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European Journal of Applied Sciences – Vol. 9, No. 5
Publication Date: October 25, 2021
DOI:10.14738/aivp.95.10956. Gu, Y. (2021). The Correlation between Rotational Gravity and Magnetic Field of Celestial Body. European Journal of Applied
Sciences, 9(5). 217-224.
Services for Science and Education – United Kingdom
The Correlation between Rotational Gravity and Magnetic Field of
Celestial Body
Ying-Qiu Gu
School of Mathematical Science, Fudan University, Shanghai 200433, China
ABSTRACT
The magnetic field of the earth plays an important role in the ecosystem, and the
magnetic field of celestial bodies is also important in the formation of cosmic large- scale structures, but the origin and evolution of the celestial magnetic field is still
an unresolved mystery. Many hypotheses to explain the origin have been proposed,
but there are some insurmountable difficulties for each one. At present, the theory
widely accepted in scientific society is the dynamo model, it says that, the movement
of magnetofluid inside celestial bodies, which can overcome the Ohmic dissipative
effect and generate persistent weak electric current and macroscopic magnetic
field. However, this model needs an initial seed magnetic field, and the true values
of many physical parameters inside the celestial body are difficult to obtain, and
there is no stable solution to the large range of fluid motion. These are all difficulties
for the dynamo model. Furthermore, it is difficult for the dynamo to explain the
correlation between the dipole magnetic field and angular momentum of a celestial
body. In this paper, by calculating the interaction between spin of particles and
gravity of celestial body according to Clifford algebra, we find that a rotational
celestial body provides a field Ω! for spins, which is similar to the magnetic field of
a dipole, and the spins of charged particles within the celestial body are arranged
along the flux line of Ω! , then a macroscopic magnetic field is induced. The
calculation shows that the strength of Ω! is proportional to the angular momentum
of the celestial body, which explains the correlation between the magnetic intensity
and angular momentum. The results of this paper suggest that further study of the
effects of internal variables such as density, velocity, pressure and temperature of
a celestial body on Ω! may provide some new insights into the origin and evolution
of celestial magnetic field.
Keywords: Earth magnetic field, celestial magnetic field, magnetic dipole, Clifford algebra,
Dirac equation, spin-gravity coupling potential, curved space-time.
INTRODUCTION
The earth magnetic field is of great significance to the ecosystem. Geomagnetism has the
function of navigation and location, and prevents the attack of solar wind against earth. The
science of geomagnetism is developed from its practical applications. From the invention of the
compass recorded by the Chinese in 250 BC to the publication of the founding book on
geomagnetism by the British doctor Gilbert in 1600, the science of geomagnetism has gone
through an early stage of 1800 years. Queen Elizabeth's private doctor, Gilbert, was the first
man to think systematically about geomagnetic phenomenon. He organized the scattered
measurement data of geomagnetic field at different points under a unified framework, and
concluded that the earth is a big magnet.
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On the origin of geomagnetism, a hundred schools of thought contend, and more than a dozen
different hypotheses have been put forward. However, there is no convincing explanation for
the origin of geomagnetism, so it is listed by Einstein as one of the five major physics problems.
Gilbert's hypothesis that the Earth is a permanent magnet, for example, faces a serious
challenge to the Curie point temperature of the material: below the depth of 20 to 30 kilometers
of the earth's crust, the temperature has exceeded the Curie point of most materials on the
earth, so the material here cannot remain enough residual magnetism. The magnetism of the
thin crustal material is far from enough to generate the observed geomagnetic field. Other
hypotheses of geomagnetism origin, such as rotating magnetic effect, rotating charge effect, Hall
effect, piezomagnetism effect and so on, are also denied due to the too small order of magnitude.
The widely accepted theory of the origin for the earth's magnetic field at present is the
geodynamo. Its basic idea is that the conductive fluid of the outer core inside the earth is
subjected to convective motion under the drive of various energy sources, and a magnetic field
is generated by the current corresponding to the convection. That is, a process in which the
driving energy is converted into the kinetic energy of the fluid, and then the kinetic energy is
converted into the magnetic energy. If the converted magnetic energy can resist Ohmic
dissipation, the magnetic field can be maintained by convective motion.
