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

...curve spacetime
(1)

...exert a pressure


...identified to a mass effect   M = f(pressure) (2)

Connecting (1) to (2) gives a consistent explanation of EFE:


Curvature of Spacetime

Mass


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
---> make a pressure on the body ---> produce a mass effect...

It is important to note that this scheme is static. Its purpose is nothing but to calculate the mass effect (please see our webpage "The mass effect").

So, the whole process is:

 
Closed volumes

Static spacetime curvature

Pressure on the body

Mass (or "mass effect")

(If necessary) Dynamic spacetime curvature


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:

  1. (red part) Calculate the static mass effect from the closed volumes, as seen in the above scheme,
  2. (green part) If necessary, calculate the spacetime curvature from the mass effect by the Kerr Metric, which is a solution of EFE.

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:

 
Closed volumes

Spacetime curvature

Pressure on the body

Mass (or "mass effect")


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).


Fig1


Without enter in the build of these two tensors, the following points must be highlighted:

  • In the constraint tensor (fig. 1a), the trace T00, T11, T22, defines the isostatic pressure.
  • In the energy-momentum tensor (fig. 1b), the trace has necessarily the same signification: a pressure.
  • Fluid mechanics, and therefore the constraint tensor, concerns volumes, not masses. Even if the mass parameter is intrinsically present in many equations of the fluid mechanics, as in the Reynolds Number, Xavier-Stokes equation, or more simply in the expression of the force [ML/T2], the fluid mechanics always applies to volumes, not to masses.

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.

fig2

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 :

  1. Attraction and concave curvature: (+ -)
  2. Attraction and convex curvature: (+ +)
  3. Pressure and concave curvature: (- -)
  4. Pressure and convex curvature: (- +)

These four combinations can be interpreted as:

  • Combination 1 (+ -) is that of Newton-Einstein. It is perfectly correct and does not need validation. However, it does not explain how a mass can curve spacetime.
  • Combinations 2 (+ +) and 3 (- -) have no physical significance and must be rejected.
  • Combination 4 (- +) is that of the proposed theory. It conducts to an identical result as the first combination: (+ -) = (- +). However, this combination is more credible than the first one because it gives a rational explanation of the spacetime curvature.

In conclusion, a close examination of the constraint and energy-momentum tensors confirms that:



  • Gravitation is a pressure force, not an attractive force,
  • The curvature of spacetime is caused by closed
    volumes, not by masses,
  • This curvature is convex, not concave.
  •  

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