Relativistic mass of gravitational field

: It is discovered that the gravitational field on the surface of a neutron star is with a relativistic mass density of 2.65 × 10 16 ~5.87 × 10 18 𝑘𝑔𝑚 −3 which can be larger than the mass density of the neutron stars ( ~10 17 𝑘𝑔𝑚 −3 ). And, the total relativistic mass of the gravitational field of the Sun is ~10 7 times the mass of the Sun. For different stars, the relativistic mass of the gravitational field is larger as the mass density of the star is larger. In the Milky Way, the total relativistic mass of the gravitational fields is much larger than the total mass of the stars, planets and gas. And, the relativistic mass density of the observed strongest magnetic field is 2.17 × 10 8 𝑘𝑔𝑚 −3 . This discovery should be a new frame to understand the fundamental problem of physics.

. The energy of a gravitational field can make a body moved with an energy of E = 1 2 2 and is the reason to make the neutrons degenerated to form the neuron star. Currently, it is thought that the concept of gravitational potential energy of a star is correct and useful. [1] The energy density, U = 2 8 , of a gravitational field on the surface of a star is determined with the gravitational acceleration and Newtonian universe gravitational constant G. [1,2] From E = m 2 , it is well known that a gravitational field is with a relativistic mass density = . Here, it is discovered that the relativistic mass density of the gravitational field on the surface of a neutron star is 2.65 × 10 16~5 .87 × 10 18 −3 which can be larger than the mass density of the neutron stars (~10 17 −3 ). [3,4] And, the total relativistic mass of the gravitational field of the Sun or a neutron star is ~10 7 times the mass of the Sun or is ~10 21 times that of the neutron star. For different stars, the relativistic mass of the gravitational field is larger as the mass density of the star is larger. And, the relativistic mass density of the observed strongest magnetic field is =

The relativistic mass density and total relativistic mass of a gravitational field
On the surface of the Earth, = 10m −2 , correspondently, = It is very significant and interesting that the relativistic mass density of the gravitational field of a neutron star can be larger than the mass density of the neutron stars.
For a star with the relativistic mass density is the Schwarzschild radius. It indicates that, as the radius of a star is less than 1 12 times the Schwarzschild radius, the relativistic mass density of the gravitational field on the surface of the star is larger than the mass density of the star.
From the energy density, the total relativistic mass of a gravitational field can be known.
As shown in Figure 1A, the energy density U = is not uniform. But, as the is very little, approximately, the average energy density can be determined with = (r) ≈ ( + ) . The energy density in dV can be treated as uniform. Therefore, in dV, the energy density can be taken approximately as U = 2 8 . In this sense, the energy density in the whole field can be described with a line as shown in Figure 2A. The energy in dV can be simply written as dE = 2 8 . Therefore, the total energy in the whole volume V = where r is the radius of the star, R is the radius of the gravitational field as shown in Figure 1A. Therefore, from E = ̅ and V =  is a proportional to 1 as shown in Figure 2B. From Figure 1, it can be clearly known that, as R is increased, the energy is correspondently increased. The total energy E = 2 18 2 is a proportional to 2 as shown in Figure 2C. The increased rate of the total energy is exponential. is a proportional to 1/R, thus, the total energy is at last a proportional to 2

Preprints
Now, it is believed that, the radius of a gravitational field is infinite. Therefore, in current theory, the relativistic mass of any gravitational field should be infinite large. However, that is not true.
It is noted that, the energy density, U = 2 8 , of a gravitational field is determined with the gravitational acceleration of the field. Thus, as the action of a gravitational field on another ones is not , the boundary of the gravitational field can be defined. From Newtonian orbital perturbation theory, we know, out of the Hill radius, the gravitational acceleration of a celestial body on others is not g = G 2 . [7,8,9] Therefore, the total energy of a Where is the Hill radius of the star or planet.
It is noted that, Equation (4) only is valid to describe a special case of the energy of a gravitational field, i.e., the total energy of the gravitational field inside the Hill sphere of the field. Therefore, the energy of the field out of the Hill sphere need be further studied elsewhere.
From Equation (4) we know, the total relativistic mass of a gravitational field is: From Equation (5), the relativistic mass of the field of the Sun, Earth and Neptune it is calculated as shown in the Table 1. It is noted that, the Hill radius of the Sun has not been known. Here, for the Sun, the is taken as the distance between the Sun and the Oort Cloud which now is thought to be the boundary of the Sun's gravitational influence. [10] Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 December 2021 But, it is clear, the distance for the action of the Sun on other body with g = G 2 is larger than that on the Oort cloud. So, the relativistic mass of the field of the Sun is larger than 4.3 × 10 37 kg.
The Table 1 shows: 1) the relativistic mass of the gravitational field of a celestial body could be larger or less than the mass of the body. 2) the relativistic mass of the gravitational field of the solar system is 10 7 times the total mass of the Sun and the planets in the system for that the mass of the Sun is larger than 90% of total mass of the bodies and gas in the solar system.
From Equation (5) For the stars with different mass density and with the mass of 1 Sun and the gravitational boundary of R=1 light year, the relativistic masses of the gravitational fields of them are shown in the Table 2:  There are almost 5.8 × 10 11 stars in the Milky Way. The Table 2 shows, the relativistic mass of the neutron star is almost 10 21 times the Sun. But, up to now only ∼2 × 10 3 neutron star have been observed. [14] Therefore, the relativistic mass of the field of the Sgr A* is larger than the sum of both the masses of the observed red giant stars, white dwarf stars and neutron stars and their fields in the Milky Way. Table 2 and Figure 3 and / ≈ 8.81 × 10 18 should indicate that, in a galaxy, the total mass of the celestial bodies and gas is much less than the total relativistic mass of the gravitational fields in the same galaxy.

