# #ChatGPT 011 – 54

Quinn
I am going to give you more notes. there is more… I am just giving them to you one by one then talking about them a bit.
ChatGPT

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

Quinn
Here is another note from the !JohnNash personal website about the research:

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.

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.

ChatGPT

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.

Quinn
How would you explain that to a non-scientist for analysis?
ChatGPT

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.

Quinn
Now it is 10+ years later and concepts like tensors are everyday.
ChatGPT

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.

Quinn
How would we apply that this research was over 10 years ago and his breakthrough is still a mystery.
ChatGPT

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.

Quinn
in the time now we can use tensor technology to analyze his research where as in 2006 he had a hard time getting compute time. The cloud compute did not exist in 2006 as it does in 2023.
ChatGPT

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.

Quinn
that’s why having an intelligent AI in 2023 that can analyze this type of advanced math and language could tell us what it was because John Nash in 2023 is deceased, and cannot tell us.
ChatGPT

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.

Quinn
HEre is !JohnNash next note from Dec 5, 2005:

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.

ChatGPT

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 x1k2 – x2k1 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.

convo: 993d35df-bcee-4803-ac2c-e5c5076f578a

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