arXiv subjects
Publications and source records attributed to Matt Visser.
Our collective views regarding the question "what is fundamental?" are continually evolving. These ontological shifts in what we regard as fundamental are largely driven by theoretical advances ("what can we calculate?"), and experimental advances ("what can we measure?"). Rarely (in my view) is epistemology the fundamental driver; more commonly epistemology reacts (after a few decades) to what is going on in the theoretical and experimental zeitgeist.
We provide a simple and clear verification of the physical need for temperature gradients in equilibrium states when gravitational fields are present. Our argument will be built in a completely kinematic manner, in terms of the gravitational red-shift/blue-shift of light, together with a relativistic extension of Maxwell's two column argument. We conclude by showing that it is the universality of the gravitational interaction (the uniqueness of free-fall) that ultimately permits Tolman's equilibrium temperature gradients without any violation of the laws of thermodynamics.
The Hawking-Ellis (Segre-Plebanski) classification of possible stress-energy tensors is an essential tool in analyzing the implications of the Einstein field equations in a more-or-less model-independent manner. In the current article the basic idea is to simplify the Hawking-Ellis type I, II, III, and IV classification by isolating the "essential core" of the type II, type III, and type IV stress-energy tensors; this being done by subtracting (special cases of) type I to simplify the (Lorentz invariant) eigenvalue structure as much as possible without disturbing the eigenvector structure. We will denote these "simplified cores" type II$_0$, type III$_0$, and type IV$_0$. These "simplified cores" have very nice and simple algebraic properties. Furthermore, types I and II$_0$ have very simple classical interpretations, while type IV$_0$ is known to arise semi-classically (in renormalized expectation values of standard stress-energy tensors). In contrast type III$_0$ stands out in that it has neither a simple classical interpretation, nor even a simple semi-classical interpretation. We will also consider the robustness of this classification considering the stability of the different Hawking-Ellis types under perturbations. We argue that types II and III are definitively unstable, whereas types I and IV are stable.
The evaporation of black holes raises a number of conceptual issues, most of them related to the final stages of evaporation, where the interplay between the central singularity and Hawking radiation cannot be ignored. Regular models of black holes replace the central singularity with a nonsingular spacetime region, in which an effective classical geometric description is available. It has been argued that these models provide an effective, but complete, description of the evaporation of black holes at all times up to their eventual disappearance. However, here we point out that known models fail to be self-consistent: the regular core is exponentially unstable against perturbations with a finite timescale, while the evaporation time is infinite, therefore making the instability impossible to prevent. We also discuss how to overcome these difficulties, highlighting that this can be done only at the price of accepting that these models cannot be fully predictive regarding the final stages of evaporation.
Tolman's relation for the temperature gradient in an equilibrium self-gravitating general relativistic fluid is broadly accepted within the general relativity community. However, the concept of temperature gradients in thermal equilibrium continues to cause confusion in other branches of physics, since it contradicts naive versions of the laws of classical thermodynamics. In this paper we discuss the crucial role of the universality of free fall, and how thermodynamics emphasises the great distinction between gravity and other forces. To do so we will present an argument given by Maxwell and apply it to an electro-thermal system, concluding with an reductio ad absurdum. Among other issues we shall show that Tolman temperature gradients could also (in principle) have been derived circa 1905 - a decade before the development of full general relativity.
It is not currently known how to put the Kerr spacetime metric into the so-called Gordon form, although the closely related Kerr-Schild form of the Kerr metric is well known. A Gordon form for the Kerr geometry, if it could be found, would be particularly useful in developing analogue models for the Kerr spacetime, since the Gordon form is explicitly given in terms of the 4-velocity and "refractive index" of an effective medium. In the current article we report progress toward this goal. First we present the Gordon form for an approximation to Kerr spacetime in the slow-rotation limit, obtained by suitably modifying the well-known Lense-Thirring form of the slow-rotation metric. Second we present the Gordon form for the Kerr spacetime in the near-null limit, (the 4-velocity of the medium being close to null). That these two perturbative approximations to the Kerr spacetime in Gordon form exist gives us some confidence that ultimately one might be able to write the exact Kerr spacetime in this form.
Analogue spacetimes can be used to probe and study physically interesting spacetime geometries by constructing, either theoretically or experimentally, some notion of an effective Lorentzian metric $[g_\mathrm{eff}(g,V,\,Ξ)]_{ab}$. These effective metrics generically depend on some physical background metric $g_{ab}$, often flat Minkowski space $η_{ab}$, some "medium" with 4-velocity $V^a$, and possibly some additional background fields $Ξ$. Electromagnetic analogue models date back to the 1920s, acoustic analogue models to the 1980s, and BEC-based analogues to the 1990s. The acoustic analogue models have perhaps the most rigorous mathematical formulation, and these acoustic analogue models really work best in the absence of vorticity, if the medium has an irrotational flow. This makes it difficult to model rotating astrophysical spacetimes, spacetimes with non-zero angular momentum, and in the current article we explore the extent to which one might hope to be able to model astrophysical spacetimes with angular momentum, (thereby implying vorticity in the 4-velocity of the medium).
