# An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally
Real Number Fields

Christian Porter111Imperial College London, United Kingdom. Corresponding
author<EMAIL_ADDRESS>Andrew Mendelsohn222Imperial College London,
United Kingdom<EMAIL_ADDRESS>
###### Abstract

Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with
discriminant and regulator $\Delta_{K},R_{K}$ respectively. In this paper,
using a similar method to van Woerden, we prove that the number of classes of
perfect unary forms, up to equivalence and scaling, can be bounded above by
$O(\Delta_{K}\exp(2n\log(n)+f(n,R_{K})))$, where $f(n,R_{K})$ is a finite
value, satisfying
$f(n,R_{K})=\frac{\sqrt{n-1}}{2}R_{K}^{\frac{1}{n-1}}+\frac{4}{n-1}\log(\sqrt{|\Delta_{K}|})^{2}$
if $n\leq 11$. Moreover, if $K$ is a unit reducible field, the number of
classes of perfect unary forms is bound above by
$O(\Delta_{K}\exp(2n\log(n)))$.

##  1 Introduction

Let $K$ be a totally real algebraic number field of degree $n$ over
$\mathbb{Q}$, with ring of integers $\mathcal{O}_{K}$. We associate to $K$ the
embeddings $\sigma_{1},\sigma_{2},\dots,\sigma_{n}$ into $\mathbb{R}$, that
necessarily fix $\mathbb{Q}$. A quadratic form $f:K^{m}\to K$ is defined by

$\displaystyle f(x_{1},\dots,x_{m})=\sum_{k,l=1}^{m}f_{kl}x_{k}x_{l}.$

We say that $f$ is positive-definite if

$\displaystyle\sigma_{i}(f)(x_{1},\dots,x_{m})=\sum_{k,l=1}^{m}\sigma_{i}(f_{k,l})x_{k}x_{l}$

is positive definite for each $i=1,\dots,n$.

Quadratic forms of the form $ax^{2}$, where $a\in K$, are called unary forms.
We say that $a$ is a totally positive element of $K$ if the form $ax^{2}$ is
positive definite, and we denote $K_{>>0}$ the set of totally positive
elements of $K$. Note that for all $a\in K_{>>0}$,
$\text{Tr}_{K/\mathbb{Q}}(ax^{2})$ corresponds to a positive-definite
quadratic form in $n$ variables with rational coefficients. This is known as
the trace-form of the unary form $ax^{2}$. For each $a\in K_{>>0}$, we will
use the notation

$\displaystyle\mu(a)=\min_{x\in\mathcal{O}_{K}\setminus\\{0\\}}\text{Tr}_{K/\mathbb{Q}}(ax^{2})$

and

$\displaystyle\mathcal{M}(a)=\\{x\in\mathcal{O}_{K}:\text{Tr}_{K/\mathbb{Q}}(ax^{2})=\mu(a)\\}.$

If $a^{\prime}=au^{2}$ for some $u\in\mathcal{O}_{K}^{\times}$, we say that
the unary form $a^{\prime}x^{2}$ is equivalent to the form $ax^{2}$, and we
denote this relation by $a\sim a^{\prime}$. Note that since the action of
$\mathcal{O}_{K}^{\times}$ fixes $\mathcal{O}_{K}$, we have
$\mu(a)=\mu(a^{\prime})$, $|\mathcal{M}(a)|=|\mathcal{M}(a^{\prime})|$ and
each element of $\mathcal{M}(a)$ is equivalent to an element of
$\mathcal{M}(a^{\prime})$, and vice versa. Finally, we say that $K$ is a
$A$-reducible if for every $a\in K_{>>0}$, if $x\in\mathcal{M}(a)$ then
$|\text{Nm}_{K/\mathbb{Q}}(x)|\leq A$. In particular, if $A=1$ we say that $K$
is a unit reducible field.

A unary form $ax^{2}$ is said to be perfect if it is uniquely determined by
$\mu(a)$ and $\mathcal{M}(a)$. It is immediately clear that if $ax^{2}$ is a
perfect unary form, then any unary form of the type $\lambda a^{\prime}x^{2}$
is a perfect unary form, where $\lambda\in\mathbb{Q}^{+}$ and $a^{\prime}\sim
a$. In this way, we may classify perfect unary forms up to scaling and
equivalence, or in other words, by their homothety classes. It is known that
for number fields of fixed dimension, the number of classes of perfect forms
can grow arbitrarily large (see e.g. [7], in which it is shown that the number
of classes of perfect unary forms for real quadratic fields can become
arbitrarily large).

In this paper, we follow a similar argument to van Woerden in [6] in order to
determine an upper bound on the number of homothety classes of perfect unary
forms for an arbitrary totally real number field. Our result is stated as
follows.

######  Theorem 1.

Let $K$ be a totally real number field with field discriminant $\Delta_{K}$,
regulator $R_{K}$ and degree $n$ over $\mathbb{Q}$. Let $n_{K}$ denote the
number of homothety classes of perfect unary forms in $K$. Then

$\displaystyle n_{K}\leq
e^{\sqrt{\eta_{K}^{2}+\theta_{K}}}|\Delta_{K}|\left(\frac{2}{\pi}\right)^{2n}\Gamma\left(2+\frac{n}{2}\right)^{4},$

where

  * •

$\eta_{K}=0$ if $K$ is a unit reducible field,

  * •

$\eta_{K}=\frac{\sqrt{n-1}}{2}R_{K}^{\frac{1}{n}}$ if $K$ is not unit
reducible and $2\leq n\leq 11$,

  * •

$\eta_{K}=\frac{\sqrt{n-1}}{2}\left(\frac{2}{\pi}\right)^{n-1}\Gamma\left(2+\frac{n-1}{2}\right)^{2}\left(\sqrt{\frac{2}{n}}\left(\frac{1}{1000}\left(\frac{\log\log
n}{\log n}\right)^{3}\right)\right)^{2-n}R_{K}$ otherwise,

and

  * •

$\theta_{K}=\frac{4\log(A)^{2}}{n-1}$, if $K$ is $A$-reducible.

We will also prove that any totally real field $K$ is $A$-reducible, where

$\displaystyle A\leq
n^{-\frac{n}{2}}\sqrt{|\Delta_{K}|}\left(\frac{2}{\pi}\right)^{\frac{n}{2}}\Gamma\left(2+\frac{n}{2}\right).$

This leads us to the following more general theorem.

######  Theorem 2.

Let $K$ be a totally real number field with field discriminant $\Delta_{K}$,
regulator $R_{K}$ and degree $n$ over $\mathbb{Q}$. Let $n_{K}$ denote the
number of homothety classes of perfect unary forms in $K$. Then

$\displaystyle n_{K}\leq
e^{\sqrt{\eta_{K}^{2}+\rho_{K}}}|\Delta_{K}|\left(\frac{2}{\pi}\right)^{2n}\Gamma\left(2+\frac{n}{2}\right)^{4},$

where $\eta_{K}$ is as defined before, and

  * •

$\rho_{K}=0$ if $K$ is unit reducible,

  * •

$\rho_{K}=\frac{4}{n-1}\log\left(n^{-\frac{n}{2}}\sqrt{|\Delta_{K}|}\left(\frac{2}{\pi}\right)^{\frac{n}{2}}\Gamma\left(2+\frac{n}{2}\right)\right)^{2}\approx\frac{4}{n-1}\log(\sqrt{|\Delta_{K}|})^{2}$
otherwise.

##  2 Proof of Theorem 1

We will define the set $K_{>>0}^{1}$ as the subset of elements in $K_{>>0}$
such that $\mu(a)=1$. Moreover, denote $\mathcal{F}_{K}$ the fundamental
region of $K_{>>0}$, that is, the set of elements such that for all $a\in
K_{>>0}$, there exists an $a^{\prime}\in\mathcal{F}_{K}$ such that $a\sim
a^{\prime}$. For any perfect unary form $ax^{2}$, we will use the notation

$\displaystyle\mathcal{V}(a)=\left\\{\sum_{x\in\mathcal{M}(a)}\lambda_{i}x_{i}^{2}:\lambda_{i}\in\mathbb{Q}^{+}\right\\},$

which is called the Voronoi cone of $a$. Denote $P_{K}$ the set of $a\in
K_{>>0}$ such that $ax^{2}$ determine all the distinct homothety classes of
perfect unary forms. It was proven by the work of Koecher [4] that

$\displaystyle\mathcal{F}_{K}=\bigcup_{a\in P_{K}}\mathcal{V}(a),$

From now on, we will use the notation $\mathcal{V}^{1}(a)=\mathcal{V}(a)\cap
K_{>>0}^{1}$ for any perfect unary form $ax^{2}$. First, we want to prove the
following useful lemma.

######  Lemma 3.

Suppose that $ax^{2}$ and $bx^{2}$ are perfect unary forms in $K$, and $a,b$
belong to distinct homothety classes. Then
$\text{Int}(\mathcal{V}^{1}(a))\cap\text{Int}(\mathcal{V}^{1}(b))=\phi$, where
Int means the interior of the cone generated by $\mathcal{V}(a)$ or
$\mathcal{V}(b)$.

###### Proof.

Suppose that
$c\in\text{Int}(\mathcal{V}^{1}(a))\cap\text{Int}(\mathcal{V}^{1}(b))$, and
$c\neq 0$. We want to show that if this holds, then $a=b$, which is a
contradiction. Since $c\in\text{Int}(\mathcal{V}^{1}(a))$, there exist
$\lambda_{x}\in\mathbb{Q}^{+}$ such that

$\displaystyle c=\sum_{x\in\mathcal{M}(a)}\lambda_{x}x^{2},$

so

$\displaystyle\text{Tr}_{K/\mathbb{Q}}(bc)=\sum_{x\in\mathcal{M}(a)}\lambda_{x}\text{Tr}_{K/\mathbb{Q}}(bx^{2})\geq\sum_{x\in\mathcal{M}(a)}\lambda_{x}=\sum_{x\in\mathcal{M}(a)}\lambda_{x}\text{Tr}_{K/\mathbb{Q}}(ax^{2})=\text{Tr}_{K/\mathbb{Q}}(ac).$

Similarly, since $c\in\text{Int}(\mathcal{V}(b))$ we can deduce that
$\text{Tr}_{K/\mathbb{Q}}(bc)\leq\text{Tr}_{K/\mathbb{Q}}(ac)$, so it must
hold that $\text{Tr}_{K/\mathbb{Q}}(ac)=\text{Tr}_{K/\mathbb{Q}}(bc)$. Then

$\displaystyle\text{Tr}_{K/\mathbb{Q}}((a-b)c)=\sum_{x\in\mathcal{M}(b)}\lambda_{x}(\text{Tr}_{K/\mathbb{Q}}(ax^{2})-1)=0.$

Given that $\lambda_{x}>0$ for all $x\in\mathcal{M}(b)$, it holds that
$\mathcal{M}(b)\subseteq\mathcal{M}(a)$. By a similar argument, we deduce that
$\mathcal{M}(a)\subseteq\mathcal{M}(b)$, and so
$\mathcal{M}(a)=\mathcal{M}(b)$. Since $\mu(a)=\mu(b)$, by the perfectness of
$ax^{2}$, it must hold that $a=b$, which is a contradiction. ∎

Denote $\mathcal{F}_{K}^{1}=\mathcal{F}_{K}\cap K_{>>0}^{1}$. Then by the
lemma above,

$\displaystyle\text{Vol}(\mathcal{F}_{K}^{1})=\text{Vol}\left(\bigcup_{a\in
P_{K}}\mathcal{V}^{1}(a)\right)=\sum_{a\in
P_{K}}\text{Vol}(\mathcal{V}^{1}(a))\geq n_{K}\min_{a\in
P_{K}}\text{Vol}(\mathcal{V}^{1}(a)),$

(here by volume, we mean the volume of the geometric object attained after
canonically embedding into the space $\mathbb{R}^{n}$). First, we will
determine an upper bound for $\text{Vol}(\mathcal{F}_{K}^{1})$. Let $T_{n}$
denote the set of elements $x$ in ${\mathbb{R}^{+}}^{n}$ such that
$|x|_{1}\leq 1$. Then

$\displaystyle\text{Vol}(\mathcal{F}_{K}^{1})\leq\text{Vol}(K_{>>0}^{1})\leq\text{Vol}(T_{n})=\frac{1}{n!},$

(since every element $a\in K_{>>0}$ will be mapped to ${\mathbb{R}^{+}}^{n}$
under the canonical embedding, by the definition of totally positive
elements).

