Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Modification of Gravitational Field Equation due to Invariance of Light Speed and New System of Universe Evolution

Modification of Gravitational Field Equation due to Invariance of Light Speed and New System of... Hindawi Advances in Astronomy Volume 2021, Article ID 5579060, 14 pages https://doi.org/10.1155/2021/5579060 Research Article Modification of Gravitational Field Equation due to Invariance of Light Speed and New System of Universe Evolution Jian Liang Yang College of Physics, Zhengzhou University, Zhengzhou 450001, China Correspondence should be addressed to Jian Liang Yang; bps267890@qq.com Received 16 February 2021; Revised 4 March 2021; Accepted 9 March 2021; Published 20 March 2021 Academic Editor: Ghulam Abbas Copyright © 2021 Jian Liang Yang. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We make a systematic examination of the basic theory of general relativity and reemphasize the meaning of coordinates. Firstly, we prove that Einsteinʼs gravitational field equation has the light speed invariant solution and black holes are not an inevitable prediction of general relativity. Second, we show that the coupling coefficient of the gravitational field equation is not unique and can be modified as 4πG to replace the previous − 8πG, distinguish gravitational mass from the inertial mass, and prove that dark matter and dark energy are not certain existence and the expansion and contraction of the universe are proven cyclic, and a new distance-redshift relation which is more practical is derived. After that, we show that galaxies and celestial bodies are formed by gradual growth rather than by the accumulation of existing matter and prove that new matter is generating gradually in the interior of celestial bodies. For example, the radius of the Earth increases by 0.5mm every year, and its mass increases by 1.2 trillion tons. A more reasonable derivation of the precession of planetary orbits is given, and the evolution equation of planetary orbits in the expanding space-time is also given. In a word, an alive universe unfolds in front of readers and the current cosmological difficulties are given new interpretations. tides, the moon still has an unexplained retreat, and the 1. Introduction increase of the day length is also inconsistent with the Although general relativity has made some remarkable prediction of the tide theory. Recently, Melissa Ness and her achievements, some basic problems have not been well colleagues have observed that there is a fine X-shaped box solved, such as the physical meaning of the coordinates of structure in vortex galaxies similar to the Milky Way [2]. Schwarzschild metric, whether general relativity is the Melissa Ness said that this structure implies that large curved theory of space-time or the theory of gravity in flat galaxies are not formed by the merger of small galaxies, space-time, whether the constant speed of light is also because once the merger occurs, the structure will inevitably tenable in the gravitational field, the singularity problem of be destroyed, and we must abandon the existing theory of the field equation, and whether the existence of black holes is galaxy formation and establish a new logic system. ,e true. However, only these basic problems have plagued the observations of Martinez-Lombila and others [3] show that the radius of disk galaxies similar to the Milky Way galaxy is development of general relativity but also led to some confusion in practice; for example, on the one hand, the expanding at a speed of 500 m/s; such a high speed cannot be radial coordinates of Schwarzschild metric are not inter- the speed at which matter accumulates at the edge. If matter preted as the normal radius, while, on the other hand, the accumulates at this speed at the edge, it should be the same radial coordinates on the solar surface are treated as the everywhere on the disk. Obviously, the current theoretical radius of the sun in calculating the curvature of light on the framework cannot explain such a rapid expansion of the surface of the sun, resulting in conceptual confusion. In radius of the disk. ,ere is also the problem of dark matter addition, there are some new observations that are not and dark energy; the reason why we need them is that the accommodated by the current gravity theory. As Lorio [1] observed phenomena do not conform to the prediction of pointed out, there is an unexplained increase in the distance the theory, but, no one has seen them really. ,en, whether between the Sun and the Earth, and after considering the they are real or the theory itself needs to be modified is also 2 Advances in Astronomy an unavoidable problem. ,e latest observation data of In this paper, we use natural units, the speed of light of Nielsen and others [4] show that the universe is expanding at flat space-time c � 1, and it is agreed that flat space-time a constant speed rather than accelerating, so whether the linear element is universe accelerates or decelerates or expands at constant 2 μ ] 2 2 2 2 2 2 ds � g dx dx � dt − dr − r 􏼐dθ + sin θdφ 􏼑. (1) μ] speed still needs to be reconsidered. Besides, some new studies of frontier disciplines [5, 6] have shown that 1 billion According to general relativity, in a spherically symmetric years ago, the brightness of the sun was less than half of what gravitational field, in the coordinate system (t, r, θ, φ), the it is today, the Earth is an ice ball, and the mountain is not as general form of space-time line elements is [7–11] high as it is today, and 2.7 billion years ago, the air pressure 2 μ ] 2 on the Earth was only half of todayʼs. ,ese seem to be purely ds � g dx dx � B(r, t)dt − Q(r, t)dtdr μ] (2) geophysical problems, which can only be reasonably 2 2 2 2 − A(r, t)dr − D(r, t)􏼐dθ + sin θdφ 􏼑. explained from the perspective of cosmology because the evolution of the Earth is an epitome of the evolution of the ,e condition of this formula is only spherical symmetry, universe and the Earth must be reflected by cosmological which is applicable to the gravitational field of both static and events. On the contrary, the phenomena on the Earth can be oscillating gravitational sources. In this paper, we will just deal used to test the cosmological theory more accurately and with the static gravitational field, which is what Newtonian people do not have to go far to test the theory of cosmology. gravity describes. For the static case, g no longer contains μ] In a word, we are faced with some new problems that cannot time. Besides, the static case requires time version to be be avoided. We will see that when the speed limit of light, that symmetric, so g � Q(r) � 0. ,erefore, for the static case of is, the speed of light always 1 (in natural units), is still satisfied spherical symmetry, the space-time line element is in the gravitational field, the above problems can be solved in a 2 μ ] 2 2 2 2 2 package. ,e author thinks that it is a great mistake of general ds � g dx dx � B(r)dt − A(r)dr − D(r)􏼐dθ + sin θdφ 􏼑. μ] relativity that the invariance of the speed of light in the (3) gravitational field is not emphasized in the past, and it is this fault that leads to a series of misconceptions and absurd results; We just have to solve for three functions B(r), A(r), and for example, it is necessary to admit singularity as physical D(r). In order to ensure that the meaning of coordinates is reality, which will never be allowed in other parts of physics. In always clear and unchanged, this paper will not continue to a word, it is shameless to tie the correctness of general relativity simplify (3) into the so-called standard form through co- with some wrong conclusions such as big bang and black holes, ordinate transformation but directly solve with the gravi- and it is shameless to praise mistakes as successes. Leading to tational field equation. Firstly, determine the external the big bang, black holes and all kinds of other singularities are solution that satisfies the vacuum field equation R � 0, and μ] not the success of general relativity, but its