Black and white holes play remarkably contrasting roles in general relativity versus observationa... more Black and white holes play remarkably contrasting roles in general relativity versus observational astrophysics. While there is observational evidence for the existence of compact objects that are "cold, dark, and heavy", which thereby are natural candidates for black holes, the theoretically viable time-reversed variants-the "white holes"-have nowhere near the same level of observational support. Herein we shall explore the theoretical possibility that the connection between black and white holes is much more intimate than commonly appreciated. We shall first construct "horizon penetrating" coordinate systems that differ from the standard curvature coordinates only in a small near-horizon region, thereby emphasizing that ultimately the distinction between black and white horizons depends only on near-horizon physics. We shall then construct an explicit model for a "black-to-white transition" where all of the nontrivial physics is confined to a compact region of spacetime-a finite-duration finite-thickness, (in principle arbitrarily small), region straddling the naïve horizon. Moreover we shall show that it is possible to arrange the "black-to-white transition" to have zero action-so that it will not be subject to destructive interference in the Feynman path integral. This then raises the very intriguing possibility that astrophysical black holes might be interpretable in terms of a quantum superposition of black and white horizons-a "gray" horizon.
International Journal of Modern Physics D, Apr 1, 2003
There are numerous derivations of the Hawking effect available in the literature. They emphasise ... more There are numerous derivations of the Hawking effect available in the literature. They emphasise different features of the process, and sometimes make markedly different physical assumptions. This article presents a "minimalist" argument, and strips the derivation of as much excess baggage as possible. All that is really necessary is quantum physics plus a slowly evolving future apparent horizon (not an event horizon). In particular, neither the Einstein equations nor Bekenstein entropy are necessary (nor even useful) in deriving Hawking radiation.
The standard energy conditions of classical general relativity are (mostly) linear in the stress-... more 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 chapter 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. 1 Draft chapter, on which the related chapter of the book Wormholes, Warp Drives and Energy Conditions (to be published by Springer) will be based.
Acoustic propagation in a moving fluid provides a conceptually clean and powerful analogy for und... more Acoustic propagation in a moving fluid provides a conceptually clean and powerful analogy for understanding black hole physics. As a teaching tool, the analogy is useful for introducing students to both General Relativity and fluid mechanics. As a research tool, the analogy helps clarify what aspects of the physics are kinematics and what aspects are dynamics. In particular, Hawking radiation is a purely kinematical effect, whereas black hole entropy is intrinsically dynamical. Finally, I discuss the fact that with present technology acoustic Hawking radiation is almost experimentally testable.
Comment: This is a draft of an introductory chapter on the Kerr spacetime that is intended for us... more Comment: This is a draft of an introductory chapter on the Kerr spacetime that is intended for use in the book "The Kerr spacetime", currently being edited by Susan Scott, Matt Visser, and David Wiltshire. This chapter is intended as a teaser and brief introduction to the mathematics and physics of the Kerr spacetime-it is not, nor is it intended to be, a complete and exhaustive survey of everything in the field. Comments and community feedback, especially regarding clarity and pedagogy, are welcome.
This is a brief survey of the current status of Stephen Hawking's "chronology protection conjectu... more This is a brief survey of the current status of Stephen Hawking's "chronology protection conjecture". That is: "Why does nature abhor a time machine?" I'll discuss a few examples of spacetimes containing "time machines" (closed causal curves), the sorts of peculiarities that arise, and the reactions of the physics community. While pointing out other possibilities, this article concentrates on the possibility of "chronology protection". As Stephen puts it: It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.
In a universe where, according to the standard cosmological models, some 97% of the total mass-en... more In a universe where, according to the standard cosmological models, some 97% of the total mass-energy is still "missing in action" it behooves us to spend at least a little effort critically assessing and exploring radical alternatives. Among possible, (dare we say plausible), nonstandard but superficially viable models, those spacetimes conformal to the standard Friedmann-Lemaître-Robertson-Walker class of cosmological models play a very special role-these models have the unique and important property of permitting large non-perturbative geometric deviations from Friedmann-Lemaître-Robertson-Walker cosmology without unacceptably distorting the cosmic microwave background. Performing a "cosmographic" analysis, (that is, temporarily setting aside the Einstein equations, since the question of whether or not the Einstein equations are valid on galactic and cosmological scales is essentially the same question as whether or not dark matter/dark energy actually exist), and using both supernova data and information about galactic structure, one can nevertheless place some quite significant observational constraints on any possible conformal mode-however there is still an extremely rich range of phenomenological possibilities for both cosmologists and astrophysicists to explore.
