Annals of Physics, Vol. 338, November 2013, Pages 186-194 , Jan 1, 2012
The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, a... more The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big Bang singularity.
In previous papers it has been proposed an equivalent form of Einstein's equation, which extends it and remains valid at an important class of singularities (including in particular the Schwarzschild, FLRW, and isotropic singularities). Here it is shown that if the Big Bang singularity is from this class, it also satisfies the Weyl curvature hypothesis.
As an application, we study a very general example of cosmological model, which generalizes the FLRW model by dropping the isotropy and homogeneity constraints. This model generalizes both the FLRW model and the isotropic singularities. We show that the Big-Bang singularity of this model is of the type under consideration, and satisfies therefore the Weyl curvature hypothesis.
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Recent advances in understanding the geometry of singularities do not require modification of General Relativity, being just non-singular extensions of its mathematics to the limit cases. They turn out to work fine for some known types of cosmological singularities (black holes and FLRW Big-Bang), allowing a choice of the fundamental geometric invariants and physical quantities which remain regular. The resulting equations are equivalent to the standard ones outside the singularities.
One consequence of this mathematical approach to the singularities in General Relativity is a special, (geo)metric type of dimensional reduction: at singularities, the metric tensor becomes degenerate in certain spacetime directions, and some properties of the fields become independent of those directions. Effectively, it is like one or more dimensions of spacetime just vanish at singularities. This suggests that it is worth exploring the possibility that the geometry of singularities leads naturally to the spontaneous dimensional reduction needed by Quantum Gravity.
In previous papers it has been proposed an equivalent form of Einstein's equation, which extends it and remains valid at an important class of singularities (including in particular the Schwarzschild, FLRW, and isotropic singularities). Here it is shown that if the Big Bang singularity is from this class, it also satisfies the Weyl curvature hypothesis.
As an application, we study a very general example of cosmological model, which generalizes the FLRW model by dropping the isotropy and homogeneity constraints. This model generalizes both the FLRW model and the isotropic singularities. We show that the Big-Bang singularity of this model is of the type under consideration, and satisfies therefore the Weyl curvature hypothesis.
general relativity.
First is introduced the singular semi-Riemannian geometry for metrics
which can change their signature (in particular be degenerate). The
standard operations like covariant contraction, covariant derivative, and
constructions like the Riemann curvature are usually prohibited by the
fact that the metric is not invertible. The things become even worse at
the points where the signature changes. We show that we can still do
many of these operations, in a different framework which we propose.
This allows the writing of an equivalent form of Einstein's equation,
which works for degenerate metric too.
Once we make the singularities manageable from mathematical viewpoint,
we can extend analytically the black hole solutions and then
choose from the maximal extensions globally hyperbolic regions. Then
we nd space-like foliations for these regions, with the implication that
the initial data can be preserved in reasonable situations. We propose
qualitative models of non-primordial and/or evaporating black holes.
We supplement the material with a brief note reporting on progress
made since this talk was given, which shows that we can analytically
extend the Schwarzschild and Reissner-Nordstrom metrics at and beyond
the singularities, and the singularities can be made degenerate
and handled with the mathematical apparatus we developed.
In this article, it is shown that for any given measurement settings, only some of all possible initial conditions of the observed system are compatible to those of the measurement apparatus. This means that there is no way unitary collapse can accommodate the previous state of a quantum system, with the outcomes of any possible experimental settings. The only way to accommodate both of them is to allow the initial state of the system to depend on what measurements will undergo in the future. The result is derived mathematically, and can be viewed as a no-go theorem, which severely restricts the hopes that a unitary collapse approach can reconcile any initial condition with any experimental setup.
We argue that this remains true in the standard formulations of quantum mechanics, both when we consider that the measurement process takes place unitarily, and when we assume non-unitary collapse. It also remains true for hidden variable theories, both deterministic and stochastic. For deterministic (including unitary) theories the condition of compatibility between the initial conditions of the observed system with those of the measurement apparatus applies indefinitely back in time.