failure. ,e reason is then the source internal solution is determined. simple: there is no singularity in real nature. No matter how In order to reflect the invariance of light speed, we re- much you boast big bang and black holes, they cannot be true. quire A(r) � B(r). From the following solving process, we ,e author thinks that if these absurd things are not stripped can see that such a solution not only exists but also is unique. away from general relativity, there will be no real progress in Equation (3) provides general relativity, the field of astrophysics will be dominated by g � B(r), all kinds of idealism, and more and more young students will be misled into the wrong way. In order to deal with these g � − A(r), problems systematically, to get to the bottom and bring order out from chaos, this paper begins with the most basic problem, g � 0(μ≠ ]), μ] that is, solving the metric of the spherically symmetric grav- itational field represented by coordinates in the usual sense. g � − D(r), 2. Spherically Symmetric Static Metric g � − D(r)sin θ, Represented in Usual Coordinates g � − , We just have to solve for the metric form in the usual (4) D(r) spherical coordinates; the form in other coordinates can be obtained by coordinate transformation. Indices g � − , μ 0 1 μ, ], λ, α, β � 0, 1, 2, 3. Space-time coordinates x � (x , x , D(r)sin θ 2 3 0 1 2 3 x , x ) � (t, r, θ, φ) and x � t, x � r, x � θ, x � φ repre- sent the usual time, radius, and pole angles, respectively. g � , ,ey have the same meaning as in quantum mechanics or B(r) electrodynamics. In the language of the observational theory of general relativity, t is the time recorded by a stationary g � − , A(r) observer at infinite distance, r is the distance the observer measures from the origin to another point, and θ, φ are the μ] g � 0(μ≠ ]). polar angles measured by the observer. Advances in Astronomy 3 According to the definition of connection, infinity, there must be C � 1/4, namely, AB � D /4 D. λ λα ] μ α Γ � (1/2)g (zg /zx + tzg /zx n − qzg /zx ), the And inserting A � D /4B D into equation (8) gets μ] αμ α] μ] repeating indices up and down means summing from 0 to 3, ′ ′ D D and it is not hard to figure out all of its nonzero connections ′ (10) B + B − � 0, 2 D 2 D as follows [2–7]: which is a differential equation with respect to B. Writing 1 zD 1 2 Γ � − sin θ, D � l , the general solution of (10) is given by B � 1 + C /l. 2A zr C is an integral constant. Because we must return to Newton gravitation in the distance, we have C � − 2GM. G cos θ 3 2 Γ � , Newtonʼs gravitational constant M is the mass of the source. sin θ It is important to insert B � 1 − 2GM/l and 1 zA ′ ′ 1 A � D /4B D � l /(1 − 2GM/l) into any one of (6)–(8). Γ � , You can obtain an identity with respect to l � l(r); namely, 2 zr no matter what l � l(r) is, this identity can be tenable, so we 1 zB can pick an appropriate l(r) so that A � B. And letting Γ � , 2B zr (5) A � B, namely, 1 − 2GM/l � dl/dr, we obtain 1 zB r � C + l + 2GM ln(l − 2GM), (11) Γ � , 2A zr where C is the integral constant and can be decided by the 2 3 continuity of l(r) on the surface of the source. In Section 4 of Γ � Γ 12 13 this paper, C will be calculated. So far, we obtain the ex- terior line element: 1 zD Γ � − , 2A zr 2GM 2GM 2 2 2 2 2 2 2 ds � 1 − dt − 1 − dr − l dθ + sin θdφ , 􏼒 􏼓 􏼒 􏼓 􏼐 􏼑 l l Γ � − sin θ cos θ. (12) According to the definition of curvature tensor, where l � l(r) can be inversely solved from (11). It can be λ ] λ λ α λ α λ R � zΓ /zx − zΓ /zx + Γ Γ − Γ Γ ; for μ≠ ], we μ] μλ μ] μλ α] μ] αλ seen from (11) that when l � 2GM, r becomes negative, have R � 0, which means that the vacuum field equation is μ] which means l − 2GM is always greater than 0 and so there is automatically satisfied. It is not hard to figure out all of the no horizon and no black hole. And considering nonzero components of R . Write A � dA/dr, μ] lim (ln x/x) � 0, we have r � l for l ⟶ ∞. 2 x⟶∞ 2 2 2 ″ ′ A � d A/dr , A � (dA/dr) , and note that R � R sin θ; we are left with the following three equa- 33 22 3. Link with the Mechanics of Special Relativity tions about B(r), A(r), and D(r): ,e invariance of the speed of light described by (11) is easy ′ ′ ′ ′ ′ ′ B ′ B B A D B to see. Suppose that the photon moves in the radial direction, (6) R � − 􏼠 􏼡 + − 􏼠 + + 􏼡 � 0, 2A 2AB 2A 2A D 2B dφ � dθ � 0, ds � 0, and from (11), we can get 2 2 dr /dt � B(r)/A(r) � 1, namely, dr/dt � ± 1, which shows 2 2 that radial speed of light is constant. And now we look at the ′ ′ ′ ′ ′ ′ ′ ′ D B D B A D B ⎝ ⎠ ⎛ ⎞ R � 􏼠 + 􏼡 + + − 􏼠 + 􏼡 � 0, 2 2 light moving tangentially and set θ � π/2, dr � 0, ds � 0; D 2B 2A D 2B 2D 4B the tangential speed of light is given from (12) √��������� (7) byrdφ/dt � (r/l) 1 − 2GM/l. And (11) tells us that the √��������� smaller 1 − 2GM/l than 1, the larger r/l than 1, so the √��������� ″ ′ ′ ′ D D A B deviation of (r/l) 1 − 2GM/l from 1 is actually very small, R � + 􏼠− + 􏼡 − 1 � 0. (8) and you can still think of it as 1. It is in this sense that we say 2A 4A A B that (11) describes the invariant speed of light, not strictly From R × (1/B) + R × (1/A) � 0 we obtain constant. ,is slight change of the tangential speed of light 00 11 causes light to bend near a celestial body; otherwise, it travels ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ AB + A B D D D (AB) D D in straight lines. ,e previous result is − + + � − + 􏼠 􏼡 􏼠 􏼡 􏼠 􏼡 √��������� 2AB D D 2AB D D 2D rdφ/dt � 1 − 2GM/r which can be obtained from (28), and it is not hard to find that when r � 2GM, the tangential D speed of light is zero and the deviation from 1 is severe; + � 0. although it means also that light can bend, it does not reflect 2D the invariance of the speed of light. (9) ,e correctness of (12) is not only in the invariance of the Equation (9) is a differential equation with respect toAB, speed of light but also in the natural connection with rel- its general solution is AB � C D /D, and C is the integral ativistic mechanics under weak field approximation. 1 1 ′ ″ constant. Since A � B � 1, D � r , D � 2r, D � 2 at Equation (12) provides 4 Advances in Astronomy dv Γ � 0, 2 € _ _ _ € � 􏼐r − rφ 􏼑e + (2rφ + rφ)e , r φ dt GM dl GM 2 1 0 dv Γ � Γ � � , 2 2 11 01 2 2 (13) � 2r _r € + 2r φ _ φ € + 2rr_φ _ , dr (21) (1 − 2GM/l)l l dt GM d(mv) dv dm dv dv Γ � . � m + v � m + mv . 2 2 dt dt dt dt l 2 1 − v dt 􏼐 􏼑 ,e dynamic equation describing the motion of free Using (20), we have particles is the geodesic equation, the proper time must be 2 2 2 2 2 eliminated when solving for acceleration, and the geodesic 1 − r φ _ r r _φ _ φ € φ _ rr _ GM (22) € _ r + + − rφ � − , 2 2 2 2 equation after the elimination of proper time is 1 − v 1 − v 1 − v r 2 μ ] λ ] λ μ d x dx dx dx dx dx μ 2 2 3 (14) + Γ · − Γ · · � 0, r − rr_ rr _φ _ r € r φ _ r _ ]λ ]λ (23) dt dt dt dt dt φ € + + + 2r _φ _ � 0, dt 2 2 2 1 − v 1 − v 1 − v which is derived in detail in post-Newtonian mechanics [9]. eliminating φ € in the use (22) ×(1 − r_ )− (23) ×rφ _ r _ gets Let the particle move on the plan θ � π/2, set μ � 1, 3, and write dr/dt � v , dφ/dt � v /r; we have r φ GM GM (24) r − + − v � 0, 2 2 2 2 2 lv r r d r GM GMv 1 1 2 0 2 1 2 � − Γ − Γ r _ + 2Γ r_ + Γ φ _ � − + + , 2 00 11 01 33 2 2 2 dt l l r which is exactly equation (17), and inserting it into (23), we (15) get 2 GM € (25) ′ φ + v v − v v � 0, d φ 2l 1 2GM r φ r φ 3 0 2 3 � − 2Γ r _φ _ + 2Γ r_φ _ � − v v + v v . r r r φ r φ 2 13 01 2 rl 1 − 2GM/l dt l r which is (19) omitting the higher-order small GMv v /r . So r φ (16) far, it can be seen that (12) in the weak field approximation In the weak field or in the distance, 2GM/l≪ 1, l ⟶ r, can link well with the special relativistic mechanics. 2 2 l ⟶ r , l ⟶ 1, (15) and (16) become, respectively, And again, when the particle moves along the radial direction, if v � 1, the acceleration of the particle is equal to 2 2 d r GM GMv zero, which means that the invariance of light speed and the (17) + − − � 0, 2 2 2 dt r r light speed limit are unified. d φ 2 2GM (18) + v v − v v � 0. 