Before finding a lower bound for $\text{Vol}(\mathcal{V}^{1}(a))$, we will
prove the following lemmas.

######  Lemma 4.

For all totally real, $A$-reducible fields $K$ with ring of integers
$\mathcal{O}_{K}$ and unit group $\mathcal{O}_{K}^{\times}$ respectively,

$\displaystyle\max_{a\in
K_{>>0}}\left(\frac{\min_{u\in\mathcal{O}_{K}^{\times}}\text{Tr}_{K/\mathbb{Q}}(au^{2})}{\mu(a)}\right)\leq
e^{\sqrt{\eta_{K}^{2}+\theta_{K}}},$

where $\eta_{K}$ and $\theta_{K}$ are defined as in Theorem 1.

###### Proof.

When $K$ is unit reducible, $\eta_{K}=0$ follows by definition. For non-unit
reducible fields, suppose that $b\in K_{>>0}$ is such that

$\displaystyle\max_{a\in
K_{>>0}}\left(\frac{\min_{u\in\mathcal{O}_{K}^{\times}}\text{Tr}_{K/\mathbb{Q}}(au^{2})}{\mu(a)}\right)=\frac{\min_{u\in\mathcal{O}_{K}^{\times}}\text{Tr}_{K/\mathbb{Q}}(bu^{2})}{\mu(b)}.$

Suppose that $x\in\mathcal{M}(b)$, and associate to $K$ the embeddings
$\sigma_{1},\dots,\sigma_{n}$ that send $K$ to $\mathbb{R}$. Then

$\displaystyle\frac{\min_{u\in\mathcal{O}_{K}^{\times}}\text{Tr}_{K/\mathbb{Q}}(bu^{2})}{\mu(b)}=\frac{\min_{u\in\mathcal{O}_{K}^{\times}}\text{Tr}_{K/\mathbb{Q}}\left(\frac{u^{2}}{x^{2}}bx^{2}\right)}{\mu(b)}\leq\frac{\min_{u\in\mathcal{O}_{K}^{\times}}\max_{i}\sigma_{i}\left(\frac{u^{2}}{x^{2}}\right)\text{Tr}_{K/\mathbb{Q}}(bx^{2})}{\mu(b)}$
$\displaystyle=\min_{u\in\mathcal{O}_{K}^{\times}}\max_{i}\sigma_{i}\left(\frac{u^{2}}{x^{2}}\right)=\min_{u\in\mathcal{O}_{K}}\max_{i}e^{|\log(\sigma_{i}(u^{2}/x^{2}))|}\leq\min_{u\in\mathcal{O}_{K}}e^{\sqrt{\sum_{i=1}^{n}(\log(|\sigma_{i}(u^{2})|)-\log(|\sigma_{i}(x^{2})|))^{2}}}.$

Consider the embedding $\operatorname{Log}:K\to\mathbb{R}^{n}$,
$\operatorname{Log}(y)=(\log(|\sigma_{1}(y)|),\log(|\sigma_{2}(y)|),\dots,\log(|\sigma_{n}(y)|)$.
Then $\operatorname{Log}(\mathcal{O}_{K}^{\times})\triangleq\Lambda_{K}$
generates a lattice of rank $n-1$ in $\mathbb{R}^{n}$ (which is known as the
log-unit lattice), such that, for any $(x_{1},\dots,x_{n})\in\Lambda_{K}$, we
have $\sum_{i=1}^{n}x_{i}=0$. Hence, any element of $\Lambda_{K}$ lies in the
plane $\sum_{i=1}^{n}x_{i}=0$.

Given that $K$ is $A$-reducible by assumption, $\operatorname{Log}(x^{2})$
lies in the plane $\sum_{i=1}^{n}x_{i}=C$, where $0\leq C\leq 2\log(A)$. Let
$m$ denote the plane $\sum_{i=1}^{n}x_{i}=0$ that lies closest to
$\operatorname{Log}(x^{2})$. Then the maximum possible distance between $m$
and $\operatorname{Log}(x^{2})$ is $\frac{2\log(A)}{\sqrt{n-1}}$, in terms of
the $l_{2}$ norm in the space $\mathbb{R}^{n}$.

The distance between $m$ and any point of $\Lambda_{K}$, in terms of the
$l_{2}$ norm in $\mathbb{R}^{n}$, cannot be greater than the covering radius
of $\Lambda_{K}$ in $\mathbb{R}^{n-1}$, which we denote $\mu(\Lambda_{K})$.
Then, using Pythagoras’s theorem, we get

$\displaystyle
e^{\sqrt{\sum_{i=1}^{n}(\log(|\sigma_{i}(u^{2})|)-\log(|\sigma_{i}(x^{2})|))^{2}}}\leq
e^{\sqrt{\mu(\Lambda_{K})^{2}+\theta_{K}}}.$

It remains to find a bound on $\mu(\Lambda_{K})$. In [2], it was proven that
for any lattice $\Lambda$ of rank $d$,
$\mu(\Lambda)\leq\frac{\sqrt{d}}{2}|\det(\Lambda)|^{\frac{1}{d}}$ for $d\leq
10$. Since $|\det(\Lambda_{K})|=R_{K}$, this yields the result stated in the
lemma for $n\leq 11$.

Finally, we need to consider the case $d>11$. Denote $\lambda_{i}$ the $i$th
successive minima of $\Lambda_{K}$, for $1\leq i\leq n-1$. It is easily shown
that

$\displaystyle\mu(\Lambda_{K})\leq\frac{\sqrt{n-1}}{2}\lambda_{n-1}.$ (1)

In [3], it was proven that

$\displaystyle\lambda_{i}\geq\lambda_{1}\geq\sqrt{\frac{2}{n}}\frac{1}{1000}\left(\frac{\log\log
n}{\log n}\right)^{3}.$ (2)

By Minkowski’s theorem, we have

$\displaystyle\prod_{i=1}^{n-1}\lambda_{i}\leq\gamma_{n-1}^{n-1}R_{K},$

where $\gamma_{n}$ denotes Hemite’s constant of rank $n$. By Blichfeldt [1],
it is known that for any $n\geq 2$,

$\displaystyle\gamma_{n}\leq\frac{2}{\pi}\Gamma\left(2+\frac{n}{2}\right)^{\frac{2}{n}},$

so combined with 2,

$\displaystyle\lambda_{n-1}\left(\sqrt{\frac{2}{n}}\frac{1}{1000}\left(\frac{\log\log
n}{\log
n}\right)^{3}\right)^{n-2}\leq\left(\frac{2}{\pi}\right)^{n-1}\Gamma\left(2+\frac{n-1}{2}\right)^{2}R_{K},$

which yields the required result after rearranging the inequality above and
using inequality 1. ∎

######  Lemma 5.

Suppose that $a\in K_{>>0}$ Then there exists an $a^{\prime}\in K_{>>0}$ such
that $a\sim a^{\prime}$ and for all $x\in\mathcal{M}(a)$,

$\displaystyle\text{Tr}_{K/\mathbb{Q}}(x^{2})\leq
e^{\sqrt{\eta_{K}^{2}+\theta_{K}}}\frac{4}{\pi^{2}}\Gamma\left(2+\frac{n}{2}\right)^{\frac{4}{n}}|\Delta_{K}|^{\frac{2}{n}},$

where $\eta_{K}$, $\theta_{K}$ are defined identically as in theorem 1.

###### Proof.

By theorem 6 in [5], and by definition of the algebraic Hermite’s constant
$\gamma_{K}$, for any $a\in K_{>>0}$ it holds that

$\displaystyle\mu(a)\leq\gamma_{K}\text{Nm}_{K/\mathbb{Q}}(a)^{\frac{1}{n}}\leq\gamma_{n}|\Delta_{K}|^{\frac{1}{n}}\text{Nm}_{K/\mathbb{Q}}(a)^{\frac{1}{n}}\leq\frac{2}{\pi}\Gamma\left(2+\frac{n}{2}\right)^{\frac{2}{n}}|\Delta_{K}|^{\frac{1}{n}}\text{Nm}_{K/\mathbb{Q}}(a)^{\frac{1}{n}}.$

Given that $a$ is totally positive, clearly $a^{-1}$ is also totally positive,
and so repeating this argument, we get

$\displaystyle\mu(a)\mu(a^{-1})\leq\frac{4}{\pi^{2}}\Gamma\left(2+\frac{n}{2}\right)^{\frac{4}{n}}|\Delta_{K}|^{\frac{2}{n}}.$

Assume without loss of generality that $\mu(a)=1$, so

$\displaystyle\mu(a^{-1})\leq\frac{4}{\pi^{2}}\Gamma\left(2+\frac{n}{2}\right)^{\frac{4}{n}}|\Delta_{K}|^{\frac{2}{n}}.$

We may also assume without loss of generality that $a$ is such that

$\displaystyle\text{Tr}_{K/\mathbb{Q}}(a^{-1})\leq
e^{\sqrt{\eta_{k}^{2}+\theta_{K}}}\mu(a^{-1}),$

as otherwise, by Lemma 4, there exists an $a^{\prime}$ such that this
inequality holds and $a\sim a^{\prime}$. Hence,

$\displaystyle\text{Tr}_{K/\mathbb{Q}}(a^{-1})\leq
e^{\sqrt{\eta_{k}^{2}+\theta_{K}}}\frac{4}{\pi^{2}}\Gamma\left(2+\frac{n}{2}\right)^{\frac{4}{n}}|\Delta_{K}|^{\frac{2}{n}}.$

Let $A$ denote the diagonal $n$-dimensional matrix with entries
$\sigma_{1}(a),\sigma_{2}(a),\dots,\sigma_{n}(a)$, and denote
$e_{1},e_{2},\dots,e_{n}$ the eigenvalues of $A$. Then

$\displaystyle\min_{i}e_{i}=\min_{\mathbf{y}\in\mathbb{R}^{n}}\frac{\mathbf{y}^{T}A\mathbf{y}}{\mathbf{y}^{T}\mathbf{y}}\leq\frac{\mathbf{x}^{T}A\mathbf{x}}{\mathbf{x}^{T}\mathbf{x}},$

where $\mathbf{x}=(\sigma_{1}(x),\sigma_{2}(x),\dots,\sigma_{n}(x))$, and
$x\in\mathcal{M}(a)$. Note that
$\text{Tr}_{K/\mathbb{Q}}(x^{2})=\mathbf{x}^{T}\mathbf{x}$ and
$\text{Tr}_{K/\mathbb{Q}}(ax^{2})=\mathbf{x}^{T}A\mathbf{x}$, so