Initial conditions seem to be in a precise state which will become an eigenstate of any observable we will decide to measure. This can be understood from the four-dimensional block universe perspective, if we require any solution to be globally self-consistent.
This indubitable fact is often taken as supporting the view that all we can know about the universe comes from the outcomes of the quantum observations. According to this view, we can even learn the physical laws, in particular the properties of the space, particles, fields, and interactions, solely from the outcomes of the quantum observations.
In this article it is shown that the unitary symmetry of the laws of Quantum Mechanics imposes severe restrictions in learning the physical laws of the universe, if we know only the observables and their outcomes.
These results follow from our research on singular semi-Riemannian geometry and singular General Relativity (arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646) (which we applied in previous articles to the black hole singularities: arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099).
In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which, under a set of conditions, allow us to define proper geometric invariants, and to write field equations, including equations which are equivalent to Einstein's at non-singular points, but remain well-defined and smooth at singularities. This class of singularities is large enough to contain isotropic singularities, warped-product singularities, including the Friedmann-Lemaitre-Robertson-Walker singularities, etc. Also a Big-Bang singularity of this type automatically satisfies Penrose's Weyl curvature hypothesis.
The Schwarzschild, Reissner-Nordstrom, and Kerr-Newman singularities apparently are not of this benign type, but we can pass to coordinates in which they become benign. The charged black hole solutions Reissner-Nordstrom and Kerr-Newman can be used to model classical charged particles in General Relativity. Their electromagnetic potential and electromagnetic field are analytic in the new coordinates - they have finite values at r=0. There are hints from Quantum Field Theory and Quantum Gravity that a dimensional reduction is required at small scale. A possible explanation is provided by benign singularities, because some of their properties correspond to a reduction of dimensionality.
Recent advances in understanding the geometry of singularities do not require modification of General Relativity, being just non-singular extensions of its mathematics to the limit cases. They turn out to work fine for some known types of cosmological singularities (black holes and FLRW Big-Bang), allowing a choice of the fundamental geometric invariants and physical quantities which remain regular. The resulting equations are equivalent to the standard ones outside the singularities.
One consequence of this mathematical approach to the singularities in General Relativity is a special, (geo)metric type of dimensional reduction: at singularities, the metric tensor becomes degenerate in certain spacetime directions, and some properties of the fields become independent of those directions. Effectively, it is like one or more dimensions of spacetime just vanish at singularities. This suggests that it is worth exploring the possibility that the geometry of singularities leads naturally to the spontaneous dimensional reduction needed by Quantum Gravity.
In previous papers it has been proposed an equivalent form of Einstein's equation, which extends it and remains valid at an important class of singularities (including in particular the Schwarzschild, FLRW, and isotropic singularities). Here it is shown that if the Big Bang singularity is from this class, it also satisfies the Weyl curvature hypothesis.
As an application, we study a very general example of cosmological model, which generalizes the FLRW model by dropping the isotropy and homogeneity constraints. This model generalizes both the FLRW model and the isotropic singularities. We show that the Big-Bang singularity of this model is of the type under consideration, and satisfies therefore the Weyl curvature hypothesis.
general relativity.
First is introduced the singular semi-Riemannian geometry for metrics
which can change their signature (in particular be degenerate). The
standard operations like covariant contraction, covariant derivative, and
constructions like the Riemann curvature are usually prohibited by the
fact that the metric is not invertible. The things become even worse at
the points where the signature changes. We show that we can still do
many of these operations, in a different framework which we propose.
This allows the writing of an equivalent form of Einstein's equation,
which works for degenerate metric too.
Once we make the singularities manageable from mathematical viewpoint,
we can extend analytically the black hole solutions and then
choose from the maximal extensions globally hyperbolic regions. Then
we nd space-like foliations for these regions, with the implication that
the initial data can be preserved in reasonable situations. We propose
qualitative models of non-primordial and/or evaporating black holes.