With the development of computer numerical technology, the dynamo model for the earth's
magnetic field has been fully developed, and a large number of numerical simulations have been
carried out. In references [1, 2], the first three-dimensional self-consistent numerical solution
of geomagneto-hydrodynamic equation with time is calculated. The equation describes the
generation of thermal convection and magnetic field in a rapidly rotating spherical fluid shell
with a solid conductive core.
As a rough simulation of the geodynamo, it is a self-sustaining supercritical generator, which
maintains three magnetic field diffusion times for about 40000 years. The maximum velocity of
the fluid in the outer core can reach 0.4 cm/s, and sometimes the magnetic field can reach 560
Gauss. Magnetic energy is usually 4000 times larger than the convective energy that maintains
it. In this system in which the fluid interacts with the magnetic field, the toroidal magnetic field
can be generated by the poloidal magnetic field(that is, the ω effect), and the poloidal magnetic
field can also be generated by the toroidal magnetic field(that is, the α effect). A variety of
dynamo models can be combined by these two basic effects.
Although great progress has been made in the numerical simulation of geodynamo, the origin
of geomagnetic field is far from being solved, such as the origin of seed magnetic field, many
properties and states of the earth's interior are also assumed, and the actual measurement is
less. The inaccessibility of the earth core enables us to rely on the observation data in the
vicinity of the surface and the space, and use the results of the laboratory simulation to infer
and guess the state and process inside the earth.
Astronomical observation shows that the existence of large-scale regular magnetic field in
rotating celestial bodies is a common phenomenon. In the solar system, the sun, Jupiter, Saturn,
Uranus, Neptune and so on, all have strong dipolar moment magnetic fields. The magnetic fields
of other distant stars, such as white dwarfs, pulsars and so on, are even greater[3, 4].
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Gu, Y. (2021). The Correlation between Rotational Gravity and Magnetic Field of Celestial Body. European Journal of Applied Sciences, 9(5). 217-
224.
URL: http://dx.doi.org/10.14738/aivp.95.10956
Even larger cosmic bodies, such as galaxy, galaxy cluster and so on, also have large-scale regular
magnetic field distribution[5, 6]. The origin of galactic magnetic field and extragalactic magnetic
field is also one of the most challenging problems in modern Astrophysics. Magnetic fields are
detected in all types of galaxy and cluster as long as appropriate observations are made. In
addition, the magnetic field also exists in the distant galaxy with cosmic redshift. It is now
widely accepted that the large-scale magnetic field in the disk galaxy is also amplified and
maintained by the αω-dynamo principle, in which the new magnetic field is regenerated
through the combination of rotation and spiral turbulence. On the contrary, the intensity of
magnetic fields in non-rotating or slow-rotating systems, such as elliptical systems and clusters,
is relative much smaller.
But the galactic dynamo model itself is incomplete because it also does not explain the origin of
the seed magnetic field used to start the dynamo. In addition, the time scale of magnetic field
amplification in the standardαω-dynamo model is too long to explain the magnetic field
intensity observed in very young galaxies.
By analysis of observational data of the magnetic field for a large number of celestial bodies, it
is found that the magnetic dipole moment of a celestial body has a strong correlation with its
angular momentum, and the so-called Schuster-Wilson-Blackett relation approximately holds
on a wide range of orders of magnitude [4, 7, 8, 9, 10].
�
� = �√�
2� , (1)
in which � and L are magnetic moment and angular momentum of the celestial body
respectively, and � ∈ O(1) is a dimensionless number. The physical reason for this relationship
was not specified at that time, so the result was not generally accepted.
In addition, in the analysis of [4], it is found that there is a significant positive correlation
between log� and log L for cold stars. But such correlation between hot stars is much smaller.
For the same kind of hot stars, log� and log L are even negatively correlated. In subsamples of
the solar system, the correlation is basically the same as the slope of the cold star. On a large
scale, log� and log L for different types of objects remain positively correlated(see Figure 9 in
[4]).