$\displaystyle\text{Tr}_{K/\mathbb{Q}}(x^{2})\leq\text{Tr}_{K/\mathbb{Q}}(ax^{2})\max_{i}\frac{1}{e_{i}}\leq\text{Tr}_{K/\mathbb{Q}}(ax^{2})\sum_{i=1}^{n}\frac{1}{e_{i}}=\mu(a)\text{Tr}_{K/\mathbb{Q}}(a^{-1})$
$\displaystyle=\text{Tr}_{K/\mathbb{Q}}(a^{-1})\leq
e^{\sqrt{\eta_{k}^{2}+\theta_{K}}}\frac{4}{\pi^{2}}\Gamma\left(2+\frac{n}{2}\right)^{\frac{4}{n}}|\Delta_{K}|^{\frac{2}{n}},$

as required. ∎

We are now ready to find a lower bound for $\mathcal{V}(a)$. Note that we can
pick $n$ elements in $\mathcal{M}(a)$ (call this set $M_{a}$) such that their
$\mathbb{Q}$-span is equal to $K$, and so the determinant of the polytope with
edges $(\sigma_{1}(x^{2}),\dots,\sigma_{n}(x^{2}))$ is greater than or equal
to $|\Delta_{K}|$, where $x\in M_{a}$. Say $M_{a}=\\{x_{1},\dots,x_{n}\\}$. We
may assume without loss of generality that $a$ is such that
$\text{Tr}_{K/\mathbb{Q}}(x^{2})\leq
e^{\sqrt{\eta_{K}^{2}+\theta_{K}}}\frac{4}{\pi^{2}}\Gamma\left(2+\frac{n}{2}\right)^{\frac{4}{n}}|\Delta_{K}|^{\frac{2}{n}}$,
for all $x\in\mathcal{M}(a)$, by Lemma 5. Then

$\displaystyle\text{Vol}(\mathcal{V}^{1}(a))\geq\text{Vol}\left(\text{conv}\left(\\{0\\}\cup\left\\{\frac{\sigma(x_{i}^{2})}{\text{Tr}_{K/\mathbb{Q}}(x_{i}^{2})}\right\\}_{i=1}^{n}\right)\right)=\frac{1}{n!}|\det(U)|,$

where

$\displaystyle
U=\left\\{\frac{\sigma(x_{i}^{2})}{\text{Tr}_{K/\mathbb{Q}}(x_{i}^{2})}\right\\}_{i=1}^{n},$

so

$\displaystyle|\det(U)|=\prod_{i=1}^{n}\frac{1}{\text{Tr}_{K/\mathbb{Q}}(x_{i}^{2})}\cdot|\det(\sigma(x_{i}^{2}))|_{i=1}^{n}\geq\frac{\Delta_{K}}{e^{n\sqrt{\eta_{k}^{2}+\theta_{K}}}\left(\frac{2}{\pi}\right)^{2n}\Gamma\left(2+\frac{n}{2}\right)^{4}|\Delta_{K}|^{2}}$
$\displaystyle=\frac{1}{e^{n\sqrt{\eta_{k}^{2}+\theta_{K}}}\left(\frac{2}{\pi}\right)^{2n}\Gamma\left(2+\frac{n}{2}\right)^{4}|\Delta_{K}|}.$

Finally, this yields

$\displaystyle n_{K}$
$\displaystyle\leq\frac{\text{Vol}(\mathcal{F}_{K}^{1}}{\min_{a\in
P_{K}}\text{Vol}(\mathcal{V}^{1}(a))}\leq\frac{\frac{1}{n!}}{\frac{1}{n!e^{n\sqrt{\eta_{k}^{2}+\theta_{K}}}\left(\frac{2}{\pi}\right)^{2n}\Gamma\left(2+\frac{n}{2}\right)^{4}|\Delta_{K}|}}$
$\displaystyle=e^{n\sqrt{\eta_{K}^{2}+\theta_{K}}}|\Delta_{K}|\left(\frac{2}{\pi}\right)^{2n}\Gamma\left(2+\frac{n}{2}\right)^{4},$

which proves Theorem 1.

##  3 Proof of Theorem 2

The only result we need to prove Theorem 2 is the following lemma.

######  Lemma 6.

Let $K$ be a totally real field and suppose that $a\in K_{>>0}$. Then

$\displaystyle\text{Nm}_{K/\mathbb{Q}}(x)\leq
n^{-\frac{n}{2}}\sqrt{|\Delta_{K}|}\left(\frac{2}{\pi}\right)^{\frac{n}{2}}\Gamma\left(2+\frac{n}{2}\right).$

###### Proof.

Suppose that $x\in\mathcal{M}(a)$. Then, using the arithmetic-geometric
inequality, the definition of the algebraic Hermite’s constant and
Leibak/Blichfeldt’s bounds,

$\displaystyle\text{Nm}_{K/\mathbb{Q}}(a)\text{Nm}_{K/\mathbb{Q}}(x^{2})=\text{Nm}_{K/\mathbb{Q}}(ax^{2})\leq\left(\frac{1}{n}\text{Tr}_{K/\mathbb{Q}}(ax^{2})\right)^{n}\leq\frac{1}{n^{n}}\left(\gamma_{K}\text{Nm}_{K/\mathbb{Q}}(a)^{\frac{1}{n}}\right)^{n}$
$\displaystyle\leq
n^{-n}\gamma_{n}^{n}|\Delta_{K}|\text{Nm}_{K/\mathbb{Q}}(a)\leq
n^{-\frac{n}{2}}\sqrt{|\Delta_{K}|}\left(\frac{2}{\pi}\right)^{\frac{n}{2}}\Gamma\left(2+\frac{n}{2}\right)\text{Nm}_{K/\mathbb{Q}}(a).$

The result follows by dividing through by $\text{Nm}_{K/\mathbb{Q}}(a)$. ∎

Theorem 2 follows immediately, as we now have an upper bound for $\theta_{K}$.

## References

  * [1] Blichfeldt, H.F.: “The minimum value of quadratic forms, and the closest packing of spheres”. Mathematische Annalen, vol. 101, pp. 605–608 (1929).
  * [2] Kathuria, L., Raka, M.: “On conjectures of Minkowski and Woods for $n=10$”. Proceedings in Mathematical Sciences, vol. 132, no. 45 (2022).
  * [3] Kessler, V.: “On the minimum of the unit lattice”. Journal de theorie des nombres de Bordeaux, vol. 3, no. 2, pp. 377–380 (1991).
  * [4] Koecher, M.: “Beiträge zu einer Reduktionstheorie in Positivitätsbereichen. I”. Math. Ann., vol. 141, pp. 384–432 (1960).
  * [5] Leibak, A.: “n additive generalization of Voronoi’s theory for algebraic number fields”. Proc. Estonian Acad. Sci. Phys. Math., vol. 54, no. 4, pp. 195–211 (2005).
  * [6] van Woerden, W.P.J.: “An upper bound on the number of perfect quadratic forms”. Advances in Mathematics, vol. 365 (2020).
  * [7] Yasaki, D.: “Perfect unary forms over real quadratic fields”. Journal de Théorie des Nombres de Bordeaux 25 (2013) pp. 759–775.


# New algorithm of measuring gravitational wave radiation from rotating binary
system

Soon-Tae Hong<EMAIL_ADDRESS>Center for Quantum Spacetime and
Department of Physics, Sogang University, Seoul 04107, Korea

###### Abstract

In order to investigate the gravitational wave (GW) radiation, without
appealing to the tensorial formalism of the linearized general relativity, we
formulate the so-called modified linearized general relativity (MLGR). As an
application of the MLGR, we construct a novel paradigm of measuring the GW
radiation from a binary system of compact objects, to theoretically interpret
their phenomenology. To do this, we formulate the mass scalar and mass vector
potentials for the merging binary compact objects, from which we construct the
mass magnetic field in addition to the mass electric one which also includes
the mass vector potential effect. Next, defining the mass Poyinting vector in
terms of the mass electric and mass magnetic fields, we construct the GW
radiation intensity profile possessing a prolate ellipsoid geometry due to the
merging binary compact objects source. At a given radial distance from the
binary compact objects, the GW radiation intensity on the revolution axis of
the binary compact objects is shown to be twice that on the equatorial plane.
Moreover, we explicitly obtain the total radiation power of the GW, which has
the same characteristic as that of the electromagnetic wave in the rotating
charge electric dipole moment. We also find that, in no distorting limit of
the merging binary compact objects, the compact objects do not yield the total
GW radiation power, consistent with the result of the linearized general
relativity.

gravitational wave radiation; modified linearized general relativity; merging
binary compact objects; mass Poyinting vector; distortion

##  I Introduction

Exploiting the tensorial formalism for the gravity, the linearized general
relativity (LGR) wald84 ; hobson06  has been investigated to describe the
gravitational wave (GW) radiation from a binary system of compact objects
(namely stars or black holes) for instance. To be specific, in order to study
the physical phenomenology in the LGR, we need to assume that the gravity
fields are weak so that the gravitational field equations can be linearized.
Next it is well known that, in vacuum of the LGR, the binary compact objects
oscillating in time emit spin-two gravitons which are quanta of the GW.
Mathematically the spin-two gravitons originate from the second rank tensor
associated with the curved spacetime metric $g_{\alpha\beta}$ pauli39 ; wald84
. In order to construct the GW which is transverse to the GW propagating
direction and yields the spin-two graviton, we have exploited the ad hoc
transverse-traceless gauge wald84 ; hobson06 . In contrast, the spin-one
photons which are quanta of the electromagnetic (EM) wave in vacuum are
delineated in terms of the vectorial quantities of the charge scalar and
charge vector potentials defined on the flat spacetime.

On the other hand, since the first GW was detected by Advanced LIGO aasi15 ,
there have been lots of progresses in an observational astrophysics. In
particular, the third generation network of ground-based detectors reitze19
is currently proposed to obtain improved sensitivities, compared to those of
the Advanced LIGO detector. Recently, in order to constrain cosmology, a
theoretical probe of cosmology which exploits population level properties of
lensed GW detections has been proposed jana23 . Next, since the gamma ray
burst (GRB) discovery was published strong73 , to elucidate the GRB associated
with the magnetar (namely highly-magnetized neutron star) for instance
hurley21 , we have observed many theoretical models including the Dirac type
relativistic massive photon model hong22 , where the anti-photon corresponding
to the negative energy solution is proposed as a candidate for an intense
radiation flare of the GRB. Next, the first neutron star binary PSR B1913+16
has been discovered showing that its orbit was losing energy due to the
emission of the GW hulse75 . The GW detector network has observed the GW
merger event originating from binary neutron star GW170817 in 2017 abbott2017a
; grb1 ; goldstein2017 . Note that the GW170817 merger event has been known to
be consistent with the GRB of GRB170817A grb1 ; goldstein2017 . To be
specific, the GW is supposed to be produced from neutron star oscillations
related with magnetar giant flares and the corresponding GRB kaspi2017 ;
ligo2023 .

In this paper, we will propose a new theoretical paradigm of the modified
linearized general relativity (MLGR), in which we will formulate the Green
retarded wald84 ; hobson06  mass scalar and mass vector potentials of a binary
system of compact objects. Using these potentials in the MLGR, we will next
construct the mass electric and mass magnetic fields. In particular, in order
to formulate the non-vanishing mass magnetic field, we will introduce the
distortion of the spirally merging binary compact objects during one cycle of
their revolution. Making use of the mass electric and mass magnetic fields, we
will find the mass Poyinting vector which is exploited to construct a
theoretical scheme of measuring the GW radiation from a binary system of
compact objects. Moreover, introducing the traceless transverse gauge, we will
study the GW of the binary compact objects in the LGR.