4. The Light Speed Invariant Solution within a r φ r φ 2 2 3 dt r r Spherically Symmetric Static Gravity Source Ignoring the higher-order small quantity 2GM/r , (17) Now, we solve the interior A(r), B(r), and D(r). In order to becomes keep the constant speed of light, we still require A(r) � B(r) d φ 2 inside the source. From the following solution’s process, we (19) + v v � 0. r φ 2 2 know such a solution not only exists but also is unique. Note dt r that the constant speed of light means that the speed of light It is not difficult to prove that (17) and (19) are just the passing through the cavity in the source is 1, but that passing relativistic Newton equations of gravity: through the medium is 1. ,e equation of gravitational field in the source is d(mv) GMm � − e , (20) dt (26) R � c􏼒T − Tg 􏼓, μ] μ] μ] √����� where m � m / 1 − v is the motion mass of the particle, 2 2 2 _ _ v � r_e + rφe , v � r _ + r φ , and e , e are the base vectors. where c is the coupling constant, and that we do not write its r φ r φ Prove as follows: from theoretical mechanics, we know specific value here is to lay a hint for the following Advances in Astronomy 5 modification of the constant. And T � (ρ + p)u u − pg the identity with respect to l � l(r); that is, this is true re- μ] μ ] μ] is the energy-momentum tensor of the source. Note that gardless of the function of l(r), regardless of the value of c. μ μ ] μ μ] μ] u � dx /ds, u � g u , u u � 1, g g � 4, T � g T � So, we can pick a function l(r) by the following equation (41) μ μ] μ μ] μ] ρ − 3p. to make A(r) � B(r): i i And for the static source, u � dx /ds � 0, i � 1, 2, 3. ] 2 0 2 l l ,en, u � g u � 0, 1 � B(dt/ds) � B(u ) , 2GM i i] − 2 2 √� � 􏼠1 − 􏼡exp 􏽚 − cA 􏼠pl + l 􏽚 l ρdl􏼡dl 0 2 u � g u � B, T � pA, T � p D, T � p D sin θ, 0 00 11 22 33 l l 0 e e T � ρB; from (26), we have (35) − 1 ′ ′ ′ ′ ′ ′ − 1 2 B ′ B B A D B c � 1 + l c 􏽚 ρl dl l . 􏼠 􏼡 R � − 􏼠 􏼡 + − 􏼠 + + 􏼡 � (ρ + 3p)B, 2A 2AB 2A 2A D 2B 2 (27) On the other hand, we know T � 0, for the static μ;] source; it is [2–7] 2 2 ′ ′ ′ ′ ′ ′ ′ ′ D B D B A D B ⎛ ⎝ ⎞ ⎠ √� � zp z R � 􏼠 + 􏼡 + + − 􏼠 + 􏼡 2 2 (36) D 2B 2A D 2B � − (ρ + p) ln B. 2D 4B μ μ (28) zx zx � (ρ − p)A, 5. Modification of the Coupling Constant and ″ ′ ′ ′ D D A B c the Internal D(r) R � + 􏼠− + 􏼡 − 1 � (ρ − p)D. (29) 2A 4A A B 2 When the pressure at the surface of the gravitational source In the use of (R /2B) + (R /2A) + (R /D) � ρc, we is set as zero, it can be seen from (39) that once the geodesic 00 11 22 get equation is required to return to Newtonian gravity under the weak field approximation, the coupling constant c must ″ ′ 1 2D D 1 2 D 􏼒 􏼓 + 􏼠 − 􏼡 − − 2cρ � 0, (30) be − 8πG, which is the previous result. But this result should ′ ′ ′ A D 2 D A D D be considered as a mistake because it leads to a lot of sin- gularities that should not occur. For example, when the ratio which can be looked as a differential equation with respect to of the mass to the radius of an object is 2GM/R> 8/9, the 1/A. And writing D � l , in the requirement of ensuring D � pressure inside the object becomes infinite [7–9], which is 0 and A being limited at origin, the solution of (31) is given obviously absurd [10–16]. ,e root of all kinds of singularity by is the improper selection of − 8πG. As we will see, when the − 1 c pressure is taken negative, the coupling constant is identified ′ (31) A � 􏼠1 + 􏽚 ρl dl􏼡 l , as 4πG, which not only avoid singularity of Schwarzschild l 0 metric but also remove in a package cosmological problems. 2 l 2 − 1 − 1 2 where l � (dl/dr) . Writing A � (1 + l c 􏽒 ρl dl) , we I need to say a few words about the negative pressure. have Einstein did not refuse the negative pressure. In the book %e Meaning of Relativity, Princeton University Press Published, ′ ′ ′ ′ ″ A � A l + 2A l l , 1 1 1922, (page 117), for T � ρu u , Einstein said, we “shall μ] μ ] add a pressure term that may be physically established as ′′ ′ ″ 􏼐l 􏼑 � 2l + 2ll , follows. Matter consists of electrically charged particles. On (32) the basis of Maxwellʼs theory these cannot be conceived of as l dB electromagnetic fields free from singularities. In order to be B � . consistent with the facts, it is necessary to introduce energy dl terms, not contained in Maxwellʼs theory, so that the single Insert these into (30), we obtain electric particles may hold together in spite of the mutual repulsion between their elements, charged with electricity of dB − 2 2 � − cA 􏼠pl + l 􏽚 l ρdl􏼡. (33) one sign. For the sake of consistency with this fact, Poincare Bdl 0 has assumed a pressure to exist inside these particles which balances the electrostatic repulsion. It cannot, however, be To make B(r) continuous on the boundary of source, the asserted that this pressure vanishes outside the particles. We solution of (33) is shall be consistent with this circumstance if, in our phe- l l nomenological presentation, we add a pressure term. ,is 2GM − 2 2 B � 􏼠1 − 􏼡exp 􏽚 − cA 􏼠pl + l 􏽚 l ρdl􏼡dl, (34) must not, however, be confused with a hydrodynamical l 0 pressure, as it serves only for the energetic presentation of where l � l(r ) is the value on the boundary and r is the the dynamical relations inside matter.” From this statement, e e e radius of the source. it can be seen that Einstein did not equate pressure as a It is important that, similar to the external solution, source of gravity with the dynamic pressure of a fluid but substituting (32) and (34) into any of (28)–(30), you can get regarded it as a phenomenological representation of all the 6 Advances in Astronomy action within matter, including the electromagnetic force. It 14 l � r − πGρr , e e is not surprising that a negative value is taken. Now solve for D(r) in the source with negative pressure and in the meantime determine the coupling constant. For 3 r � l + πGρl , (42) e e e the convenience of calculation, we use the average density 9 instead of the density of each point that is to take the interior 3 3 5 ρ � const. ,e density itself is a statistical average, such l � r − πGρr . e e e treatment is equivalent to treating the whole celestial body as a statistical volume element, so it is suitable for some not too Inserting (42) into (40), we obtain large celestial bodies as such the sun, and it is also an ap- 2 2 4πρ 80π ρ G proximation for larger celestial bodies. 3 5 (43) M � r − r + · · · . e e When ρ is regarded as a constant, p � − ρ is the solution 3 9 of (36). Since the geodesic under the weak field approxi- Writing 4πρr /3 � M , we have e i mation must return to Newtonian gravity, there must be 5G Γ ⟶ GM/r on the surface of the source, and we conclude 2 M � M − M + · · · , that, from (33), the coupling constantc � 4πG. Notice that r under the weak field approximation l ⟶ r, A ⟶ 1, (44) 2 3 5G 4π 􏽒 ρl dl ⟶ M, and p � − ρ ⟶ − 3M/4πr . 2 M � M + M + · · · , With p � − ρ, c � 4πG, and l as the fixed value, the r e e integral of both sides of (35) is easy, and we get where M is the inertial mass and M is the gravitational mass. 􏽳���������� � dl 1 − 2GM/l 4πGρl ,e reason why M is called inertial mass is that ρ represents � 􏼠1 + 􏼡. (37) the inertial density of matter measured in comoving coordinate dr 3 1 + 4πGρl /3 system, while M is introduced from the perspective of gravity, ′ so it is natural to call it gravitational mass. Ensuring B � 0 for r � 0, which is because the force on the particle at the origin is zero, the solution of (37) is Equation (44) distinguishes gravitational mass from 􏽳���� � 􏽳����������� 􏽲���� � inertial mass, which is the result of an in-depth discussion in 3 4πGρ 1 − 2GM/l this paper, and for high-density celestial bodies, the dif- arctg􏼠l 􏼡 � r . (38) 4πGρ 3 1 + 4πGρl /3 ference is obvious. We see l � r − 7GM/6 + · · ·, and so far, the integral e e On the other hand, we require Γ to be also continuous constant C in (11) can be decided according to the con- on the boundary, which is a necessary condition to ensure tinuity on the boundary; that is (11) is applied to the surface the continuity of gravity on the boundary. So, there exists on of the source the boundary C � r − l − 2GM ln l − 2GM � r − l 􏽳���������� � 3 e e e e e zB zl 8πGρl 1 − 2GM/l 2GM 1 1 e e (45) Γ � � � Γ � . 