Write4U
Valued Senior Member
Mathematics describes the mathematical imperatives physical forces must obey. Change the value of the force and the mathematics require a new equation in behaviors.Mathematics don't make things move, nor do they make things move in specific, repeatable ways.
What makes things move in specific, repeatable ways is forces. There are four. When a given mass encounters a given force, it will always result in the same outcome.
Mathematics describes how we see things move due to forces.
Is gravity a physical object or a geometric tensor?
Quantum Geometric Tensor (Fubini-Study Metric) in Simple Quantum
INTRODUCTION; The most intriguing feature of modern physics is the introduction of geometrical concepts describing fundamental principles of nature [1]. One one hand, the gravity emerges as the local space-time symmetry, where comparison between nearby local frames naturally gives rise to the concept of Christoffel connection
On the other hand, electroweak and strong interactions are unified by Yang-Mills theory, which identifies the gauge interactions as local symmetries of internal degrees of freedom.
https://arxiv.org/pdf/1012.1337.pdfSimilarly, comparison between nearby frames of the internal spaces (e.g., for SU(2), the three isospin axis) introduces the gauge connection. In electromagnetic theory, this reduces to the Weyl’s principle where the gauge connection is the usual four-potential Aµ. The nature resumes all the observed interactions by simply obeying space-time and gauge symmetries
i.e. the relative values in any action dictate the mathematical functions of that action.
Christoffel symbols
From Wikipedia, the free encyclopedia
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.[1] The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.
However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor.
Abstractly, one would say that the manifold has an associated (orthonormal)
frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold.
https://en.wikipedia.org/wiki/Christoffel_symbolsThe Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.