(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.)
I wanted to share with you something I stumbled upon about Hilbert spaces and Newton's formula for gravitation.
Newton's formula for gravitation is represented by
$$
F(a,b) = G \cdot \frac{m_a \cdot m_b }{| x_a – x_b|^2}.
$$
However, this can be equivalently expressed as
$$k(a,b) := \exp\left( -\frac{|x_a-x_b|^2}{G} \right) = \exp\left( -\frac{ m_a \cdot m_b }{ F(a,b)} \right)$$
which is known as the radial basis function kernel (RBF kernel) and has applications in machine learning and statistics.
Interestingly, $k(a,b)$ can be shown to be the inner product $\langle a,b\rangle$ to an infinite dimensional Hilbert space, specifically the Hilbert Sphere:
$$
S_\infty = \{ x \in H \mid k(x,x) = \langle x,x\rangle = 1 \}.
$$
By solving the last equality for $F(a,b)$, we can define the force in any Hilbert sphere with inner product $k(a,b)$ for $a \neq b$ as:
$$
F(a,b) := – \frac{m_N(a) \cdot m_N(b) }{ \log( k(a,b) )},
$$
where $m_N(x)$ denotes the mass of $x$.
Now, if we consider only kernels that satisfy the axioms of force that implement Mach's principle (A. K. Assis' axioms), we could create something I would like to name Weber–Assis–Mach-space (WAM-space for short) that adds more dynamics to the Hilbert sphere. The axioms from A. Assis are as follows:
-
Force is a vectorial quantity.
-
The force that a material body A exerts on a material body B is equal and opposite to the force that B exerts on A.
-
The sum of all forces on any material body is zero.
However, I am unsure if the Weber law for gravity, as described by Assis, allows one to define such a kernel. Additionally, from a physics standpoint, the geodesic distance in Hilbert sphere is given by $\arccos(k(a,b))$, which results in
$$
d(a,b) = \arccos \exp\left( -\frac{|x_a-x_b|^2}{G} \right) = \arccos\exp\left( -\frac{ m_a \cdot m_b }{ F(a,b)} \right).
$$
I am imagining a so called WAM-space, where you can define objects having $a, b, c, d, \dotsc$ living in a Hilbert sphere with inner product $k(a,b)$ having a certain 'mass' $m_N(a), m_N(b), \dotsc$ which are non-zero real numbers, and between two of these objects, there is a force:
$$
F(a,b) := – \frac{m_N(a) \cdot m_N(b) }{ \log( k(a,b) )}.
$$
subject to the axioms of A. Assis, which can be found in his book Relational Mechanics and Implementation of Mach’s Principle with Weber’s Gravitational Force.
I'm not entirely sure how this Hilbert space hypothesis could be tested experimentally, or if it's just a crazy idea, but I wanted to share it with the community and see where it leads. Let me know what you think!
This approach described above, might maybe be used to find a Hilbert sphere, where the forces of Newton's gravity and Coulomb's electricity are combined to one force. Indeed, suppose on the material body $a$ act two forces of different quality such as the gravity force of Newton and the electric force of Coulomb. In physics, if they act upon the same body, then the sum of the forces does again by Newtons law, corresponds to a new force, hence:
$$
F(a,b) = F_N(a,b) + F_C(a,b) \label{1}\tag{$\ast$}
$$
where $F_N$ is the Newton gravity force, and $F_C$ is the electric Coulomb force. The "masses" or let us better say, "quantities" are given by:
$$
m(a) = \big(m_N(a),m_C(a)\big),\quad
m(b) = \big(m_N(b),m_C(b)\big),
$$
where $m_N(x)$ denotes, as above, the mass of $x$ while $m_C(x)$ denotes the electric charge of $x$. Hence $m(x)$ is a two dimensional vector.