The main theoretical explanations of Schuster-Wilson-Blackett relation are as follows: in [12],
under Thoms-Fermi approximation the calculation shows that every ultra-dense material
element in a gravitational field obtains a very small positive charge. But the celestial bodies as
a whole are electrically neutral, because the negative charges are concentrated on their surface.
In order of magnitude, the positive volume charge is very small (only 10!"#e), but this is
sufficient to explain the occurrence of the celestial magnetic field, as well as the existence of
discrete spectra of the steady-state values of mass of stars and pulsars.
The explanation proposed in [10] is that, the function Φ$% = (�⁄�&) �'(�'($% satisfies the
Maxwell equations and can be used as a function to determine the electromagnetic properties
of a rotating star, where �'($% is the Riemann tensor, which determines the gravity field of the
object, �& is the gravitational radius, and η is a constant that must be determined by observation.
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The field Φ$% describes the observed correlation of magnetic moments with angular
momentum. It also explains the stability of the magnetic field of white dwarf and neutron stars
under the Ohmic dissipation.
Among many suggestions on the origin of the earth's magnetic field in [8], a relatively
satisfactory one is to make a slight modification to the law of electrodynamics. Electroneutral
materials are considered to be a mixture of a large number of positive and negative charges,
and their electrical and magnetic effects are usually balanced. If this balance is not precise, it
will have small residual effects, including gravity and the Earth's magnetic field. Under this
assumption, we can expect the moving matter to generate a magnetic field similar to the moving
current, and we expect that there is a certain relationship between the magnetic field produced
by the moving material and its gravitational action. In references [11], it is assumed that the
moving mass can directly produce electromagnetic induction, so the rotating star is equivalent
to a magnetic dipole.
NEW EXPLANATION FOR THE ORIGIN OF CELESTIAL MAGNETIC FIELD
Before expounding the author's point of view, let's review the concept of magnetic dipoles. The
magnetic dipole model is a very small planar current-carrying coil. Its magnetic moment is
defined as �⃗ = ��⃗, I is the current, S is the coil loop area, and the direction of �⃗ has a right-hand
spiral relationship with the current direction. The vector potential generated by the magnetic
dipole is given by,
�⃗ (�⃗) = )!
*+," (�⃗ × �⃗), (�- = 4� × 10!.(N A/ ⁄ )), (2)
�- is vacuum permeability, �⃗ is the position vector from the center of the dipole to the
measuring point. The magnetic field intensity of the magnetic dipole is calculated by,
�C⃗ = ∇ × �⃗ = �-
4��0 (3(�⃗ ∙ �̂)�̂− �⃗) +
2�-�⃗
3 �0(�⃗), (3)
in which �0(�⃗) is the Dirac-δ. Since the Dirac-δ= 0 for any �⃗ ≠ 0, the Dirac-δ vanishes when
computing the field strength in the region r > 0. However, in quantum mechanics at the atomic
scale, this term will make an important contribution. The Dirac- δ function of the dipolar
magnetic field causes the atomic energy level to split, thus forming the hyperfine structure. In
astronomy, the hyperfine structure of hydrogen atoms gives a 21cm spectra, which is the
broadest electromagnetic radiation in the universe except 3K background radiation in the radio
wave range of electromagnetic radiation. In the spherical coordinate system, the magnetic flux
equation of (3) is as follows
��⃗
�� = �C⃗ ⇒
��
�� = 2� cos �
sin �
⇔ � = �sin/�. (4)
When there are multiple magnetic dipoles, according to the superposition principle, the total
magnetic field is the total vector sum of the magnetic field of each magnetic dipole. In order to
calculate the magnetic moment and magnetic field of a planet, according to the structure of the
planet, the ring or sphere can be selected as the magnetic dipole element, and then the whole
magnetic moment and magnetic field of the planet can be obtained by integral. The distribution
of magnetic fields outside the planet is very close to that produced by a single magnetic dipole.
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Gu, Y. (2021). The Correlation between Rotational Gravity and Magnetic Field of Celestial Body. European Journal of Applied Sciences, 9(5). 217-
224.