Both traversable wormholes and warp drives, concepts originally developed within the context of s... more Both traversable wormholes and warp drives, concepts originally developed within the context of science fiction, have now (for some 30 odd years) been studied, debated, and carefully analyzed within the framework of general relativity. An overarching theme of the general relativistic analysis is unavoidable violations of the classical point-wise energy conditions. Another science fiction trope, now over 80 years old, is the tractor beam and/or pressor beam. We shall discuss how to formulate both tractor beams and/or pressor beams, and a variant to be called a stressor beam, within the context of reverse engineering the spacetime metric. (While such reverse engineering is certainly well beyond our civilization's current capabilities, we shall be more interested in asking what an arbitrarily advanced civilization might be able to accomplish.) We shall see that tractor beams and/or pressor beams can be formulated by suitably modifying the notion of warp drives, and that, as for wormholes and warp drives, violations of the classical point-wise energy conditions are utterly unavoidable.
We consider the characteristics of nonlinear energy conditions and of quantum extensions of these... more We consider the characteristics of nonlinear energy conditions and of quantum extensions of these and the usual energy conditions. We show that they are satisfied by some quantum vacuum states that violate the usual energy conditions.
So-called average subsystem entropies are defined by first taking partial traces over some pure s... more 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 tripartite and multi-partite analyses is that this framework allows one to introduce an "environment" for small subsystems to couple to.
Based on various string theoretic constructions, and various string-inspired generalizations ther... more Based on various string theoretic constructions, and various string-inspired generalizations thereof, there have been repeated suggestions that the areas of black hole event horizons might be quantized in a quite specific manner, in terms of linear combinations of square roots of positive integers. It is important to realise that there are significant physical constraints on such integer-based proposals when one (somewhat speculatively) attempts to extend them outside their original extremal and supersymmetric framework. Specifically, in their most natural and direct physical in-terpretations, some of the more speculative integer-based proposals for the quantization of horizon areas fail for the ordinary Kerr–Newman black holes in (3+1) dimensions, essentially because the fine structure constant is not an integer. A more baroque inter-pretation involves asserting the fine structure constant is the square root of a rational number; but such a proposal has its own problems. Insofar as...
Geodesics (by definition) have an intrinsic 4-acceleration zero. However, when expressed in terms... more Geodesics (by definition) have an intrinsic 4-acceleration zero. However, when expressed in terms of coordinates, the coordinate acceleration d 2 x i /dt 2 can very easily be non-zero, and the coordinate velocity dx i /dt can behave unexpectedly. The situation becomes extremely delicate in the near-horizon limit-for both astrophysical and idealised black holes-where an inappropriate choice of coordinates can quite easily lead to significant confusion. We shall carefully explore the relative merits of horizon-penetrating versus horizon-non-penetrating coordinates, arguing that in the near-horizon limit the coordinate acceleration d 2 x i /dt 2 is best interpreted in terms of horizon-penetrating coordinates.
In a universe where, according to the standard cosmological models, some 97% of the total mass-en... more In a universe where, according to the standard cosmological models, some 97% of the total mass-energy is still "missing in action" it behooves us to spend at least a little effort critically assessing and exploring radical alternatives. Among possible, (dare we say plausible), nonstandard but superficially viable models, those spacetimes conformal to the standard Friedmann-Lemaître-Robertson-Walker class of cosmological models play a very special role-these models have the unique and important property of permitting large non-perturbative geometric deviations from Friedmann-Lemaître-Robertson-Walker cosmology without unacceptably distorting the cosmic microwave background. Performing a "cosmographic" analysis, (that is, temporarily setting aside the Einstein equations, since the question of whether or not the Einstein equations are valid on galactic and cosmological scales is essentially the same question as whether or not dark matter/dark energy actually exist), and using both supernova data and information about galactic structure, one can nevertheless place some quite significant observational constraints on any possible conformal mode-however there is still an extremely rich range of phenomenological possibilities for both cosmologists and astrophysicists to explore.
