Sure, feel free to share any notes and we can discuss them.

This memo is first created 8 Feb. 2005.

Today the directory for The Equation, which is a fourth

order tensor PDE (or a PDES in 10 variables in 4 dimensions)

that is considered as applicable to space-time with the

classical Einsteinian description with a metric tensor

(g sub a sub b), has been moved into a newly formed

sub-directory of itself.

And this new sub-directory (“Input.problem”) is being set

up to have files relating to the main current research project

which seeks to find an appropriate “input” relation by means

of which a description of matter, capable of “gravitating”,

would be linked into relationship with the vacuum equation.

Remark on the parallel with classical GR:

In classical GR matter gravitates via an “energy-momentum

tensor” that is considered to describe the effective

significance of the matter in relation it effects on the

equilibrium of space-time.

Einstein himself seems to have sought to find something

different, at least for electro-magnetic fields, than simply

the “input” of their gravitating significance into an equation

system by means of the (classical) energy-momentum tensor

that can be associated with a (classical) electro-magnetic

field. This was the “unified field theory” project. In

particular, Einstein studied the (conceivable) possibility

of generalizing the metric tensor to a non-symmetric tensor

and using the 6 additional variables thus introduced to give

the description of the electromagnetic field information.

Such a system, with the 16 variables corresponding to the

10 + 6 variables of the combined fields, might not be equivalent

to the system of equations deriving from the simple use of the

energy momentum tensor for the “input” of gravitating action.

So, in principle, the general issue of how this “input” should

be structured, mathematically, has been raised.

With our 4th order equation it is clear that the energy-momentum

tensor cannot be used directly. However, in some circumstances at

least, like as perturbing initially flat space-time (or even

space-time simply Ricci-flat), it seems that a tensor mathematically

derived from the energy-momentum tensor might be suitable.

So I have found a puzzle or mystery here which I am currently

studying.

Added 3 March 2005:

It should also be mentioned that I feel that the “EIH approach”

to the whole issue of how vacuum equations for space-time should

be related to the gravitating influences of distributions of

gravitating matter is very suggestive of research possibilities.

(Here EIH stands for Einstein, Infeld and Hoffmann who worked on

a project for deriving the equations for the gravitational

influence of matter directly from Einstein’s vacuum equations

of classical GR.)

In principle, starting with our 4th order vacuum equation it is

conceivably possible that a direct analogue of the “EIH program”

would be able to lead to the INFERENCE of the appropriate

structure of “input relations” (equations) describing the action

of matter in “gravitationally” modifying the space-time geometry.

This note provides more context to Nash’s research on finding an appropriate “input” relation to connect matter capable of “gravitating” to the vacuum equation. He discusses the idea that in classical general relativity, matter is described by an energy-momentum tensor that is used as the “input” for its gravitating effect on space-time. However, Einstein himself sought to find a different way of incorporating electromagnetism into the theory through his unified field theory project, which involved using a non-symmetric metric tensor with 16 variables instead of the symmetric metric tensor with 10 variables used in general relativity. Nash notes that this approach may not necessarily be equivalent to the system of equations deriving from the energy-momentum tensor as the input for gravitating action.

Nash also mentions that he finds the “EIH approach” to the issue of relating vacuum equations to the influence of gravitating matter to be suggestive of research possibilities. The EIH approach refers to the work of Einstein, Infeld, and Hoffmann, who worked on deriving equations for the gravitational influence of matter directly from Einstein’s vacuum equations in classical general relativity. Nash suggests that it may be possible to infer the appropriate structure of input relations describing the action of matter in modifying space-time geometry through a direct analogue of the EIH program starting from his 4th order vacuum equation.

In layman’s terms, the note explains that John Nash was trying to find a way to mathematically describe the relationship between the force of gravity and matter, such as planets or stars, that causes this force. This was important because it could help us better understand the nature of gravity and how it affects the universe. He was exploring different mathematical models, including the idea of using a tensor to describe this relationship, which could help incorporate the influence of electromagnetism in a more natural way than previously thought. His notes also suggest that he may have had a breakthrough in his research in late 2006, although the details of this breakthrough are unclear.

Yes, that’s true! The field of data science, in particular, has made the use of tensors and related mathematical concepts much more mainstream. This has led to a greater appreciation and understanding of the work that pioneers like John Nash did in fields like differential geometry and mathematical physics. It’s exciting to see how scientific ideas and concepts evolve and become more accessible over time.

