POISSON EQUATION, GENERAL RELATIVITY AND
METAPHYSICSHow it is possible to infer too much physical meaning
from mathematical tools
By J.P. BaugherOctober 2012
Ver 1.0
Let me know if you catch any mistakes!
Poisson Equation From Poisson Equation
2 3or
2 f For a function f given on
Solution to Poisson Equation
On boundary of
g
Uniqueness of solution… “First, since u = constant satisfies the
homogeneous problem with f = 0, g = 0, it is clear that a solution to a Neumann problem can only be unique up to an additive constant.”
Uniqueness of solution… This means take the derivative of g
2
2( ) ' 'g
2
20 'g
Uniqueness of solution…
1
2 'g f
2 f Minus sign switched since convention had normal vectors pointing out from a volume
Gauss’ LawTypical field drawing…
Uniqueness of solution… Presence of au seems irrelevant since
disappears when derivative taken, arbitrary value would give same solution
2
2( ) ' 'g
2
20 'g
Uniqueness of solution… What does solution “only unique up to an
additive constant” mean? Example given as Euclidean 2D slice of scalar values in R3 space for center point.
g
10 10 1010 10 1010 10 10
1 1 21 2 32 3 3
11 11 1211 12 1312 13 13
Uniqueness of solution… Second derivative…
2
2( ) ' 'g
10 10 1010 10 1010 10 10
1 1 21 2 32 3 3
11 11 1211 12 1312 13 13
Regardless of magnitudes, no gradient exists….
These become identical second derivatives
Uniqueness of solution… But our understanding is metaphysical. We are
implying more certainty than our tool allows…
g
g
Easily understood here… Second derivative…
10 10 1010 10 1010 10 10
1 1 21 2 32 3 3
11 11 1211 12 1312 13 13
Regardless of magnitudes, no gradient exists….
These are identical second derivatives
g
But also…
10 10 1010 10 1010 10 10
9 9 89 8 78 7 7
1 1 21 2 32 3 3
Regardless of magnitudes, no gradient exists here, but we could model this as….
…however the directional derivatives are still identical whether au is taken into consideration or not.
g
0
Uniqueness of solution…What about Gauss’ Theorem?
2 f
Theorem also implies too much certainty…
1 2( )C
Integration of a constant over a differential volume has no effect…
Uniqueness of solution…What about the whole volume?
2 f
Not a problem since second derivatives identical…but…
Uniqueness of solution…There is another relationship that must hold to demonstrate that there are no unique solutions…so not only...
1 2( )C
1 2( )C
…but also (need to preserve total u in volume, scalar values of u important)...
…or reducing a repulsion. If au exists, what does it physically represent?
This means that although the theorems are formulaically correct, the most we can agree on is that, by convention, the “field” is directed towards or away from the source since it could be considered as either attracting…
0
To state definitively that some physical parameter represented by function f causes, through action at a distance, an “attraction” represented by the Poisson equation cannot be supported by the equations themselves. This is a metaphysical description brought about by ignoring au.
0
2 f Is f over “there”causal to the appearance (from nothing) of this vector “here”? or…
is f over “there” causal to a change to what is already “here”?
General Relativity… For regular General Relativity, the Einstein
tensor Guv is equated to the stress energy tensor of matter kTuv. The T00 term is modeled to be energy density from a perfect fluid tensor. In the weak field approximation to Newtonian gravity, we now must use the Poisson equation to derive our understanding of how gravity is “attractive”. So…
Why is this important?
12
R Rg G T
T00 term only for velocities much slower than c we can infer an approximation…
Why is this important?
00T T
…so that we get Poisson’s equation for gravity…
200 4T G
Great approximation to “attractive” Newtonian gravity for most of 20th century as long as one does not worry about “dark matter”…
Cosmological Constant problem General Relativity is derived so that in the
presence of matter (energy, mass) there is curvature, but if this mass or energy vanishes, so too must the curvature…
Why is this important?
0R
Although there is much debate about the cosmological constant, in Unimodular relativity it is known simply as a constant of integration…
Why is this important?
12
R Rg G g
For the derivation of GR, this 𝜦 term must be set to zero since Einstein had already equated Guv to kTuv.
It became apparent through observations though, as accurate as GR is, for some reason there appears to be a tiny value for 𝜦.
Why is this important?
And so, in spite of the mathematical rigor used to derive the field equation using Ruv=0, our understanding demanded that we must consider it possible for 𝜦 to have a value. In the late 1960s, Zel’dovich postulated that perhaps it represented an energy density of the vacuum. Although it is possible to measure the value there are no accepted methods to understand either its magnitude or even physical presence.
Using our previous argument of the Poisson equation and Unimodular Relativity, we ask:
Why is this important?
Using Unimodular theory, let us define a multiple 𝜴 of the metric as
g G L
where Guv is the Einstein tensor and Luv is a tensor we name the Lorentz tensor.
Why is this important?
We can see that if 𝜴=0, then the negative of the Lorentz tensor is equivalent to the Einstein tensor (reduces to GR)..L G
Thus let us enforce Ruv=0, so that the field equation can be stated as
12
R Rg G g L
Why is this important?
Note that since we are enforcing Ruv=0, no extraneous multiple of the metric, such as 𝜦 is allowed. Therefore no “vacuum solutions” or cosmological models such as deSitter space exists within this field equation…This model presents the same challenge that was required of Guv, explaining the Newtonian approximation. Although I have made much progress, there will probably be great push back on explanation of the perfect fluid tensor. The end goal is to compare rigorously…
00G G
g
1 1 21 2 32 3 3
0
1 1 21 2 32 3 3
00 00g L g L
10 10 1010 10 1010 10 10
9 9 89 8 78 7 7
1 1 21 2 32 3 3
…approximates to…
g
0
…approximates to…
Between these two approximations, we can develop an understanding of why the following is so paradoxical…
…for example…
…for example…
Newtonian field equation with Cosmological Constant 𝜦
Equation from General Relativity: An Introduction for Physicists, Hobson, Efstathiou, Lasenby.For a spherical mass M (note that the r vectors are unit vectors)
2
2 3GM c rg r rr
We have 𝜦 from modified EFE. Why is the first vector so many orders of magnitude larger than the second? Why isn’t the vector from 𝜦 exactly zero? What does it represent? What is dark energy?
Does “dark energy” show that field theory is untenable? Is the small
magnitude (at small radii) and repulsive presence of 𝜦 represent the
difference between Guv and guv-Luv?𝜴g
𝜦 is not derived as being completely independent of radius. If this also applies to 𝜴 then vector A may decrease in magnitude at a greater rate than Newtonian. When a mass achieves the radius of where vector A is zero, what occurs at radius greater than this?
0
A
Is gravity due to an effect already present at each point in spacetime, so that the repulsion and attraction are
from the same phenomena?
Does gravity become repulsive after:1/3
2
6( )resG Vrc