The classical energy conditions are known to not be fundamental physics-they are typically violat... more The classical energy conditions are known to not be fundamental physics-they are typically violated by semiclassical quantum effects. Consequently, some effort has gone into finding possible semiclassical replacements for the classical energy conditions-the most well developed being the nonlocal time-integrated Ford-Roman quantum inequalities. In the current article we shall instead develop classical and quantum versions of a local "flux energy condition" (FEC and QFEC) based on the notion of constraining the possible fluxes measured by timelike observers. The naive classical FEC will be seen to be satisfied in some situations, and even for some quantum vacuum states, while its quantum analogue (the QFEC) is satisfied (for naturally defined quantum vacuum states) under a rather wide range of conditions. The situation for completely general (non-vacuum) quantum states is less clear.
The 'theoretical' existence of traversable Lorentzian wormholes in the classical, macroscopic wor... more The 'theoretical' existence of traversable Lorentzian wormholes in the classical, macroscopic world is plagued by the violation of the well-known energy conditions of General Relativity. In this brief article we show : (i) how the extent of violation can be quantified using certain volume integrals (ii) whether this 'amount of violation' can be minimised for some specific cut-and-paste geometric constructions. Examples and possibilities are also outlined.
Area products for multi-horizon stationary black holes often have intriguing properties, and are ... more Area products for multi-horizon stationary black holes often have intriguing properties, and are often (though not always) independent of the mass of the black hole itself (depending only on various charges, angular momenta, and moduli). Such products are often formulated in terms of the areas of inner (Cauchy) horizons and outer (event) horizons, and sometimes include the effects of unphysical "virtual" horizons. But the conjectured mass-independence sometimes fails. Specifically, for the Schwarzschild-de Sitter [Kottler] black hole in (3+1) dimensions it is shown by explicit exact calculation that the product of event horizon area and cosmological horizon area is not mass independent. (Including the effect of the third "virtual" horizon does not improve the situation.) Similarly, in the Reissner-Nordstrom-anti-de Sitter black hole in (3+1) dimensions the product of inner (Cauchy) horizon area and event horizon area is calculated (perturbatively), and is shown to be not mass independent. That is, the mass-independence of the product of physical horizon areas is not generic. In spherical symmetry, whenever the quasi-local mass m(r) is a Laurent polynomial in aerial radius, r = A/4π, there are significantly more complicated mass-independent quantities, the elementary symmetric polynomials built up from the complete set of horizon radii (physical and virtual). Sometimes it is possible to eliminate the unphysical virtual horizons, constructing combinations of physical horizon areas that are mass independent, but they tend to be considerably more complicated than the simple products and related constructions currently being mooted in the literature.
The future of theoretical physics and cosmology: …, 2002
8 The quantum physics of chronology protection Matt Visser Washington University, St. Louis Simpl... more 8 The quantum physics of chronology protection Matt Visser Washington University, St. Louis Simply put, chronology protection is the assertion that nature abhors a time machine. In the words of Stephen Hawking [1]: It seems that there is a Chronology Protection Agency which ...
Finding affine parameters for null geodesics is often of considerable physical importance, especi... more Finding affine parameters for null geodesics is often of considerable physical importance, especially when studying the paths swept out by null geodesics or dealing with conservation laws and/or averaged energy conditions. But explicitly finding null affine parameters is also often quite tedious and can sometimes even be somewhat tricky. Herein we shall demonstrate that the existence of a conformally related spacetime containing a conformal Killing vector, timelike in the domain of outer communication, is quite sufficient to define a preferred set of spatial three-slices-on which a well-defined "affine" three-metric can be introduced to capture the notion of affine null parameter-before explicitly finding the null geodesics. The construction depends on the properties of conformal transformations and on the conserved quantity associated with the conformal Killing vector. Having the affine null parameter in hand before attempting to find the actual null geodesics often quite radically simplifies other parts of the analysis. We emphasize that the successful identification of affine null parameters is a general-purpose tool of wide applicability in both general relativistic and astrophysical settings.
