Weyl's postulate
Encyclopedia
In relativistic cosmology
, Weyl's postulate stipulates that in a fluid
cosmological model, the world line
s of the fluid particles, which act as the source of the gravitational field and which are often taken to model galaxies, should be hypersurface orthogonal. That is, the world lines should be everywhere orthogonal to a family of spatial hyperslices.
Sometimes, the additional hypothesis is added that the world lines form timelike geodesic
s.
we introduce a family of spatial hyperslices. This allows us to think of the geometry of "space" as evolving over "time". This is an attractive viewpoint, but in general no such family of hyperslices will be physically preferred. The Weyl hypothesis can be understood as the assumption that we should consider only cosmological models in which there is such a preferred slicing, namely the one given by taking the unique hyperslices orthogonal to the world lines of the fluid particles.
One consequence of this hypothesis is that if it holds true, we can introduce a comoving chart such that the metric tensor
contains no terms of form dt dx, dt dy, or dt dz.
The additional hypothesis that the world lines of the fluid particles be geodesics is equivalent to assuming that no body forces act within the fluid. In other words, the fluid has zero pressure, so that we are considering a dust solution
.
corresponding to the fluid particles should be hypersurface orthogonal is by no means assured. A generic congruence does not possess this property, which is in fact mathematically equivalent to stipulating that the congruence of world lines should be vorticity-free. That is, they should not be twisting about one another, or in other words, the fluid elements should not be swirling about their neighbors in the manner of the fluid particles in a stirred cup of coffee.
This coordinate system is called the system of comoving coordinates.
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
, Weyl's postulate stipulates that in a fluid
Fluid solution
In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid....
cosmological model, the world line
World line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
s of the fluid particles, which act as the source of the gravitational field and which are often taken to model galaxies, should be hypersurface orthogonal. That is, the world lines should be everywhere orthogonal to a family of spatial hyperslices.
Sometimes, the additional hypothesis is added that the world lines form timelike geodesic
Geodesic (general relativity)
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational, force is a particular type of geodesic...
s.
Intuitive significance
In the ADM formalismADM formalism
The ADM Formalism developed in 1959 by Richard Arnowitt, Stanley Deser and Charles W. Misner is a Hamiltonian formulation of general relativity...
we introduce a family of spatial hyperslices. This allows us to think of the geometry of "space" as evolving over "time". This is an attractive viewpoint, but in general no such family of hyperslices will be physically preferred. The Weyl hypothesis can be understood as the assumption that we should consider only cosmological models in which there is such a preferred slicing, namely the one given by taking the unique hyperslices orthogonal to the world lines of the fluid particles.
One consequence of this hypothesis is that if it holds true, we can introduce a comoving chart such that the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
contains no terms of form dt dx, dt dy, or dt dz.
The additional hypothesis that the world lines of the fluid particles be geodesics is equivalent to assuming that no body forces act within the fluid. In other words, the fluid has zero pressure, so that we are considering a dust solution
Dust solution
In general relativity, a dust solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid which has positive mass density but vanishing pressure...
.
Relation to vorticity
The condition that the congruenceCongruence (general relativity)
In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime...
corresponding to the fluid particles should be hypersurface orthogonal is by no means assured. A generic congruence does not possess this property, which is in fact mathematically equivalent to stipulating that the congruence of world lines should be vorticity-free. That is, they should not be twisting about one another, or in other words, the fluid elements should not be swirling about their neighbors in the manner of the fluid particles in a stirred cup of coffee.
This coordinate system is called the system of comoving coordinates.
See also
- Comoving coordinatesComoving distanceIn standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
- Congruence (general relativity)Congruence (general relativity)In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime...
, for the relatation between vorticity and hypersurfaces. - Friedmann-Lemaître-Robertson-Walker for the best known cosmological model (which does obey Weyl's hypothesis)
- Shape of the UniverseShape of the UniverseThe shape of the universe is a matter of debate in physical cosmology over the local and global geometry of the universe which considers both curvature and topology, though, strictly speaking, it goes beyond both...
- Weyl curvature hypothesisWeyl curvature hypothesisThe Weyl curvature hypothesis, which arises in the application of Albert Einstein's general theory of relativity to physical cosmology, was introduced by the British mathematician and theoretical physicist Sir Roger Penrose in an article in 1979 in an attempt to provide explanations for two of the...
, for a different cosmological hypothesis due to Roger PenroseRoger PenroseSir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
.