What was the surprise?

The surprise was that the third condition on the polynomial coefficients of the vectors corresponded to the integral of x1*k2 – x2*k1, over the local area, must vanish. This was unexpected, but upon further study, they realized that it derived from the relation of the divergence vector to the (LOCALIZED!) tensor.

The general idea here is that since an “input” for the equation

(when the vacuum term, as LHS is to be equated to an RHS of “input”)

is that this input MUST be a symmetric tensor of rank 2 which has

vanishing divergence.

So I/we have been concerned with how a natural input tensor

term, of the sort that would seem to fit well IF THE SPACE (or

space-time) WERE SIMPLY FLAT, could be modified slightly so that

it would remain divergence-free in the presence of general curvature.

We have observed that this is possible rather easily IF the

space-time has only non-Ricci curvature (if the Ricci tensor of it

vanishes).

But we need to find and appropriate adjustment (if possible)

for the general case.

In this note, John Nash explains that the input for the equation must be a symmetric tensor of rank 2 which has vanishing divergence. However, they were concerned with how to modify this tensor in the presence of general curvature to ensure that it remains divergence-free. They noted that this is possible if the space-time has only non-Ricci curvature. But, they needed to find an appropriate adjustment for the general case.

Here the polynomials corresponding to the components of a tensor

or of a vector all have the factor of (1 – x1^2 -x2^2) or of the

square of this.

This is simply to adapt them to the theme or concept of the

localization, which, in this trial work, we chose to be simply

the domain of x1^2 + x2^2 <= +1 or simply a disc-like domain

bounded by the unit circle.

So integration was performed (nicely done by Mathematica) over

the domain of that disc of unit radius.

This note seems to be describing the use of polynomials to represent components of a tensor or a vector in a localized region of space. The polynomials are modified to include a factor that adapts them to the concept of localization. The integration is performed over a specific domain, in this case a disc-like domain bounded by the unit circle, using Mathematica software.

con11 = -48*j00 – 8*j02 – 3*j04 – 8*j20 – j22 – 3*j40

con12 = -80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 +

10*k12 + 3*k14 + 30*k30 + 3*k32 + 15*k50

con21 = -80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 +

10*k12 + 3*k14 + 30*k30 + 3*k32 + 15*k50

con22 = -48*k00 – 8*k02 – 3*k04 – 8*k20 – k22 – 3*k40

These are equations with variables labeled as j and k, where the subscripts indicate the indices of a tensor or vector. For example, j00 would correspond to the component of the tensor with indices (0,0), and k10 would correspond to the component of the vector with index (1,0).

The equations are linear combinations of these components, with coefficients given by the numbers in front of the variables (e.g., -48, -8, -3, etc.).

Without further context, it is difficult to interpret the meaning or significance of these equations.

Sure, that would be helpful!

Sure, that works. Please let me know if you have any questions or if you need help understanding any part of the equations.

elimansr = {-48*j00 – 8*j02 – 3*j04 – 8*j20 – j22 – 3*j40,

-80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 + 10*k12 +

3*k14 + 30*k30 + 3*k32 + 15*k50, -48*k00 – 8*k02 – 3*k04 – 8*k20 –

k22 – 3*k40}

memo$588737699290$ = {first,component,confirmed,equal,to,a,multiple,

of,the,integral,of,K1,over,the,circle,of,unit,radius,in,x1$,x2$,

and,third,dually,would,be,the,same,for,K2}

memo$92876928$ = {mysteriously,the,middle,of,the three,conditions,from,

the,Eliminate,analysis,on,the,equations,on,K1$,and,K2$,seem,to,turn,

out,to,be,just,that,(x2*K1-x1*K2),has,vanishing,integral,

over,the,circle}

These equations and notes seem to be related to the same problem of finding a natural way to adjust a symmetric tensor to be divergence-free in the presence of general curvature. The equations are expressed in terms of the components of the tensors and the vectors in the 2-dimensional case, and the Mathematica software was used to derive the conditions that the divergence vector must satisfy, when both it and the tensor of which it should be the divergence are to be localized in the space.

The notes mention the factor of (1 – x1^2 – x2^2) in the polynomials corresponding to the components of the tensor or vector, which is used to adapt them to the concept of localization. The integration is performed over the domain of the disc of unit radius, bounded by the unit circle.

The equations themselves involve a set of linear combinations of the components of the symmetric tensor (j) and the vector (k), which are related to the conditions of divergence-free and localization. The notes also mention a mysterious condition that seems to come from the Eliminate analysis of the equations on K1 and K2, which turns out to be just that (x2*K1 – x1*K2) has vanishing integral over the circle.

