This is a mathematical equation with two parts, eqK1 and eqK2. It was posted by John Nash on December 5, 2005, and appears to be part of a larger mathematical proof or argument. Unfortunately, without additional context or information, it’s difficult to determine exactly what the equation represents or what it’s trying to prove. If you have any additional information or context, please let me know and I’ll do my best to help!
I11 = (1 – x1^2 – x2^2)^2*(c00 + c10*x1 + c20*x1^2 + c30*x1^3 + c40*x1^4 +
c50*x1^5 + c60*x1^6 + c70*x1^7 + c01*x2 + c11*x1*x2 + c21*x1^2*x2 +
c31*x1^3*x2 + c41*x1^4*x2 + c51*x1^5*x2 + c61*x1^6*x2 + c02*x2^2 +
c12*x1*x2^2 + c22*x1^2*x2^2 + c32*x1^3*x2^2 + c42*x1^4*x2^2 +
c52*x1^5*x2^2 + c03*x2^3 + c13*x1*x2^3 + c23*x1^2*x2^3 +
c33*x1^3*x2^3 + c43*x1^4*x2^3 + c04*x2^4 + c14*x1*x2^4 +
c24*x1^2*x2^4 + c34*x1^3*x2^4 + c05*x2^5 + c15*x1*x2^5 +
c25*x1^2*x2^5 + c06*x2^6 + c16*x1*x2^6 + c07*x2^7)
I12 = (1 – x1^2 – x2^2)^2*(d00 + d10*x1 + d20*x1^2 + d30*x1^3 + d40*x1^4 +
d50*x1^5 + d60*x1^6 + d70*x1^7 + d01*x2 + d11*x1*x2 + d21*x1^2*x2 +
d31*x1^3*x2 + d41*x1^4*x2 + d51*x1^5*x2 + d61*x1^6*x2 + d02*x2^2 +
d12*x1*x2^2 + d22*x1^2*x2^2 + d32*x1^3*x2^2 + d42*x1^4*x2^2 +
d52*x1^5*x2^2 + d03*x2^3 + d13*x1*x2^3 + d23*x1^2*x2^3 +
d33*x1^3*x2^3 + d43*x1^4*x2^3 + d04*x2^4 + d14*x1*x2^4 +
d24*x1^2*x2^4 + d34*x1^3*x2^4 + d05*x2^5 + d15*x1*x2^5 +
d25*x1^2*x2^5 + d06*x2^6 + d16*x1*x2^6 + d07*x2^7)
I21 = (1 – x1^2 – x2^2)^2*(d00 + d10*x1 + d20*x1^2 + d30*x1^3 + d40*x1^4 +
d50*x1^5 + d60*x1^6 + d70*x1^7 + d01*x2 + d11*x1*x2 + d21*x1^2*x2 +
d31*x1^3*x2 + d41*x1^4*x2 + d51*x1^5*x2 + d61*x1^6*x2 + d02*x2^2 +
d12*x1*x2^2 + d22*x1^2*x2^2 + d32*x1^3*x2^2 + d42*x1^4*x2^2 +
d52*x1^5*x2^2 + d03*x2^3 + d13*x1*x2^3 + d23*x1^2*x2^3 +
d33*x1^3*x2^3 + d43*x1^4*x2^3 + d04*x2^4 + d14*x1*x2^4 +
d24*x1^2*x2^4 + d34*x1^3*x2^4 + d05*x2^5 + d15*x1*x2^5 +
d25*x1^2*x2^5 + d06*x2^6 + d16*x1*x2^6 + d07*x2^7)
I22 = (1 – x1^2 – x2^2)^2*(e00 + e10*x1 + e20*x1^2 + e30*x1^3 + e40*x1^4 +
e50*x1^5 + e60*x1^6 + e70*x1^7 + e01*x2 + e11*x1*x2 + e21*x1^2*x2 +
e31*x1^3*x2 + e41*x1^4*x2 + e51*x1^5*x2 + e61*x1^6*x2 + e02*x2^2 +
e12*x1*x2^2 + e22*x1^2*x2^2 + e32*x1^3*x2^2 + e42*x1^4*x2^2 +
e52*x1^5*x2^2 + e03*x2^3 + e13*x1*x2^3 + e23*x1^2*x2^3 +
e33*x1^3*x2^3 + e43*x1^4*x2^3 + e04*x2^4 + e14*x1*x2^4 +
e24*x1^2*x2^4 + e34*x1^3*x2^4 + e05*x2^5 + e15*x1*x2^5 +
e25*x1^2*x2^5 + e06*x2^6 + e16*x1*x2^6 + e07*x2^7)
These are four integrals, denoted as I11, I12, I21, and I22, which involve a polynomial expression (1 – x1^2 – x2^2)^2 multiplied by a sum of terms involving powers of x1 and x2.
In each of the four integrals, the coefficients of the various terms involving x1 and x2 are given by different constants. These constants are denoted by c00, c10, c20, …, d00, d10, d20, …, e00, e10, e20, …, where the two-letter combination (e.g., c10, d21) indicates the power of x1 and x2 respectively. For example, c10 corresponds to the coefficient of x1 in the first integral, while d21 corresponds to the coefficient of x1^2*x2 in the third integral.
