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Wolfram Physics Project: Working Session Tuesday, Apr. 20, 2021 [Dimension Decay]

Wolfram・2 minutes read

A working session on a physics project related to the early universe aims to determine homogeneous isotropic dimension changes using the Riemann tensor and Friedman equations. The focus is on deriving the Riemann tensor in different dimensional spaces, exploring geometric parameters, and constructing the sectional curvature tensor, with discussions on dimension estimators and the evolution of household dimension in hypergraphs.

Insights

Homogeneity and isotropy in the early universe are defined by symmetry conditions on the metric tensor in general relativity, with homogeneity involving scalar invariance and isotropy involving tensor invariance under rotations.
The construction of the Riemann tensor for hypographs involves the sectional curvature tensor, which is interchangeable with the Riemann curvature, ultimately aiming to achieve Riemannian geometry.
The method for computing curvature involves comparing lengths of geodesics rather than volumes, allowing for dimension-independent curvature measurement applicable even in undefined dimensions.
The conversation explores the relationship between the sectional curvature tensor and the shape of cross-sections of geodesic bundles in a curved space, with a focus on constructing the tensor directionally and its consistency under parallel transport.

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Recent questions

What is homogeneity in physics?
Homogeneity involves scalar invariance in physics.

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Summary

00:00

"Early Universe Physics Project: Homogeneity and Isotropy"

The live stream involves a working session with Jonathan Gorad and another individual on a physics project related to the early universe.
The goal is to determine if the early universe exhibited homogeneous isotropic dimension changes.
Homogeneity and isotropy are typically defined with respect to symmetry conditions on the metric tensor in general relativity.
Homogeneity involves scalar invariance, while isotropy involves tensor invariance under rotations.
The condition for homogeneity is that all components of the metric tensor are independent of position.
The Riemann tensor must be invariant under the Euclidean group for full homogeneity and isotropy.
In the ordinary Friedman universe, a single quantity, 'a,' characterizes the Riemann tensor along with the curvature parameter.
The construction of the Riemann tensor for hypographs involves the sectional curvature tensor, which is interchangeable with the Riemann curvature.
The goal is to obtain Riemannian geometry ultimately.
In different dimensional spaces, the Friedman equations for the scale parameter change, but the k parameter remains constant, with distinctions between hyperbolic, elliptic, and parabolic curvature types independent of dimension.

28:17

Dimensionless Hausdorff and Topological Tensors in Geometry

The discussion revolves around dimensions, with a focus on a dimension less than one.
Topological dimension is distinguished from Hausdorff dimension in equations.
A Hausdorff dimension less than one implies a topological dimension of zero, affecting the Riemann tensor.
The absence of indices in a tensor with a dimension less than one leads to uncertainty in interpretation.
The conversation shifts to deriving the Riemann tensor from a hypergraph using specific rules.
The process involves removing directionality information from tangent vectors to compute a four-component tensor.
The discrepancy in distances between endpoints and centers yields the sectional curvature tensor.
The derivation aims to mimic the Romanian case of the sectional curvature tensor.
Clarification is sought on the definition of perpendicularity and inner product in the context of a hypergraph.
The discussion delves into the scales of discretization, curvature, and tangent space in relation to geometric parameters.

44:32

"Curvature computation in non-linear continuum geometry"

The discussion revolves around defining concepts like the dot product and orthogonality in non-linear scales, particularly in continuum geometry.
Tangent vectors on different coordinate patches and the angle between them are explored, especially in scenarios like the Earth's surface.
The complexity increases when dealing with non-small distances, like shortest paths on irregularly shaped asteroids.
An approach is detailed for scales where curvature is relevant but the change in curvature is minimal, applicable to spheres or hyperbolic spaces.
The process involves finding pairs of orthogonal geodesics and computing the discrepancy between their endpoints, akin to dropping perpendiculars.
The discrepancy helps compute the sectional curvature tensor, analogous to the richie tensor construction in ordinary geometry.
The directed version of the richie tensor construction involves pairs of orthogonal geodesics and the average distance between their endpoints.
The construction is dimension-independent, unlike other methods that rely on defining dimensions a priori.
The method computes curvature by comparing lengths rather than volumes, eliminating the need to know the dimension beforehand.
The discussion delves into the relationship between volume and vertices in the context of defining volumes in geometric constructions.

01:01:19

"Curvature Measurement in Multi-Dimensional Spaces"

The Richie scalar allows defining curvature without dimension, recoverable from the Richie tensor.
A model space is assumed in the Richie scalar/tensor construction for comparing geodesic ball sizes.
A bijection between points on two balls minimizes average distance, regardless of volume.
Comparing average surface distance and center-to-center distance reveals tidal deformation.
Tidal tensor measures the same quantity as the old Richie tensor construction.
Positive curvature means longer geodesic paths, negative curvature shorter paths.
Curvature measurement is independent of dimension, applicable even in undefined dimensions.
Thick geodesic bundles recover the Richie scalar from the tensor construction.
Dimension estimators may differ when using directional dimension tensor approximations.
Varying curvature over length scales affects dimension assumptions in constructions.