URL: http://dx.doi.org/10.14738/aivp.95.10956
We know that the properties of electrons and protons are described by quantum mechanics, so
strictly investigating the interaction between Dirac spinor and gravitational field may be the
key to unravel the secrets of celestial magnetic field. Denote the element of space-time by
�� = �)��) = �)��) = �2��2 = �2��2. (5)
In which the tetrad satisfies Clifford algebra Cl(1,3)
�2�3 + �3�2 = 2�23, �)�4 + �4�) = 2�)4, (6)
where �23 = diag(1, −1, −1, −1) is Lorentz metric. The relation between the tetrad coefficient
and the metric is given by
�) = � 2
)�2, �) = �) 2�2, �) 2� 3
) = �3
2, �) 2� 2
4 = �)
4, (7)
� 2
)� 3
)�23 = �)4, �) 2�4 3�23 = �)4. (8)
In [13, 14], by straightforward calculation we have the following results.
�, �2, �23 = �
2 �2356�56�7, �235 = ��2356�6�7, �-"/0 = −��7. (9)
�)�4 = �)4 + �)4, �)4�8 = �)�48 − �4�)8 + �)48. (10)
For Dirac equation in curved space-time without torsion, we have
�)(�∇) − ��))� = ��, ∇)� = (�) + Γ))�, (11)
in which the spinor connection is given by
Γ) ≡
1
4 �4�;)
4 = 1
4 �4�4;) = 1
4 �4n�)�4 − Γ)4
: �:o. (12)
The total spinor connection �)Γ) = Λ" ∪ Λ0. By (9), (10) and Clifford calculus, we have
�)Γ) = 1
4 �)�4�;)
4 = Υ)�) +
1
4Ω)�)�7, (13)
in which
Υ) = 1
2 � 2
4 (�) �4
2 − �4 �)
2), Ω: = 1
4 � 6
: � 2
)� 3
4 �) �4
;�2356�5;. (14)
Substituting it into (11) and multiplying the equation by �-, we get the Dirac equation in the
natural unit system,
�)�̂
)ф + �w
)Ω)ф = ��-ф, (15)
where �) is current operator, �̂
) is momentum operator and �w
) spin operator. They defined
respectively as
�) = diag(�), �y)), �̂
) = �n�) + Υ)o − ��), �w) = 1
2 diag(�), −�y)), (16)
where �) = h 2
) �2, �y) = h 2
) �y2 is the Pauli matrices in curved space-time. Υ) is Keller
connection. Professor Jaime Keller was the late former editor in chief of “Advances in Applied
Clifford Algebras”. He founded this journal and made outstanding contributions to promote the
research and application of Clifford algebra. The Gu-Nester potential Ω) is a new pseudo vector
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describing the interaction between spinor spin and gravity. For the LU decomposition of metric,
we have
Ω: = 1
4 �6235� 6
: � 2
<�35
)4
�<�)4, �23
)4 ≡
1
2 n� 2
) � 3
4 + � 2
4 � 3
) osign(� − �) = − �32
)4. (17)
In the Hamiltonian of a spinor we derived a spin-gravity coupling potential �w
)Ω). If the metric
can be orthogonalized, we have Ω) ≡ 0, and then the spin and gravity are decoupled.
If the gravitational field is generated by a rotating ball, the corresponding metric, like the Kerr
metric, cannot be diagonalized. At this time, the spin-gravity coupling term have non-zero
coupling effect. Similarly to the case of charged particles in a magnetic field, the spins of spinors
will be automatically arranged along the fluxes of Ω). If the spins of all charged particles are
arranged regularly along these fluxes, a macroscopic magnetic field will be induced. In order to
clarify whether this magnetic field is related to the magnetic field of celestial bodies, we
examine the flux of Ω) field of a rotating star. The metric produced by the rotating sphere is
similar to the Kerr metric, and in the asymptotically flat space-time we have the line element in
quasi-spherical coordinate system[15].
