An extendable Space-Time Curvature Mode model

Einstein calculated the curvature of space-time from a Field equation based on an equation of motion derived by the variation of a geodetic path and a second rank Metric tensor. In this study a Space-Time Curvature Mode (STCM) model is calculated from a series of coupled Field equations derived from the variation of a Taylor series expansion of a Lagrangian potential of the STCM. The Taylor series expansion coefficients are Metric tensors of increasing rank. The STCM model can be extended to any number of Field equations. The coupled Field equations of the STCM describe the warping of space-time by masses and energy fields, and the effect of the curved space-time on the motion of masses and energy fields. This results in a description of space coordinates that have locally positive and negative slopes as a function of the time coordinate and can describe a Universe with at times an accelerating expansion and at later times a contracting Universe. This study replaces the postulated concepts of Einstein’s Cosmological constant and postulated concepts of Dark Mass and Dark Energy. It is shown that the Hamiltonian is conserved.


Introduction and summary
In this study, an extendable Space-Time Curvature Mode (STCM) model using the variation of a Taylor series expansion of a Lagrangian potential of the STCM was derived. This resulted in a series of Metric tensors of increasing rank to describe the STCM. In the STCM, the spatial coordinates can have at times a positive slope, and at later times a negative slope, as a function of the time coordinate. Here is a brief history which led to the derivation of the STCM.
The 1916 General Relativity Theory (GRT) [1, 2] was derived from basic principles by the variation of a geodetic path using a second rank Metric tensor. It has a parabolic curved space-time solution. This describes a Universe with either a slowing expansion or a shrinking Universe. Since the Universe at that time, was thought to be constant, Einstein added a constant multiplied by the Metric tensor to the GRT Field Equation. Hubble [3] in 1929 observed that the universe is expanding. Thinking this proved his original derivation of the GRT is correct, Einstein removed the Cosmological constant. In 1998 astronomical observations showed that the Universe's expansion is accelerating and the rotation curve of stars farther than 5 kpc or 16.3078 light years from the central Black Hole of the Milky Way Galaxy is faster than predicted by calculations of the Kepler Newton Theory. This contradicted the GRT which described gravity causing masses being attracted to each other. To explain this surprising observation, Dark Matter (DM) and Dark Energy (DE) [4][5][6] were postulated. For more than 20 years, experiments were conducted to measure DM or DE. All these experiments failed consistently to find any trace of DM or DE. Perhaps, DM and DE are like the Aether of the early 20 th century through which light waves were supposed to propagate. Michelson's and Morley's experiment [7], and Einstein's theory disproved the existence of the Aether.
In this study, an alternate STCM model is derived from basic principles by the variation of a Taylor series expansion of a Lagrangian potential of the STCM to replace the postulated concepts of Einstein's cosmological constant and the DM and DE. This results in a wave Equation with a series of Metric tensors with increasing rank. The coupled Field Equations of the STCM have solutions of a multi curved space-time. The curvature of the space coordinates can have negative and positive local slopes as a function of the time coordinate. Therefore, the STCM can describe a Universe with at times an accelerating expansion and at later times a contracting expansion. This mathematical model reverts to the GRT and also Newton's model of Nature in limits. The Lagrangian potential is not a function of the energies of an object, it is a function of the STCM.
We first hypothesize that the STCM are the coordinates themselves, where time, multiplied by the speed of light, is one of the four coordinates. The STCM consists of an infinite number of sets of values of the four coordinates. The STCM is described by each of the coordinates being a function of the other three coordinates. There are no other external physical coordinates with respect to which the STCM can change. We postulate that the STCM can not be observed from the outside because the observers coordinates must also be part of the STCM.
We secondly hypothesize that if the STCM could change, then according to the theory of relativity the effects causing the STCM to change would also cause the measurements to change the same way, and therefore it would appear that the STCM does not change, it remained constant.
However, the motion inside the STCM can be described by selecting one coordinate of the four, and then observing the change in the remaining three coordinates for different consecutive values of the selected coordinate. If the selected coordinate is the time coordinate, we observe a change in the other coordinates, which are the three spatial coordinates, with time. Here, the function of the spatial coordinates as a function of time is derived from the STCM instead of Newton's Equations.
We thirdly hypothesize that only actions described in the STCM can cause space-time to warp. Just as there are no coordinates external to the STCM, there are no external interactions. All interactions are described by the STCM. We will need to distinguish between 'flat space-time' and 'curved space-time'. The total STCM is constant. If there are no interactions that will cause space-time to warp then space-time is constant. As described before both the coordinates and interactions are in the STCM.
Our fourth hypothesis is that all derivatives of functions of the coordinates in constant space-time are equal to zero.
The wave Equation is derived from the variation of the STCM which in the second hypothesis was shown to be constant and thus, its variation is equal to zero.