In Sec. 2 we will formulate the MLGR. In Sec. 3, as an application of the
MLGR, we will investigate the phenomenology of the merging binary compact
objects. To do this, we will explicitly formulate the physical quantities such
as the mass scalar and mass vector potentials, and the mass Poyinting vector.
We will also study the traceless transverse gauge in the LGR. Sec. 4 includes
conclusions. In Appendix A, we will pedagogically study the EM radiation from
the rotating charge electric dipole moment.

##  II Set up of MLGR algorithm

In this section, we will formulate the MLGR which includes explicitly the mass
current degrees of freedom (DOF). To do this, in the linearized general
relativity wald84 ; hobson06 , we first assume that the deviation
$h_{\alpha\beta}$ of the spacetime metric $g_{\alpha\beta}$ from a flat metric
$\eta_{\alpha\beta}$ is small:
$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$. After some algebra we
can find the linearized Einstein equation wald84

$-\frac{1}{2}\square\bar{h}_{\alpha\beta}+\partial^{\gamma}\partial_{(\alpha}\bar{h}_{\beta)\gamma}-\frac{1}{2}\eta_{\alpha\beta}\partial^{\gamma}\partial^{\delta}\bar{h}_{\gamma\delta}=\frac{8\pi
G}{c^{4}}T_{\alpha\beta},$ (2.1)

where $\square=\partial^{\beta}\partial_{\beta}$ with the metric
$(-1,+1,+1,+1)$ and $T_{\alpha\beta}$ is the stress-energy tensor. Here the
trace reversed perturbation $\bar{h}_{\alpha\beta}$ is given by
$\bar{h}_{\alpha\beta}=h_{\alpha\beta}-\frac{1}{2}\eta_{\alpha\beta}h$ and
$h=\eta^{\alpha\beta}h_{\alpha\beta}$. Next exploiting (2.1) and the Lorentz
gauge condition (LGC) in the gravity

$\partial^{\beta}\bar{h}_{\alpha\beta}=0,$ (2.2)

we arrive at

$\square\bar{h}_{\alpha\beta}=-\frac{16\pi G}{c^{4}}T_{\alpha\beta}.$ (2.3)

Now we consider the MLGR where we have the non-vanishing $\bar{h}^{00}$ and
$\bar{h}^{0i}$ $(i=1,2,3)$, together with

$\bar{h}^{ij}=0.$ (2.4)

We define in the MLGR the time-time component of the trace reversed
perturbation $\bar{h}^{00}$ and the corresponding stress energy tensor
$T^{00}$ as

$\bar{h}^{00}=-\frac{4}{c}A_{M}^{0},~{}~{}~{}T^{00}=c^{2}\rho_{M},$ (2.5)

where $\rho_{M}$ is the mass density. Here the subscript $M$ denotes the mass
related quantities and will be applied to the gravitational interaction
physical quantities from now on. Inserting (2.5) into (2.3) yields

$\square A_{M}^{0}=\frac{4\pi G}{c}\rho_{M},$ (2.6)

implying that we have the dynamical DOF in the MLGR. Note that, in the static
case, (2.6) and $\phi_{M}\equiv cA_{M}^{0}$ yields $\nabla^{2}\phi_{M}=4\pi
G\rho_{M}$ which is consistent with the Newtonian gravity goldstein80 ;
marion04 ; wald84 .

Next in the MLGR we define the time-space components of the trace reversed
perturbation $\bar{h}^{0i}$ and the stress energy tensor $T^{0i}$ as

$\bar{h}^{0i}=-\frac{4}{c}A^{i}_{M},~{}~{}~{}T^{0i}=cJ^{i}_{M},$ (2.7)

where $J^{i}_{M}\equiv\rho_{M}v^{i}$ is the mass current density associated
with the mass current velocity $v^{i}$. In the MLGR, inserting (2.7) into
(2.3) produces

$\square A^{i}_{M}=\frac{4\pi G}{c^{2}}J^{i}_{M}.$ (2.8)

Combining (2.6) and (2.8), we find the covariant form of the equations of
motion

$\square A^{\alpha}_{M}=\frac{4\pi G}{c^{2}}J^{\alpha}_{M},$ (2.9)

where $A_{M}^{\alpha}=\left(\frac{\phi_{M}}{c},A_{M}^{i}\right)$ and the four
mass current density is defined by $J^{\alpha}_{M}=(c\rho_{M},J^{i}_{M})$.
Note that the LGC in (2.2) is rewritten as

$\partial_{\alpha}A^{\alpha}_{M}=0.$ (2.10)

Exploiting $A^{\alpha}_{M}$ in (2.5) and (2.7), we can find

$A^{\alpha}_{M}=-\frac{c}{4}\bar{h}^{0\alpha},$ (2.11)

to yield the equation of motion

$\square\bar{h}^{0\alpha}=-\frac{16\pi G}{c^{3}}J^{\alpha}_{M}.$ (2.12)

Note that the LGC in (2.10) becomes

$\partial_{\alpha}\bar{h}^{0\alpha}=0.$ (2.13)

Now we investigate the GW defined in the vacuum. To do this, inserting
$J^{\alpha}_{M}=0$ into (2.12) and exploiting $\bar{h}^{0\alpha}$ in (2.11),
we obtain the wave equation for the non-vanishing field $\bar{h}^{0\alpha}$
(or $A_{M}^{\alpha}$)

$\square\bar{h}^{0\alpha}=\square A_{M}^{\alpha}=0.$ (2.14)

Note that the wave equations in (2.14) in the MLGR describe a massless spin-
one graviton propagating in the flat spacetime, similar to a massless spin-one
photon propagating in the flat spacetime. To be specific, (2.14) explains the
graviton of gravitational field.

Next it seems appropriate to comment on the spin of the graviton in the MLGR.
In vacuum with $T_{\alpha\beta}=0$ in (2.3), we end up with the wave equation
$\square\bar{h}_{\alpha\beta}=0$ which could describe a massless spin-two
graviton propagating in the flat spacetime pauli39 ; wald84 . However, if we
consider only the physically meaningful wave equation for $A^{\alpha}_{M}$ in
(2.14) obtained from that for $\bar{h}^{0\alpha}$ $(\alpha=0,1,2,3)$, we
mathematically find a massless spin-one graviton whose properties are well
defined in the MLGR. Note that the condition $\bar{h}^{ij}=0$ in (2.4) reduces
the number of independent components of $\bar{h}^{\alpha\beta}$ to four. The
independent components $(\bar{h}^{00},\bar{h}^{0i})$ thus correspond to the
four components of $A^{\alpha}_{M}$. Next we explicitly find the solution of
the GW field equation in (2.14) as follows

$\bar{h}^{0\alpha}=\epsilon^{\alpha}e^{ik_{\mu}x^{\mu}},$ (2.15)

where $k^{\mu}$ is the GW four-vector. Here $\epsilon^{\alpha}$ is the
constant amplitude vector corresponding to the GW polarizations. Inserting
(2.15) into (2.14), we find $k_{\alpha}k^{\alpha}=0$ to yield

$k^{\alpha}=(k,0,0,k),$ (2.16)

which describes the massless spin-one graviton. Making use of (2.15) and the
LGC in (2.13), we obtain

$k_{\alpha}\epsilon^{\alpha}=0,$ (2.17)

to produce $\epsilon^{0}=\epsilon^{3}$.

Now we exploit the fact that the LGC in (2.13) is preserved by the gauge
transformation
$\bar{h}^{0\alpha}\rightarrow\bar{h}^{0\alpha}+\partial^{\alpha}\psi$ if
$\square\psi=0$ wald84 ; hobson06 . We assume $\psi=\epsilon
e^{ik_{\mu}x^{\mu}}$ and $\bar{h}^{\prime
0\alpha}=\epsilon^{\prime\alpha}e^{ik_{\mu}x^{\mu}}$, and then we exploit the
gauge transformation equation $\bar{h}^{\prime
0\alpha}=\bar{h}^{0\alpha}+\partial^{\alpha}\psi$ to yield

$\epsilon^{\prime\alpha}=\epsilon^{\alpha}+i\epsilon k^{\alpha}.$ (2.18)

Using (2.16) and (2.18), we explicitly find the non-trivial relation
$\epsilon^{\prime 0}=\epsilon^{0}+i\epsilon k$. Next, exploiting the ansatz
$\epsilon=\frac{i}{k}\epsilon^{0}$, we find $\epsilon^{\prime 0}=0$. Dropping
the primes in $\epsilon^{\prime\alpha}$ we obtain

$\epsilon^{\alpha}=(0,\epsilon^{1},\epsilon^{2},0),$ (2.19)

implying that the GW is transverse to the propagation direction $k^{\alpha}$
in (2.16). Note that $\epsilon^{\alpha}$ in (2.19) thus satisfies the identity
$k_{\alpha}\epsilon^{\alpha}=0$ in (2.17) as expected.

Next we have several comments on the GW phenomenology in terms of
$\epsilon^{\alpha}$ in (2.19). First, in the cases of
$\epsilon^{\alpha}=(0,\epsilon^{1},0,0)$ and
$\epsilon^{\alpha}=(0,0,\epsilon^{2},0)$, as the GW passes a free positive
test charge, the test charge will oscillate in the $x$ and $y$ directions with
a magnitude changing sinusoidally with time, respectively, following the GW
wave relation in (2.15). Second, in the cases of
$\epsilon^{\alpha}=(0,\epsilon^{1},i\epsilon^{1},0)$ and
$\epsilon^{\alpha}=(0,\epsilon^{1},-i\epsilon^{1},0)$, the GW polarizations
yield left-circularly and right-circularly polarized GWs, and then the test
charge will move in the corresponding circles, respectively. Third, in the
general cases of $\epsilon^{\alpha}=(0,\epsilon^{1},i\bar{\epsilon}^{1},0)$
and $\epsilon^{\alpha}=(0,\epsilon^{1},-i\bar{\epsilon}^{1},0)$
$(\epsilon^{1}\neq\bar{\epsilon}^{1})$, the GW polarizations produce left-
elliptic and right-elliptic polarized GWs, respectively, and thus the test
charge will move in the corresponding ellipses.

Now we investigate the properties of the massive graviton parenthetically. As
in the case of the massive photon hong22 , we find the identity
$k_{\alpha}\epsilon^{\alpha}\neq 0$ for the massive graviton, since for the
massive graviton we have longitudinal component in addition to transverse
ones, similar to the phonon associated with massive particle lattice
vibrations phonon . Note that $\epsilon^{\alpha}$ is a polarization vector
possessing the spacetime index $\alpha$ $(\alpha=0,1,2,3)$ which is the same
as the massive graviton spin index and is needed to incorporate minimally the
spin DOF for the massive graviton. Note also that, as in the case of the GRB
in the Dirac type relativistic massive photon model hong22 , the massive
graviton could be interpreted as a spin-one GRB-like particle grb1 ;
goldstein2017 . The GRB-like graviton corresponding to its negative energy
solution could then be regarded as an anti-graviton, which could be a
candidate for an intense radiation flare of the GRB of GRB170817A grb1 ;
goldstein2017  associated with the binary compact objects merger event
GW170817. Note that the massive anti-graviton could be described in terms of
the massive gauge boson possessing a finite radius as in the massive photon in
the stringy photon model hong211 .