Black and white holes play remarkably contrasting roles in general relativity versus observationa... more Black and white holes play remarkably contrasting roles in general relativity versus observational astrophysics. While there is observational evidence for the existence of compact objects that are "cold, dark, and heavy", which thereby are natural candidates for black holes, the theoretically viable time-reversed variants-the "white holes"-have nowhere near the same level of observational support. Herein we shall explore the theoretical possibility that the connection between black and white holes is much more intimate than commonly appreciated. We shall first construct "horizon penetrating" coordinate systems that differ from the standard curvature coordinates only in a small near-horizon region, thereby emphasizing that ultimately the distinction between black and white horizons depends only on near-horizon physics. We shall then construct an explicit model for a "black-to-white transition" where all of the nontrivial physics is confined to a compact region of spacetime-a finite-duration finite-thickness, (in principle arbitrarily small), region straddling the naïve horizon. Moreover we shall show that it is possible to arrange the "black-to-white transition" to have zero action-so that it will not be subject to destructive interference in the Feynman path integral. This then raises the very intriguing possibility that astrophysical black holes might be interpretable in terms of a quantum superposition of black and white horizons-a "gray" horizon.
International Journal of Modern Physics D, Apr 1, 2003
There are numerous derivations of the Hawking effect available in the literature. They emphasise ... more There are numerous derivations of the Hawking effect available in the literature. They emphasise different features of the process, and sometimes make markedly different physical assumptions. This article presents a "minimalist" argument, and strips the derivation of as much excess baggage as possible. All that is really necessary is quantum physics plus a slowly evolving future apparent horizon (not an event horizon). In particular, neither the Einstein equations nor Bekenstein entropy are necessary (nor even useful) in deriving Hawking radiation.
The standard energy conditions of classical general relativity are (mostly) linear in the stress-... more 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 chapter 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. 1 Draft chapter, on which the related chapter of the book Wormholes, Warp Drives and Energy Conditions (to be published by Springer) will be based.
Acoustic propagation in a moving fluid provides a conceptually clean and powerful analogy for und... more Acoustic propagation in a moving fluid provides a conceptually clean and powerful analogy for understanding black hole physics. As a teaching tool, the analogy is useful for introducing students to both General Relativity and fluid mechanics. As a research tool, the analogy helps clarify what aspects of the physics are kinematics and what aspects are dynamics. In particular, Hawking radiation is a purely kinematical effect, whereas black hole entropy is intrinsically dynamical. Finally, I discuss the fact that with present technology acoustic Hawking radiation is almost experimentally testable.
Comment: This is a draft of an introductory chapter on the Kerr spacetime that is intended for us... more Comment: This is a draft of an introductory chapter on the Kerr spacetime that is intended for use in the book "The Kerr spacetime", currently being edited by Susan Scott, Matt Visser, and David Wiltshire. This chapter is intended as a teaser and brief introduction to the mathematics and physics of the Kerr spacetime-it is not, nor is it intended to be, a complete and exhaustive survey of everything in the field. Comments and community feedback, especially regarding clarity and pedagogy, are welcome.
This is a brief survey of the current status of Stephen Hawking's "chronology protection conjectu... more This is a brief survey of the current status of Stephen Hawking's "chronology protection conjecture". That is: "Why does nature abhor a time machine?" I'll discuss a few examples of spacetimes containing "time machines" (closed causal curves), the sorts of peculiarities that arise, and the reactions of the physics community. While pointing out other possibilities, this article concentrates on the possibility of "chronology protection". As Stephen puts it: It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.
In a universe where, according to the standard cosmological models, some 97% of the total mass-en... more In a universe where, according to the standard cosmological models, some 97% of the total mass-energy is still "missing in action" it behooves us to spend at least a little effort critically assessing and exploring radical alternatives. Among possible, (dare we say plausible), nonstandard but superficially viable models, those spacetimes conformal to the standard Friedmann-Lemaître-Robertson-Walker class of cosmological models play a very special role-these models have the unique and important property of permitting large non-perturbative geometric deviations from Friedmann-Lemaître-Robertson-Walker cosmology without unacceptably distorting the cosmic microwave background. Performing a "cosmographic" analysis, (that is, temporarily setting aside the Einstein equations, since the question of whether or not the Einstein equations are valid on galactic and cosmological scales is essentially the same question as whether or not dark matter/dark energy actually exist), and using both supernova data and information about galactic structure, one can nevertheless place some quite significant observational constraints on any possible conformal mode-however there is still an extremely rich range of phenomenological possibilities for both cosmologists and astrophysicists to explore.
