Einstein Field Equations (EFE) and The Energy-Momentum Tensor Introduction We have demonstrated that gravitation is a pressure force produced by the curvature of spacetime. Here we show that this point of view is in perfect accordance with EFE and the energy-momentum tensor. Einstein Field Equations (EFE) EFE connect Mass-Energy to the Curvature of Spacetime by the simplified equation: Gµν = 8πG/c4 Tµν EFE are mathematically correct, but no one can explain, with few simple words, how can a mass make a spacetime curvature. The proposed theory gives the solution to this enigma. Since the spacetime curvature comes exclusively from closed volumes, interpretation of EFE is:
Closed volumes Connecting (1) to (2) gives a consistent explanation of EFE:
Curvature of Spacetime However, this scheme does not agree general relativity. The next paragraph covers this inconsistency. Mass vs. spacetime curvature
We could think that the spacetime curvature (C) depends on mass (m) since the expression of the energy-momentum
tensor is C = f(m): ??? ---> Mass ---> Spacetime curvature This assertion is not true. In reality, the red part is missing since no one can explain where the mass comes from. Since the entire proposition is incomplete, we must avoid any misinterpretation of the energy-momentum tensor. The present theory proposes to replace ??? by:
Closed volumes ---> curve spacetime So, the whole process is:
Once the mass effect calculated, if we need to know the spacetime behaviour in a particular context, we must use EFE with correct parameters. This situation mainly exists for dynamic calculations. For example, for a rotating sphere, we must proceed in two steps:
The Schwarzschild Metric To calculate the spacetime curvature with a static body having a spherical symmetry, we must use the Schwarzschild Metric. In that particular case, the second part (in green) isn't necessary since the spacetime curvature has been already calculated from the closed volumes of the body. We have:
In our webpage "The Schwarzschild Metric", the spacetime curvature has been calculated from closed volumes, not from EFE. This calculation confirms the sequence proposed here. Note : In the calculation of the Schwarzschild Metric in our Webpage, result is function of mass, m. That might suggest that nothing has changed. In reality, fig. 1 of the Schwarzschild metric Webpage leaves no doubt on this subject. The purpose of this Webpage was to demonstrate that the Schwarzschild metric may be derived from the concept of closed volumes instead of masses. Mass Einstein devised his energy-momentum tensor (fig. 1b) starting with the constraint tensor of the 1850's fluid mechanics (fig. 1a).
Without enter in the build of these two tensors, the following points must be highlighted:
These three points shows that the energy-momentum tensor is related to a pressure on a volume, not to a mass. This conclusion perfectly matches the proposed theory: the origin of mass is a pressure on closed volumes. Finally, the proposed theory is nothing but an extension of the fluid mechanics, as demonstrated by Einstein in 1910's, but keeping the original significations: We have no reason to replace (closed) volume by mass. In other words, if the constraint tensor applies to volumes, the energy-momentum tensor must do likewise. Gravitation In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of a force. Does the proposed theory match EFE ? Yes, for the following reason. In physics, we consider that a normal attracting constraint is positive by convention, and a pressure constraint is negative. This means that gravitation is "+" and the pressure exerted by spacetime on closed volumes (the proposed theory) is "-" On the other hand, the curvature of spacetime is considered concave by physicists, as shown in fig. 2.
We can consider, by convention, that a concave curvature has the "-" sign, and a convex curvature (the proposed theory) has the "+" sign. This conducts to four combinations :
These four combinations can be interpreted as:
In conclusion, a close examination of the constraint and energy-momentum tensors confirms that:
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