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To further clarify the understanding of the foundations of our new Integral Geometry theory, a counterargument to General Relativity (GR), we seek to find an exact point where we diverge. This point is our current goal of seeking an equation with which all can agree philosophically and subsequent GR equations for which we do not. It is the equations upon which we agree and do not agree that we view as two branching notational/philosophical paths.
We will initially present a short history on the path of General Relativity, emphasizing alternate theories and disputes, and where GR finds itself in 2016. We then give a basic presentation for our proposed starting point, the geometric explanation given by Bernard Riemann in On the Hypotheses which lie at the Bases of Geometry (Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.), and our fundamental disagreement. It is the goal of this presentation to familiarize the audience with both the basis for our dispute as well as historical disputes concerning the development and interpretation of General Relativity.

Parallel line segments are the basic graphical foundation for geometrical field theories such as General Relativity. Although the concept of parallel and curved lines have been well researched for over a century as a description of gravity, certain controversial issues have persisted, namely point singularities (Black Holes) and the physical interpretation of a scalar multiple of the metric Λ, commonly known as a Cosmological Constant. We introduce a graphical and notational analysis system which we will refer to as Integral Geometry. Through variational analysis of perpendicular line segments we derive equations that ultimately result from the changes in the area bounded by them. Based upon changing area bounded by relative and absolute line segments we attempt to prove the following hypothesis: General Relativity can-not be derived from Integral Geometry. We submit that examination of the notational differences between GR and IG in order to accept the hypothesis could lead to evidence that the inability to merge General Relativity and Quantum Physics may be due to notational and conceptual flaws concerning area inherent in the equations describing them.

By 1914, a scalar Lorentz covariant gravitational theory advanced by Finnish physicist Gunnar Nordstr?m had been developed, with the assistance of Einstein and Fokker, to the point it seemed to profoundly challenge the developing General Theory of Relativity.  One major difference though was that it did not seem to predict the gravitational deflection of light. After this phenomenon was empirically observed, the theory was relegated mainly to the history books as General Relativity took off.

We present in this paper an argument based upon our fundamental understandings of Lagrangian mechanics, the Action Principle and Calculus that the scalar argument used by Nordstr?m and Einstein is fundamentally and indefensibly restrictive. Using plots of linearized gravity we demonstrate that the cosmological constant problem, touted as the worst theoretical prediction in physics, can most plausibly be solved by a change of our understanding of the magnitude relationships between vacuum and baryonic energy density. These plots will also demonstrate our argument that derivatives of functions are a notational shortcut for tangents of integrals, and that "tangents of lines" (differential topology) are a basic misunderstanding of the Fundamental Theorem of Calculus. The D'alembertian equation of this, using the same arguments of hydrodynamics which led to Max von Laue's stress-energy tensor, seems to naturally lead to a Lorentz covariant aether theory that should be able to account for General Relativity, Quantum Field Theory and plausibly "dark energy". We also consider whether this aether theory can present a second gradient via a spatial change in vacuum energy density which is similar in effect to MOND in order to account for "dark matter" effects.

The Cosmological Constant Λ within the modified form of the Einstein Field Equation (EFE) is now thought to best represent a ?dark energy? responsible for a repulsive gravitational effect, although there is no accepted argument for its magnitude or even physical presence. In this work we compare the origin of the Λ argument with the concept of unimodular gravity. A metaphysical interpretation of the Poisson equation during introduction of Λ could account for the confusion.

In a previous web conference in December, we outlined a rough field theory reinterpretation of gravity based upon inverted integrals and the Poisson equation. This presentation is an update on the research. Between 1912 and 1914, Gunnar Nordstr?m theorized that a covariant scalar theory of gravity could be had from simply extending Newtonian gravity into four dimensions. Before the paper was even published though, Einstein notified Nordstr?m that he had already attempted this previously and had given up on it as an impossible dead end. To Einstein?s complete surprise, Nordstr?m then modified the equation by using the d?Alembertian operator instead. This new formulation was at the time exciting since it apparently presented a covariant field equation that could be interpreted as the trace of the stress-energy tensor R=kT. Although this was the last covariant scalar theory to present a serious alternative to General Relativity, it is now considered as nothing more than an occasional teaching aid in the development of GR.

In this presentation, we will demonstrate that efficient calculus notation is based on Euclidean geometry (derivative-of-a-function notation is simpler than writing out the redundant derivative-of-an-integral). We will show there is strong evidence that one cannot base non-Euclidean mathematics upon this efficient notation and that differential topology, the foundation of General Relativity (derivatives/tangents of lines), is most likely based upon a fundamental misunderstanding of non-Euclidean integrals and should be discarded after reviewing Riemann sums and the covariant d?Alembertian operator (Nordstr?m?s second theory). As preliminary evidence of this, we will demonstrate that whereas differential topology cannot combine the linearized gravitational equation ((1-2𝝓)?, 1-2𝝓, 𝝓->0 at infinity) with the cosmological constant Ʌ (constant of integration under unimodular conditions), the tangent ((C-f)?) of a linearized and normalized non-Euclidean integral (C-f=C(1-f/C), f->C at infinity) most certainly can and does not lead to mathematical artifacts such as black holes. Based upon this, we will consider that the scalar equation attempted in Nordstr?m?s theory was arbitrarily restricted and that the best physical model is to consider baryonic energy density as a delta of vacuum energy density leading to a tensor, rather than two separate tensors summed together. These physical model ?densities? will be founded upon the same perfect fluid analogies that lead to Max von Laue?s stress-energy tensor, but in this case is sensibly called the Aether. In other words, differential topology (derivatives of lines) only works because it mimics the derivative of an integral, and that the need for a mysterious tiny vacuum energy that ?repulses? attractive gravity stretches the credibility of mainstream models beyond the breaking point. Based upon this logical solution to the ?worst theoretical prediction in physics?, we will be proposing that empirical evidence overwhelmingly suggests most physical mathematical laws are ?backwards? due to this fundamental misunderstanding of the derivation of calculus notation.