We supplement the material with a brief note reporting on progress
made since this talk was given, which shows that we can analytically
extend the Schwarzschild and Reissner-Nordstrom metrics at and beyond
the singularities, and the singularities can be made degenerate
and handled with the mathematical apparatus we developed.
In this article, it is shown that for any given measurement settings, only some of all possible initial conditions of the observed system are compatible to those of the measurement apparatus. This means that there is no way unitary collapse can accommodate the previous state of a quantum system, with the outcomes of any possible experimental settings. The only way to accommodate both of them is to allow the initial state of the system to depend on what measurements will undergo in the future. The result is derived mathematically, and can be viewed as a no-go theorem, which severely restricts the hopes that a unitary collapse approach can reconcile any initial condition with any experimental setup.
We argue that this remains true in the standard formulations of quantum mechanics, both when we consider that the measurement process takes place unitarily, and when we assume non-unitary collapse. It also remains true for hidden variable theories, both deterministic and stochastic. For deterministic (including unitary) theories the condition of compatibility between the initial conditions of the observed system with those of the measurement apparatus applies indefinitely back in time.
Initial conditions seem to be in a precise state which will become an eigenstate of any observable we will decide to measure. This can be understood from the four-dimensional block universe perspective, if we require any solution to be globally self-consistent.
This indubitable fact is often taken as supporting the view that all we can know about the universe comes from the outcomes of the quantum observations. According to this view, we can even learn the physical laws, in particular the properties of the space, particles, fields, and interactions, solely from the outcomes of the quantum observations.
In this article it is shown that the unitary symmetry of the laws of Quantum Mechanics imposes severe restrictions in learning the physical laws of the universe, if we know only the observables and their outcomes.
These results follow from our research on singular semi-Riemannian geometry and singular General Relativity (arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646) (which we applied in previous articles to the black hole singularities: arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099).
In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which, under a set of conditions, allow us to define proper geometric invariants, and to write field equations, including equations which are equivalent to Einstein's at non-singular points, but remain well-defined and smooth at singularities. This class of singularities is large enough to contain isotropic singularities, warped-product singularities, including the Friedmann-Lemaitre-Robertson-Walker singularities, etc. Also a Big-Bang singularity of this type automatically satisfies Penrose's Weyl curvature hypothesis.
The Schwarzschild, Reissner-Nordstrom, and Kerr-Newman singularities apparently are not of this benign type, but we can pass to coordinates in which they become benign. The charged black hole solutions Reissner-Nordstrom and Kerr-Newman can be used to model classical charged particles in General Relativity. Their electromagnetic potential and electromagnetic field are analytic in the new coordinates - they have finite values at r=0. There are hints from Quantum Field Theory and Quantum Gravity that a dimensional reduction is required at small scale. A possible explanation is provided by benign singularities, because some of their properties correspond to a reduction of dimensionality.
The primary motivation of this research, which will be pursued in forthcoming articles, is the construction of invariants in Singular Semi-Riemannian Geometry, especially those related to the curvature. It turns out that the operations constructed in this article are enough for this purpose. Such invariants can be applied to the study of singularities in the Theory of General Relativity.
In this article I show that, despite these problems, there are quantities related to the curvature which remain finite, allowing us to rewrite the equation in a form which remains valid even when the metric becomes degenerate. These quantities are the densitized versions of the stress-energy tensor, the Ricci curvature tensor and the scalar curvature. The resulting densitized Einstein equation makes sense even when the metric is degenerate, and it reduces to Einstein's equation when the metric is non-degenerate.
This opens the possibility to resolve the problems usually associated to the presence of singularities, such as the breakdown of General Relativity and the information loss, without changing the Theory of General Relativity, just by dropping some implicit assumptions about the metric tensor.