Feynman’s iϵ prescription for quantum field theoretic propagators has a quite natural reinterpret... more Feynman’s iϵ prescription for quantum field theoretic propagators has a quite natural reinterpretation in terms of a slight complex deformation of the Minkowski space-time metric. Though originally a strictly flat-space result, once reinterpreted in this way, these ideas can be naturally extended first to semi-classical curved-spacetime QFT on a fixed background geometry and then, (with more work), to fluctuating spacetime geometries. There are intimate connections with variants of the weak energy condition. We shall take the Lorentzian signature metric as primary, but note that allowing the complex deformation to become large leads to a variant of Wick rotation, and more importantly leads to physically motivated constraints on the configuration space of acceptable off-shell geometries to include in Feynman’s functional integral when attempting to quantize gravity. Ultimately this observation allows one to connect the discussion back to recent ideas on “acceptable” complex metrics, ...
Despite some 60 years of work on the subject of the Kerr rotating black hole there is as yet no w... more Despite some 60 years of work on the subject of the Kerr rotating black hole there is as yet no widely accepted physically based and pedagogically viable ansatz suitable for deriving the Kerr solution without significant computational effort. (Typically involving computer-aided symbolic algebra.) Perhaps the closest one gets in this regard is the Newman–Janis trick; a trick which requires several physically unmotivated choices in order to work. Herein we shall try to make some progress on this issue by using a non-ortho-normal tetrad based on oblate spheroidal coordinates to absorb as much of the messy angular dependence as possible, leaving one to deal with a relatively simple angle-independent tetrad-component metric. That is, we shall write g a b = g A B e A a e B b seeking to keep both the tetrad-component metric g AB and the non-ortho-normal co-tetrad e A a relatively simple but non-trivial. We shall see that it is possible to put all the mass dependence into g AB , while the n...
Herein we explore the non-equatorial constant-r (“quasi-circular”) geodesics (both timelike and n... more Herein we explore the non-equatorial constant-r (“quasi-circular”) geodesics (both timelike and null) in the Painlevé–Gullstrand variant of the Lense–Thirring spacetime recently introduced by the current authors. Even though the spacetime is not spherically symmetric, shells of constant-r geodesics still exist. Whereas the radial motion is (by construction) utterly trivial, determining the allowed locations of these constant-r geodesics is decidedly non-trivial, and the stability analysis is equally tricky. Regarding the angular motion, these constant-r orbits will be seen to exhibit both precession and nutation — typically with incommensurate frequencies. Thus this constant-r geodesic motion, though integrable in the precise technical sense, is generically surface-filling, with the orbits completely covering a symmetric equatorial band which is a segment of a spherical surface, (a so-called “spherical zone”), and whose latitudinal extent is governed by delicate interplay between th...
Feynman’s iǫ prescription for quantum field theoretic propagators has a natural reinterpretation ... more Feynman’s iǫ prescription for quantum field theoretic propagators has a natural reinterpretation in terms of a slight complex deformation of the Minkowski spacetime metric. Though originally a strictly flat-space result, once reinterpreted in this way, these ideas can be naturally extended first to semi-classical curved-spacetime QFT on a fixed background geometry and then, (with more work), to fluctuating spacetime geometries. There are intimate connections with with variants of the weak energy condition. We shall take the Lorentzian signature metric as primary, but note that allowing the complex deformation to become large leads to a variant of Wick rotation, and more importantly leads to physically motivated constraints on the configuration space of acceptable off-shell geometries to include in Feynman’s functional integral when attempting to quantize gravity. Ultimately this allows one to connect the discussion back to recent ideas on “acceptable” complex metrics, in the Louko–S...
Normally one thinks of the observed cosmological constant as being so small that it can be utterl... more Normally one thinks of the observed cosmological constant as being so small that it can be utterly neglected on typical astrophysical scales, only affecting extremely large-scale cosmology at Gigaparsec scales. Indeed, in those situations where the cosmological constant only has a quantitative influence on the physics, a separation of scales argument guarantees the effect is indeed negligible. The exception to this argument arises when the presence of a cosmological constant qualitatively changes the physics. One example of this phenomenon is the existence of outermost stable circular orbits (OSCOs) in the presence of a positive cosmological constant. Remarkably the size of these OSCOs are of a magnitude to be astrophysically interesting. For instance: for galactic masses the OSCOs are of order the inter-galactic spacing, for galaxy cluster masses the OSCOs are of order the size of the cluster.
Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Len... more Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Lense–Thirring spacetime, which has some particularly elegant features, including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics—the “rain” geodesics. At the linear level in the rotation parameter, this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton–Jacobi equation. Furthermore, we shall show that the Klein–Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing–Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.
arXiv: General Relativity and Quantum Cosmology, 2018
Physically reasonable stationary axisymmetric spacetimes can (under very mild technical condition... more Physically reasonable stationary axisymmetric spacetimes can (under very mild technical conditions) be put into Boyer-Lindquist form. Unfortunately a metric presented in Boyer-Lindquist form is not well-adapted to the "quasi-Cartesian" meta-material analysis we developed in our previous article on "bespoke analogue spacetimes" (arXiv:1801.05549 [gr-qc]). In the current article we first focus specifically on spacetime metrics presented in Boyer-Lindquist form, and determine the equivalent meta-material susceptibility tensors in a laboratory setting. We then turn to analyzing generic stationary spacetimes, again determining the equivalent meta-material susceptibility tensors. While the background laboratory metric is always taken to be Riemann-flat, we now allow for arbitrary curvilinear coordinate systems. Finally, we reconsider static spherically symmetric spacetimes, but now in general spherical polar rather than quasi-Cartesian coordinates. The article provides...
The Newman–Janis trick is a procedure, (not even really an ansatz), for obtaining the Kerr spacet... more The Newman–Janis trick is a procedure, (not even really an ansatz), for obtaining the Kerr spacetime from the Schwarzschild spacetime. This 50 years old trick continues to generate heated discussion and debate even to this day. Most of the debate focusses on whether the Newman–Janis procedure can be upgraded to the status of an algorithm, or even an inspired ansatz, or is it just a random trick of no deep physical significance. (That the Newman–Janis procedure very quickly led to the discovery of the Kerr–Newman spacetime is a point very much in its favor.) In the current paper, we will not answer these deeper questions, we shall instead present a much simpler alternative variation on the theme of the Newman–Janis trick that might be easier to work with. We shall present a 2-step version of the Newman–Janis trick that works directly with the Kerr–Schild “Cartesian” metric presentation of the Kerr spacetime. That is, we show how the original 4-step Newman–Janis procedure can, (using ...
We argue that an arbitrary general relativistic static anisotropic fluid sphere, (static and sphe... more We argue that an arbitrary general relativistic static anisotropic fluid sphere, (static and spherically symmetric but with transverse pressure not equal to radial pressure), can nevertheless be successfully mimicked by suitable linear combinations of theoretically attractive and quite simple classical matter: a classical (charged) isotropic perfect fluid, a classical electromagnetic field and a classical (minimally coupled) scalar field. While the most general decomposition is not unique, a preferred minimal decomposition can be constructed that is unique. We show how the classical energy conditions for the anisotropic fluid sphere can be related to energy conditions for the isotropic perfect fluid, electromagnetic field, and scalar field components of the model. Furthermore, we show how this decomposition relates to the distribution of both electric charge density and scalar charge density throughout the model. The generalized TOV equation implies that the perfect fluid component ...
The energy conditions of Einsteinian gravity (classical general relativity) do not require one to... more The energy conditions of Einsteinian gravity (classical general relativity) do not require one to fix a specific equation of state. In a Friedmann-Robertson-Walker universe where the equation of state for the cosmological fluid is uncertain, the energy conditions provide simple, model-independent, and robust bounds on the behavior of the density and look-back time as a function of red shift. Current observations suggest that the “strong energy condition” was violated sometime between the epoch of galaxy formation and the present. This implies that no possible combination of “normal” matter is capable of fitting the observational data.
Cosmology is most typically analyzed using standard co-moving coordinates, in which the galaxies ... more Cosmology is most typically analyzed using standard co-moving coordinates, in which the galaxies are (on average, up to presumably small peculiar velocities) “at rest”, while “space” is expanding. But this is merely a specific coordinate choice; and it is important to realise that for certain purposes other, (sometimes radically, different) coordinate choices might also prove useful and informative, but without changing the underlying physics. Specifically, herein we shall consider the k= 0 spatially flat FLRW cosmology but in Painlevé-Gullstrand coordinates — these coordinates are very explicitly not co-moving: “space” is now no longer expanding, although the distance between galaxies is still certainly increasing. Working in these Painlevé-Gullstrand coordinates provides an alternate viewpoint on standard cosmology, and the symmetries thereof, and also makes it somewhat easier to handle cosmological horizons. With a longer view, we hope that investigating these Painlevé-Gullstrand...