The Taylor series expansion coefficients are higher-order derivatives of the Lagrangian potential. These derivatives are Metric tensors of increasing rank. The odd expansion terms of the Taylor series are zero because of symmetry. The Lagrangian potential expressed by a Taylor series is a sequence of scalar products of Metric tensors and coordinate increments.
The zeroth-order term of the Taylor series is a scalar. The expansion coefficients are derivatives of the scalar Lagrangian potential with respect to the coordinates, and therefore are components of covariant tensors. The second-order expansion coefficient is defined to be equal to the conventional second-rank metric tensor components of Einstein's GRT [1,2]. This is a condition that has to be included with the solution. The higherorder expansion terms of the Taylor series are the higher rank Metric tensor components.
The variation of the zero-order term of the Taylor series is the Euler Lagrange equation of Classical Mechanics [8]. The variation of the second-order terms form Einstein's equation of motion of the GRT [1, 2]. The variation of the higher-order expansion coefficients form higher-order terms including a higher-order Christoffel symbol. The higher-order Christoffel symbol, together with the terms that follow it form a higherorder addition to the GRT. All the calculations in this paper are derived from basic principles by the use of the Calculus of Variation.
The Equation of motion is interpreted here as a wave equation for the portions of the Space-Time Curvature Mode rather than a description of the motion of an object. Here, the function of the spatial coordinates as a function of time coordinate is derived from the STCM instead of Newton's Equations. It describes the everyday world. These also describe gravity waves.
It would be interesting to calculate the Kretschmann invariant [9] which is a scalar, to observe the effect of the additional terms of the Taylor series would have on the various singularities in the extended GRT.
A Hamiltonian potential for the Lagrangian potential including the effect of the higher-order terms of the Taylor series was derived. The Hamiltonian potential is the Legendre transform of the Lagrangian potential. The change of the Hamiltonian is equal to zero, and thus, the Hamiltonian is conserved.

Wave equation and Christoffel symbols
In a flat space, and only in a rectangular coordinate system is each coordinate independent of the other three coordinates. In a flat space, in general curvilinear coordinates, each coordinate can be a function of any number of the other coordinates. To avoid confusion with the curvature of the space-time coordinates, the discussion will be restricted to rectangular coordinates. This derivation is based on a Lagrangian potential of the Space-Time Curvature Mode (STCM). Our fifth hypothesis is that the Lagrangian potential and its derivatives are continuous and analytic in all of space-time. The Lagrangian potential  is a function of the Minkowski space-time coordinates Î m  x (4). The space-time coordinates ( ) l m x in turn, are functions of a parameter ( ) l Î  1 . The Lagrangian potential is not an explicit function of the parameter l. The parameter l is similar to the way time is used in the theory of Euler and Lagrange. The parameter facilitates the calculation and it does not have a physical meaning, however, here it has a dimension of time. Expanding the Lagrangian potential  and a constraint C, in a Taylor series in the coordinates Î a  x (4). The constraint C, in this case, describes that a path length in space-time is constant and uniform.
Because only a variation of the constraint is used, the length of the path in space-time is arbitrary. The length of the path in space-time is chosen to be zero, The first term  0 of equation (2) is a scalar. The 2nd and 3rd expansion coefficients of equation (2) are fourdimensional covariant second-rank metric tensor components ab g .The 4th term of equation (2) are the components of a four-dimensional fourth-rank covariant metric tensor abgd h and the 5th term is equal to zero.
The h ab is a component of the Minkowski tensor. It is possible to have even higher rank metric tensors with components such as abgdef h etc. Because the Lagrangian potential  0 has dimensions of velocity squared, the expansion coefficient 4 is a component of a higher-rank metric tensor, with dimensions of reciprocal velocity squared. Because the form of the Lagrangian potential is not specified at this point of the calculation, it can be required that the tensor with components ab g have properties consistent with the Metric tensor of the GRT [1, 2]. Here a Lorentzian manifold with a signature −+++is used. e det g det f det g 1 g det g det g det h det g det 1 5 1 Since derivatives with respect to different variables such as l x , and x are used, the derivatives are now shown in a form such as Here ( ) ab g det denotes the determinate of ba g .According to our fourth hypothesis that in flat space-time all derivatives with respect to the coordinates are zero the derivatives of the Lagrangian in flat space are zero. Thus the tensor with components ab g reverts to the Minkowski tensor with components h ab . The tensor with components abgd h , and any higher expansion coefficients go to zero. The Euler-Lagrange equations can be used to calculate the wave equation of the STCM, and it is equivalent to the derivation from the Calculus of Variation. However, here we prefer to use the more basic derivation using the Variation of the Taylor series expansion.