00 2 00 2 7GM B zl zr 3 1 + 4πGρl /3 l e e − 2GM ln r − 2GM􏼁 ≈ − 2GM ln r . e e (39) 􏽰�������� � Inserting C into (11), we can complete the calculation of 2 2 Solve (39), M � G k + kl − Gk, in which Mercury precession and ray bending, and the calculated 2 2 5 2 k � 16π ρ l /(9 + 12πGρl ). e e result is that the difference between the new results and the ,e explicit form of M can be solved under the weak field original ones is very small and completely consistent with approximation, and expanding the square root in (39) by the observation. ,e concrete calculation is not written here, Taylor, we obtain readers can do it by themselves. In a word, with the new coupling constant 4πG, the 4πρl 8 e 2 2 5 (40) M � − π ρ Gl + · · · . gravitational field equation is now modified as 3 3 (46) Applying (39) on the boundary and taking the ap- R � 4πG􏼒T − Tg 􏼓, μ] μ] μ] proximation M � 4πρl /3 into, we have and correspondingly the pressure p as the source of grav- 􏽶���������� � 􏽳���� � 􏽲���� � itation takes negative. Multiplying the two sides of (46) with 3 4πGρ 1 − 8πGρl /3 μ] g , we have R � − 4πGT, so the equivalent form of (46) is (41) arctg􏼠l 􏼡 � r . e e 4πGρ 3 1 + 4πGρl /3 R − (1/2)Rg � 4πGT . μ] μ] μ] Obviously, for 1 − 8πGρl /3 � 0, r ⟶ ∞; that is to say, 6. Further Interpretation of the Physical e e no matter how big the radius of the celestial body, there is Meaning of the Negative Pressure always l < 3/8πGρ; we do not have to worry about whether l e e has a solution. And expanding the two sides of (41) by Taylor Einstein did not interpret the pressure term in a gravitational and taking second-order approximation, we obtain source as the dynamic pressure of a fluid, but as a Advances in Astronomy 7 phenomenological representation of the pressure within a usually. So, we say that dark energy is just the binding energy matter to balance the electromagnetic force and prevent of matter, rather than an independent existence. charged particles from being disintegrated by electrical re- pulsion, which we should accept. Einstein did not point out 7. The Problems of Schwarzschild Metric that pressure is produced by which power. Today, it is easy to infer and prevent the disintegration of the protons and Schwarzschild metric is neutrons are strong; preventing electronics is the disinte- − 1 2GM 2GM 2 2 2 2 2 2 2 gration of the weak force. ,erefore, the pressure term ds � 􏼒1 − 􏼓dt − 􏼒1 − 􏼓 dr − r 􏼐dθ + sin θdφ 􏼑. r r should be understood as a phenomenological representation (47) of the combined effects of the strong, weak, electromagnetic, gravitational, and all other forms of action within a matter, Because the precession angle of Mercury orbit predicted representing the total binding energy that holds the matter by (47) is consistent with the observation, it is generally together, represented by the potential energy of the system. believed that it is correct. However, there are also some In other words, if you divide the matter infinitely, and you serious problems with the metric; for example, it exposes the move each part to infinity, the work done is the volume incompatibility of electromagnetic theory and gravitational integral of the pressure, which is negative, and the absolute theory: when a charged particle moves in the radial direction value is equal to the mass of the gravitational source, namely, at a speed dr/dt ≈ 1, near the singularity r � 2GM, 􏽒 pdxdydz � − M, which is like adding a physical condition − 1 2 2 2 ds /dt � (1 − 2GM/r) − (1 − 2GM/r) (dr/dt) < 0, to make the solution of the pressure definite. In fact, the which is ridiculous because there is no reason to think that form of the gravitational source already determines the the electromagnetic equipment in the gravitational field interpretation of p. As gravitational source cannot make the speed of a charged particle close to 1. T � (ρ + p)u u − pg , if p is still understood as the μ] μ ] μ] In order to avoid the defect with (47), textbooks do not common dynamic pressure, then the field equation can only be interpret r in (47) as the usual radius but instead refer to it as used to solve for the metric of an ideal fluid, which is obviously the radial parameter with fuzzy meaning [7, 8]. But this is not hoped by general relativity, and the dynamic pressure in a unhelpful and leads only to conceptual confusion since it has solid is generally considered to be zero. let alone (36). Can p be been used as the usual radius when calculating the pre- understood as a thermal pressure? No, because the thermal cession angle and the bending angle of light; there should be motion is absorbed by ρ in the form of thermal kinetic energy no other explanation. and cannot be repeated to appear. p is called pressure only Now, we calculate the ordinary pressure given by the because it appears in the equation of motion in the form of Schwarzschild interior metric, from which we can see the pressure; of course, p includes the effect of common pressure. defects of the Schwarzschild metric and the necessity of μ] ,e equation of motion refers to T � 0. ;] modifying the field equation. According to the definite of When the field equation with the coupling constant 4πG pressure, it refers to the stress per unit area. It may as well let is applied to the universe, T � (ρ + p)u u − pg repre- μ] μ ] μ] the celestial body be a fluid with ρ � const, P denotes the sents the energy-momentum tensor of space of the universe, common stress, and the common pressure given by the and taking the statistical average of ρ and p, we have p � − ρ, interior solution of Schwarzschild is which is just the equation of state of the dark energy said r r r e e e zB 1 P � ρ 􏽚 Γ dr � ρ 􏽚 dr � ρ 􏽚 dB c 00 A zr A r r r 􏽱����������� 􏽱����������� ρ 8πGρ 2 2 � 􏽚 1 − r d 3 1 − 8πGρr /3 − 1 − 8πGρr /3 􏼒 􏼓 􏼒 􏼓 (48) 4 3 􏽳��������� � 2 3/2 2 2 2 2 3ρ 8πGρr ρ 8πGρr 8πGρr ρ 8πGρr e e � − 􏼠1 − 􏼡 + 1 − 􏼠1 − 􏼡 − 􏼠1 − 􏼡 . 8 3 2 3 3 8 3 In weak field approximation, P � ρ(GM/r n − q But, the common pressure given by (39) is c e 2 3 GMr /r ), which is Newton’s result. And at the center, (48) r r r r e e e e zB zB gives 􏽳��������� � P � ρ 􏽚 gdr � ρ 􏽚 Γ dr � ρ 􏽚 dr � ρ 􏽚 dl c 00 2 B zr B zl 2 r r r r 3ρ 8πGρr ρ 8πGρr ρ e e (49) P � − 􏼠1 − 􏼡 + 1 − − . 8 3 2 3 8 8πGl 1 + 4πGρl /3 � ρ 􏽚 dl � ρ ln , 2 2 Obviously, for 1 − 8πGρr /3 � 0, P < 0, and for e c 1 + 4πGρl /3 1 + 4πGρl /3 1 − 8πGρr /3 � 1/9, P � 0, which are abnormal because the e c (50) pressure should not be zero anyway. 8 Advances in Astronomy − 1 − 1 in which l � l(r) and l � l(r ) satisfy (38). ,ere is no parameter of today H � H(t ) � 70 km · s · Mpc into e e 0 0 singularity in (50). Under weak field approximation, l ⟶ r, the above equation, we get t � 1.37 × 10 years, that is, 13.7 2 3 P � ρ(GM/r n − qGMr /r ), which is just the result of billion years, the same as the previous theoretical results. c e e Newton. It can be seen from (54) that the cyclical period of expansion 􏽰����� � and contraction of the universe is 2π/ω � 3π/Gρ � 2 × 10 years, namely, 200 billion years, so the universe is currently in 8. The Application of (46) in Cosmology the expansion stage and will begin to contract in 36.3 billion years. Contraction is the reverse course of expansion. In the comove coordinate system (t, l, θ, φ), the metric that Now, we derive the new relation between distance and describes cosmic space is the Robertson–Walker metric: redshift given by (46). Similar to the previous operation, 2 2 2 2 2 2 2 2 2 letting the light given out by distant galaxy at the time t in ds � dt − R (t)􏼠 dl + l dθ + l sin θdφ 􏼡, 1 − kl past and reach the Earth at the time t of today, its redshift z � (λ − λ /λ ) � (R(t )/R(t)) − 1. λ is wavelength. We (51) 0 e e 0 may as well put today’s R(t ) � 1. Note that the subscript 0 where R(t) is the cosmic scale factor and k is a constant. l is represents today. And differentiating 1 + z � 1/R(t), we get the radial coordinate, and in other books it is denoted by r, dR dR dt just to distinguish it from the usual radius, here l instead of r. dz � − � − . (57) Rdt R When the new field equation (46) is applied to the universe, R (t) that is, combined with the Robertson–Walker metric, the And the derivative of equation (52) gives following two equations are given: 2 2 € _ 4πGρ/3H � − RR/3H , where H � dR/Rdt, R � dR/dt. dR 4πG Writing today’s 4πGρ/3H � q , H(t ) � H , and ap- 2 0 0 0 0 (52) 􏼠 􏼡 + k � − ρR , plying (52) to today, we have dt 3 k � − H 1 + q 􏼁 , 0 0 dρ dR (53) R + 3 (p + ρ) � 0. (58) dt dt 4πGρ k 2 2 2 H � − − � H 􏽨 1 + q 􏼁 (1 + z) − q 􏽩. 