Definition of Mass and Production of Equations of General Relativity

In this paper, the general relations that determine conic sections in a curved differential manifold M is applied by observers of the 3D space (free observers). From this process, they will find equations of the same form as Einstein’s equations of general relativity, without using the principle of equivalence. So, those equations, free from the equivalence principle, are used in all regions of physics and not only to gravity. In the founding of Classical Mechanics (Krikos, 2019) it was referred that a point circle of R2 is depicted as a material point in 3D space generating a local Citation: Krikos C (2020) Definition of Mass and Production of Equations of General Relativity. Open Science Journal 5(3)


Definition of a differential manifold
In this paper, the general relations that determine conic sections in a curved differential manifold M is applied by observers of the 3D space (free observers).
From this process, they will find equations of the same form as Einstein's equations of general relativity, without using the principle of equivalence. So, those equations, free from the equivalence principle, are used in all regions of physics and not only to gravity.
In the founding of Classical Mechanics (Krikos, 2019) it was referred that a point circle of R 2 is depicted as a material point in 3D space generating a local From the measurements of the anisotropies of the cosmic background radiation at the present time, we get a value for the density parameter (Ω(t)) near to unit, i.e.
. The value of the density parameter determines if the Universe is open (Ω(t) < 1), flat (Ω(t) = 1) or closed (Ω(t) > 1)). This result forces us to assume that the boundary of the Universe is a 2D flat space, i.e. the R 2 , since its interior is a 3D space as we conceive it. The R 2 space is characterized by isotropy and homogeneity. It is a simply connected space and that it does not exhibit any particular characteristic anywhere. These attributes are expressed by a circle of an infinite radius in the center of which is an observer, at every point in the Universe. Since circle is the geometric object from which all other conic sections produced, then we shall examine the equations that characterize them and the consequences of their mappings in the interior of the Universe through one to one correspondence. anisotropy in its neighborhood, due to the mass attributed to it. In other words, the neighborhood of this material point exhibits curvature. In order to calculate the various physical quantities, a coordinate system that is only locally approached by a Cartesian coordinate system should be defined.
If a second material point located in the neighborhood of the first, with a different mass from that of the first, correlates with the first one. In the case of Classical Mechanics it was considered that 3D space does not change its basic characteristics, that is, that it is still isotropic and homogeneous. The second material point creates its own neighborhood in which also a curvature, other than that of the neighborhood of the first material point, is displayed. Thus a different coordinate system other than the coordinate system of first material point, is required, in order to calculate the different physical quantities in this neighborhood. Since these two material points are correlated, then this correlation should be expressed through the correlations of both coordinate systems.
A collection of such material points defines a differential manifold M whose basic characteristic is that it is covered by a set of coordinate neighborhoods that each neighborhood has the same number of coordinates as each other. The basic property of two different coordinate systems is that in a common region they are related to a differentiable transformation of a class not less than 1. (Willmore, 2012).

Equations of conic sections in a differential manifold
The following relations are the relations through which vector fields L and R are defined on a 3D space, leading to Maxwell's equations in vacuum (Krikos, 2018). These relations will be generalized, writing them as relations between differential forms. The reason we make this generalization is to find their expressions on a flat 3D space, and then to have them transferred to a curved 3D differential manifold.