Feynman’s iϵ prescription for quantum field theoretic propagators has a quite natural reinterpret... more Feynman’s iϵ prescription for quantum field theoretic propagators has a quite natural reinterpretation in terms of a slight complex deformation of the Minkowski space-time metric. Though originally a strictly flat-space result, once reinterpreted in this way, these ideas can be naturally extended first to semi-classical curved-spacetime QFT on a fixed background geometry and then, (with more work), to fluctuating spacetime geometries. There are intimate connections with variants of the weak energy condition. We shall take the Lorentzian signature metric as primary, but note that allowing the complex deformation to become large leads to a variant of Wick rotation, and more importantly leads to physically motivated constraints on the configuration space of acceptable off-shell geometries to include in Feynman’s functional integral when attempting to quantize gravity. Ultimately this observation allows one to connect the discussion back to recent ideas on “acceptable” complex metrics, ...
Despite some 60 years of work on the subject of the Kerr rotating black hole there is as yet no w... more Despite some 60 years of work on the subject of the Kerr rotating black hole there is as yet no widely accepted physically based and pedagogically viable ansatz suitable for deriving the Kerr solution without significant computational effort. (Typically involving computer-aided symbolic algebra.) Perhaps the closest one gets in this regard is the Newman–Janis trick; a trick which requires several physically unmotivated choices in order to work. Herein we shall try to make some progress on this issue by using a non-ortho-normal tetrad based on oblate spheroidal coordinates to absorb as much of the messy angular dependence as possible, leaving one to deal with a relatively simple angle-independent tetrad-component metric. That is, we shall write g a b = g A B e A a e B b seeking to keep both the tetrad-component metric g AB and the non-ortho-normal co-tetrad e A a relatively simple but non-trivial. We shall see that it is possible to put all the mass dependence into g AB , while the n...
Herein we explore the non-equatorial constant-r (“quasi-circular”) geodesics (both timelike and n... more Herein we explore the non-equatorial constant-r (“quasi-circular”) geodesics (both timelike and null) in the Painlevé–Gullstrand variant of the Lense–Thirring spacetime recently introduced by the current authors. Even though the spacetime is not spherically symmetric, shells of constant-r geodesics still exist. Whereas the radial motion is (by construction) utterly trivial, determining the allowed locations of these constant-r geodesics is decidedly non-trivial, and the stability analysis is equally tricky. Regarding the angular motion, these constant-r orbits will be seen to exhibit both precession and nutation — typically with incommensurate frequencies. Thus this constant-r geodesic motion, though integrable in the precise technical sense, is generically surface-filling, with the orbits completely covering a symmetric equatorial band which is a segment of a spherical surface, (a so-called “spherical zone”), and whose latitudinal extent is governed by delicate interplay between th...
Feynman’s iǫ prescription for quantum field theoretic propagators has a natural reinterpretation ... more Feynman’s iǫ prescription for quantum field theoretic propagators has a natural reinterpretation in terms of a slight complex deformation of the Minkowski spacetime metric. Though originally a strictly flat-space result, once reinterpreted in this way, these ideas can be naturally extended first to semi-classical curved-spacetime QFT on a fixed background geometry and then, (with more work), to fluctuating spacetime geometries. There are intimate connections with with variants of the weak energy condition. We shall take the Lorentzian signature metric as primary, but note that allowing the complex deformation to become large leads to a variant of Wick rotation, and more importantly leads to physically motivated constraints on the configuration space of acceptable off-shell geometries to include in Feynman’s functional integral when attempting to quantize gravity. Ultimately this allows one to connect the discussion back to recent ideas on “acceptable” complex metrics, in the Louko–S...
Normally one thinks of the observed cosmological constant as being so small that it can be utterl... more Normally one thinks of the observed cosmological constant as being so small that it can be utterly neglected on typical astrophysical scales, only affecting extremely large-scale cosmology at Gigaparsec scales. Indeed, in those situations where the cosmological constant only has a quantitative influence on the physics, a separation of scales argument guarantees the effect is indeed negligible. The exception to this argument arises when the presence of a cosmological constant qualitatively changes the physics. One example of this phenomenon is the existence of outermost stable circular orbits (OSCOs) in the presence of a positive cosmological constant. Remarkably the size of these OSCOs are of a magnitude to be astrophysically interesting. For instance: for galactic masses the OSCOs are of order the inter-galactic spacing, for galaxy cluster masses the OSCOs are of order the size of the cluster.
Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Len... more Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Lense–Thirring spacetime, which has some particularly elegant features, including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics—the “rain” geodesics. At the linear level in the rotation parameter, this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton–Jacobi equation. Furthermore, we shall show that the Klein–Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing–Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.
arXiv: General Relativity and Quantum Cosmology, 2018
Physically reasonable stationary axisymmetric spacetimes can (under very mild technical condition... more Physically reasonable stationary axisymmetric spacetimes can (under very mild technical conditions) be put into Boyer-Lindquist form. Unfortunately a metric presented in Boyer-Lindquist form is not well-adapted to the "quasi-Cartesian" meta-material analysis we developed in our previous article on "bespoke analogue spacetimes" (arXiv:1801.05549 [gr-qc]). In the current article we first focus specifically on spacetime metrics presented in Boyer-Lindquist form, and determine the equivalent meta-material susceptibility tensors in a laboratory setting. We then turn to analyzing generic stationary spacetimes, again determining the equivalent meta-material susceptibility tensors. While the background laboratory metric is always taken to be Riemann-flat, we now allow for arbitrary curvilinear coordinate systems. Finally, we reconsider static spherically symmetric spacetimes, but now in general spherical polar rather than quasi-Cartesian coordinates. The article provides...
The Newman–Janis trick is a procedure, (not even really an ansatz), for obtaining the Kerr spacet... more The Newman–Janis trick is a procedure, (not even really an ansatz), for obtaining the Kerr spacetime from the Schwarzschild spacetime. This 50 years old trick continues to generate heated discussion and debate even to this day. Most of the debate focusses on whether the Newman–Janis procedure can be upgraded to the status of an algorithm, or even an inspired ansatz, or is it just a random trick of no deep physical significance. (That the Newman–Janis procedure very quickly led to the discovery of the Kerr–Newman spacetime is a point very much in its favor.) In the current paper, we will not answer these deeper questions, we shall instead present a much simpler alternative variation on the theme of the Newman–Janis trick that might be easier to work with. We shall present a 2-step version of the Newman–Janis trick that works directly with the Kerr–Schild “Cartesian” metric presentation of the Kerr spacetime. That is, we show how the original 4-step Newman–Janis procedure can, (using ...
We argue that an arbitrary general relativistic static anisotropic fluid sphere, (static and sphe... more We argue that an arbitrary general relativistic static anisotropic fluid sphere, (static and spherically symmetric but with transverse pressure not equal to radial pressure), can nevertheless be successfully mimicked by suitable linear combinations of theoretically attractive and quite simple classical matter: a classical (charged) isotropic perfect fluid, a classical electromagnetic field and a classical (minimally coupled) scalar field. While the most general decomposition is not unique, a preferred minimal decomposition can be constructed that is unique. We show how the classical energy conditions for the anisotropic fluid sphere can be related to energy conditions for the isotropic perfect fluid, electromagnetic field, and scalar field components of the model. Furthermore, we show how this decomposition relates to the distribution of both electric charge density and scalar charge density throughout the model. The generalized TOV equation implies that the perfect fluid component ...
The energy conditions of Einsteinian gravity (classical general relativity) do not require one to... more The energy conditions of Einsteinian gravity (classical general relativity) do not require one to fix a specific equation of state. In a Friedmann-Robertson-Walker universe where the equation of state for the cosmological fluid is uncertain, the energy conditions provide simple, model-independent, and robust bounds on the behavior of the density and look-back time as a function of red shift. Current observations suggest that the “strong energy condition” was violated sometime between the epoch of galaxy formation and the present. This implies that no possible combination of “normal” matter is capable of fitting the observational data.
Cosmology is most typically analyzed using standard co-moving coordinates, in which the galaxies ... more Cosmology is most typically analyzed using standard co-moving coordinates, in which the galaxies are (on average, up to presumably small peculiar velocities) “at rest”, while “space” is expanding. But this is merely a specific coordinate choice; and it is important to realise that for certain purposes other, (sometimes radically, different) coordinate choices might also prove useful and informative, but without changing the underlying physics. Specifically, herein we shall consider the k= 0 spatially flat FLRW cosmology but in Painlevé-Gullstrand coordinates — these coordinates are very explicitly not co-moving: “space” is now no longer expanding, although the distance between galaxies is still certainly increasing. Working in these Painlevé-Gullstrand coordinates provides an alternate viewpoint on standard cosmology, and the symmetries thereof, and also makes it somewhat easier to handle cosmological horizons. With a longer view, we hope that investigating these Painlevé-Gullstrand...
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