As described before the variation is performed in curved space-time, the STCM. The STCM  is a function of the coordinates, and the coordinates are functions of the one-dimensional parameter As previously explained the STCM  is constant and the variation of the STCM  is equal to zero. To facilitate this variation a dummy variable This is a Hamiltonian process is applied to the STCM. The variation of the integral of equation (6)  It is hypothesized that both the Lagrangian potential  and the tensor components ab g and abgd h and their derivatives are analytic and continuous in the interval between l a to l b of the parameter λ, and assuming that the variation of the coordinates such d dx m x is equal to zero at the ends of the integration at λ equal to l a and to l . b The 3rd, 4th, 6th, 7th, 8th, and 9th terms of equation (7) must be integrated by parts. Expanding the third term in the bracket of equation (7b) Expanding the fourth term in the bracket of equation  The 7th, 8th, and 9th terms of equation (7b) can be expanded similarly to the way the 3rd, 4th and the 6th terms are expanded. 3 Substituting these resulting 6 Equations back into equation (7b).
The first square bracket is equal to zero because the variation d dx m x was assumed to be equal to zero at the limits of integration at l l and .
x x x even higher order terms 0 12 0 Changing indices for example the index β, to the index f of the quasi accelerations such as̈b x in equation (12). For instance, we change the index β to f in the scalar products terms following it are higher-order extensions of the GRT. All these terms were derived from basic principles using the Calculus of Variation.
Collecting terms and factoring ouẗf x .
x x x even higher order terms 0 14 The first bracket with three terms in equation (14), The tensor with components q mf which augments the metric tensor with components mf g is dimensionless. Substituting equation (15) into equation (14) yields to be equal to a component d m r of a delta function.
The 16 components of the tensor with components W rm can be calculated from the 16 Equations represented by equation (18c).
Defining the components of the modified Christoffel symbols of the second kind: The Christoffel symbol G ab r is symmetric G = G Setting the curvature f l d d 2 2 in the l parameter space to be equal to a scalar product of a second-rank tensor and l space quasi-velocities.
The calculations that follow are independent whether equation (26) derived from f or y is used. In curved space-time the second-rank tensor is an extension of the firstrank tensor or vector b V .
Changing indices in scalar products and collecting terms.

Conclusion
The spatial part of the solution of the Field Equation of the GRT has a parabolic form for space as a function of time. It describes a Universe with either a slowing expansion or a contracting Universe. In 1916 when the GRT was written it was thought that the Universe was flat. Therefore, Einstein added a constant multiplied by the Metric tensor. In 1929 Hubble [3] observed that the Universe was expanding. He noticed that the farther celestial objects are from the earth, the faster they move away. Thinking this proved that the original derivation of the GRT was correct, Einstein removed the Cosmological constant.
In 1998 a surprising astronomical observation showed that the Universe's expansion is accelerating. The latest astronomical observations have shown that the rotation curve of stars farther than 5 kpc or 16.3078 light years from the central Black Hole of the Milky Way Galaxy is faster than predicted by calculations from the Kepler Newton Theory. This contradicts the theory of the GRT because the GRT describes masses being attracted to each other by gravity. To explain this surprising observation, Dark Matter (DM) and Dark Energy (DE) [4][5][6] were postulated. For more than 20 years, experiments were conducted to measure DM or DE. All these experiments failed consistently to find any trace of DM or DE. Perhaps, DM and DE are like the Ether of the early 20 th century through which light waves were supposed to propagate. Michelson's and Morley's experiment [7], and Einstein's theory disproved the existence of the Ether.
In this study, an alternate model is derived from basic principles, to replace the postulated concepts of Einstein's Cosmological constant, Black Mass and Black Energy. This model is calculated from the variation of a Taylor series expansion of the Lagrangian potential. The expansion coefficients are derivatives of the Lagrangian potential with respect to the coordinate and therefore are covariant Metric tensors. Thus, the Taylor series expansion coefficients are Metric tensors of increasing rank. The Lagrangian is not a function of the energies of an object but is a function of the STCM.
The variation of the Taylor series expansion results in an Equation for the space-time wave mode with a series of Metric tensors of increasing rank. Coupled Field Equations are derived by a method similar to the one used to derive the GRT Field Equation [11], but here, the wave Equation with a series of Metric tensors is used. This study describes the reciprocal interaction of warped space-time and masses and energy fields. The masses and energy fields cause space-time to warp, and the curvature of space-time affects the motion of masses and energy, etc.
The coupled Field Equations of the STCM have solutions of a multi-curved space as a function of time with negative and positive local slopes. Therefore, the STCM model can describe a Universe that can, at times, have an acceleration expansion and can at later times, have a decelerating expansion. This mathematical model reverts to the GRT and also Newton's model of Nature in appropriate limits.
The variational method used here is similar to the derivation of the Euler Lagrange Equation of Classical Mechanics and Einstein's Equation of motion in the GRT. All have successfully described real physical events.
Einstein's Field Equation represents ten simultaneous Equations. These can be solved with considerable effort without the aid of a computer. However, the Field equations of the STCM represent approximately 100 simultaneous Equations. These will have to be solved with the aid of a computer, using Symbolic Algebra Software [12] now in the year 2022.