�� = �-√�(�� + ���) + √�(�"�� + �/���) + �0√�!" �sin���, (18)
��/ = �(�� + ���)/ − �(��/ + �/��/) − �!"�/sin/���/, (19)
in which (�, �, �) is just functions of (�, �). As � → ∞, we have
� = 1 − 2�
� , � →
4�
�
sin/�, � → 1 +
2�
� , (20)
where (m, L) are mass and angular momentum of the star respectively. For common stars and
planets we always have � ≫ � ≫ � . For example, we have � ≈ 3 km for the sun. For LU
decomposition of metric (19), the nonzero tetrad coefficients are given by
⎩
⎪
⎨
⎪
⎧�=
- = √�, �,
" = √�, �>
/ = �√�, �?
0 = � sin �
√� , �?
- = √��,
� -
= = 1
√�
, � "
, = 1
√�
, � /
> = 1
�√�
, � 0
? = √�
�sin�
, � 0
= = −√��
�sin� .
(21)
Substituting it into (17) we get
Ω: = � -
= � "
, � /
>� 0
?n0, �>�=?, −�,�=?, 0o
= (��/sin�)!"(0, �>(��), −�,(��), 0)
→
4�
�* (0, 2�cos�, sin�, 0). (22)
By (22) we find that, the intensity of Ω: is proportional to the angular momentum of the star,
that is to say, the absolute value of the spin-gravitational coupling potential of charged particles
is proportional to the angular momentum of the star. Now we check the flux line of Ω:. By (22)
we have
��)
�� = Ω) ⇒
��
�� = 2� cos �
sin �
⇔ � = �sin/�. (23)
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Gu, Y. (2021). The Correlation between Rotational Gravity and Magnetic Field of Celestial Body. European Journal of Applied Sciences, 9(5). 217-
224.
URL: http://dx.doi.org/10.14738/aivp.95.10956
(23) shows that, the flux lines of -ff and the magnetic fluxes (4) of the magnetic dipole (3)
coincide with each other. According to the above conclusions, we know that the spin-gravity
coupling potential of charged particles will certainly induce a macroscopic dipolar magnetic
field for the star, and it should be approximately in accordance with the Schuster-Wilson- Blackett relation (1).
So far, we have two more questions to explain for the magnetic fields of the star and planet: The
first one is how to understand that, the direction of the magnetic dipole of a planet always
deviates a little from the direction of angular momentum? The second is how to understand the
negative correlation between the magnetic dipoles and angular momentum of the same type of
hot stars (see figure 6, 7, 8 in [4]). In the above discussion, we only consider a simplified model
with concentrated parameters, that is, only the total mass m and total angular momentum L of
the star are considered, but the distribution of variables such as mass density, temperature, and
velocity are ignored. These factors obviously have significant influences on the intensity and
distribution of magnetic field of a star. For example, temperature reflects the speed of particle
motion, and high temperature will inevitably reduce the order of spin arrangement, and then
reduce the magnetic dipole intensity of a star, so the magnetic field of the star will be relatively
weakened with the increase of temperature. So a more detailed study should include physical
variables inside the star such as mass density, speed of fluid, temperature, and pressure. By
introducing these parameters, we will get a more accurate model for the magnetic field of
celestial body. Furthermore, the metric of a rotating celestial body is non-diagonal, which will
produce some dynamic effect. The precession of the planet magnetic dipole relative to the
rotational pole should be a relativity effect, so in order to clarify these effect we need more
detailed dynamic analysis. After considering these factors, we may be able to answer the above
two questions.
DISCUSSION AND CONCLUSION
The origin and evolution of celestial magnetic field is a complex and difficult problem.
Compared with the existing hypotheses and theories, the explanation proposed in this paper
seems to be more natural and reasonable, and may be closer to the deep essence. The rotating
planet provides a weak gravitational field for particle spin like the magnetic dipole magnetic
field, which is a somewhat unexpected discovery, and one has to admire the magic and subtlety
of creation. As Professor James Nester encouraged me when I studied the decomposition of
spinor connections (14), “There may be a deep wisdom that we haven't appreciated”. The spin- gravity coupling potential is equivalent to equip each particle with a pair of eyes of navigation
and location functions. The results of this paper suggest that further study of the effects of
internal parameters such as density, velocity, pressure and temperature of a celestial body on
Ω: field may play an important role in further understanding the magnetic field of celestial
bodies. But this is also a complex systematic project, which requires the joint efforts of multi- disciplinary colleagues.
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