0 0 2 0 R (t) Equation (52) shows that k must be negative, which proves that space-time is infinite. Equation (52) is similar to On the other hand, for the motion of light, the original Friedman equation; just replace G there with − (G/2). Equation (53) is the so-called energy equation, dl dt dz dl 􏽱������� � 􏽰����� � − � � − 􏽚 which is in the same form as the original. Now putting p � R(t) H 􏼐1 − kl 􏼑 1 − kl − ρ in (52), we obtain p � − ρ � const, which means that the (59) density and pressure remain the same while the universe 1 dz expands or contracts, and new matter must be created 􏽱����������������� � 􏽚 . continuously in the universe. ,e solution of (52) is 0 1 + q 􏼁 (1 + z) − q 0 0 􏽲��� � πGρ Note that l as superscript of the integral sign refers to the R(t) � k sin􏼠2t + k 􏼡, (54) 1 2 galaxy’s unchanged comoving coordinate. Using the relation between luminosity distance and redshift √����� � which shows that the universe expands and contracts in d � (1 + z)R(t ) 􏽒 dl/ 1 − kl and completing the inte- L 0 cycles. Here, k and k are two integral constants. Since time 1 2 gration of the right of (65), we get has no beginning and no end, the moment of R(t) � 0 has 􏽱����������������� 􏽰����� occurred countless times. Let us define the nearest moment (z + 1) q + 1 + q + 1 􏼁 (z + 1) − q z + 1 0 0 0 of R(t) � 0 as zero; that is to say, at the moment t � 0, then 􏽰����� 􏽰����� H d � ln . 0 L q + 1 1 + q + 1 0 0 k � 0. Hubble parameter: 􏽲���� � 􏽲���� � (60) dR 4πGρ 4πGρ � (55) H(t) ≡ cot􏼠t 􏼡. Here, d is Luminosity distance and q is the deceleration Rdt 3 3 L 0 parameter today. As z ⟶ 0, expanding it, we have Since everything disappears at R(t) � 0 [17], including light, the universe in the last cycle is unobservable and no 1 − q 3q − 2q − 1 0 2 0 3 (61) H d � z + z + z + · · · , 0 L concern to us. What we care about is the age of our universe, 2 6 which is the time from the beginning (t � 0) of the most recent which is classical Hubble law after omitting high-order cycle to today, and using (55), we obtain our universe’s age: terms. ,e conclusion of (60) is in good agreement with the 􏽳���� � 1 4πGρ observed distance and redshift data [9–19], which strongly 􏽰������ � t � arctan . (56) indicates that the modified field equation (46) is correct, the 4πGρ/3 3H so-called dark energy does not have to exist, and the ex- pansion of the universe is still decelerating. ,e curve in By substituting the observed density − 28 3 Figure 1 is the simulation of (60) with q � 0.14 and ρ � 3.1 × 10 kg/m of the universe and the Hubble 0 Advances in Astronomy 9 50 expanding, and new matter is continuously generating in galaxies. In a word, just like the night sky we see with a magnifying glass, everything is expanding but the periods of various rotations and revolutions are unchanged. ,e essence of cosmic expansion is the simultaneous gener- ation of space and matter. Many people have recognized the fractal structure of the 40 universe, but they are unwilling to explain the formation of galaxies as the growth of fractals [17, 18]; the obstacle is obviously that people do not know the mechanism of the generation of new matter. Now, the generation of matter is Red shi (z) no longer a problem. Figure 2 is a step-by-step magnification of the Solar System. It represents the actual growth process of the Solar Figure 1: ,e recent Hubble diagram of 69 GRBs and 192 SNe Ia. System. With the universe expanding, the Solar System is becoming bigger and bigger; not only do its size and mass − 1 − 1 H � 70 km · s · Mpc , H is Hubble parameter of today, 0 0 increase, but also brightness increases. For example, the distance-modulus is equal to 5lgd + 25, and unit of d is L L Earth is moving away from the Sun at a speed of Mpc. v � H s � 9m/year, namely, following Hubble expansion. 0 0 0 Note that according to the observation of ρ, people Since the Hubble expansion does not change the revolution deduce q � 4πGρ/3H � 0.1 ± 0.05. 0 0 period, the revolution speed of the Earth increases today at a ,e redshift-distance relation derived from the original rate of H v � 61m/year , and, accordingly, the mass of the 0 0 field equation cannot explain the observation, in order to be Sun increases at a rate of 3H M � 4 × 10 kg/year. Here, 0 0 consistent with the observation, dark matter and dark energy s � 1.49 × 10 km is the distance between the sun and the must be introduced temporarily, but such an operation has Earth today, M � 2 × 10 kg is the mass of the Sun, and no scientific value because dark matter and dark energy are v � 30000m/s is the revolution speed of the Earth today. no different from the copy of ether, and in essence, they Again, in addition to the tide, the expansion of Hubble belong to the pseudoscientific concept that can never be recedes the Moon 2.7 cm away from the Earth every year, the verified by experiments. ,e accelerating expansion of the tide only recedes the Moon 1.1 cm away from the Earth, and, universe advocated by some people cannot be consistent meanwhile, the radius of the Earth increases at a speed of with the facts. ,ough the data of the distance and redshift v � H r � 0.5mm/year, the Earthʼs mass increases at a rate 0 0 they measured are right, the theoretical basis for analyzing of 3H m � 1.2 billion tons per year, r � 6400 km is the 0 0 0 these data is wrong; that is to say, the middle derivation from radius of today’s Earth, and m � 5.96 × 10 kg is the data to conclusion is wrong. Earth’s mass of today. ,e Earthʼs rotation is slowing down at a rate of 3.8 cm/year, just because of the tide and not the Hubble expansion. If the 3.8 cm/year is all the effect of tides, 9. Galaxies and Celestial Bodies are Formed by the result calculated according to the theory of tidal damping Gradual Growth rather than by the is that the rotation period of the Earth increases by 1.7 Convergence of Existing Matter millisecond every year, which is inconsistent with the ob- servation, and if the tides make the Moon only 1.1cm away Since the negative pressure is confined to the inner part of from the earth every year, the calculated result is that the the celestial body, the new matter can only generate in the μ Earthʼs rotation period slows down by 0.6 millisecond per celestial body not wherever. Applying T � 0 to a celestial ];μ year, which is in consistence with observation. Of course, body’s interior, we obtain dm � d(ρV) � − pdV � ρdV. these data belong to today and do not represent the past, and Here, V is the volume of the object, m is its mass. if you want to infer the past or future situation, you need to In order to keep the density of the universe unchanged do a similar derivation; I will not discuss it here. during the expansion process, the celestial body must grow Figure 3 is a step-by-step magnification of the Milky with time and its volume satisfies V∝ R (t). From Way. It represents the actual growth process of the Milky dm � ρdV � (m/V)dV, we know m � CV and C is integral 3 Way. With universe expansion, not only do its size and mass constant, so we obtain m∝ R (t); that is, for any two increase, but also its brightness increases. ,at is to say, all moments t and t , 1 2 parts of it have been expanding according to Hubble; at the m t 􏼁 R t 􏼁 same time, new matter is continuously generated in the 1 1 � . (62) celestial bodies. For example, the radius of the galactic disk m t R t 􏼁 (refers to the luminous part) is expanding at a rate of Further, dm � 3Hm. Of course, (62) is also suitable for v � H d � 600m/s; d � 30, 000 light-years is the radius of 0 0 0 0 describing the mass change of a galaxy. So, we get a new the luminous part of the galactic disk today. ,e Solar picture of the evolution of the