the quantum mechanical measurement scheme breaks down for quantum field theory, and offer a reason why the cosmological constant of the present epoch might have a magnitude of around 10−120 in natural units.

Finitary substitute for continuous topology

Finite topological spaces are combinatorial structures that can serve as replacements for, or approximations to, bounded regions within continuous spaces such as manifolds. In this spirit, the present paper studies the approximation of general topological spaces by finite ones, or really by “finitary” ones in case the original space is unbounded. It describes how to associate a finitary spaceF with any locally finite covering of aT1-spaceS; and it shows howF converges toS as the sets of the covering become finer and more numerous. It also explains the equivalent description of finite topological spaces in order-theoretic language, and presents in this connection some examples of posetsF derived from simple spacesS. The finitary spaces considered here should not be confused with the so-called causal sets, but there may be a relation between the two notions in certain situations.

Gravitational action with null boundaries

The present paper provides a complete treatment of boundary terms in general relativity to include cases with lightlike boundary segments along with the usual spacelike and timelike ones. Applications of this exhaustive treatment includes a recent conjecture on computational complexity in the context of AdS/CFT.

Entropy of self-gravitating radiation

We examine the entropy of self-gravitating radiation confined to a spherical box of radiusR in the context of general relativity. We expect that configurations (i.e., initial data) which extremize total entropy will be spherically symmetric, time symmetric distributions of radiation in local thermodynamic equilibrium. Assuming this is the case, we prove that extrema ofS coincide precisely with static equilibrium configurations of the radiation fluid. Furthermore, dynamically stable equilibrium configurations are shown to coincide with local maxima ofS. The equilibrium configurations and their entropies are calculated and their properties are discussed. However, it is shown that entropies higher than these local extrema can be achieved and, indeed, arbitrarily high entropies can be attained by configurations inside of or outside but arbitrarily near their own Schwarzschild radius. However, if we limit consideration to configurations which are outside their own Schwarzschild radius by at least one radiation wavelength, then the entropy is bounded and we find Smax ≲ MR, whereM is the total mass. This supports the validity for self-gravitating systems of the Bekenstein upper limit on the entropy to energy ratio of material bodies.

The electromagnetic field on a simplicial net

The ’’Regge calculus’’ approach is extended to the electromagnetic case. To this end an ’’affine’’ tensor formalism and associated exterior calculus are developed. The simplicial approach to linear field equations is illustrated by the two‐dimensional scalar wave equation, on which also a discussion of the treacherous character of the continuum limit is based.

Variational Aspects of Relativistic Field Theories, with Application to Perfect Fluids

Abstract By investigating perturbations of classical field theories based on variational principles we develop a variety of relations of interest in several fields, general relativity, stellar structure, fluid dynamics, and superfluid theory. The simplest and most familiar variational principles are those in which the field variations are unconstrained. Working at first in this context we introduce the Noether operator, a fully covariant generalization of the socalled canonical stress energy tensor, and prove its equivalence to the symmetric tensor Tμν. By perturbing the Noether operators definition we establish our fundamental theorem, that any two of the following imply the third (a) the fields satisfy their field equations, (b) the fields are stationary, (c) the total energy of the fields is an extremum against all perturbations. Conversely, a field theory which violates this theorem cannot be derived from an unconstrained principle. In particular both Maxwells equations for Fμν and Eulers equations for the perfect fluid have stationary solutions which are not extrema of the total energy [(a) + (b) (c)]. General relativity is a theory which does have an unconstrained variational principle but the definition of Noether operator is more ambiguous than for other fields. We define a pseudotensorial operator which includes the Einstein and Landau-Lifschitz complexes as special cases and satisfies a certain criterion on the asymptotic behavior. Then our extremal theorem leads to a proof of the uniqueness of Minkowski space: It is the only asymptotically flat, stationary, vacuum solution to Einsteins equations having R 4 global topology and a maximal spacelike hypersurface. We next consider perfect fluid dynamics. The failure of the extremal-energy theorem elucidates why constraints have always been used in variational principles that lead to Eulers equations. We discuss their meaning and give what we consider to be the “minimally constrained” principle. A discussion of one constraint, “preservation of particle identity,” from the point of view of path-integral quantum mechanics leads to the conclusion that it is inapplicable to degenerate Bose fluids, and this gives immediately the well-known irrotational flow of such fluids. Finally, we develop a restricted extremal theorem for the case of perfect fluids with self-gravitation, which has the same form as before except that certain perturbations are forbidden in (c). We show that it is a generalization of the Bardeen-Hartle-Sharp variational principle for relativistic stellar structure. It may be useful in constructing nonaxisymmetric stellar models (generalized Dedekind ellipsoids). We also give the Newtonian versions of the main results here, and we show to what extent the extremal theorems extend to fields that may not even have a variational principle.

Boundary terms in the action for the Regge calculus

The boundary terms in the action for Regges formulation of general relativity on a simplicial net are derived and compared with the boundary terms in continuum general relativity.

Fundamental quantum optics experiments conceivable with satellites : reaching relativistic distances and velocities

Physical theories are developed to describe phenomena in particular regimes, and generally are valid only within a limited range of scales. For example, general relativity provides an effective description of the Universe at large length scales, and has been tested from the cosmic scale down to distances as small as 10 m (Dimopoulos 2007 Phys. Rev. Lett. 98 111102; 2008 Phys. Rev. D 78 042003). In contrast, quantum theory provides an effective description of physics at small length scales. Direct tests of quantum theory have been performed at the smallest probeable scales at the Large Hadron Collider, ~10−20 m, up to that of hundreds of kilometres (Ursin et al 2007 Nature Phys. 3 481–6). Yet, such tests fall short of the scales required to investigate potentially significant physics that arises at the intersection of quantum and relativistic regimes. We propose to push direct tests of quantum theory to larger and larger length scales, approaching that of the radius of curvature of spacetime, where we begin to probe the interaction between gravity and quantum phenomena. In particular, we review a wide variety of potential tests of fundamental physics that are conceivable with artificial satellites in Earth orbit and elsewhere in the solar system, and attempt to sketch the magnitudes of potentially observable effects. The tests have the potential to determine the applicability of quantum theory at larger length scales, eliminate various alternative physical theories, and place bounds on phenomenological models motivated by ideas about spacetime microstructure from quantum gravity. From a more pragmatic perspective, as quantum communication technologies such as quantum key distribution advance into space towards large distances, some of the fundamental physical effects discussed here may need to be taken into account to make such schemes viable.

Testing Born's Rule in Quantum Mechanics with a Triple Slit Experiment

In Mod. Phys. Lett. A 9, 3119 (1994), one of us (R.D.S) investigated a formulation of quantum mechanics as a generalized measure theory. Quantum mechanics computes probabilities from the absolute squares of complex amplitudes, and the resulting interference violates the (Kolmogorov) sum rule expressing the additivity of probabilities of mutually exclusive events. However, there is a higher order sum rule that quantum mechanics does obey, involving the probabilities of three mutually exclusive possibilities. We could imagine a yet more general theory by assuming that it violates the next higher sum rule. In this paper, we report results from an ongoing experiment which sets out to test the validity of this second sum rule by measuring the interference patterns produced by three slits and all the possible combinations of those slits being open or closed. We use an attenuated laser light combined with single photon counting to confirm the particle character of the measured light.