While no mathematical background is necessary to enjoy this blog, mathematical interest is.
This particular blog will discuss the mathematics of relativity. Later blogs will discuss the mathematics of quantum mechanics, quantum field theory, string theory, complexity science, economics, finance and actuarial science.
Descriptions like that which follows can be found in many relativity publications. In this blog we will be following the formalism laid out in Relativity Demystified.
The mathematical structure of relativity is as straightforward as it is complex. Einstein's special and general theories of relativity change the way we think about our world. Many phenomena of common sense fall within the mathmatics of relativity.
The basic idea flows from a series of mathematical relationships. In 1865, physicist James Clerk Maxwell published his equations which posited that electromagnetic (em) waves exist, and move at the speed of light. Electromagnetic fields can be shown to satisfy wave equations using simple vector calculus.
This is important, because wave equations are used extensively in physics to model physical events. It is important in the context of relativity because it can be shown that em waves (light) always travel at the same speed. It is the constancy of the speed of light in all frames of reference which leads to many of the unique elements of relativity theory.
In Newtonian physics, distances and times are fixed. When it was discovered that waves don't need a medium in which to travel, a mathematical tool called a Lorentz Transformation was applied to explain the results.
Newton used two equations to describe gravity. The first describes the path of a particle through space. The second describes how mass acts as a source of gravity.
Einstein's equations have similar form. In general relativity, the energy-momentum tensor acts as a source of gravitation, and relates to the curvature of spacetime via the Einstein equation.
Symmetries lead to conservation laws. In relativity, Killing vectors can be used to tease out symmetries. The motions of particles and light waves in spacetime involves Lagrangian methods which we will deal with in the math of quantum mechanics blog.
In terms of these blogs, it is important to note that understanding the mathematics of relativity is not just an abstract exercise. Harmonious Universe posits that a metaphysics based on mathematics and science is at least a possibility.
From the mathematics of relativity, particle physics, chaos theory, complexity science and actuarial science, patterns can be seen which point to the possibility that all of reality can be understood at a deep level. The evolution of the universe itself is mathematiclly derivable.
These ideas can be followed without understanding the math, but understanding the math enhances the insights.
Getting back to the math of relativity, now that we have a basic framework, we can begin to build the mathematical structure. The study of relativity involves what most physics does, the analysis of events.
In physics, events occur in a framework. That framework usually involves space, time, matter, and energy. Elements of matter and energy interact in space and time and reality emerges from these interactions.
Galilean Transformations were used to study how phenomena appear to different observers in Newtonian Mechanics. In Relativity Theory, Lorentz Transformations apply.
Among the tools used in developing some of the key elements of Relativity theory are, vectors, tensors, metrics, and manifolds.
Vectors are simply symbols which represent magnitude and direction. Tensors map from vectors to the real numbers. Metrics describe three dimensional space. Manifolds are continuous spaces of points that can be curved globally, but locally are Euclidean.
Many tools of mathematics have as their primary role the simplification of calculation and analysis. Transformations often accomplish this quite directly.
One example is a gauge transformation which is quite prominent in quantum theory, and is used in relativity to tease out wave equations.
In relativity, much of the theory is nonlinear, but can be worked on using linear methods which are much more manageable.
Lie derivatives, Killing vectors, Riemann tensors, Ricci tensors, Cartan Equations, and Einstein's field equations are all in the relativity tool box.
The most famous equation of all, e=mc**2 came out of these elements
From relativity theory came a new vision of the world and a new way of living.
Einstein's relativity theories changed the landscape of the human enterprise. Combined with advances in particle physics, it changed metaphysics itself.
Lee
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