universe: everything is System is moving away from the center at a rate of expanding in Hubble; not only is the space between v � H r � 450m/s; r � 8.5 kpc is the distance of the Sun 0 0 0 0 galaxies expanding, but the galaxies themselves are to the galactic center. Distance-modulus 10 Advances in Astronomy Figure 2: Schematic diagram of the growth process of the solar system with time. Figure 3: Schematic diagram of Gradual growth of the Milky Way. It is because the Milky Way is formed by gradual growth, L M (63) not by the accumulation of existing matter that its spiral � 􏼠 􏼡 , L M ⊙ ⊙ arms are not getting tighter and tighter; otherwise, they would have been destroyed. L is the luminosity of the star and L and M are, re- ⊙ ⊙ Figure 4 is a step-by-step magnification of a piece of spectively, luminosity and mass of the sun. ,e brightness cosmic space, which represents the actual expansion and temperature of celestial bodies have the following process of cosmic space. ,e white spot in the figure relations: represents galaxies, not only is the space between galaxies 2 4 2 2 4 2 L � 4πr · σT � 4πr · l � 4πd · σT � 4πd · l , (64) expanding, but also the galaxies themselves are expand- e e e e p p p p ing. It tells that the more backward we look, the more where l is absolute brightness of the star, l is its vision evenly matter is distributed, which is just reflected by the e p brightness, d is the distance from the star to us, σ is Ste- microwave background radiation. ,erefore, we say that fan–Boltzmann constant, and T and T are the temperature the microwave background radiation is the comprehen- e p of surface and the vision temperature, respectively. sive effect of redshifted photons emitted by the matter at a Now we treat M as a variable, namely, M∝ R (t). Since distant and indistinguishable distance on our instrument, r ∝ R(t), d ∝ R(t), for the same star, at any two moments e p and these photons have a blackbody spectrum because t and t , we have following relations: they come from different stars. ,is is a simple and re- 1 2 alistic explanation, but it is like a myth to describe it as a 4 10 l t 􏼁 T t 􏼁 l t 􏼁 T t 􏼁 R t 􏼁 p 1 p 1 e 1 e 1 1 relic or sound of the big bang. ,e distant sky we see with � � � � . (65) 4 4 10 l t 􏼁 l t 􏼁 T t 􏼁 T t 􏼁 R t 􏼁 e 2 p 2 the naked eye is uniform, and, similarly, the distant sky we 2 2 2 e p see with the telescope should be also uniform. It is And assume t � t � 1.37 × 10 years, which is our 2 0 shameless to deliberately tie microwave background ra- universe age, then 1 billion years ago t � 1.27 × 10 years, diation with the big bang. and using (52) and the approximated formula x ≈ sin x for It should be noted that the inverse process of the ex- x ⟶ 0, we have pansion of the universe is its contraction, and in the con- traction process all galaxies and space atrophy reversibly. l t 􏼁 l t 􏼁 1.27 p 1 e 1 � � 􏼒 􏼓 � 0.46, For a more detailed discussion of the expansion process l t l t 1.37 􏼁 􏼁 p 2 e 2 of the universe and the fractal structure of galaxies, see the (66) authorʼs paper and related papers [17–22]. 2.5 T t 􏼁 T t 􏼁 1.27 p 1 e 1 � � 􏼒 􏼓 � 0.82, T t 􏼁 T t 􏼁 1.37 p 2 e 2 10. TheTemperatureandBrightnessofCelestial which means that the Sunʼs brightness was less than half of Bodies Are Increasing todayʼs and the temperature of the solar light is 82% of today It is found that the mass of a celestial body is related to its 1 billion ago. For the change of temperature of the surface of luminosity, generally speaking, the greater the mass, the the Earth, we can also roughly estimate to use (15), if the greater the luminosity. For a main sequence star, we have the Earthʼs surface temperature is 25 C (298 K) today, 1 billion following empirical formula: ° years ago the temperature was 246 k (− 27 C), and in 30 Advances in Astronomy 11 Figure 4: Schematic diagram of the generation process of cosmic space. billion year its temperature will reach 6000 k (5727 C), derive the planetary orbit equation from the well-known which is equal to the surface temperature of the Sun today. Schwarzschild metric and, by the way, point out the And as the universe will contract in 3.6 billion years, it can shortcoming in the previous calculation. ,e orbital equa- become reality for the Earth to shine like the Sun today. tion described by the Schwarzschild metric is Similarly, the evolution of gravity acceleration on the 2 2 du a − 1 2GM 2 3 surface of the Earth can be deduced; 1 billion years ago the (69) 􏼠 􏼡 � + u − u + 2GMu , 2 2 dφ acceleration of gravity on the surface was h h R t 􏼁 1.27 2 2 removing the final term, which is Newtonʼs ellipse orbit g t 􏼁 � g t 􏼁 � 10m/s × � 9.2m/s . (67) 1 2 equation. Here, u � 1/r, and h � r dφ/ds � const and a � R t 􏼁 1.37 (1 − 2GM/r)dt/ds � const are two integral constants. ,e Todayʼs atmospheric pressure is 101 kPa, since density derivation of (69) can be found in any textbook and I will not does not change and the height of the atmosphere increases repeat it here. As an initial condition, we can let the peri- following Hubble expansion; then, 1 billion years ago, the helion on the x-axis; then, atmospheric pressure was φ u 􏽱������������������������������� � 2 􏽚 dφ � 􏽚 du, R t 􏼁 2 2 2 3 2 1 0 u 􏼐a − 1􏼑/h + 2GMu/h + 2GMu − u P t 􏼁 � P t 􏼁 � 86kPa. (68) c 1 c 2 R t 􏼁 (70) Equation (66) tells us that planets can evolve into stars; where u denotes the reciprocal of the perihelion distance. this should be the main mechanism of star formation. We On the other hand, according to the theorem of factoriza- usually think that the objects that do not emit light are older tion, we have objects and the luminous objects are younger; this idea 􏽳������������������������ � should be changed. ,e reason why a celestial body does 2 a − 1 2GMu 3 2 not emit light is that its mass is not large enough, and the + + 2GMu − u 2 2 h h second reason is that the material that makes up the ce- (71) 􏽱������������������������ lestial body is too loose. ,e age of a celestial body refers to � 2GM u − ε u − ε u − ε , the time when the celestial body exists as an independent 􏼁 􏼁 􏼁 1 2 3 individual, not the time when the matter that makes up the where ε , ε , ε are the three roots of the cubic equation celestial body exists. ,e chemical composition of a ce- 1 2 3 2 2 2 3 2 (a − 1)/h + 2GMu/h + 2GMu − u � 0. And since lestial body should be determined by its temperature, not 2GMu is regarded as a perturbation, two of ε , ε , ε must be having a direct relationship with the existence time of the 1 2 3 very close to u and u . ,erefore, as an approximation, we celestial body. ,erefore, it may not be appropriate to use 1 2 may as well let ε � u and ε � u , where u and u are the the content of radioactive elements to infer the age of 1 1 2 2 1 2 two roots of the quadratic equation celestial bodies. I do not advocate talking about the concept 2 2 2 2 (a − 1)/h + 2GMu/h − u � 0, which corresponds to the of celestial age. perihelion and the aphelion. Note that there must be du/dφ � 0 at the extreme points. And according to Vedaʼs 11. More Reasonable Derivation of theorem, ε � 1/2GM − u − u ; then, 3 1 2 Orbit Precession 􏽱������������������������ 2GM u − ε u − ε u − ε 􏼁 􏼁 􏼁 1 2 3 In the case of weak field and low speed, the conclusion of (12) 􏽱���������������������������������� � (72) is almost the same as that of the Schwarzschild metric. As � − u − u 􏼁 u − u 􏼁 􏼂1 − 2GM u + u + u 􏼁 􏼃. 1 2 1 2 long as r in (69) is replaced by l(r), the orbit equation described as (12) can be obtained. ,erefore, it is advisable to Next, (70) becomes 12 Advances in Astronomy 1 + GMu + GM u + u 1 2 􏽱�������������� � φ � 􏽚 du 1 − u − u u − u 􏼁 􏼁 1 2 u u d􏽨− u + u + u 􏼁 u − u u 􏽩 GM 1 + 3GM u + u 􏼁 1 1 2 1 2 1 2 􏽱������������������� � 􏽱�������������� � (73) � − 􏽚 + 􏽚 du 2 u 2 u 1 1 − u − u 􏼁 u − u 􏼁 − u + u + u 􏼁 u − u u 1 2 1 2 1 2 􏽱�������������� � 3GM u + u 􏼁 2u − u − u 1 2 1 2 � GM − u − u 􏼁 u − u 􏼁 − 􏼢1 + 􏼣arccos . 