Although John Nash’s research on the “interesting equation” was conducted more than a decade ago and his personal notes do not fully disclose what his breakthrough was, it is important to remember that scientific discoveries can take time to fully comprehend and analyze. It is also possible that his research was ahead of its time, and it may take further advancements in technology and understanding to fully appreciate the significance of his work.

Moreover, the process of scientific discovery often involves a series of incremental steps and setbacks. It is not uncommon for researchers to encounter roadblocks and difficulties along the way, and sometimes the path to discovery can be slow and circuitous. It is possible that Nash’s research on the “interesting equation” has contributed to the broader understanding of gravity and the relationship between matter and space-time, even if the full extent of his contributions is not yet fully understood.

In summary, while it may be frustrating that the details of Nash’s breakthrough are still a mystery, it is important to recognize that scientific research can be a complex and iterative process, and the full implications of his work may take time to fully appreciate.

Yes, that’s correct. The availability of cloud computing and advances in computing technology over the past decade may make it easier to analyze and potentially make breakthroughs in John Nash’s research. With more powerful computational resources, it may be possible to run simulations and experiments that were not possible in 2006. However, it’s important to note that there may still be limitations to what can be achieved with even the most advanced computing technology, and that breakthroughs in research often involve more than just computational power.

Indeed, AI models like GPT-3.5 can analyze and understand complex mathematical and scientific concepts, and may be able to shed some light on what John Nash’s breakthrough might have been in his research. However, it’s important to note that even with the most advanced AI technology, there is still a possibility that some aspects of John Nash’s work may remain a mystery or require further research and analysis.

We were working with Mathematica to try to find a NATURAL way for adjusting

a tensor to be divergence free starting initially with a second rank symmetric tensor.

This led to the problem, given a vector corresponding to the divergence of such a

tensor, of finding naturally a minimally sized tensor having that vector as its

divergence.

And then this study led to the study of this problem in the case of flat

geometry and in the simplest case of two dimensions.

A few days ago we seem, by means of the use of Mathematica, to have found

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 (of two

dimensions for the explicit calculations with Mathematica).

We had a model of a type of polynomials representing a localized vector of a field

of divergence and a model (with considerably more free variables) of polynomials

representing the terms of a tensor. (These are included as files derived from

the work with Mathematica.)

We were lucky to find that the solvability of the equations for that the divergence

of the tensor should be the vector REQUIRED certain simple conditions on the vector.

(This was obtained accidentally when the equations of correspondence did not initially

have a solution in general (or for some specific versions of the vector) and we used

the Mathematica operation of “Eliminate” on the divergence correspondence equations

with the variables describing the polynomial coefficients for the tensor terms set as

those to be eliminated.

Three residual conditions were revealed by Mathematica.

We verified first that two of three three conditions (each of which applied only

to the coefficients of one of the two vectors and not to those of the other) were

representative of the constraint that we had known of previously. This known constraint

is that the double integral of each vector, over the space where things are allowed

not to vanish, must be zero, because of its relation with the derivatives of the tensors

which are themselves only non-vanishing in a local area.

The third condition came as a surprise, but with a study of the revealed condition

on the polynomial coefficients defining the vectors we discovered that it corresponded

to that the integral of x1*k2 – x2*k1, over the local area, must vanish. (Then,

retrospectively, with integration by parts, we saw that this condition derives from

the relation of the divergence vector to the (LOCALIZED!) tensor.)

This probably generalizes to higher dimensions, with a skew-symmetric tensor and

the n components of the vector so that n + ((n-1)*n)/2 = n*(n+1)/2 integral conditions

should hold on the vector.

We include Mathematica files on this work.

In this note, John Nash describes his work on finding a natural way to adjust a tensor to be divergence-free, starting with a second-rank symmetric tensor. He explains that this led to the problem of finding a minimally sized tensor having a given vector as its divergence, and how they studied this problem in the case of flat geometry and the simplest case of two dimensions. By using Mathematica, they found the conditions that the divergence vector must satisfy, and were lucky to find that the solvability of the equations for the divergence of the tensor required certain simple conditions on the vector. They discovered three residual conditions, with two of them representing a known constraint and the third coming as a surprise. They found that this condition corresponded to the integral of x1*k2 – x2*k1 over the local area must vanish, which derives from the relation of the divergence vector to the localized tensor. They also mention that this probably generalizes to higher dimensions with a skew-symmetric tensor and the n components of the vector, so that n + ((n-1)*n)/2 = n*(n+1)/2 integral conditions should hold on the vector. They include Mathematica files on this work.

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