Lorentz Transformation and General Covariance Principle

I tell about different mathematical tool that is important in general relativity. The text of the book includes definition of geometrical object, concept of reference frame, geometry of metric-affinne manifold. Using this concept I learn few physical applications: dynamics and Lorentz transformation in gravitational fields, Doppler shift. A reference frame in event space is a smooth field of orthonormal bases. Every reference frame is equipped by anholonomic coordinates. Using anholonomic coordinates allows to find out relative speed of two observers and appropriate Lorentz transformation. Synchronization of a reference frame is an anholonomic time coordinate. Simple calculations show how synchronization influences time measurement in the vicinity of the Earth. Measurement of Doppler shift from the star orbiting the black hole helps to determine mass of the black hole. We call a manifold with torsion and nonmetricity the metrichyph affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection. The analysis of the Frenet transport leads to the concept of the Cartan transport and an introduction of the connection compatible with the metric tensor. The dynamics of a particle follows to the Cartan transport. We need additional physical constraints to make a nonmetricity observable. Learning how torsion influences on tidal force reveals similarity between tidal equation for geodesic and the Killing equation of second type.

Diagram of Representations of Universal Algebras

Theory of representations of universal algebra is a natural development of the theory of universal algebra. In the book, I considered representation of universal algebra, diagram of representations and examples of representation. Morphism of the representation is the map that conserve the structure of the representation. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation.

Differential Equation over Banach Algebra

In the book, I considered differential equations of order $1$ over Banach $D$Hyph algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. I considered examples of differential equations in quaternion algebra. In order to study homogeneous system of linear differential equations, I considered vector space over division $D$-algebra, solving of linear equations over division $D$-algebra and the theory of eigenvalues in non commutative division $D$-algebra.

Introduction into Calculus over Banach algebra

Let $A$, $B$ be Banach $D$-algebras. The map $f:Arightarrow B$ is called differentiable on the set $Usubset A$, if at every point $xin U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =frac{d f(x)}{d x}circ dx +o(dx)$$ where $$frac{d f(x)}{d x}:Arightarrow B$$ is linear map and $o:Arightarrow B$ is such continuous map that $$lim_{arightarrow 0}frac{|o(a)|_B}{|a|_A}=0$$ Linear map $displaystylefrac{d f(x)}{d x}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $omegainmathcal{LA}(D;Arightarrow B)$ is defined by map $g:Arightarrow Botimes B$, $omega=gcirc dx$. If the map $g$, is derivative of the map $f:Arightarrow B$, then the map $f$ is called indefinite integral of the map $g$ $$f(x)=int g(x)circ dx=intomega$$ Then, for any $A$-numbers $a$, $b$, we define definite integral by the equality $$int_a^bomega=int_{gamma}omega$$ for any path $gamma$ from $a$ to $b$.

Quadratic Equation over Associative D-Algebra

In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $ain R$, $a<0$, then the equation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Vietes theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.

Derivative of Map of Banach algebra

Let $A$ be Banach algebra over commutative ring $D$. The map $f:Arightarrow A $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=partial f(x)circ a+o(a)$$ where the Gateaux derivative $partial f(x)$ of map $f$ is linear map of increment $a$ and $o$ is such continuous map that $$ lim_{arightarrow 0}frac{|o(a)|}{|a|}=0 $$ Assuming that we defined the Gateaux derivative $partial^{n-1} f(x)$ of order $n-1$, we define $$ partial^n f(x)circ(a_1otimes...otimes a_n) =partial(partial^{n-1} f(x)circ(a_1otimes...otimes a_{n-1}))circ a_n $$ the Gateaux derivative of order $n$ of map $f$. Since the map $f(x)$ has all derivatives, then the map $f(x)$ has Taylor series expansion $$ f(x)=sum_{n=0}^{infty}(n!)^{-1}partial^n f(x_0)circ(x-x_0)^n $$

Linear Map of $D$-Algebra

Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more general point of view I started the book from consideration of Cartesian product of representations. Polymorphism of representations is a map of Cartesian product of representations which is a morphism of representations with respect to each separate independent variable. Reduced morphism of representations allows us to simplify the study of morphisms of representations. However a representation has to satisfy specific requirements for existence of reduced polymomorphism of representations. It is possible that Abelian group is only $Omega$-algebra, such that representation in this algebra admits polymorphism of representations. However, today, this statement has not been proved. Multiplicative $Omega$-group is $Omega$-algebra in which product is defined. The definition of tensor product of representations of Abelian multiplicative $Omega$-group is based on properties of reduced polymorphism of representations of Abelian multiplicative $Omega$-group. Since an algebra is a module in which the product is defined, then we can use this theory to study linear map of algebra. For instance, we can study the set of linear transformations of $D$-algebra $A$ as representation of algebra $Aotimes A$ in algebra $A$.

Division in Associative $D$-Algebra

From the symmetry between definitions of left and right divisors in associative $D$-algebra $A$, the possibility to define quotient as $Aotimes A$-number follows. In the paper, I considered division and division with remainder. I considered also definition of prime $A$-number.

Integral of Map into Abelian $Omega$-group

The common in ring, module and algebra is that they are Abelian group with respect to addition. This property is enough to study integration. I treat integral of measurable map into normed Abelian $Omega$-group. Theory of integration of maps into $Omega$-group has a lot of common with theory of integration of functions of real variable. However I had to change some statements, since they implicitly assume either compactness of range or total order in $Omega$-group.

Normed Omega-Group

Since sum which is not necessarily commutative is defined in Omega-algebra A, then Omega-algebra A is called Omega-group. I also considered representation of Omega-group. Norm defined in Omega-group allows us to consider continuity of operations and continuity of representation.