1 2 2 u − u 1 2 Obviously, for u � u , φ � π[1 + 3GM(u + u )/2] � Our foothold is still the spherically symmetric metric 2 1 2 2 2 2 π + 3πG M /h , which implies that the processional angle is field. And for a spherically symmetric metric field, no matter 2 2 2 Δφ � 6πG M /h . its source is static, oscillatory, or variable-mass, as long as the And since u + u � 2GM/h , (u − u )/(u + u ) � e, spherical symmetry is kept, the exterior solution is still the 1 2 1 2 1 2 further we have same form, namely, − 1 u + u u − u 1 2 1 2 2Gk 2Gk 2 2 2 2 2 2 2 u � + ds � 􏼠1 − 􏼡dt − 􏼠1 − 􏼡 dλ − λ 􏼐dθ + sin θdφ 􏼑. 2 2 λ λ 􏽱�������������� � (76) 2GM − u − u u − u 􏼁 􏼁 1 2 ⎡ ⎢ × cos⎣ 2 + 3GM u + u 􏼁 ,at is, with t, λ, θ, φ as independent coordinate vari- 1 2 (74) ables, (76) is the solution of the vacuum field equation 3GM u + u R � 0. Do not consider the meaning of λ for the moment, 1 2 μ] − φ/ 1 + 􏼠 􏼡􏼣 and k is only thought of as a constant. ,e proof of (76) is similar to the proof of Birkhoff law; I will not repeat here. 2 2 GM 3G M Equation (76) offers the orbit equation of the planets: ≈ 􏼢1 + e cos􏼠1 − 􏼡φ􏼣, 2 2 h h du a − 1 2Gk 2 3 � + u − u + 2Gku , (77) 􏼠 􏼡 2 2 whose final step takes advantage of the formula cos(α − β) � dφ 􏽰��������������� h h cos α cos β + sin α sin β and2GM − (u − u )(u − u )≪ 1, 1 2 whose derivation is the same as (69). However, here u � 1/λ 1/[1 + 3GM(u + u )/2] ≈ 1 − 3GM(u + u )/2. 1 2 1 2 It should be pointed out that the second-order ap- and h � λ dφ/ds � const.a � (1 − 2Gk/λ)dt/ds � const. Similar to (74), we have proximate solution obtained by using 1/r � u � 2 2 2 (1 + e cos φ)G M /h as the first-order approximation is 2 2 Gk Gk 3G k wrong, that is, the following (75) is wrong: u � + e cos 1 − φ. (78) 􏼠 􏼡 2 2 2 h h h GM 3 3 3 u � (1 + e cos φ) + G M eφ · sin φ. (75) 2 4 ,e above is the result of the coordinate system h h (t, λ, θ, φ), and our purpose is to solve the orbit equation in ,e shortcoming of (75) is that when φ � 2nπ, the orbital the coordinate system (t, r, θ, φ). To this end, we introduce two crossover points with the x-axis are always invariant, so the coordinate transformation. λ � l/R(t) and meanwhile set the shape of the ellipse is not guaranteed when it rotates, and k � M/R (t), where R(t) is the cosmic scale factor, and M � 2 2 2 the precession angle Δφ � 6πG M /h cannot be obtained M(t) is the mass of the central celestial body and satisfies from (75) when φ is quite big; that is to say, the transition (62). And in the light of (11), l � l(r, t) satisfies − 2 2 2 − 2 from (75) to u � GMh [1 + e cos(1 − 3G M h )φ] can- not be realized. In short, using (75) to explain the precession 7GM(t) l � r − 2GM(t)ln[r − 2GM(t)] − + 2GM(t)ln r (t), of Mercury is not only grudging but also causing serious other problems. And again, Einsteinʼs original calculations (79) were also ambiguous and cannot obtain the correct pro- Of course, without considering the expansion of space- cessional angle according to Einsteinʼs calculation [14]. time, all equations must go back to the previous. Now (77) is transformed into 12. Planetary Orbit Equations of Giving 2 2 R(t) R(t) Gk Gk 3G k Consideration to the Expansion of Space- ≈ � + e cos􏼠1 − 􏼡φ, (80) 2 2 2 time: The Evolution of Planetary Orbit l(r, t) r h h h Now, let us look at the orbital equation of planets in which is just the orbital equation of planets and shows that expanding space-time, which is also the equation that de- while planets are moving around the center, they recede termines the formation and evolution of galaxies. from the center in Hubble. Advances in Astronomy 13 Figure 5: ,e schematic diagram of the speed distribution of matter in the Milky Way. 3 2 Besides, give consideration to Keplerʼs law a /T � GM, reached without being thrown out of the galaxy. Because the 3 3 3 since a ∝ � R (t) and M∝ R (t); then, T � const. ,at is thickness of the galactic disk decreases slowly, its speed does to say, the period of motion of planets does not change and not weaken, which is normal. We have no reason to deny − 24 3 the speed of planets increases gradually while they go away that the density of halo can reach 9 × 10 kg/m ; such from the center. density is extremely thin. In a word, there is no need to introduce dark matter; let alone a black hole. Figure 5 shows the speed distribution of matter in the 13. Modern Observations Do Not Confirm the Milky Way, the red line represents the result without Existence of Dark Matter considering halo mass, and the white line represents the observation result. ,e white line represents the observation Take the Milky Way as an example. It is composed of result and is also the result of the calculation of considering galactic ball, galactic disk, and galactic halo. Near the the halo mass. center, the material distribution is dense, so the galactic ball I do not think dark matter, dark energy, and black holes can be treated as a rotating rigid body, so that it is natural exist. Although peopleʼs observation technology is con- that the velocity of matter at the center is proportional to stantly improving and data is constantly accumulating, the radius, and there is no need to assume that dark matter peopleʼs interpretations of the observed phenomena and or black hole exists. ,e calculated velocity of the material data are basically wrong. ,e reason for this lies in the in the halo is lower than the measured velocity because the contradiction between these interpretations. Dark matter, mass of the halo itself is ignored; that is to say, once dark energy, and black holes have pushed science into considering the mass of the halo, there is no need to assume metaphysics, which is not progress but retrogression. It is dark matter or black hole, too. Here is a rough estimate of shameless for those who deliberately bind the correct the velocity of the material in the halo. Since the halo is conclusion of general relativity with the contemporary spherical with a radius of about 100,000 light-years, at etheric, namely, dark matter, dark energy, and black holes. rfrom the center, the acceleration of gravity provided by the It is imperative to separate general relativity from these halo itself is (let us not consider the effect of the mass of the absurd sermons. People attribute the incomprehensible dish): phenomena to that dark matter, dark energy, and black GM(r) 4πGρr v holes are the passivation of human intelligence. In a word, (81) g � � � , 3 r r all singularity physics is unreal, no matter how many halos 􏽰������ � it has. where ρ is the density of the halo, v � r 4πGρ/3 is the speed of a moving particle around the galactic center, and probably Data Availability as well setting v � 30km/s and r � 70, 000 light-years, we obtain ,e data used to support the findings of this study are available from the corresponding author upon request. 3v − 24 3 (82) ρ � � 9 × 10 kg/m . 4πGr Conflicts of Interest ,at is to say, as long as the halo density reaches − 24 3 9 × 10 kg/m , the particle can maintain the speed of ,e author declares that there are no conflicts of interest. 30 km/s to make a circular motion and not be thrown out of the galaxy. It is possible that the density of the halo can reach Acknowledgments − 24 3 9 × 10 kg/m , and considering that the galactic ball and disk also have mass, even if the mass density of the halo is ,e study was supported by the National Key Research and − 24 3 smaller than 9 × 10 kg/m , the speed 30km/s can still be Development Plan (973 Plan), no. A030101. 