Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

The classical uncertainty principle inequalities were imposed as a mathematical constraint over the general relativity geodesic equation. In this way, the uncertainty principle was reformulated in terms of the proper space-time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirmed the necessity for a minimum length for the space-time line element in the geodesic, dependent on a geodesic-derived scalar which made the expression Lorentz-covariant. In agreement with quantum gravity theories, GeUP required the imposition of a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone was found around the singularity where uncertainty in space-time diverged to infinity.

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Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

10.20944/preprints202107.0646.v2 ◽

2021 ◽

Author(s):

David Escors ◽

Grazyna Kochan

Keyword(s):

Mathematical Model ◽

General Relativity ◽

Black Hole ◽

Quantum Gravity ◽

Uncertainty Principle ◽

Zero Point ◽

Space Time ◽

Planck Length ◽

Exclusion Zone ◽

Curvature Perturbation

General relativity is a theory for gravitation based on Riemannian geometry, difficult to compatibilize with quantum mechanics. This is evident in relativistic problems in which quantum effects cannot be discarded. For example in quantum gravity, gravitation of zero-point energy or events close to a black hole singularity. Here, we set up a mathematical model to select general relativity geodesics according to compatibility with the uncertainty principle. To achieve this, we derived a geometric expression of the uncertainty principle (GeUP). This formulation identified proper space-time length with Planck length by a geodesic-derived scalar. GeUP imposed a minimum allowed value for the interval of proper space-time which depended on the particular space-time geometry. GeUP forced the introduction of a “zero-point” curvature perturbation over flat Minkowski space, caused exclusively by quantum uncertainty but not to gravitation. When applied to the Schwarzschild metric and choosing radial-dependent geodesics, our mathematical model identified a particle exclusion zone close to the singularity, similar to calculations by loop quantum gravity. For a 2 black hole merger, this exclusion zone was shown to have a radius that cannot go below a value proportional to the energy/mass of the incoming black hole multiplied by Planck length.

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Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

10.20944/preprints202107.0646.v1 ◽

2021 ◽

Author(s):

David Escors ◽

Grazyna Kochan

Keyword(s):

Mathematical Model ◽

General Relativity ◽

Black Hole ◽

Quantum Gravity ◽

Uncertainty Principle ◽

Zero Point ◽

Space Time ◽

Planck Length ◽

Exclusion Zone ◽

Curvature Perturbation

General relativity is a theory for gravitation based on Riemannian geometry, difficult to compatibilize with quantum mechanics. This is evident in relativistic problems in which quantum effects cannot be discarded. For example in quantum gravity, gravitation of zero-point energy or events close to a black hole singularity. Here, we set up a mathematical model to select general relativity geodesics according to compatibility with the uncertainty principle. To achieve this, we derived a geometric expression of the uncertainty principle (GUP). This formulation identified proper space-time length with Planck length by a geodesic-derived scalar. GUP imposed a minimum allowed value for the interval of proper space-time which depended on the particular space-time geometry. GUP forced the introduction of a “zero-point” curvature perturbation over flat Minkowski space, caused exclusively by quantum uncertainty but not to gravitation. When applied to the Schwarzschild metric and choosing radial-dependent geodesics, our mathematical model identified a particle exclusion zone close to the singularity, similar to calculations by loop quantum gravity. For a 2 black hole merger, this exclusion zone was shown to have a radius that cannot go below a value proportional to the energy/mass of the incoming black hole multiplied by Planck length.

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Covariant Space-Time Line Elements in the Friedmann-Lemaitre-Robertson-Walker Geometry

10.20944/preprints202110.0452.v2 ◽

2021 ◽

Author(s):

David Escors ◽

Grazyna Kochan

Keyword(s):

Quantum Gravity ◽

Uncertainty Principle ◽

Lorentz Invariance ◽

Space Time ◽

Planck Length ◽

Time Line ◽

Geometric Uncertainty ◽

Expected Length ◽

Gravity Theories ◽

Line Elements

Most quantum gravity theories endow space-time with a discreet nature by space quantization on the order of Planck length (lp ). This discreetness could be demonstrated by confirmation of Lorentz invariance violations (LIV) manifested at length scales proportional to lp. In this paper, space-time line elements compatible with the uncertainty principle are calculated for a homogeneous, isotropic expanding Universe represented by the Friedmann-Lemaitre-Robertson-Walker solution to General Relativity (FLRW or FRW metric). To achieve this, the covariant geometric uncertainty principle (GeUP) is applied as a constraint over geodesics in FRW geometries. A generic expression for the quadratic proper space-time line element is derived, proportional to Planck length-squared and dependent on two contributions. The first is associated to the energy-time uncertainty, and the second depends on the Hubble function. The results are in agreement with space-time quantization on the expected length orders, according to quantum gravity theories and experimental constraints on LIV.