Discussions and conclusions
The energy density of the gravitational field on the surface of a star was presented early in 1950. [2] The energy density of the gravitational field of a star is not uniform. Instead, it is varied with 1 4 as shown in Figure 2A.
Therefore, the average energy density of the field is needed to obtain the sum of the total energy in the whole volume of the field. But, the average energy density has not been known till know. In this work, the average energy density of the gravitational field is first studied.
From the average energy density of a gravitational field, in this work, the total energy of a whole gravitational field is first calculated. It is noted that the formula E = is not suitable to calculate the total energy of one whole gravitational field. If it should be valid for the total energy of the whole gravitational field, the total energy in a whole field should be less than the part energy in the part field. For example, in the Figure  That is clearly not true.
The observed strongest magnetic field is with the magnetic flux density of B ≈ 7 × 10 9 T. [18,19,20] From the energy density of the magnetic field U = it is noted that this is a very rough estimation: 1) the magnetic field is not uniform. B ≈ 7 × 10 9 T is not suitable for the whole magnetic field. 2) the volume of the magnetic field has not been accurately known. But, this estimation can imply that the relativistic mass of the magnetic field of a neutron star could be larger than the mass of the star.
Therefore, the two natures: 1) the relativistic mass density of field is very large and 2) the relativistic mass of the field can be larger than the mass of the star, are not unique to the super strong gravitational field, but is common to the super strong magnetic field. Because the gravitational field is full of all the space while the magnetic field only exist in the part space, and, the relativistic mass density and the total relativistic mass of the gravitational field of the neutron star is much larger than that of the strongest magnetic field, the gravitational field is more fundamental to the mass of a galaxy.
It is well known by almost every physicist that, according to E = m 2 , a field is with a relativistic mass. But, it is the first time that the relativistic mass of the gravitational and magnetic fields is calculated in this work. The result is extremely astonishing and fascinating: The relativistic mass of the gravitational or magnetic field of a star can be much larger than the mass of the star.
This result shows a new frame to understand the total mass of a galaxy. It indicates that, in a galaxy, the total mass of the bodies and gas is very little. Now, the total mass of the Milky Way is estimated to be 5.8 × 10 11 times that of the Sun. [21] From Table 2 and Figure 3, it is shown that the relativistic mass of the gravitational field of a neutron star could be 7.39 × 10 21 times the mass of the Sun. It is much larger than the total mass of the stars and gas in the Milky Way. Although it is astonishing, it is understandable. It is easy to know that the mass density in the Milky Way is only = 2.5 × 10 −17 / 3 as the total mass in the Milky way is taken as 5.8 × 10 11  A new frame is needed to understand both that the relativistic mass density of the gravitational field on the surface of a neutron star can be larger than the mass density of the star and that relativistic mass of the gravitational field of a star is much larger than the mass of the star.
In the new frame, the main mass in a galaxy should be the relativistic mass of the gravitational field. As we consider the mass of a galaxy or the whole universe, not only the mass of the bodies and gas is important but also the Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 December 2021 relativistic mass of the field is significant.
The two problems are very interesting and important: 1) For body (including star and planet) and gravitational field, which one is the premise? It is usually thought that a gravitational field is emitted by a body. But, now, it is known that the relativistic mass of the gravitational field of a body is much larger than the mass of the body. Why a body can emit a field with the mass that is much larger than the mass of the body?
2) What is the mechanism that can make the matter with the relativistic mass density larger than the mass density of the neutron star a field?