Definition of mass
Since the neighborhood of a point of a differential manifold M resembles with a neighborhood of the Euclidean space, then a depiction between them can be defined. In the neighborhood of this point of M is also depicted the relation (2.4), that is In the case of applying this relation to a circle in , the quantity dV expresses the correlation of the center of the circle with a point on its periphery. This correlation results from the change of the tangent on two infinitesimally close points of the circumference of the circle, defining the "acceleration" of each point. The quantity D gives the measure of this correlation for the free observers, because it is the depiction of the term which contains the k from which the mass or charge arises. Based on these observations, free observers in a curved differential manifold, follow a corresponding procedure as follows: On each point of a curved differential manifold, a matrix V is attached, whose elements are expressed by the basis (dx i ) of the 1-forms, resulting from the coordinate system (x i ), that is associated to this point. The matrix elements of V, are expressed as (Willmore, 2012) The matrix V of a point of a curved differential manifold is called affine connection. The coefficients belong to the set Λ 0 (M ), where the set Λ 0 (M ) includes all functions of M ,are called coefficients of the affine connection. On each point of a curve of a neighborhood, is attached a matrix V , whose elements are homogeneous of first degree in dx k , which is resulted from an isomorphic mapping between the tangent spaces of two points belonging to this curve. The matrix V is the mapping of an "affine connection" of a point of , to an affine connection that is attached to a point of a differential manifold M. The elements of dV in equation (2.4) are 2-forms, because they are expressed, as The quantity dV corresponds to the variation of the tangent, that is, the acceleration that correlates the center of a circle and a point on its circumference and characterizes that point. We will now examine how , in the right-hand member of equation (2.4), is expressed in each neighborhood of a differential manifold.
Free observers call this neighborhood of a differential manifold central neighborhood in correspondence with the circle. At each point of the central neighborhood, the affine connection V and its variation dV have already been attached. We consider all the possible curves passing through every point of this neighborhood that connect it to all points of this neighborhood. Since the points of each neighborhood of the manifold M are characterized by a matrix V and correlate with each other, then the affine connections of the points of the central neighborhood are correlated with each other.
In order those correlations to be expressed mathematically, we construct a space with base elements dx r dx q , in which are defined 2-forms. This base of 2forms, results from the composition of the dual spaces with bases 1-forms, of the tangent spaces of neighboring points of a central neighborhood. Accordingly, the

Equations of General Relativity or general equations
Having determine the matrix elements of the terms dV, of the right member of equation ( Initially, is contracted the left member of equation (4.11), with respect to the indices i, l and then from this expression, is lifted the index j so, we get (Pathria, 2003) In this relation, the indices j, k are contracted so, we get where R is the scalar curvature. From this relation we define the Einstein's tensor as or for covariant components, is taken the relation So, if is a conserved quantity, then are derived Einstein's equations for gravity or From the equation (4.8), we will find the equations of Einstein or general equations. We write equation (4.8) as The quantity in parenthesis should be a constant, i.e.
For the covariant components of the tensors of the equation (4.15) we obtain the equations The constant Λ is called Λ-cosmological constant. Equations (4.14) or (4.16) are raised through the reformulation of the definition of mass without using the principle of equivalence. The equations (4.16), are called general, because they can be applied in any physical scale, and not only to gravity. In gravity's scale, the equations (4.14), which are named Einstein's equations of General Relativity, are applied. The constant k depends on the particular problem we are facing. We can apply these equations in the case of strong interactions, finding the equations that govern the motion of quarks inside a nucleon.
Add and subtract to the right-hand side of this equation the quantity where R is the scalar curvature, so we get or where the quantity becomes .
We define the following quantities and The equations (4.19) are called general equations because they apply to all physical scales and not only to gravity and the constant Λ is called the cosmological constant.

If
then we obtain the Einstein's equations with Λ = 0 , that is, If then we obtain the general equations with , that is, Equation (4.19) was derived from a member of the Planck patch, that is, from a conic section, so by virtue of self-similarity it would also express the evolution equation of the Planck patch at each stage of its evolution.

Conclusions
In this work the geometric definition of mass and the production of the equations of General Relativity are achieved through a one-to-one mapping of relations that determine the geometric shape expressing a flat two-dimensional space into a curved space. Through this process we do not need to invoke any principle of equivalence to find equations in curved spaces but to study the plane two-dimensional space and transfer relations from it to a curved space. This process is general and was applied in the case of foundation of equations of Maxwell (Krikos 2018). Through this process the flat two-dimensional boundary of the Universe can be considered as the common place of development of all the theories of Physics where through a holographic representation they are transferred to the three-dimensional curved space that we realize. This view can unify General Relativity and Quantum Mechanics since both theories are defined in the border of Universe.