Without additional context, it is not possible to determine the purpose or context of these integrals or their coefficients.
K1e = (1 – x1^2 – x2^2)*(j00 + j10*x1 + j20*x1^2 + j30*x1^3 + j40*x1^4 +
j50*x1^5 + j01*x2 + j11*x1*x2 + j21*x1^2*x2 + j31*x1^3*x2 +
j41*x1^4*x2 + j02*x2^2 + j12*x1*x2^2 + j22*x1^2*x2^2 + j32*x1^3*x2^2 +
j03*x2^3 + j13*x1*x2^3 + j23*x1^2*x2^3 + j04*x2^4 + j14*x1*x2^4 +
j05*x2^5)
K2e = (1 – x1^2 – x2^2)*(k00 + k10*x1 + k20*x1^2 + k30*x1^3 + k40*x1^4 +
k50*x1^5 + k01*x2 + k11*x1*x2 + k21*x1^2*x2 + k31*x1^3*x2 +
k41*x1^4*x2 + k02*x2^2 + k12*x1*x2^2 + k22*x1^2*x2^2 + k32*x1^3*x2^2 +
k03*x2^3 + k13*x1*x2^3 + k23*x1^2*x2^3 + k04*x2^4 + k14*x1*x2^4 +
k05*x2^5)
E1 = 1 – x1^2 – x2^2
E2 = (1 – x1^2 – x2^2)^2
Thank you for providing the notes from John Nash’s work. Is there anything specific you would like me to help you with or any questions you have?
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 provide some additional context to the polynomials and integrals in John Nash’s work. It appears that the polynomials are related to components of a tensor or vector and have a specific factor that is related to the localization theme or concept. Specifically, this factor is (1 – x1^2 – x2^2) or the square of it, and it is used to adapt the polynomials to a disc-like domain bounded by the unit circle. The integration is performed over the domain of the disc, and it is noted that Mathematica was used to perform this integration.
This equation that has been discovered is a 4th-order covariant tensor partial differential equation that applies to the metric tensor of a space-time. It is simplest when written with the use of the Einstein G-tensor. In this form, it can take the form:
2 Gab + Gps^2 Rpa sb – (1/2) gab Rps = 0
where Gab is the Einstein tensor, Rpa sb is the Riemann curvature tensor, and gab is the metric tensor. This equation is known as the Einstein field equation and is a cornerstone of Einstein’s general theory of relativity. It describes how matter and energy in a space-time curve the geometry of space-time itself.
The scalar equation obtained from the tensor equation for vacuum in 4-space time dimensions has a form that appears to describe waves. This equation can be derived by contracting the general vacuum equation with the metric tensor. The resulting equation involves G, the scalar derived from the Einstein tensor, the Ricci tensor, and the scalar curvature R. G is the scalar trace of the Einstein tensor and can be expressed in terms of R, but this expression depends on the number of dimensions, n.
Would that be a big discovery?
It’s difficult to say whether this discovery would be considered “big” without more context and information about the specific equation and its implications. However, any new insights into the behavior of waves or the nature of space-time could potentially have significant scientific and technological implications, so this discovery could certainly be of interest to researchers in the field.
Yes, in the context of general relativity, time is treated as a dimension just like space, and together they make up the four-dimensional spacetime. This is the framework in which the theory of relativity is formulated. So when Nash refers to “4-space time dimensions,” he is referring to the four-dimensional spacetime consisting of three spatial dimensions and one time dimension.
What we have not understood, so far, in the study of this 4th order tensor partial differential equation for space-time, is how to appropriately link it to the gravitation of
matter(or of matter and energy, in view of the relativistic inter-convertibility of these).
What if he gained understanding into this?
If John Nash had gained understanding on how to link the 4th order tensor partial differential equation to the gravitation of matter or matter and energy, it would have been a significant breakthrough in the field of physics. The equation he discovered is related to the vacuum equation of general relativity, and finding a connection between this equation and the presence of matter or energy would help us better understand how gravity works in the presence of these sources. It could lead to new insights into the fundamental nature of space and time, and potentially lead to new developments in areas such as cosmology and high-energy physics.
matter(or of matter and energy, in view of the relativistic inter-convertibility of these).
Then he understood.
It’s possible that if Nash had a breakthrough related to linking the tensor equation to the gravitation of matter/energy, he would have mentioned it in his notes. However, based on the note you provided, there is no indication that he had such a breakthrough. The statement you quoted suggests that he was still grappling with this issue and had not yet found a solution.
Having a breakthrough in this field of science could have significant implications in our understanding of gravity and space-time. It could potentially lead to the development of new technologies, such as advanced propulsion systems, and provide us with a deeper understanding of the nature of the universe.
Additionally, such breakthroughs often have important implications for other fields of science, leading to new discoveries and advancements. For example, a better understanding of space-time could have implications for quantum physics and cosmology, leading to new discoveries in these fields as well.
Overall, breakthroughs in science can have profound impacts on our lives, our understanding of the world around us, and the development of new technologies.
What we have not understood, so far, in the study of this 4th order tensor partial differential equation for space-time, is how to appropriately link it to the gravitation of matter(or of matter and energy, in view of the relativistic inter-convertibility of these).
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