##  III Gravitational wave radiation from merging binary compact objects

Figure 1: (a) The schematic diagram for a binary system of compact objects
$M_{1}$ and $M_{2}$ of equal mass $M$ which co-rotate in circular orbits of
radius $a$ about their common center of mass located at the origin, with an
angular frequency $\omega$. (b) The diagram for the geometry of $R_{\pm}$ for
the retarded mass scalar potentials $\phi_{M}^{I}$ in (3.21) and
$R_{+}^{\prime}$ for the retarded mass vector potentials $\vec{A}_{M}^{I}$ in
(3.25). Here
$\vec{x}=r(\hat{x}\sin\theta\cos\phi+\hat{y}\sin\theta\sin\phi+\hat{z}\cos\theta)$
denotes a radial vector from the origin $o$ to an observation point.

In this section, in the MLGR we investigate a binary system of merging compact
objects which possesses masses $M_{1}$ and $M_{2}$ rotating on the $x$-$y$
plane. We assume that for simplicity the two compact objects $M_{1}$ and
$M_{2}$ of equal mass $M_{1}=M_{2}=M$ co-rotate in circular orbits of radius
$a$ about their common center of mass with an angular frequency $\omega$, as
shown in Figure 1(a). Note that the rotating binary compact objects can be
thought of as the superposition of two oscillating binary compact objects, one
along the $x$-axis and the other along the $y$-axis. The rotating mass vector
$\vec{M}(t)$ on the $x$-$y$ plane is given by

$\vec{M}(t)=M(\hat{x}\cos\omega t+\hat{y}\sin\omega t).$ (3.20)

For the masses on the $x$-axis and $y$-axis, exploiting (3.20) the Green
retarded wald84 ; hobson06  mass scalar potentials (or gravitational
potentials) $\phi_{M}^{I}$ and $\phi_{M}^{II}$ are respectively given by

$\displaystyle\phi_{M}^{I}(\vec{x},t)$ $\displaystyle=$
$\displaystyle-G\left(\frac{M\cos[\omega(t-R_{+}/c)]}{R_{+}}+\frac{M\cos[\omega(t-R_{-}/c)]}{R_{-}}\right),$
$\displaystyle\phi_{M}^{II}(\vec{x},t)$ $\displaystyle=$
$\displaystyle-G\left(\frac{M\sin[\omega(t-\bar{R}_{+}/c)]}{\bar{R}_{+}}+\frac{M\sin[\omega(t-\bar{R}_{-}/c)]}{\bar{R}_{-}}\right),$
(3.21)

where

$\displaystyle R_{\pm}$ $\displaystyle=$ $\displaystyle
r\left(1\mp\frac{2a}{r}\sin\theta\cos\phi+\frac{a^{2}}{r^{2}}\right)^{1/2}=r\left(1\mp\frac{a}{r}\sin\theta\cos\phi\right),$
$\displaystyle\bar{R}_{\pm}$ $\displaystyle=$ $\displaystyle
r\left(1\mp\frac{2a}{r}\sin\theta\sin\phi+\frac{a^{2}}{r^{2}}\right)^{1/2}=r\left(1\mp\frac{a}{r}\sin\theta\sin\phi\right).$
(3.22)

Here we have used approximation $a\ll r$. For more details of $R_{\pm}$, see
the geometry in Figure 1(b). After some algebra we arrive at the total
contributions to the mass scalar potentials
$\phi_{M}=\phi_{M}^{I}+\phi_{M}^{II}$

$\displaystyle\phi_{M}(\vec{x},t)$ $\displaystyle=$
$\displaystyle-\frac{2GM}{r}\left(\cos\left[\omega\left(t-\frac{r}{c}\right)\right]+\sin\left[\omega\left(t-\frac{r}{c}\right)\right]-\frac{\omega
a^{2}}{cr}\sin^{2}\theta\cos^{2}\phi\sin\left[\omega\left(t-\frac{r}{c}\right)\right]\right.$
(3.23) $\displaystyle\left.+\frac{\omega
a^{2}}{cr}\sin^{2}\theta\sin^{2}\phi\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\right),$

where we have ignored the much smaller terms exploiting approximation
$\frac{1}{r}\ll\frac{\omega}{c}$. Note that the wavelength
$\lambda\left(=\frac{2\pi c}{\omega}\right)$ of the GW is also much smaller
than $r$ to yield $\lambda\ll r$.

Next we construct the mass vector potential $\vec{A}_{M}$ which is related
with the mass current and is missing in the Newtonian gravity. Making use of
(3.20) for the binary compact objects co-rotating on the $x$-$y$ plane, we
find the mass current

$\vec{I}_{M}(t)=-M\omega(\hat{x}\sin\omega t-\hat{y}\cos\omega t).$ (3.24)

We then explicitly obtain the retarded mass vector potentials
$\vec{A}_{M}^{I}$ and $\vec{A}_{M}^{II}$ for the mass current on the $x$-axis
and $y$-axis

$\displaystyle\vec{A}_{M}^{I}(\vec{x},t)$ $\displaystyle=$
$\displaystyle-\frac{GM\omega}{c^{2}}\left(\int_{-a}^{0}\frac{\sin[\omega(t-R_{-}^{\prime}/c)]}{R_{-}^{\prime}}dx^{\prime}-\int_{0}^{\bar{a}}\frac{\sin[\omega(t-R_{+}^{\prime}/c)]}{R_{+}^{\prime}}dx^{\prime}\right)\hat{x},$
$\displaystyle\vec{A}_{M}^{II}(\vec{x},t)$ $\displaystyle=$
$\displaystyle-\frac{GM\omega}{c^{2}}\left(-\int_{-a}^{0}\frac{\cos[\omega(t-\bar{R}_{-}^{\prime}/c)]}{\bar{R}_{-}^{\prime}}dy^{\prime}+\int_{0}^{\bar{a}}\frac{\cos[\omega(t-\bar{R}_{+}^{\prime}/c)]}{\bar{R}_{+}^{\prime}}dy^{\prime}\right)\hat{y},$
(3.25)

where $R_{\pm}^{\prime}$ and $\bar{R}_{\pm}^{\prime}$ are given by

$R_{\pm}^{\prime}=r\left(1\mp\frac{x^{\prime}}{r}\sin\theta\cos\phi\right),~{}~{}~{}\bar{R}_{\pm}^{\prime}=r\left(1\mp\frac{y^{\prime}}{r}\sin\theta\sin\phi\right).$
(3.26)

Here we also have exploited approximation $a\ll r$. For more details of
$R_{+}^{\prime}$, see the geometry in Figure 1(b). Note that $\bar{a}$ in the
upper bound of the second integral in (3.25) is the reduced radius from the
origin after the mass $M_{1}(=M)$ travels during the half revolution. The
distance $\bar{a}$ is defined by

$\delta a\equiv 2(a-\bar{a}),$ (3.27)

so that $\delta a$ can measure the distortion of the spirally merging binary
compact objects during one cycle of their revolution. After some algebra
associated with the definite integrations, we arrive at the total retarded
mass vector potential $\vec{A}_{M}=\vec{A}_{M}^{I}+\vec{A}_{M}^{II}$

$\displaystyle\vec{A}_{M}(\vec{x},t)$ $\displaystyle=$
$\displaystyle-\frac{GM\omega\delta
a}{2c^{2}r}\left[\left(\sin\left[\omega\left(t-\frac{r}{c}\right)\right]+\frac{\omega
a}{c}\sin\theta\cos\phi\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\right)\hat{x}\right.$
(3.28)
$\displaystyle\left.+\left(-\cos\left[\omega\left(t-\frac{r}{c}\right)\right]+\frac{\omega
a}{c}\sin\theta\sin\phi\sin\left[\omega\left(t-\frac{r}{c}\right)\right]\right)\hat{y}\right],$

where we have again used the approximations $a\ll r$ and $\lambda\ll r$ with
$\lambda$ being the GW wavelength.

Figure 2: The characteristic diagram for a surface of the GW radiation
intensity $\frac{dP_{M}}{d\Sigma}$ in (3.32), as a function of the polar angle
$\theta$, showing a prolate ellipsoid geometry in the corresponding intensity
profile. Here the binary compact objects reside on the equatorial $x$-$y$
plane as shown in Figure 1(a). Note that, at a given radial distance
$r=|\vec{x}|$ from the origin of the binary system of compact objects, the GW
radiation intensity on the revolution $z$ axis of the binary compact objects
is twice that on the equatorial plane.

Now exploiting (3.23) and (3.28), we find the mass electric field which also
includes the mass vector potential effect
$\vec{E}_{M}\equiv-\nabla\phi_{M}-\frac{\partial\vec{A}_{M}}{\partial
t}=E_{M}^{r}\hat{r}+E_{M}^{\theta}\hat{\theta}+E_{M}^{\phi}\hat{\phi}$ and the
mass magnetic one
$\vec{B}_{M}\equiv\nabla\times\vec{A}_{M}=B_{M}^{\theta}\hat{\theta}+B_{M}^{\phi}\hat{\phi}$

$\displaystyle E_{M}^{r}$ $\displaystyle=$
$\displaystyle\frac{2GM\omega}{cr}\left(\sin\left[\omega\left(t-\frac{r}{c}\right)\right]-\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\right),$
$\displaystyle E_{M}^{\theta}$ $\displaystyle=$ $\displaystyle
cB_{M}^{\phi}=\frac{GM\omega^{2}\delta
a}{2c^{2}r}\cos\theta\left(\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\cos\phi+\sin\left[\omega\left(t-\frac{r}{c}\right)\right]\sin\phi\right),$
$\displaystyle E_{M}^{\phi}$ $\displaystyle=$ $\displaystyle-
cB_{M}^{\theta}=\frac{GM\omega^{2}\delta
a}{2c^{2}r}\left(-\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\sin\phi+\sin\left[\omega\left(t-\frac{r}{c}\right)\right]\cos\phi\right),$
(3.29)

where we have included the relevant leading order terms. Exploiting (3.29), we
find $\vec{B}_{M}=\frac{1}{c}(\hat{r}\times\vec{E}_{M})$ which implies that
the $\vec{E}_{M}$ and $\vec{B}_{M}$ fields are mutually perpendicular, and are
spherical waves (not plane waves) and their amplitudes decrease like
$\frac{1}{r}$ as they propagate. However, for large $r$, the GW waves are
approximately plane waves over small regions, similar to the EM waves in (A.7)
discussed in Appendix A. Note that the $\vec{E}_{M}$ field possesses
$E_{M}^{r}\hat{r}$ component originated from the gravitational attraction from
the binary compact objects. This characteristic is different from the rotating
charge electric dipole moment system where $E_{Q}^{r}\hat{r}$ component
vanishes as in (A.7). Note also that, in the GW radiation intensity
$\frac{dP_{M}}{d\Sigma}$ in (3.32), we have a vanishing contribution of
$E_{M}^{r}\hat{r}$ to $\frac{dP_{M}}{d\Sigma}$ by averaging in time over a
complete cycle as shown below.

