In 1823, the Cauchy stress tensor was defined in three-dimensional geometric space as:
Fxx Fxy Fxz
Fyx Fyy Fyz
Fzx Fzy Fzz . . (1)
The Cauchy stress tensor is clearly incomplete because it does not contain time and must be modified to take the 4x4 form:
Fxx Fxy Fxz Fxt
Fyx Fyy Fyz Fyt
Fzx Fzy Fzz Fzt
Ftx Fty Ftz Ftt . . . (2)
Or,
∇2xx ∇2xy ∇2xz ∇2xt [U]
∇2yx ∇2yy ∇2yz ∇2yt [U]
∇2zx ∇2zy ∇2zz ∇2zt [U]
∇2tx ∇2ty ∇2tz ∇2tt [U] . . . (2*)
Where t is time in the 4-dimensional unit space xyzt.
and Fxy= ∇^2yx U(x,y,z,t). . ., etc.
This tensor is called the Cairo-Cauchy tensor to distinguish it from the steady-state Cauchy tensor.
Note that:
1- Formula 2 reduces to Formula 1 as dU/dt)partial approaches zero.
2- The stationary Cauchy tensor 1 cannot handle time-dependent cosmic events, unlike the Cairo Cauchy tensor 2.
The Cauchy stress tensor in a three-dimensional geometric space x, y, z is defined as follows:
Fxx Fxy Fxz
Fyx Fyy Fyz
Fzx Fzy Fzz . . (1)
Where Fxy, Fxx, etc. are the forces acting per unit normal area.
The corresponding curvature tensor Cij = dij/Lij is defined by the following equation:
Fxx Fxy Fxz
Fyx Fyy Fyz
Fzx Fzy Fzz
X
Cxx Cxy Cxz
Cyx Cyy Cyz
Czx Czy Czz
=I (3x3) . . (2)
where I is the identity matrix, Cxy = dx dy/x y, Cxx = dx dx/x^2, and so on.
Equation 2 produces three simultaneous algebraic equations:
Fxx Cxx + Fxy Cxy + Fxz Cxz = 1
Fyx Cyx + Fyy Cyy + Fyz Cyz = 1
Fzx Czx + Fzy Czy + Fzz Czz = 1 ... (3)
The system of simultaneous algebraic equations 3 reduces to a simple system of equations with a simple solution when all shear forces Fij (i.NE.j) are zero, as follows:
Fxx Cxx = 1
Fyy Cyy = 1
Fzz Czz = 1 ... (4)
Which means that for a uniform material Cxx=Cyy=Czz = Young's modulus of elasticity E.
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To be continued.
Answer II continued
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It is clear that the Cauchy stress tensor is incomplete and can be misleading because it does not contain time and must be modified to take the unitary 4D xyzt form as follows,
Fxx Fxy Fxz Fxt
Fyx Fyy Fyz Fyt
Fzx Fzy Fzz Fzt
Ftx Fty Ftz Ftt . . . ( 6)
Or,∇2xx ∇2xy ∇2xz ∇2xt [U
]∇2yx ∇2yy ∇2yz ∇2yt [U]
∇2zx ∇2zy ∇2zz ∇2zt [U]
∇2tx ∇2ty ∇2tz ∇2tt [U] . . . (6*)
When shear forces are neglected, equation 6* reduces to,
Fxx Cxx = 1
Fyy Cyy = 1
Fzz Czz = 1
Ftt Ctt = 1 ... (7)
Where t is time in the 4-dimensional unit space xyzt.
and Fxy = ∇^2yx U(x,y,z,t). . ., etc.
This tensor 6 is called the Cairo Cauchy tensor to distinguish it from the steady-state Cauchy tensor.
Note that:
1- Formula 6 reduces to Formula 5 when dU/dt)partial tends to zero.
2- The steady-state Cauchy tensor 4 cannot handle time-dependent cosmic events, unlike the Cairo Cauchy tensor 5. When all shear forces Fij (i.NE.j) are zero,
This means that:
i- for a uniform material, Cxx = Cyy = Czz = Young's modulus of elasticity E.
ii- Ctt = d^2/dt ^2 U(x,y,x,t)
The Cauchy problem and related concepts are now obsolete.
This is why Einstein did not use the Cauchy tensor in his theory of general relativity; instead, he used the Einstein stress-strain tensor (based on the Ricci tensor) to describe the curvature of spacetime, along with the stress-energy tensor (also called the energy-momentum tensor).
However, the Einstein tensor and the stress-energy tensor remain inferior to the stress-strain tensor derived from the statistical theory of Cairo techniques.
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Yo be continued.
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The Cauchy curvature tensor in three-dimensional geometric space is expressed as follows:
Fxx Fxy Fxz
Fyx Fyy Fyz
Fzx Fzy Fzz . . (1)
The extension of this tensor in unitary four-dimensional space is:
Fxx Fxy Fxz Fxt
Fyx Fyy Fyz Fyt
Fzx Fzy Fzz Fzt
Ftx Fty Ftz Ftt . . . (2)
The above force tensor can also be expressed by the Laplacian formula:
∇2xx ∇2xy ∇2xz ∇2xt [U]
∇2yx ∇2yy ∇2yz ∇2yt [U]
∇2zx ∇2zy ∇2zz ∇2zt [U]
∇2tx ∇2ty ∇2tz ∇2tt [U] . . . (3)
Where t is time in the 4-dimensional unit space xyzt.
and Fxy = ∇^2yx U(x,y,z,t), Ftx = ∇^2yt U(x,y,z,t). ., etc.
This tensor is called the Cairo tensor to distinguish it from the stationary Riemann tensor. Note that:
Equation 3 multiplied by the curvature tensor is general relativity in a single sentence.
Formula 2 reduces to Cauchy's formula 1 as dU/dt)partial approaches zero.
The time-dependent Rieman tensor R relates the curvature of spacetime to the energy density U in a strange way:
Curvature C is equal to or proportional to the energy density (C = const1 .E . . . 4).
This relationship is false and misleading, because the correct relationship should be: curvature is inversely proportional to U ( .U x C = Const2 . . . 5), in accordance with the universal law of the 4D Lorentz transformation, which implies that the unitary space xyzt is constant or conserved during its motion. But here is the satanic trap into which Einstein irretrievably fell by following or trying to prove the wrong relation 4 instead of the correct one 5. The author assumes that Einstein did not understand physics to the point of assuming that the relationship between space curvature and energy density is one of proportionality, where space curvature should be inversely proportional to energy density, as implied by the universal Lorentz law in 4D physics xyzt:
Spacetime is constant or conserved in motion.
But here's the satanic trap Einstein fell into, irretrievably, by following or attempting to prove the erroneous relation 4 instead of the correct relation 5.
The author assumes that Einstein was so lacking in physics that he assumed the relationship between space curvature and energy density was one of proportionality, where space curvature should be inversely proportional to energy density, as suggested by Lorentz's universal law in 4D xyzt:
Spacetime is constant or conserved in motion.
However, Einstein resorted to his black magic, remembered a thought experiment called black magic, and wrongly proved his false rule in 1915: curvature induces energy density, not the other way around.
The question arises: if Einstein's procedure is flawed, how come his results are perfect? The answer is simple: both equations 4 and 5 can give the same results if the square of the curvature tensor is I (C^2=I), which is the second part of the Satanic trap.