Both traversable wormholes and warp drives, concepts originally developed within the context of s... more Both traversable wormholes and warp drives, concepts originally developed within the context of science fiction, have now (for some 30 odd years) been studied, debated, and carefully analyzed within the framework of general relativity. An overarching theme of the general relativistic analysis is unavoidable violations of the classical point-wise energy conditions. Another science fiction trope, now over 80 years old, is the tractor beam and/or pressor beam. We shall discuss how to formulate both tractor beams and/or pressor beams, and a variant to be called a stressor beam, within the context of reverse engineering the spacetime metric. (While such reverse engineering is certainly well beyond our civilization's current capabilities, we shall be more interested in asking what an arbitrarily advanced civilization might be able to accomplish.) We shall see that tractor beams and/or pressor beams can be formulated by suitably modifying the notion of warp drives, and that, as for wormholes and warp drives, violations of the classical point-wise energy conditions are utterly unavoidable.
We consider the characteristics of nonlinear energy conditions and of quantum extensions of these... more We consider the characteristics of nonlinear energy conditions and of quantum extensions of these and the usual energy conditions. We show that they are satisfied by some quantum vacuum states that violate the usual energy conditions.
So-called average subsystem entropies are defined by first taking partial traces over some pure s... more 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 tripartite and multi-partite analyses is that this framework allows one to introduce an "environment" for small subsystems to couple to.
Based on various string theoretic constructions, and various string-inspired generalizations ther... more Based on various string theoretic constructions, and various string-inspired generalizations thereof, there have been repeated suggestions that the areas of black hole event horizons might be quantized in a quite specific manner, in terms of linear combinations of square roots of positive integers. It is important to realise that there are significant physical constraints on such integer-based proposals when one (somewhat speculatively) attempts to extend them outside their original extremal and supersymmetric framework. Specifically, in their most natural and direct physical in-terpretations, some of the more speculative integer-based proposals for the quantization of horizon areas fail for the ordinary Kerr–Newman black holes in (3+1) dimensions, essentially because the fine structure constant is not an integer. A more baroque inter-pretation involves asserting the fine structure constant is the square root of a rational number; but such a proposal has its own problems. Insofar as...
Geodesics (by definition) have an intrinsic 4-acceleration zero. However, when expressed in terms... more Geodesics (by definition) have an intrinsic 4-acceleration zero. However, when expressed in terms of coordinates, the coordinate acceleration d 2 x i /dt 2 can very easily be non-zero, and the coordinate velocity dx i /dt can behave unexpectedly. The situation becomes extremely delicate in the near-horizon limit-for both astrophysical and idealised black holes-where an inappropriate choice of coordinates can quite easily lead to significant confusion. We shall carefully explore the relative merits of horizon-penetrating versus horizon-non-penetrating coordinates, arguing that in the near-horizon limit the coordinate acceleration d 2 x i /dt 2 is best interpreted in terms of horizon-penetrating coordinates.
In a universe where, according to the standard cosmological models, some 97% of the total mass-en... more In a universe where, according to the standard cosmological models, some 97% of the total mass-energy is still "missing in action" it behooves us to spend at least a little effort critically assessing and exploring radical alternatives. Among possible, (dare we say plausible), nonstandard but superficially viable models, those spacetimes conformal to the standard Friedmann-Lemaître-Robertson-Walker class of cosmological models play a very special role-these models have the unique and important property of permitting large non-perturbative geometric deviations from Friedmann-Lemaître-Robertson-Walker cosmology without unacceptably distorting the cosmic microwave background. Performing a "cosmographic" analysis, (that is, temporarily setting aside the Einstein equations, since the question of whether or not the Einstein equations are valid on galactic and cosmological scales is essentially the same question as whether or not dark matter/dark energy actually exist), and using both supernova data and information about galactic structure, one can nevertheless place some quite significant observational constraints on any possible conformal mode-however there is still an extremely rich range of phenomenological possibilities for both cosmologists and astrophysicists to explore.