Next we formulate the mass Poyinting vector in terms of $\vec{E}_{M}$ and
$\vec{B}_{M}$ in (3.29)

$\vec{S}_{M}\equiv\frac{c^{2}}{4\pi
G}(\vec{E}_{M}\times\vec{B}_{M})=\frac{c}{4\pi
G}\left[(E_{M}^{\perp})^{2}\hat{r}-E_{M}^{r}\vec{E}_{M}^{\perp}\right],$
(3.30)

where

$\displaystyle\vec{E}_{M}^{\perp}$ $\displaystyle=$ $\displaystyle
E_{M}^{\theta}\hat{\theta}+E_{M}^{\phi}\hat{\phi},$
$\displaystyle(E_{M}^{\perp})^{2}$ $\displaystyle=$
$\displaystyle(E_{M}^{\theta})^{2}+(E_{M}^{\phi})^{2}=\left(\frac{GM\omega^{2}\delta
a}{2c^{2}r}\right)^{2}\left(1-\sin^{2}\theta\cos^{2}\left[\omega\left(t-\frac{r}{c}\right)-\phi\right]\right).$
(3.31)

Note that the normalization factor $\frac{c^{2}}{4\pi G}$ in (3.30) is
systematically determined by using the proportionality constants
$-\frac{G}{c^{2}}$ of $\vec{A}_{M}^{I}$ and $\vec{A}_{M}^{II}$ in (3.25), and
$-G$ of $\phi_{M}^{I}$ and $\phi_{M}^{II}$ in (3.21) which is consistent with
the Newtonian gravity goldstein80 ; marion04 . Next using (3.30) and (3.31) we
find the GW radiation intensity, namely the GW radiation power per surface
$\Sigma$, which is obtainable by averaging in time over a complete cycle

$\frac{dP_{M}}{d\Sigma}\equiv\langle\vec{S}_{M}\rangle=\frac{c}{4\pi
G}\left(\frac{GM\omega^{2}\delta
a}{2c^{2}r}\right)^{2}\left(1-\frac{1}{2}\sin^{2}\theta\right)\hat{r}.$ (3.32)

Note that in (3.32) the $\vec{E}_{M}^{\perp}$-component of $\vec{S}_{M}$ in
(3.30) vanishes, after averaging in time over a complete cycle and then using
$\langle E_{M}^{r}\rangle=0$. Note also that $\frac{dP_{M}}{d\Sigma}$ has a
maximum (minimum) value at $\theta=0$ ($\theta=\frac{\pi}{2}$) to yield the
invariant relation independent of the radial distance $r$

$\frac{dP_{M}}{d\Sigma}(\theta=0)=2\frac{dP_{M}}{d\Sigma}\left(\theta=\frac{\pi}{2}\right),$
(3.33)

implying that, at a given radial distance $r$ from the origin of the binary
system of compact objects, the GW radiation intensity on the revolution $z$
axis of the compact objects is twice that on the equatorial $x$-$y$ plane
where the compact objects locate. The characteristic invariant in (3.33) also
occurs in the EM radiation intensity of the rotating charge electric dipole
moment in (A.10). The geometrically invariant surface of the GW radiation
intensity $\frac{dP_{M}}{d\Sigma}$ then yields a prolate ellipsoid geometry in
the corresponding intensity profile as shown in Figure 2.

The total GW radiation power constructed by integrating
$\langle\vec{S}_{M}\rangle$ over a sphere $\Sigma_{r}$ of radius $r$ is then
given by

$P_{M}\equiv\int\langle\vec{S}_{M}\rangle\cdot
d\vec{\sigma}=\frac{GM^{2}\omega^{4}(\delta a)^{2}}{6c^{3}},$ (3.34)

where $d\vec{\sigma}$ is an area element vector perpendicular to the surface
$\Sigma_{r}$. As expected, the dimensionality of $P_{M}$ is that of power,
which is the same as the dimensionality of $P_{Q}$ in (A.11). This aspect
implies that the normalization factor $\frac{c^{2}}{4\pi G}$ in (3.30) is well
defined. Note that in the ideal binary compact objects without any distortion
($\delta a=0$) we obtain the vanishing total GW radiation power. In contrast,
in the physically merging binary compact objects source we have the total
power in (3.34) which is proportional to $(\delta a)^{2}$. Here $\delta a$ is
defined in (3.27). These phenomenological aspects are consistent with the
second rank tensor approach to the merging binary compact objects wald84 ;
hobson06  in the LGR.

Now, it seems appropriate to comment on the second rank tensor approach in the
LGR, which is distinct from the MLGR defined in (2.5) and (2.7). For the LGR
associated with merging binary compact objects of equal mass $M$ which co-
rotate in circular orbits of radius $a$ about their common center of mass with
an angular frequency $\omega$ as shown in Figure 1(a), we construct the
symmetric second rank tensor for the GW radiation hobson06

$\displaystyle\bar{h}_{rad}^{00}$ $\displaystyle=$ $\displaystyle
0,~{}~{}~{}\bar{h}_{rad}^{0i}=0,$ $\displaystyle\bar{h}^{ij}_{rad}$
$\displaystyle=$
$\displaystyle\frac{8GM\omega^{2}a^{2}}{c^{4}r}\left(\begin{array}[]{ccc}\cos[2\omega(t-r/c)]&\sin[2\omega(t-r/c)]&0\\\
\sin[2\omega(t-r/c)]&-\cos[2\omega(t-r/c)]&0\\\
0&0&0\end{array}\right),~{}~{}~{}(i,j=1,2,3).$ (3.38)

In order to investigate the GW from binary compact objects in the LGR, we
introduce the traceless transverse gauge wald84 ; hobson06 , and thus the GW
originates from the second rank tensor $\bar{h}_{rad}^{\alpha\beta}$ in
(3.38). Note that the GW radiation from the binary compact objects in the MLGR
can be described in terms of the mathematically and physically independent
vector $\bar{h}^{0\alpha}=-\frac{4}{c}A_{M}^{\alpha}$ in (2.5) and (2.7),
without appealing to the second rank tensor $\bar{h}_{rad}^{ij}$ in (3.38) in
the LGR. Note also that the graviton possesses spin-one in the MLGR, but spin-
two in the LGR, since the MLGR and LGR are delineated in terms of the vector
$\bar{h}^{0\alpha}$ related with $\bar{h}^{ij}=0$ in (2.4), and the second
rank tensor $\bar{h}_{rad}^{\alpha\beta}$ (especially the non-vanishing space-
space components $\bar{h}_{rad}^{ij}$ of the tensor in (3.38)), respectively.
Moreover in the LGR the second rank tensor $\bar{h}_{rad}^{ij}$ does not
include $(\theta,\phi)$ dependences of $A_{M}^{i}=-\frac{c}{4}\bar{h}^{0i}$ in
(3.28) in the MLGR, which are related with those of the GW radiation intensity
profile having a prolate ellipsoid geometry in Figure 2.

##  IV Conclusions

In summary, we have formulated the MLGR where the mass scalar potential has
the additional dynamical DOF. In the static gravitational field limit of the
MLGR, the mass scalar potential is consistent with the well-established
gravitational potential in the Newtonian gravity. Next we also have found that
the MLGR allows the mass vector potential which originates from the mass
current and is missing in the Newtonian gravity. As an application of the
MLGR, we have investigated the phenomenological aspects of the merging binary
compact objects which co-rotate in circular orbits about their common center
of mass with a constant angular frequency. To be specific, we have constructed
the mass scalar and mass vector potentials of the merging binary compact
objects at an observation point located far away from the compact objects. We
then have formulated the mass electric and mass magnetic fields, and the mass
Poyinting vector. Next we have found the GW radiation intensity profile having
a prolate ellipsoid geometry due to the merging binary compact objects source,
and then explicitly obtained the total GW radiation power due to the source.
We also have observed that, in no distorting limit of the binary compact
objects, they do not yield the total GW radiation power, which is consistent
with the second rank tensor approach in the LGR. One of the main points of
this paper is that, in the MLGR we have found the mathematically and
physically well defined spin-one graviton, distinct from the second rank
tensor approach in the LGR which could allow the spin-two graviton. Note that
the status of the graviton existence is still hypothetical. It will be
interesting to search for the spin-one graviton which could be related with
the well-established spin-one photon phenomenology such as the photoelectric
effect and Compton scattering for instance. Once this is done, the MLGR
algorithm associated with the graviton could give some progress impacts on the
realistic precision astrophysics including the merging binary compact objects
phenomenology which is detected in the experimental instruments. Assuming the
possibility of the massive spin-one graviton, we have observed that the
massive graviton would be interpreted as a GRB-like graviton, which could then
be regarded as an anti-graviton. This anti-graviton could be
phenomenologically a candidate for an intense radiation flare of the
GRB170817A related with the binary compact objects merger event GW170817.

###### Acknowledgements.

The author was supported by Basic Science Research Program through the
National Research Foundation of Korea funded by the Ministry of Education,
NRF-2019R1I1A1A01058449.

##  Appendix A EM wave radiation from rotating charge electric dipole moment

In this appendix we pedagogically investigate the charge electric dipole
moment jackson99 ; baylis02 ; griffiths99  which possesses charges $+Q$ and
$-Q$ rotating on the $x$-$y$ plane. Here the charge electric dipole moment is
not merging to yield no distortion. We assume that for simplicity the two
charges co-rotate in circular orbits of radius $a$ with an angular frequency
$\omega$, as shown in Figure 3. Note that the rotating charge electric dipole
can be thought of as the superposition of two oscillating charge electric
dipoles, one along the $x$-axis and the other along the $y$-axis griffiths99 .
The rotating charge vector $\vec{Q}(t)$ on the $x$-$y$ plane is given by

$\vec{Q}(t)=Q(\hat{x}\cos\omega t+\hat{y}\sin\omega t).$ (A.1)

For the charges on the $x$-axis and $y$-axis, exploiting (A.1) we obtain the
Green retarded charge scalar potential $\phi_{Q}^{I}$ and $\phi_{Q}^{II}$,
respectively

$\displaystyle\phi_{Q}^{I}(\vec{x},t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\epsilon_{0}}\left(\frac{Q\cos[\omega(t-R_{+}/c)]}{R_{+}}-\frac{Q\cos[\omega(t-R_{-}/c)]}{R_{-}}\right),$
$\displaystyle\phi_{Q}^{II}(\vec{x},t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\epsilon_{0}}\left(\frac{Q\sin[\omega(t-\bar{R}_{+}/c)]}{\bar{R}_{+}}-\frac{Q\sin[\omega(t-\bar{R}_{-}/c)]}{\bar{R}_{-}}\right),$
(A.2)

where $R_{\pm}$ and $\bar{R}_{\pm}$ are given by (3.22). Here the subscript
$Q$ stands for the charge related quantities. Following the technical
procedures in Section 3, we find the total contributions to the charge scalar
potentials $\phi_{Q}=\phi_{Q}^{I}+\phi_{Q}^{II}$

$\phi_{Q}(\vec{x},t)=-\frac{Q\omega
a}{2\pi\epsilon_{0}}\left(\frac{\sin\theta}{cr}\right)\left(\cos\phi\sin\left[\omega\left(t-\frac{r}{c}\right)\right]-\sin\phi\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\right),$
(A.3)

where we have used approximations $a\ll r$ and
$\frac{1}{r}\ll\frac{\omega}{c}$, implying that the wavelength
$\lambda\left(=\frac{2\pi c}{\omega}\right)$ of the EM wave is much smaller
than $r$ to yield $\lambda\ll r$.

Figure 3: The schematic diagram for a rotating charge electric dipole moment
consisting of charges $+Q$ and $-Q$ which co-rotate in circular orbits of
radius $a$, with an angular frequency $\omega$.