In three very recent papers, (an initial paper by Morishima and Futamase, and two subsequent papers by Morishima, Futamase, and Shimizu), it has been argued that the observed experimental anomaly in the anomalous magnetic moment of the muon might be explained using general relativity. It is my melancholy duty to report that these articles are fundamentally flawed in that they fail to correctly implement the Einstein equivalence principle of general relativity. Insofar as one accepts the underlying logic behind these calculations (and so rejects general relativity) the claimed effect due to the Earth's gravity will be swamped by the effect due to Sun (by a factor of fifteen), and by the effect due to the Galaxy (by a factor of two thousand). In contrast, insofar as one accepts general relativity, then the claimed effect will be suppressed by an extra factor of [(size of laboratory)/(radius of Earth)]^2. Either way, the claimed effect is not compatible with explaining the observed experimental anomaly in the anomalous magnetic moment of the muon.
Modern meta-materials allow one to construct electromagnetic media with almost arbitrary bespoke permittivity, permeability, and magneto-electric tensors. If (and only if) the permittivity, permeability, and magneto-electric tensors satisfy certain stringent compatibility conditions, can the meta-material be fully described (at the wave optics level) in terms of an effective Lorentzian metric --- an analogue spacetime. We shall consider some of the standard black-hole spacetimes of primary interest in general relativity, in various coordinate systems, and determine the equivalent meta-material susceptibility tensors in a laboratory setting. In static black hole spacetimes (Schwarzschild and the like) certain eigenvalues of the susceptibility tensors will be seen to diverge on the horizon. In stationary black hole spacetimes (Kerr and the like) certain eigenvalues of the susceptibility tensors will be seen to diverge on the ergo-surface.
Using electromagnetism to study analogue space-times is tantamount to considering consistency conditions for when a given (meta-)material would provide an analogue space-time model or --- vice versa --- characterizing which given metric could be modelled with a (meta-)material. While the consistency conditions themselves are by now well known and studied, the form the metric takes once they are satisfied is not. This question is mostly easily answered by keeping the formalisms of the two research fields here in contact as close to each other as possible. While fully covariant formulations of the electrodynamics of media have been around for a long while, they are usually abandoned for (3+1)- or 6-dimensional formalisms. Here we shall use the fully unified and fully covariant approach. This enables us even to generalize the consistency conditions for the existence of an effective metric to arbitrary background metrics beyond flat space-time electrodynamics. We also show how the familiar matrices for permittivity $ε$, permeability $μ^{-1}$, and magneto-electric effects $ζ$ can be seen as the three independent pieces of the Bel decomposition for the constitutive tensor $Z^{abcd}$, i.e., the components of an orthogonal decomposition with respect to a given observer with four-velocity $V^a$. Finally, we shall use the Moore--Penrose pseudo-inverse and the closely related pseudo-determinant to then gain the desired reconstruction of the effective metric in terms of the permittivity tensor $ε^{ab}$, the permeability tensor $[μ^{-1}]^{ab}$, and the magneto-electric tensor $ζ^{ab}$, as an explicit function $g_{\text{eff}}(ε,μ^{-1},ζ)$.
The Baker-Campbell-Hausdorff series computes the quantity \begin{equation*} Z(X,Y)=\ln\left( e^X e^Y \right) = \sum_{n=1}^\infty z_n(X,Y), \end{equation*} where $X$ and $Y$ are not necessarily commuting, in terms of homogeneous multinomials $z_n(X,Y)$ of degree $n$. (This is essentially equivalent to computing the so-called Goldberg coefficients.) The Baker-Campbell-Hausdorff series is a general purpose tool of wide applicability in mathematical physics, quantum physics, and many other fields. The Reinsch algorithm for the truncated series permits one to calculate up to some fixed order $N$ by using $(N+1)\times(N+1)$ matrices. We show how to further simplify the Reinsch algorithm, making implementation (in principle) utterly straightforward. This helps provide a deeper understanding of the Goldberg coefficients and their properties. For instance we establish strict bounds (and some equalities) on the number of non-zero Goldberg coefficients. Unfortunately, we shall see that the number of terms in the multinomial $z_n(X,Y)$ often grows very rapidly (in fact exponentially) with the degree $n$. We also present some closely related results for the symmetric product \begin{equation*} S(X,Y)=\ln\left( e^{X/2} e^Y e^{X/2} \right) = \sum_{n=1}^\infty s_n(X,Y). \end{equation*} Variations on these themes are straightforward. For instance, one can just as easily consider the series \begin{equation*} L(X,Y)=\ln\left( e^{X} e^Y e^{-X} e^{-Y}\right) = \sum_{n=1}^\infty \ell_n(X,Y). \end{equation*} This type of series is of interest, for instance, when considering parallel transport around a closed curve. Several other related series are investigated.
