Vacuum Quantum Fluctuations in Curved Space and the Theory of Gravitation

Sakharov, A. D.

doi:10.1023/a:1001947813563

Cited by 207 publications

(183 citation statements)

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“…Interestingly, regularization of field theories in curved space is known to induce in the action a series in the curvature tensor, see [20]. We now see that if this series possesses a finite radius of convergence it induces a bandwidth cutoff and sampling theorem for the metric itself, i.e., a cutoff on the curvature of space-time.…”

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confidence: 67%

Covariant Information-Density Cutoff in Curved Space-Time

Kempf

2004

Phys. Rev. Lett.

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In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity.PACS numbers: 04.60.-m, 03.67.-a, 02.90.+p It is generally assumed that the notion of distance loses operational meaning at the Planck scale, l P ≈ 10 −35 m (assuming 3+1 dimensions), due to the combined effects of general relativity and quantum theory. Namely, if one tried to resolve a spatial region with an uncertainty of less than a Planck length, then the corresponding momentum uncertainty should randomly curve and thereby significantly disturb the very region in space that was meant to be resolved. It is expected, therefore, that the existence of a smallest possible length, area or volume, at the Planck scale or above, plays a central role in the yet-to-be-found theory of quantum gravity.In the literature, no consensus has been reached as to whether this implies that space-time is discrete. On the one hand, quantization literally means discretization, and space-time discreteness is indeed naturally accommodated within the functional analytic framework of quantum theory, see, e.g., [1]. Also, most interacting quantum field theories are mathematically well-defined only on lattices. On the other hand, within the mathematical framework of general relativity, space-time is naturally described as a differentiable manifold and deep principles such as local Lorentz invariance would appear to be violated if space-time were discrete.There is the possibility that the cardinality of spacetime is between discrete and continuous, but it is strongly restricted by results of Gödel and Cohen. In [2], they proved that both are consistent with conventional (ZF) set theory: to adopt an axiom claiming the existence of sets with intermediate cardinality or to adopt an axiom claiming their non-existence. Therefore, it is not possible to explicitly construct any set of cardinality between discrete and continuous infinity from the axioms of conventional set theory. If space-time is describable as a set and if this set is of intermediate cardinality, then its description cannot be constructive and requires mathematics beyond conventional set theory.In this Letter, we consider a simpler possibility. In a concrete sense, space-time could be simultaneously discrete and continuous. Namely, in the simplest case, physical fields could be differentiable functions which possess merely a finite density of degrees of freedom. If such a field's amplitude is sam...

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confidence: 67%

Covariant Information-Density Cutoff in Curved Space-Time

Kempf

2004

Phys. Rev. Lett.

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“…One of the more fascinating approaches to “quantum gravity” is the suggestion, typically attributed to Sakharov [540, 628], that gravity itself may not be “fundamental physics”. Indeed it is now a relatively common opinion, maybe not mainstream but definitely a strong minority opinion, that gravity (and in particular the whole notion of spacetime and spacetime geometry) might be no more “fundamental” than is fluid dynamics.…”

Section: Towards a Theory Of Quantum Gravity?mentioning

confidence: 99%

Analogue Gravity

Barceló

Liberati

Visser

2011

Living Rev. Relativ.

724

1,080

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Analogue gravity is a research programme which investigates analogues of general relativistic gravitational fields within other physical systems, typically but not exclusively condensed matter systems, with the aim of gaining new insights into their corresponding problems. Analogue models of (and for) gravity have a long and distinguished history dating back to the earliest years of general relativity. In this review article we will discuss the history, aims, results, and future prospects for the various analogue models. We start the discussion by presenting a particularly simple example of an analogue model, before exploring the rich history and complex tapestry of models discussed in the literature. The last decade in particular has seen a remarkable and sustained development of analogue gravity ideas, leading to some hundreds of published articles, a workshop, two books, and this review article. Future prospects for the analogue gravity programme also look promising, both on the experimental front (where technology is rapidly advancing) and on the theoretical front (where variants of analogue models can be used as a springboard for radical attacks on the problem of quantum gravity).

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“…One of the more fascinating approaches to “quantum gravity” is the suggestion, typically attributed to Sakharov [332, 393], that gravity itself may not be “fundamental physics”. Indeed it is now a relatively common opinion, maybe not mainstream but definitely a strong minority opinion, that gravity (and in particular the whole notion of spacetime and spacetime geometry) might be no more “fundamental” than is fluid dynamics.…”

Section: Future Directionsmentioning

confidence: 99%

“…Sakharov had in mind a specific model in which gravity could be viewed as an “elasticity” of the spacetime medium, and was “induced” via one-loop physics in the matter sector [332, 393]. In this way Sakharov had hoped to relate the observed value of Newton’s constant (and the cosmological constant) to the spectrum of particle masses.…”

Section: Future Directionsmentioning

confidence: 99%

Analogue Gravity

Barceló

Liberati

Visser

2005

Living Rev. Relativ.

947

992

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Analogue models of (and for) gravity have a long and distinguished history dating back to the earliest years of general relativity. In this review article we will discuss the history, aims, results, and future prospects for the various analogue models. We start the discussion by presenting a particularly simple example of an analogue model, before exploring the rich history and complex tapestry of models discussed in the literature. The last decade in particular has seen a remarkable and sustained development of analogue gravity ideas, leading to some hundreds of published articles, a workshop, two books, and this review article. Future prospects for the analogue gravity programme also look promising, both on the experimental front (where technology is rapidly advancing) and on the theoretical front (where variants of analogue models can be used as a springboard for radical attacks on the problem of quantum gravity).

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Vacuum Quantum Fluctuations in Curved Space and the Theory of Gravitation

Cited by 207 publications

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Covariant Information-Density Cutoff in Curved Space-Time

Covariant Information-Density Cutoff in Curved Space-Time

Analogue Gravity

Analogue Gravity

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