Let us define in this case:
$$
F(a,b) = \frac{\langle m(a) , m(b)\rangle}{ \log( k(a,b) )}.
$$
where we want to determine $k(a,b)$ in such a way, that \eqref{1} is fulfilled
$$
\begin{split}
F_N(a,b) & = \frac{\langle m_N(a),m_N(b)\rangle}{\log(k_N(a,b))},\\ \langle m_N(a),m_N(b)\rangle &:= m_N(a)\cdot m_N(b),\\
k_N(a,b) &:= \exp\left(-\frac{|x_a-x_b|^2}{G}\right)
\end{split}
$$
where $G$ is the Gravity constant and
$$
\begin{split}
F_C(a,b) &= \frac{\langle m_C(a),m_C(b)\rangle}{\log(k_C(a,b))},\\ \langle m_C(a),m_C(b)\rangle &:= m_C(a)\cdot m_C(b),\\
k_C(a,b) &:= \exp\left(-\frac{|x_a-x_b|^2}{C}\right)
\end{split}
$$
where $C$ is Coulomb's constant.
Using \eqref{1} and solving for $k(a,b)$ we find that for $a\neq b$:
$$
k(a,b) = \exp\left( \frac{\langle m(a),m(b)\rangle}{ F_C(a,b)+F_N(a,b) } \right)
$$
and $k(a,a) := 1$.
The mathematical question, which comes to my mind is this:
Why should the kernel defined by $k(a,b)$ as above, be positive definite?
I have tested this idea with some Sagemath code empirically, but am not sure why it always should be positive definite, if it is at all:
def k1(a,b):
G = 1
return exp(-abs(a-b)**2/G)
def k2(a,b):
C = 2
return exp(-abs(a-b)**2/C)
def m1(a):
return len(prime_divisors(a))+1
def m2(a):
return sum(valuation(a,p) for p in prime_divisors(a))+1
def km(a,b):
return m1(a)*m1(b)+m2(a)*m2(b)
def kk(a,b):
if a==b:
return 1
return exp((km(a,b))/(m1(a)*m1(b)/log(k1(a,b))+m2(a)*m2(b)/log(k2(a,b))))
for n in range(1,100):
M=(matrix([[kk(a,b) for a in range(1,n+1)] for b in range(1,n+1)],ring=RDF))
if not M.is_positive_definite():
print(n,M.eigenvalues())
else:
print(n)
Here is a Sagecell link for the code above, in case you do want to test it online.
Best Answer
I will try to answer my own question, and allow me to change a little bit the notation and meaning, since it was not so difficult as first thought, it was.
First let us look at the situation where two material bodies $a,b$ which might posses mass, electric charge and magnetic charge, which exert force on each other, by the Newtons gravity law and Columb's law for electric charges and magnetic charges. By Newton's law, we can build the sum of these forces:
$$F(a,b) \cdot r/|r| = (F_N(a,b) + F_C(a,b) + F_M(a,b) ) r/|r|$$
which is equal to:
$$F(a,b) \cdot r/|r| = (\mu_N(a)\mu_N(b)G+\mu_C(a)\mu_C(b)k_e+\mu_M(a)\mu_M(b)k_m)/|x_a-x_b|^2)\cdot r/|r|$$
and this might be written as:
$$F(a,b) \cdot r/|r| = \frac{\left < \mu(a),\mu(b)\right>}{|x_a-x_b|^2} \cdot r/|r|$$
which again might be written as:
$$F(a,b) = -\frac{\left < \mu(a),\mu(b)\right>}{\log(k(a,b))}$$
where we have put:
$$\mu(x) = (\mu_N(x),\mu_C(x),\mu_M(x))$$
and here $\mu_N(x),\mu_C(x),\mu_M(x) \equiv $ mass, electric charge, magnetic charge, $k(a,b) := \exp( - |x_a-x_b|^2)$ and $$\left < \mu(a),\mu(b) \right> := \mu_N(a)\mu_N(b)G+\mu_C(a)\mu_C(b)k_e+\mu_M(a)\mu_M(b)k_m$$ is an inner product on the measurable quantities.
Observe also, that by puting two of the quantities to $0$ the other quantity will give the force previously started with.
We observe that the function: $$k(a,b) = \exp(-|x_a-x_b|^2) $$ is indeed a positive definite function, since it is a Gaussian kernel, which is known to have this property.