Making use of (A.1) for the charges co-rotating on the $x$-$y$ plane, we find
the charge current

$\vec{I}_{Q}(t)=-Q\omega(\hat{x}\sin\omega t-\hat{y}\cos\omega t).$ (A.4)

Next we find the retarded charge vector potentials $\vec{A}_{Q}^{I}$ and
$\vec{A}_{Q}^{II}$ for the charge currents on the $x$-axis and the $y$-axis,
respectively

$\displaystyle\vec{A}_{Q}^{I}(\vec{x},t)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}Q\omega}{4\pi}\left(-\int_{-a}^{0}\frac{\sin[\omega(t-R_{-}^{\prime}/c)]}{R_{-}^{\prime}}dx^{\prime}-\int_{0}^{a}\frac{\sin[\omega(t-R_{+}^{\prime}/c)]}{R_{+}^{\prime}}dx^{\prime}\right)\hat{x},$
$\displaystyle\vec{A}_{Q}^{II}(\vec{x},t)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}Q\omega}{4\pi}\left(\int_{-a}^{0}\frac{\cos[\omega(t-\bar{R}_{-}^{\prime}/c)]}{\bar{R}_{-}^{\prime}}dy^{\prime}+\int_{0}^{a}\frac{\cos[\omega(t-\bar{R}_{+}^{\prime}/c)]}{\bar{R}_{+}^{\prime}}dy^{\prime}\right)\hat{y},$
(A.5)

where $R_{\pm}^{\prime}$ and $\bar{R}_{\pm}^{\prime}$ are given by (3.26).
Here we have used the approximations $a\ll r$ and
$\frac{1}{r}\ll\frac{\omega}{c}$. Note that in (A.2) and (A.5) we have used
the charges $(+Q,-Q)$, distinct form the binary system of compact objects
where we have exploited the masses $(M,M)$ in (3.21) and (3.25). Exploiting
the charge vector potentials in (A.5), we obtain the total retarded charge
vector potential $\vec{A}_{Q}=\vec{A}_{Q}^{I}+\vec{A}_{Q}^{II}$

$\displaystyle\vec{A}_{Q}(\vec{x},t)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}Q\omega a}{2\pi
r}\left[-\left(\sin\left[\omega\left(t-\frac{r}{c}\right)\right]+\frac{\omega
a}{2c}\sin\theta\cos\phi\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\right)\hat{x},\right.$
(A.6)
$\displaystyle\left.+\left(\cos\left[\omega\left(t-\frac{r}{c}\right)\right]-\frac{\omega
a}{2c}\sin\theta\sin\phi\sin\left[\omega\left(t-\frac{r}{c}\right)\right]\right)\hat{y}\right].$

Now using (A.3) and (A.6), we find the charge electric field
$\vec{E}_{Q}\equiv-\nabla\phi_{Q}-\frac{\partial\vec{A}_{Q}}{\partial
t}=E_{Q}^{\theta}\hat{\theta}+E_{Q}^{\phi}\hat{\phi}$ and charge magnetic one
$\vec{B}_{Q}\equiv\nabla\times\vec{A}_{Q}=B_{Q}^{\theta}\hat{\theta}+B_{Q}^{\phi}\hat{\phi}$

$\displaystyle E_{Q}^{\theta}$ $\displaystyle=$ $\displaystyle
cB_{Q}^{\phi}=\frac{\mu_{0}Q\omega^{2}a}{2\pi
r}\cos\theta\left(\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\cos\phi+\sin\left[\omega\left(t-\frac{r}{c}\right)\right]\sin\phi\right),$
$\displaystyle E_{Q}^{\phi}$ $\displaystyle=$ $\displaystyle-
cB_{Q}^{\theta}=\frac{\mu_{0}Q\omega^{2}a}{2\pi
r}\left(-\cos\left[\omega\left(t-\frac{r}{c}\right)\right]\sin\phi+\sin\left[\omega\left(t-\frac{r}{c}\right)\right]\cos\phi\right),$
(A.7)

where we have included the relevant leading terms. From (A.7), we obtain
$\vec{B}_{Q}=\frac{1}{c}(\hat{r}\times\vec{E}_{Q})$. Note that $\vec{E}_{Q}$
does not have the $E_{Q}^{r}$ component, different form the $\vec{E}_{M}$ of
the binary compact objects which possesses the $E_{M}^{r}$ component in
(3.29).

Next we formulate the charge Poyinting vector in terms of $\vec{E}_{Q}$ and
$\vec{B}_{Q}$ in (A.7)

$\vec{S}_{Q}\equiv\frac{1}{\mu_{0}}(\vec{E}_{Q}\times\vec{B}_{Q})=\frac{1}{\mu_{0}c}(E_{Q})^{2}\hat{r}$
(A.8)

where

$(E_{Q})^{2}=(E_{Q}^{\theta})^{2}+(E_{Q}^{\phi})^{2}=\left(\frac{\mu_{0}Q\omega^{2}a}{2\pi
r}\right)^{2}\left(1-\sin^{2}\theta\cos^{2}\left[\omega\left(t-\frac{r}{c}\right)-\phi\right]\right).$
(A.9)

Next, using (A.8) and (A.9) and averaging in time over a complete cycle, we
arrive at the EM radiation intensity, namely the EM radiation power per
surface $\Sigma$

$\frac{dP_{Q}}{d\Sigma}\equiv\langle\vec{S}_{Q}\rangle=\frac{1}{\mu_{0}c}\left(\frac{\mu_{0}Q\omega^{2}a}{2\pi
r}\right)^{2}\left(1-\frac{1}{2}\sin^{2}\theta\right)\hat{r}.$ (A.10)

Note that the total EM radiation power constructed by integrating
$\langle\vec{S}_{Q}\rangle$ over a sphere $\Sigma_{r}$ of radius $r$ is given
by

$P_{Q}\equiv\int\langle\vec{S}_{Q}\rangle\cdot
d\vec{\sigma}=\frac{2\mu_{0}Q^{2}\omega^{4}a^{2}}{3\pi c},$ (A.11)

where $d\vec{\sigma}$ is an area element vector perpendicular to the surface
$\Sigma_{r}$. Note that the total EM radiation power $P_{Q}$ in (A.11) for the
rotating charge electric dipole moment on the $x$-$y$ plane is twice that for
the oscillating charge electric dipole moment along the $x$-direction,
assuming the same time dependence possessing an angular frequency $\omega$ of
these rotating and oscillating cases.

## References

  * (1) R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
  * (2) M.P. Hobson, G. Efstathiou and A.N. Lasenby, General Relativity: An Introduction for Physicists (University of Chicago Press, Chicago, 2006).
  * (3) M. Fierz and W. Pauli, Relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173, 211 (1939).
  * (4) J. Aasi et al. [LIGO Scientific Collaboration], Advanced LIGO, Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547.
  * (5) D. Reitze et al., Cosmic explorer: the U.S. contribution to gravitational-wave astronomy beyond LIGO, Bull. Am. Astron. Soc. 51, 35 (2019), arXiv:1907.04833.
  * (6) S. Jana, S.J. Kapadia, T. Venumadhav and P. Ajith, Cosmography using strongly lensed gravitational waves from binary black holes, Phys. Rev. Lett. 130, 261401 (2023), arXiv:2211.12212.
  * (7) R.W. Klebesadel, I.B. Strong and R.A. Olson, Observations of gamma-ray bursts of cosmic origin, Astrophys. J. Lett. 182, L85 (1973).
  * (8) D. Svinkin et al., A bright gamma-ray flare interpreted as a giant magnetar flare in NGC 253, Nature 589, 211 (2021), arXiv:2101.05104.
  * (9) S.T. Hong, Dirac type relativistic quantum mechanics for massive photons, Nucl. Phys. B 980, 115852 (2022), arXiv:2108.07299.
  * (10) R.A. Hulse and J.H. Taylor, Discovery of a pulsar in a binary system, Astrophys. J. Lett. 195, L51 (1975).
  * (11) B.P. Abbott et al. [LIGO Scientific and Virgo Collaborations], GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119, 161101 (2017), arXiv:1710.05832.
  * (12) B.P. Abbott et al. [LIGO Scientific and Virgo and Fermi-GBM and INTEGRAL Collaborations], Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017), arXiv:1710.05834.
  * (13) A. Goldstein et al., An ordinary short gamma-ray burst with extraordinary implications: fermi-GBM detection of GRB 170817A, Astrophys. J. Lett. 848, L14 (2017), arXiv:1710.05446.
  * (14) V.M. Kaspi and A. Beloborodov, Magnetars, Ann. Rev. Astron. Astrophys. 55, 261 (2017), arXiv:1703.00068.
  * (15) R. Abbott et al. [LIGO Scientific and VIRGO and KAGRA Collaborations], Search for gravitational-wave transients associated with magnetar bursts in Advanced LIGO and Advanced Virgo data from the third observing run, (2022), arXiv:2210.10931.
  * (16) H. Goldstein, Classical Mechanics (Addison-Wesley, London, 1980).
  * (17) S.T. Thornton and J.B. Marion, Classical Dynamics of Particles and Systems (Thomson, Belmont, 2004).
  * (18) N.W. Ashcroft and N.D. Mermin, Solid State Physics (Brooks/Cole, London, 1976).
  * (19) S.T. Hong, Photon intrinsic frequency and size in stringy photon model, Nucl. Phys. B 976, 115720 (2022), arXiv:2111.11852.
  * (20) J.D. Jackson, Classical Electrodynamics (John Wiley & Sons, Danvers, 1999).
  * (21) W.E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhäuser, Boston, 2002).
  * (22) D.J. Griffiths, Introduction to Electrodynamics (Prentice Hall, Upper Saddle River, 1999).


# Interpretation of the power spectrum of the quiet Sun photospheric
turbulence

Itzhak Goldman1,2
1 Department of Physics, Afeka College, Tel Aviv, Israel
2 Department of Astrophysics, Tel Aviv University, Tel Aviv, Israel
<EMAIL_ADDRESS>
###### Abstract

Observational power spectra of the photospheric magnetic field turbulence, of
the quiet-sun, were presented in a recent paper by Abramenko & Yurchyshyn.
Here I focus on the power spectrum derived from the observations of the Near
InfraRed Imaging Spectrapolarimeter (NIRIS) operating at the Goode Solar
Telescope. The latter exhibits a transition from a power law with index $-1.2$
to a steeper power law with index $-2.2$, for smaller spatial scales. The
present paper presents an interpretation of this change. Furthermore, this
interpretation provides an estimate for the effective width of the turbulent
layer probed by the observations. The latter turns out to be practically equal
to the depth of the photosphere.

###### keywords:

turbulence-solar photosphere- magnetic field

††pubyear: 2020††pagerange: Interpretation of the power spectrum of the quiet
Sun photospheric turbulence–References

##  1 Introduction

Quite recently, Abramenko & Yurchyshyn (2020) measured the line-of-sight
magnetic fields and derived magnetic power spectra of the quiet sun
photosphere. The data was obtained from magnetograms of the Helioseismic and
Magnetic Imager (HMI) on-board the Solar Dynamic Observatory (SDO), and the
Near Infrared Imaging Spectrapolarimeter (NIRIS) operating at the Goode Solar
Telescope (GST) of the Big Bear Solar Observatory(BBSO). The authors computed
1-dimensional power spectra of the observed line-of-sight turbulent magnetic
field. For data from both observatories, the power spectra inertial range were
considerably shallower than the Kolmogorov spectrum characterized by an index
of $-5/3$.(Kolmogorov (1941)).

An interesting feature is revealed in the NIRIS power spectrum (Abramenko &
Yurchyshyn (2020), Fig. 6) derived from measurements using the Fe I 15650Å.
While the inertial range on spatial scales larger than ($0.8\div 0.9$) Mm has
a logarithmic slope of $-1.2$, the logarithmic slope on smaller scales is
$-2.2$. The NIRIS data extends down to spatial scale of $\sim 0.3$ Mm. The
above transition is not present in the HMI data, as the HMI resolution is
about 2.4 Mm.

The magnetometer data originate from an integral along the line-of-sight of
the emitted near infra-red radiation. The Zeeman splitting of emission lines
is used to obtain the line-of-sight magnetic field. For more details on such
measurements see Abramenko et al. (2001) and references therein. The focus of
the present paper is on the two logarithmic slopes in the power spectrum of
the NIRIS data.