The classical energy conditions are known to not be fundamental physics-they are typically violat... more The classical energy conditions are known to not be fundamental physics-they are typically violated by semiclassical quantum effects. Consequently, some effort has gone into finding possible semiclassical replacements for the classical energy conditions-the most well developed being the nonlocal time-integrated Ford-Roman quantum inequalities. In the current article we shall instead develop classical and quantum versions of a local "flux energy condition" (FEC and QFEC) based on the notion of constraining the possible fluxes measured by timelike observers. The naive classical FEC will be seen to be satisfied in some situations, and even for some quantum vacuum states, while its quantum analogue (the QFEC) is satisfied (for naturally defined quantum vacuum states) under a rather wide range of conditions. The situation for completely general (non-vacuum) quantum states is less clear.
The 'theoretical' existence of traversable Lorentzian wormholes in the classical, macroscopic wor... more The 'theoretical' existence of traversable Lorentzian wormholes in the classical, macroscopic world is plagued by the violation of the well-known energy conditions of General Relativity. In this brief article we show : (i) how the extent of violation can be quantified using certain volume integrals (ii) whether this 'amount of violation' can be minimised for some specific cut-and-paste geometric constructions. Examples and possibilities are also outlined.
Area products for multi-horizon stationary black holes often have intriguing properties, and are ... more Area products for multi-horizon stationary black holes often have intriguing properties, and are often (though not always) independent of the mass of the black hole itself (depending only on various charges, angular momenta, and moduli). Such products are often formulated in terms of the areas of inner (Cauchy) horizons and outer (event) horizons, and sometimes include the effects of unphysical "virtual" horizons. But the conjectured mass-independence sometimes fails. Specifically, for the Schwarzschild-de Sitter [Kottler] black hole in (3+1) dimensions it is shown by explicit exact calculation that the product of event horizon area and cosmological horizon area is not mass independent. (Including the effect of the third "virtual" horizon does not improve the situation.) Similarly, in the Reissner-Nordstrom-anti-de Sitter black hole in (3+1) dimensions the product of inner (Cauchy) horizon area and event horizon area is calculated (perturbatively), and is shown to be not mass independent. That is, the mass-independence of the product of physical horizon areas is not generic. In spherical symmetry, whenever the quasi-local mass m(r) is a Laurent polynomial in aerial radius, r = A/4π, there are significantly more complicated mass-independent quantities, the elementary symmetric polynomials built up from the complete set of horizon radii (physical and virtual). Sometimes it is possible to eliminate the unphysical virtual horizons, constructing combinations of physical horizon areas that are mass independent, but they tend to be considerably more complicated than the simple products and related constructions currently being mooted in the literature.
The future of theoretical physics and cosmology: …, 2002
8 The quantum physics of chronology protection Matt Visser Washington University, St. Louis Simpl... more 8 The quantum physics of chronology protection Matt Visser Washington University, St. Louis Simply put, chronology protection is the assertion that nature abhors a time machine. In the words of Stephen Hawking [1]: It seems that there is a Chronology Protection Agency which ...
Finding affine parameters for null geodesics is often of considerable physical importance, especi... more Finding affine parameters for null geodesics is often of considerable physical importance, especially when studying the paths swept out by null geodesics or dealing with conservation laws and/or averaged energy conditions. But explicitly finding null affine parameters is also often quite tedious and can sometimes even be somewhat tricky. Herein we shall demonstrate that the existence of a conformally related spacetime containing a conformal Killing vector, timelike in the domain of outer communication, is quite sufficient to define a preferred set of spatial three-slices-on which a well-defined "affine" three-metric can be introduced to capture the notion of affine null parameter-before explicitly finding the null geodesics. The construction depends on the properties of conformal transformations and on the conserved quantity associated with the conformal Killing vector. Having the affine null parameter in hand before attempting to find the actual null geodesics often quite radically simplifies other parts of the analysis. We emphasize that the successful identification of affine null parameters is a general-purpose tool of wide applicability in both general relativistic and astrophysical settings.
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