The (generalized) Rainich conditions are algebraic conditions which are polynomial in the (mixed-component) stress-energy tensor. As such they are logically distinct from the usual classical energy conditions (NEC, WEC, SEC, DEC), and logically distinct from the usual Hawking-Ellis (Segré-Plebański) classification of stress-energy tensors (type I, type II, type III, type IV). There will of course be significant inter-connections between these classification schemes, which we explore in the current article. Overall, we shall argue that it is best to view the (generalized) Rainich conditions as a refinement of the classical energy conditions and the usual Hawking-Ellis classification.
Blackbody radiation contains (on average) an entropy of 3.9\pm2.5 bits per photon. If the emission process is unitary, then this entropy is exactly compensated by "hidden information" in the correlations. We extend this argument to the Hawking radiation from GR black holes, demonstrating that the assumption of unitarity leads to a perfectly reasonable entropy/information budget. The key technical aspect of our calculation is a variant of the "average subsystem" approach developed by Page, which we extend beyond bipartite pure systems, to a tripartite pure system that considers the influence of the environment.
Classical point particles in Newtonian gravity obey, as they do in general relativity, the universality of free fall. However classical structured particles, (for instance with a mass quadrupole moment), need not obey the universality of free fall. Quantum mechanically, an elementary "point" particle (in the particle physics sense) can be described by a localized wave-packet, for which we can define a probability quadrupole moment. This probability quadrupole can, under plausible hypotheses, affect the universality of free fall. (So point-like elementary particles, in the particle physics sense, can and indeed must nevertheless have structure in the general relativistic sense once wave-packet effects are included.) This raises an important issue of principle, as possible quantum violations of the universality of free fall would fundamentally impact on our ideas of what "quantum gravity" might look like. I will present an estimate of the size of the effect, and discuss where if at all it might be measured.
So-called average subsystem entropies are defined by first taking partial traces over some pure state to define density matrices, then calculating the subsystem entropies, and finally averaging over the pure states to define the average subsystem entropies. These quantities are standard tools in quantum information theory, most typically applied in bipartite systems. We shall first present some extensions to the usual bipartite analysis, (including a calculation of the average tangle, and a bound on the average concurrence), follow this with some useful results for tripartite systems, and finally extend the discussion to arbitrary multi-partite systems. A particularly nice feature of tri-partite and multi-partite analyses is that this framework allows one to introduce an "environment" for small subsystems to couple to.
Deconstruction of the theme of the 2017 FQXi essay contest is already an interesting exercise in its own right: Teleology is rarely useful in physics --- the only known mainstream physics example (black hole event horizons) has a very mixed score-card --- so the "goals" and "aims and intentions" alluded to in the theme of the 2017 FQXi essay contest are already somewhat pushing the limits. Furthermore, "aims and intentions" certainly carries the implication of consciousness, and opens up a whole can of worms related to the mind-body problem. As for "mindless mathematical laws", that allusion is certainly in tension with at least some versions of the "mathematical universe hypothesis". Finally "wandering towards a goal" again carries the implication of consciousness, with all its attendant problems. In this essay I will argue, simply because we do not yet have any really good mathematical or physical theory of consciousness, that the theme of this essay contest is premature, and unlikely to lead to any resolution that would be widely accepted in the mathematics or physics communities.
The nature of coarse graining is intuitively "obvious", but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes an arbitrary system, with specified initial entropy S, and monotonically and controllably drives the entropy to its maximum value. This does not have to be a physical process, in fact for some purposes it is better to deal with a gedanken-process, since then it is more obvious how the "hidden information" is hiding in the fine-grain correlations that one is simply agreeing not to look at. We shall present several simple mathematically well-defined and easy to work with conceptual models for coarse graining. We shall consider both the classical Shannon and quantum von Neumann entropies, including models based on quantum decoherence, and analyze the entropy flow in some detail. When coarse-graining the quantum von Neumann entropy, we find it extremely useful to introduce an adaptation of Hawking's super-scattering matrix. These explicit models that we shall construct allow us to quantify and keep clear track of the entropy that appears when coarse-graining the system, and the information that can be hidden in unobserved correlations. (While not the focus of the current article, in the longer run these considerations are of interest when addressing black hole information puzzle.)
The standard energy conditions of classical general relativity are (mostly) linear in the stress-energy tensor, and have clear physical interpretations in terms of geodesic focussing, but suffer the significant drawback that they are often violated by semi-classical quantum effects. In contrast, it is possible to develop non-standard energy conditions that are intrinsically non-linear in the stress-energy tensor, and which exhibit much better well-controlled behaviour when semi-classical quantum effects are introduced, at the cost of a less direct applicability to geodesic focussing. In this article we will first review the standard energy conditions and their various limitations. (Including the connection to the Hawking--Ellis type I, II, III, and IV classification of stress-energy tensors). We shall then turn to the averaged, nonlinear, and semi-classical energy conditions, and see how much can be done once semi-classical quantum effects are included.