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Covariant Space-Time Line Elements in the Friedmann-Lemaitre-Robertson-Walker Geometry

10.20944/preprints202110.0452.v1 ◽

2021 ◽

Author(s):

David Escors ◽

Grazyna Kochan

Keyword(s):

Quantum Gravity ◽

Uncertainty Principle ◽

Lorentz Invariance ◽

Space Time ◽

Planck Length ◽

Time Line ◽

Geometric Uncertainty ◽

Expected Length ◽

Gravity Theories ◽

Line Elements

Most quantum gravity theories endow space-time with a discreet nature by space quantization on the order of Planck length (lp ). This discreetness could be demonstrated by confirmation of Lorentz invariance violations (LIV) manifested at length scales proportional to lp. In this paper, space-time line elements compatible with the uncertainty principle are calculated for a homogeneous, isotropic expanding Universe represented by the Friedmann-Lemaitre-Robertson-Walker solution to General Relativity (FLRW or FRW metric). To achieve this, the covariant geometric uncertainty principle (GeUP) is applied as a constraint over geodesics in FRW geometries. A generic expression for the quadratic proper space-time line element is derived, proportional to Planck length-squared and dependent on two contributions. The first is associated to the energy-time uncertainty, and the second depends on the Hubble function. The results are in agreement with space-time quantization on the expected length orders, according to quantum gravity theories and experimental constraints on LIV.

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SPACE–TIME ENERGY–MOMENTUM, THE OBSERVER AND UNCERTAINTY IN GENERAL RELATIVITY

International Journal of Modern Physics D ◽

10.1142/s0218271810018487 ◽

2010 ◽

Vol 19 (14) ◽

pp. 2353-2359 ◽

Cited By ~ 3

Author(s):

F. I. COOPERSTOCK ◽

M. J. DUPRE

Keyword(s):

General Relativity ◽

Gravitational Field ◽

Uncertainty Principle ◽

Ricci Tensor ◽

Space Time ◽

Energy Integral ◽

New Approach ◽

Extended Space ◽

Energy And Momentum

In this essay, we introduce a new approach to energy–momentum in general relativity. Space–time, as opposed to space, is recognized as the necessary arena for its examination, leading us to define new extended space–time energy and momentum constructs. From local and global considerations, we conclude that the Ricci tensor is the required element for a localized expression of energy–momentum to include the gravitational field. We present and rationalize a fully invariant extended expression for space–time energy, guided by Tolman's well-known energy integral for an arbitrary bounded stationary system. This raises fundamental issues which we discuss. The role of the observer emerges naturally and we are led to an extension of the uncertainty principle to general relativity, of particular relevance to ultra-strong gravity.

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A Role of the Uncertainty Principle in General Relativity and the Limiting Size of Collapsing Fermion Spheres

Progress of Theoretical Physics ◽

10.1143/ptp.46.984 ◽

1971 ◽

Vol 46 (3) ◽

pp. 984-989 ◽

Cited By ~ 2

Author(s):

Noboru Hokkyo

Keyword(s):

General Relativity ◽

Uncertainty Principle

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WEYL GEOMETRY AS CHARACTERIZATION OF SPACE-TIME

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194511001176 ◽

2011 ◽

Vol 03 ◽

pp. 87-97 ◽

Cited By ~ 5

Author(s):

F. P. POULIS ◽

J. M. SALIM

Keyword(s):

General Relativity ◽

Scalar Field ◽

Riemannian Geometry ◽

Axiomatic Approach ◽

Space Time ◽

Weyl Geometry ◽

Test Particles ◽

Variational Formalism ◽

Physical Fields

Motivated by an axiomatic approach to characterize space-time it is investigated a reformulation of Einstein's gravity where the pseudo-riemannian geometry is substituted by a Weyl one. It is presented the main properties of the Weyl geometry and it is shown that it gives extra contributions to the trajectories of test particles, serving as one more motivation to study general relativity in Weyl geometry. It is introduced its variational formalism and it is established the coupling with other physical fields in such a way that the theory acquires a gauge symmetry for the geometrical fields. It is shown that this symmetry is still present for the red-shift and it is concluded that for cosmological models it opens the possibility that observations can be fully described by the new geometrical scalar field. It is concluded then that this reformulation, although representing a theoretical advance, still needs a complete description of their objects.

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