It is worth noting that several authors addressed the issue of power spectra
of quantities which are the result of integration along the line-of-sight (see
e.g. Stutzki et al. (1998), Goldman (2000), Lazarian & Pogosyan (2000),
Miville-Deschênes, Levrier, & Falgarone (2003)). They concluded that when the
lateral spatial scale is much smaller than the depth of the layer, the
logarithmic slope steepens exactly by $-1$ compared to its value when the
lateral scale is much larger than the depth.

This behavior was indeed found in observational power spectra of Galactic and
extra Galactic turbulence (e.g. Elmegreen, Kim, & Staveley-Smith (2001),
Miville-Deschênes et al. (2003), Block et al. (2010) and Contini & Goldman
(2011)).

Here, the same turbulence phenomenon is addressed in the context of the solar
photosphere, where the measured magnetic field follows from the sum of the
infra-red radiation emanating from various depths.

In section 2 the theoretical 1-D power spectrum of a, line-of-sight integrated
data, is explicitly derived.

In section 3 the asymptotic behavior of the power spectrum in the small and
large wavenumber limits is obtained analytically. In section 4., the numerical
results for the specific case under consideration here, are obtained. One
interesting implication is an estimate of effective width of the photosspheric
turbulent layer that is probed by the detector. Conclusions are presented in
section 5.

##  2  The 1-D power spectrum of line-of-sight integrated data

We are interested in the power spectrum of a quantity $n(x,D)$ (where x is
straight line in a lateral direction) which is an integral along the line-of
-sight $z$ of a underlying physical quantity, $f(x,z)$.

$n(x,D)=\int_{0}^{D}f(x,z)dz$$$ (1)

with $D$ denoting the effective width of the region that contributes to the
observed $n(x,D)$. It is termed "effective" because it is tacitly assumed in
equation (1) that the weight of $f(x,z)$ is independent of $z$ . An ideal case
is when the observed quantity is some radiation intensity emanating from a
optically thin slab of width $D$. It is also "effective" because in real
situations $D$ can differ for different $x$ values.

We are interested in the 1-dimensional power spectrum of $n(x,D)$ which
depends also on the value of the depth $D$.

$E(k_{x},D)=\int_{-\infty}^{\infty}e^{ik_{x}x}C_{n}(x,D)dx$ (2)

with $C_{n}(x,D)$ being the 2-point autocorrelation of the fluctuating
$n(x,D)$ ( mean of $n(x,D)=0$).

$\displaystyle C_{n}(x,D)=<n(x^{\prime},D)n(x^{\prime}+x,D)>=$ (3)
$\displaystyle\int_{0}^{D}\int_{0}^{D}<f(x^{\prime},z^{\prime})(f(x^{\prime}+x,z)>dzdz^{\prime}$

with the $<>$ brackets denoting ensemble average. Assuming isotropy and
homogeneity the 2-point autocorrelation of $f(x,z)$ is
$C_{f}(x,z-z^{\prime})=<f(x^{\prime},z^{\prime})(f(x^{\prime}+x,z)>$.

So that

$\displaystyle
C_{n}(x,D)=\int_{0}^{D}\int_{0}^{D}C_{f}(x,z-z^{\prime})dzdz^{\prime}=$ (4)
$\displaystyle\int_{0}^{D}\int_{0}^{D}\left(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}P_{2}(k_{x},k_{z})e^{-ik_{x}x-ik_{z}(z-z^{\prime})}dk_{x}dk_{z}\right)dzdz^{\prime}$

where $P_{2}(k_{x}.k_{y})$ is the 2-dimensional power spectrum. Interchanging
the integration order yields,

$\displaystyle
C_{n}(x,D)=D^{2}\int_{-\infty}^{\infty}e^{-ik_{x}x}\int_{-\infty}^{\infty}\left(\frac{\sin(k_{z}D/2)}{k_{z}D/2}\right)^{2}P_{2}(k_{x},k_{z})$
(5) $\displaystyle dk_{z}dk_{x}$

Consider a 2-dimensional power spectrum that is isotropic, homogeneous and in
the form of a power law

$P_{2}(k_{x},k_{y})=A\left(k_{x}^{2}+k_{z}^{2}\right)^{-(m+1)/2}$ (6)

where A is a constant and $m$ is the index of the 1-D power spectrum.

Combining equations (2, 5, 6) one easily identifies

$\displaystyle
E(k_{x},D)=M\int_{0}^{\infty}\left(k_{x}^{2}+k_{z}^{2}\right)^{-(m+1)/2}\left(\frac{\sin(k_{z}D/2)}{k_{z}D/2}\right)^{2}dk_{z}$
(7)

with M a constant. Using a variable $\eta=k_{z}D/2$, equation (7) takes the
form

$\displaystyle
E(k_{x},D)=N\int_{0}^{\infty}\left((k_{x}D/2)^{2}+\eta^{2}\right)^{-(m+1)/2}\left(\frac{\sin(\eta}{\eta}\right)^{2}d\eta$
(8)

with $N$ a constant.

##  3 analytical results

It follows from equation (8) that for given values of $m$ and $N$,
$E(k_{x},D)$ is a function of the dimensionless product $k_{x}D/2$. We are
interested in the shape of the power spectrum and not its absolute
normalization so the actual value of $N$ does not matter. In what follows we
show analytically that the asymptotic shapes of the power spectrum are

$\displaystyle E(k_{x},D)\propto(k_{x}D/2)^{-m};\ \ k_{x}D/2<<1$ (9)
$\displaystyle E(k_{x},D)\propto(k_{x}D/2)^{-m-1};\ \ k_{x}D/2>>1$

To that end note that the $(\sin(\eta)/\eta)^{2}$ term implies that the
integration upper limit is $\sim\pi/2$. Therefore,

$\displaystyle E(k_{x},D)\sim
N\int_{0}^{\pi/2}\left((k_{x}D/2)^{2}+\eta^{2}\right)^{-(m+1)/2}d\eta$ (10)
$\displaystyle=N(k_{x}D/2)^{-m}\int_{0}^{\pi/(k_{x}D)}\left((1+\mu^{2}\right)^{-(m+1)/2}d\mu$

with $\mu=\eta/(k_{x}D/2)$.

In the limit $k_{x}D/2<<1$ the upper integration limit tends to infinity and
the integral is independent of $k_{x}D/2$. In the limit $k_{x}D/2>>1$ the
integral is proportional to $(k_{x}D/2)^{-1}$.

The transition occurs at a transition wavenumber, $k_{t}$, such that
$k_{t}D/2=O(1)$. The exact value of $k_{t}D/2$ depends on $m$ and is required
in order to find the value of $D$. To get the exact value, for the present
case, $m=1.2$, equation (8) is solved numerically, using WolframMathematica.

Figure 1:  Thick red curve: $E(k_{x}D/2)$ in arbitrary units as function of
$k_{x}D/2$. blue line : slope of -1.2. green line: slope of -2.2. yellow line
: slope of -1.7.

##  4 numerical Results

The power spectrum $E(k_{x}D/2)$ for $m=1.2$, obtained by the numerical
solution of equation (8), is displayed in Fig. 1.

It is seen that for $k_{x}D/2<<1$ the power spectrum has a logarithmic slope
of $-1.2$ while for $k_{x}D/2>>1$ the logarithmic slope is $-2.2$, as expected
from the analytical analysis presented above.

A line with a logarithmic slope of $-1.7$, the mean of the former two, which
is tangent to the curve defines the "point of transition". One should note
that the transition is a smooth one, and not a sharp break at a point.

From Fig. 1 it follows that the transition wavenumber, $k_{t}$, satisfies

$k_{t}D/2=2.25\pm 0.05$ (11)

So a given observational value of $k_{t}$ determines $D$. Expressing $k_{t}$by
$x_{t}=2\pi/k_{t}$ , the observational transition spatial scale, leads to

$D=(2.25\pm 0.05)\frac{x_{t}}{\pi}$ (12)

Abramenko & Yurchyshyn (2020)found $x_{t}=0.85\pm 0.05$ Mm. Thus,

$D=(0.57\pm 0.05){\rm\ Mm}$

##  5 Concluding Remarks

The paper by Abramenko & Yurchyshyn (2020)provided a very neat power spectrum
of the photosphere magnetic field of the quiet Sun. The high spatial
resolution of the NIRIS data enabled a power spectrum down to spatial scales
as small as $0.3$ Mm. This power spectrum exhibits what appears as two
inertial ranges with logarithmic slopes of $-1.2$ and $-2.2$, for a spatial
scale larger/smaller than $(0.85\pm 0.05)$ Mm, respectively.

The logarithmic slope of -1.2 corresponds to a 3- dimensional power spectrum
with a logarithmic slope of -3.2. This is significantly shallower that the
-3.67 value corresponding to Kolmogorov turbulence. The very interesting issue
of the origin of such a power spectrum was not addressed here.

The present paper demonstrates that two inertial ranges actually correspond to
a turbulence with a single inertial range with logarithmic slope equaling
-1.2. The two ranges are manifestation of the fact that the observed infra-red
emission is an integral along the line of sight. Since the value of $D$ is
within the span of the lateral spatial scales probed, the transition shows up.

Interestingly, the value of the derived effective width is comparable to the
width of the photosphere itself. This value of $D$ implied by the NIRIS power
spectrum of Abramenko & Yurchyshyn (2020) is an observational constraint to be
faced with theoretical models of near IR lines in the quiet solar photosphere.

In this context, a numerical model by Rueedi et al. (1998) yielded that the
FeI15648 Å line is formed in the lower part of the photosphere and extends
over a vertical span of about $0.1$ Mm. Their work refers to a sun spot and
not to the quiet sun, and also assumes an absence of macro velocity
turbulence. The situation here is different.

## Acknowledgments

I thank Prof. Valentina Abramenko, the referee, for very constructive remarks
that helped to improve and clarify the paper. This research has been supported
by the Afeka College Research Fund.

## data availability

The article presents a theoretical research. It does apply the observational
results of Abramenko & Yurchyshyn (2020) to the derived theoretical results.

## References

  * Abramenko & Yurchyshyn (2020) Abramenko V. I., Yurchyshyn V. B., 2020, MNRAS.tmp, doi:10.1093/mnras/staa2427
  * Abramenko et al. (2001) Abramenko V., Yurchyshyn V., Wang H., Goode P. R., 2001, SoPh, 201, 225
  * Block et al. (2010) Block D. L., Puerari I., Elmegreen B. G., Bournaud F., 2010, ApJ, 718, L1
  * Contini & Goldman (2011) Contini M., Goldman I., 2011, MNRAS, 411, 792
  * Elmegreen, Kim, & Staveley-Smith (2001) Elmegreen B. G., Kim S., Staveley-Smith L., 2001, ApJ, 548, 749
  * Kolmogorov (1941) Kolmogorov A., 1941, DoSSR, 30, 301 edu
  * Goldman (2000) Goldman I., 2000, ApJ, 541, 701
  * Lazarian & Pogosyan (2000) Lazarian A., Pogosyan D., 2000, ApJ, 537, 720
  * Miville-Deschênes, Levrier, & Falgarone (2003) Miville-Deschênes M.-A., Levrier F., Falgarone E., 2003, ApJ, 593, 831
  * Miville-Deschênes et al. (2003) Miville-Deschênes M.-A., Joncas G., Falgarone E., Boulanger F., 2003, A&A, 411, 109
  * Rueedi et al. (1998) Rueedi I., Solanki S. K., Keller C. U., Frutiger C., 1998, A&A, 338, 1089
  * Stutzki et al. (1998) Stutzki J., Bensch F., Heithausen A., Ossenkopf V., Zielinsky M., 1998, A&A, 336, 697