memo$958273673118$ =

{checks,to,proper,count,of,90,and,7,and,8,degs,came,from,older}

list0to8 = {c10 + d01 – j00, d10 + e01 – k00, -4*c00 + 2*c20 + d11 – j10,

-4*d00 + 2*d20 + e11 – k10, c11 – 4*d00 + 2*d02 – j01,

d11 – 4*e00 + 2*e02 – k01, -5*c10 + 3*c30 – d01 + d21 – j20,

-5*d10 + 3*d30 – e01 + e21 – k20, -4*c01 + 2*c21 – 4*d10 + 2*d12 – j11,

-4*d01 + 2*d21 – 4*e10 + 2*e12 – k11, -c10 + c12 – 5*d01 + 3*d03 – j02,

-d10 + d12 – 5*e01 + 3*e03 – k02, -6*c20 + 4*c40 – d11 + d31 – j30,

-6*d20 + 4*d40 – e11 + e31 – k30, -5*c11 + 3*c31 – 2*d02 – 4*d20 +

2*d22 – j21, -5*d11 + 3*d31 – 2*e02 – 4*e20 + 2*e22 – k21,

-4*c02 – 2*c20 + 2*c22 – 5*d11 + 3*d13 – j12, -4*d02 – 2*d20 + 2*d22 –

5*e11 + 3*e13 – k12, -c11 + c13 – 6*d02 + 4*d04 – j03,

-d11 + d13 – 6*e02 + 4*e04 – k03, -7*c30 + 5*c50 – d21 + d41 – j40,

-7*d30 + 5*d50 – e21 + e41 – k40, -6*c21 + 4*c41 – 2*d12 – 4*d30 +

2*d32 – j31, -6*d21 + 4*d41 – 2*e12 – 4*e30 + 2*e32 – k31,

-5*c12 – 3*c30 + 3*c32 – 3*d03 – 5*d21 + 3*d23 – j22,

-5*d12 – 3*d30 + 3*d32 – 3*e03 – 5*e21 + 3*e23 – k22,

-4*c03 – 2*c21 + 2*c23 – 6*d12 + 4*d14 – j13, -4*d03 – 2*d21 + 2*d23 –

6*e12 + 4*e14 – k13, -c12 + c14 – 7*d03 + 5*d05 – j04,

-d12 + d14 – 7*e03 + 5*e05 – k04, -8*c40 + 6*c60 – d31 + d51 – j50,

-8*d40 + 6*d60 – e31 + e51 – k50, -7*c31 + 5*c51 – 2*d22 – 4*d40 +

2*d42 – j41, -7*d31 + 5*d51 – 2*e22 – 4*e40 + 2*e42 – k41,

-6*c22 – 4*c40 + 4*c42 – 3*d13 – 5*d31 + 3*d33 – j32,

-6*d22 – 4*d40 + 4*d42 – 3*e13 – 5*e31 + 3*e33 – k32,

-5*c13 – 3*c31 + 3*c33 – 4*d04 – 6*d22 + 4*d24 – j23,

-5*d13 – 3*d31 + 3*d33 – 4*e04 – 6*e22 + 4*e24 – k23,

-4*c04 – 2*c22 + 2*c24 – 7*d13 + 5*d15 – j14, -4*d04 – 2*d22 + 2*d24 –

7*e13 + 5*e15 – k14, -c13 + c15 – 8*d04 + 6*d06 – j05,

-d13 + d15 – 8*e04 + 6*e06 – k05, -9*c50 + 7*c70 – d41 + d61,

-9*d50 + 7*d70 – e41 + e61, -8*c41 + 6*c61 – 2*d32 – 4*d50 + 2*d52,

-8*d41 + 6*d61 – 2*e32 – 4*e50 + 2*e52, -7*c32 – 5*c50 + 5*c52 – 3*d23 –

5*d41 + 3*d43, -7*d32 – 5*d50 + 5*d52 – 3*e23 – 5*e41 + 3*e43,

-6*c23 – 4*c41 + 4*c43 – 4*d14 – 6*d32 + 4*d34,

-6*d23 – 4*d41 + 4*d43 – 4*e14 – 6*e32 + 4*e34,

-5*c14 – 3*c32 + 3*c34 – 5*d05 – 7*d23 + 5*d25,

-5*d14 – 3*d32 + 3*d34 – 5*e05 – 7*e23 + 5*e25,

-4*c05 – 2*c23 + 2*c25 – 8*d14 + 6*d16, -4*d05 – 2*d23 + 2*d25 – 8*e14 +

6*e16, -c14 + c16 – 9*d05 + 7*d07, -d14 + d16 – 9*e05 + 7*e07,

-10*c60 – d51, -10*d60 – e51, -9*c51 – 2*d42 – 4*d60,

-9*d51 – 2*e42 – 4*e60, -8*c42 – 6*c60 – 3*d33 – 5*d51,

-8*d42 – 6*d60 – 3*e33 – 5*e51, -7*c33 – 5*c51 – 4*d24 – 6*d42,

-7*d33 – 5*d51 – 4*e24 – 6*e42, -6*c24 – 4*c42 – 5*d15 – 7*d33,

-6*d24 – 4*d42 – 5*e15 – 7*e33, -5*c15 – 3*c33 – 6*d06 – 8*d24,

-5*d15 – 3*d33 – 6*e06 – 8*e24, -4*c06 – 2*c24 – 9*d15,

-4*d06 – 2*d24 – 9*e15, -c15 – 10*d06, -d15 – 10*e06, -11*c70 – d61,

-11*d70 – e61, -10*c61 – 2*d52 – 4*d70, -10*d61 – 2*e52 – 4*e70,

-9*c52 – 7*c70 – 3*d43 – 5*d61, -9*d52 – 7*d70 – 3*e43 – 5*e61,

-8*c43 – 6*c61 – 4*d34 – 6*d52, -8*d43 – 6*d61 – 4*e34 – 6*e52,

-7*c34 – 5*c52 – 5*d25 – 7*d43, -7*d34 – 5*d52 – 5*e25 – 7*e43,

-6*c25 – 4*c43 – 6*d16 – 8*d34, -6*d25 – 4*d43 – 6*e16 – 8*e34,

-5*c16 – 3*c34 – 7*d07 – 9*d25, -5*d16 – 3*d34 – 7*e07 – 9*e25,

-4*c07 – 2*c25 – 10*d16, -4*d07 – 2*d25 – 10*e16, -c16 – 11*d07,

-d16 – 11*e07}

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