01:18:24

"Sectional Curvature Tensor: Geodesics and Invariance"

The sectional curvature tensor is constructed by looking at the Gaussian curvature of unit speed geodesics emanating along a given tangent plane.
The sectional curvature tensor relates to the shape of a cross-section of a bundle of geodesics in a curved space.
To construct the sectional curvature tensor, transform the Ricci tensor construction to be directional, involving pairs of tangent vectors.
The sectional curvature tensor has four indices, representing the directions of the tangent vectors.
The sectional curvature tensor is related to parallel transport and satisfies consistency conditions.
The universe considered is one with constant dimension and curvature at every point in the hypergraph.
A random sprinkling of a homogeneous and isotropic manifold will be homogeneous and isotropic.
The conformal invariance of the universe's hypersurface is maintained due to the scale parameter and curvature scaling.
The light cone structure of the universe's hypersurface remains constant due to the characteristic structure of the solutions of the equations.
The invariance of the conformal structure is derived from the homogeneity feature of the equations, ensuring the characteristic lines do not change with time.

01:38:02

"Exploring Equations, Graphs, and Cosmology Dimensions"

The speaker discusses their advantage of having experience in various fields but admits a long gap since last computing characteristic lines for equations.
They express surprise at the lack of a function for computing Lagrange-Chocolate equations and consider looking it up.
A potential approach involving the scale parameter and average distance between random points in a manifold is introduced.
The relationship between the scale parameter 'a' and the average distance between vertices in a graph is discussed, linking it to the dimension 'd'.
The speaker suggests setting 'a' equal to a constant times 'n' to the power of 1 over 'd' to track changes in 'n' and 'd'.
The concept of 'a' being directly computed from changes in dimension and the number of vertices in a graph is explained.
A discussion ensues regarding the Hubble radius, 'a', and their relationship, with a focus on cosmology questions.
The conversation shifts to the volume of the universe, different interpretations of volume, and the relationship between 'z' and 'n'.
The speaker proposes a direct proportionality between 'z' and 'a', leading to equations for 'a dot' and 'a double dot' concerning changes in 'n' and 'd'.
The dialogue delves into the topological dimension, household dimension, and their relevance to the hypergraph's evolution and dimension estimation.

01:56:24

"Fractal Universe: Dimensions and Challenges Explored"

Mandelbrot's book discusses the night sky not being white due to fractal explanations.
Inflationary modification to lambda CDM resolves paradoxes in the universe's structure.
The universe's structure is not infinite, either in space or time, leading to non-uniform distribution of stars.
Different definitions of fractals exist, including topological and computational definitions.
The definition of fractals involves a nested structure or a topological dimension exceeding the household dimension.
The topological dimension is challenging to define for hypergraphs compared to the household dimension.
The goal is to find an equation describing the evolution of the household dimension for hypergraphs.
Challenges include defining scales, cutoffs, and characteristic scales for the household dimension.
The computation of limits and the infinite size limit are crucial for understanding hypergraph dimensions.
Taking the limit with respect to the graph radius is essential for defining the household dimension accurately.

02:12:46

"Hypergraph Evolution and Dimension Stability Experiment"

The graph's radius must remain smaller than 20 to avoid measurement issues with the Judy's sick ball.
The hypograph evolves from a given node, reaching various nodes at different graph radii.
The claim suggests that as the hypergraph evolves, the function v of r becomes shallower over time.
The question poses a challenge due to the discrete quantity computation in a continuous geometry.
Different curves in the plots represent various time steps in the hypergraph evolution.
The logarithmic differences of v of r are plotted against r for different time steps.
The behavior of v of r suggests stability in graphs where the dimension remains constant.
The goal is to observe dimension changes over time, not convergence to a fixed value.
The experiment involves rules that maintain size and dimension stability.
The process includes defining a cutoff radius and observing changes in the dimension estimator.

02:36:15

Dimension Computation and Graph Behavior Analysis

The property to specify is "all dimensions" and a cutoff of 10 is mentioned.
The states list is discussed, with a mention of an undirected edge graph.
The dimension at each vertex is being computed with a fixed cutoff size of 10.
Changing the cutoff size to 400 affects the dimension computation.
The process leads from a high-dimensional initial state to a fragmented lower-dimensional state.
The behavior involves untangling to end up with a low-dimensional outcome.
The discussion delves into the continuum approximation of the process.
Diffusion equations and their behavior on graphs versus manifolds are compared.
The concept of diffusion equations and relaxation equations on graphs is explored.
The operation of rules on individual nodes and edges versus their impact on GDC balls is considered.

02:55:54

Dimensional Effects of Segment Exchange in Systems

The fanning out rate determines the effect of dimension in a system involving gd6 segments.
Analogous to diffusion, gd6 segments exchange randomly, affecting the dimension.
The horizontal versus vertical translational distance is crucial in determining dimension changes.
Segment exchange can occur in the fan-like or orthogonal direction, impacting dimension.
Cross-fiber motion does not alter dimension, while fan-like motion does.
The probability of a fan-like change is influenced by the topological dimension.
The volume element is defined by the Hausdorff dimension, not the topological dimension.
The probability of a fan-like change is inversely related to the effective topological dimension.
The equation for the probability of a fan-like change involves the dimension and time.
Deriving an elliptic equation involves considering the radial and angular dependence of element shifts.

03:16:46

"Exploring Kinetic Equations in Dimension Change"

The discussion revolves around the second derivative, referred to as the diffusion equation in standard kinetic theory.
The focus shifts to finding an analogous concept in a different context.
Progress is noted in the exploration, with a mention of potential derivation work.
The conversation touches on the idea of a dynamical equation for dimension curvature.
The potential for a continuum equation for dimension is discussed, emphasizing its independence of curvature.
The derivation of a continuum equation for curvature is considered, highlighting gauge dependencies.
The concept of a kinetic equation for the exchange of segments is introduced.
The discussion delves into conformal invariance and its implications in the context of dimension change.
The exploration concludes with a consideration of the causal graph in the scenario of dimension change.

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