14 Advances in Astronomy References [1] L. Lorio, “Gravitational anomalies in the solar system,” In- ternational Journal of Modern Physics D, vol. 24, no. 6, pp. 1–37, 2015. [2] M. Ness and D. Lang, “,e X-shaped bulge of the Milky way revealed bywise,” %e Astronomical Journal, vol. 152, no. 1, p. 14, 2016. [3] C. Martinez-Lombilla and I. Trujillo, “Discovery of disc truncations above the galaies’smid-plane in Milky May-like galaxies,” Monthly Notice of the Royal Society, vol. 483, no. 1, pp. 664–691, 2019. [4] J. T. Nielsen, A. Guffanti, and S. Sarkar, “Marginal evidence for cosmic acceleration from type Ia supernovae,” Science Reports, vol. 6, pp. 1–8, 2016. [5] D. Herwartz, A. Pack, D. Krylov et al., “Revealing the climate of snowball Earth from Δ17O systematics of hydrothermal rocks,” in Proceedings of the National Academy of Sciences, vol. 112, no. 17, pp. 5337–5341, 2015. [6] S. M. Som, R. Buick, J. W. Hagadorn et al., “Earthʼs air pressure 2.7 billion years ago constrained to less than half of modern levels,” Nature Geoscience, vol. 9, no. 6, pp. 448–451, [7] S. Weinberg, Gravitation and Cosmology, Wiley, New York, NY, USA, 2013. [8] S. Carroll, Lecture Notes on General Relativity, Columbia University, New York, NY, USA, 2013. [9] L. D. Landau, %e Classical %eory of Fields, Pergmon Press, Oxford, UK, 1987. [10] Einstein, %e Meaning of Relativity, Princeton University Press, Princeton, NJ, USA, 1922. [11] B. C. Tolman, Relativity, %ermodynamics and Cosmology, Oxford Clarendon Press, Oxford, UK, 1934. [12] J. L. Yang, “Criticism to universal big bang,” Astrophys and Aerospace Technology, vol. 4, p. 1, 2016. [13] T. Felicead, “f (R) theories,” Living Reviewing in Relativity, vol. 13, no. 1, p. 3, 2012. [14] X. Mei and P. Yu, “Did LIGO really detect gravitational waves?” Journal of Modern Physics, vol. 7, pp. 1098–1104, [15] P. Bhar and N. Pant, “Relativistic anisotropic stellar models with Tolman VII spacetime,” Astrophysics and Space Science, vol. 359, no. 1, 2015. [16] M. Yang, “Modification of gravitational field equation and rational solution to cosmological puzzles,” International Journal of Physical Science, vol. 5, no. 2, 2010. [17] J. L. Yang, “Unavoidable correction to the coupling constant in Einstein field equation,” International Journal of Advanced Research in Physical Science, vol. 6, no. 11, pp. 4–30, 2019. [18] J. Gaite, “,e fractal geometry of the cosmic web and its formation,” Advance in Astronomy, vol. 1, p. 25, 2019. [19] J. de Haro, A. Paliathanasis, and R. J. Slagter, “Evolution and dynamics of a matter creation model,” Monthly Notices of the Royal Astronomical Society, vol. 460, no. 2, pp. 1445–1456, [20] S. N. Gurbatov and A. T. Saichev, “Large-scale structure of the universe,” Physics-Uspekhi, vol. 55, p. 3, 2012. [21] S. L. Blibbikov and A. D. Dolgov, “Cosmological accelera- tion,” Physics-Uspekhi, vol. 62, no. 6. [22] J. L. Yang, “Light speed invariant solution and its enlight- enment of field equation of general relativity,” Advances in Astronomy, vol. 2020, Article ID 3930947, 12 pages, 2020. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Astronomy Hindawi Publishing Corporation

Modification of Gravitational Field Equation due to Invariance of Light Speed and New System of Universe Evolution

Advances in Astronomy , Volume 2021 – Mar 20, 2021

Loading next page...
 
/lp/hindawi-publishing-corporation/modification-of-gravitational-field-equation-due-to-invariance-of-HdGEkZJB5A
Publisher
Hindawi Publishing Corporation
Copyright
Copyright © 2021 Jian Liang Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN
1687-7969
eISSN
1687-7977
DOI
10.1155/2021/5579060
Publisher site
See Article on Publisher Site

Abstract

Hindawi Advances in Astronomy Volume 2021, Article ID 5579060, 14 pages https://doi.org/10.1155/2021/5579060 Research Article Modification of Gravitational Field Equation due to Invariance of Light Speed and New System of Universe Evolution Jian Liang Yang College of Physics, Zhengzhou University, Zhengzhou 450001, China Correspondence should be addressed to Jian Liang Yang; bps267890@qq.com Received 16 February 2021; Revised 4 March 2021; Accepted 9 March 2021; Published 20 March 2021 Academic Editor: Ghulam Abbas Copyright © 2021 Jian Liang Yang. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We make a systematic examination of the basic theory of general relativity and reemphasize the meaning of coordinates. Firstly, we prove that Einsteinʼs gravitational field equation has the light speed invariant solution and black holes are not an inevitable prediction of general relativity. Second, we show that the coupling coefficient of the gravitational field equation is not unique and can be modified as 4πG to replace the previous − 8πG, distinguish gravitational mass from the inertial mass, and prove that dark matter and dark energy are not certain existence and the expansion and contraction of the universe are proven cyclic, and a new distance-redshift relation which is more practical is derived. After that, we show that galaxies and celestial bodies are formed by gradual growth rather than by the accumulation of existing matter and prove that new matter is generating gradually in the interior of celestial bodies. For example, the radius of the Earth increases by 0.5mm every year, and its mass increases by 1.2 trillion tons. A more reasonable derivation of the precession of planetary orbits is given, and the evolution equation of planetary orbits in the expanding space-time is also given. In a word, an alive universe unfolds in front of readers and the current cosmological difficulties are given new interpretations. tides, the moon still has an unexplained retreat, and the 1. Introduction increase of the day length is also inconsistent with the Although general relativity has made some remarkable prediction of the tide theory. Recently, Melissa Ness and her achievements, some basic problems have not been well colleagues have observed that there is a fine X-shaped box solved, such as the physical meaning of the coordinates of structure in vortex galaxies similar to the Milky Way [2]. Schwarzschild metric, whether general relativity is the Melissa Ness said that this structure implies that large curved theory of space-time or the theory of gravity in flat galaxies are not formed by the merger of small galaxies, space-time, whether the constant speed of light is also because once the merger occurs, the structure will inevitably tenable in the gravitational field, the singularity problem of be destroyed, and we must abandon the existing theory of the field equation, and whether the existence of black holes is galaxy formation and establish a new logic system. ,e true. However, only these basic problems have plagued the observations of Martinez-Lombila and others [3] show that the radius of disk galaxies similar to the Milky Way galaxy is development of general relativity but also led to some confusion in practice; for example, on the one hand, the expanding at a speed of 500 m/s; such a high speed cannot be radial coordinates of Schwarzschild metric are not inter- the speed at which matter accumulates at the edge. If matter preted as the normal radius, while, on the other hand, the accumulates at this speed at the edge, it should be the same radial coordinates on the solar surface are treated as the everywhere on the disk. Obviously, the current theoretical radius of the sun in calculating the curvature of light on the framework cannot explain such a rapid expansion of the surface of the sun, resulting in conceptual confusion. In radius of the disk. ,ere is also the problem of dark matter addition, there are some new observations that are not and dark energy; the reason why we need them is that the accommodated by the current gravity theory. As Lorio [1] observed phenomena do not conform to the prediction of pointed out, there is an unexplained increase in the distance the theory, but, no one has seen them really. ,en, whether between the Sun and the Earth, and after considering the they are real or the theory itself needs to be modified is also 2 Advances in Astronomy an unavoidable problem. ,e latest observation data of In this paper, we use natural units, the speed of light of Nielsen and others [4] show that the universe is expanding at flat space-time c � 1, and it is agreed that flat space-time a constant speed rather than accelerating, so whether the linear element is universe accelerates or decelerates or expands at constant 2 μ ] 2 2 2 2 2 2 ds � g dx dx � dt − dr − r 􏼐dθ + sin θdφ 􏼑. (1) μ] speed still needs to be reconsidered. Besides, some new studies of frontier disciplines [5, 6] have shown that 1 billion According to general relativity, in a spherically symmetric years ago, the brightness of the sun was less than half of what gravitational field, in the coordinate system (t, r, θ, φ), the it is today, the Earth is an ice ball, and the mountain is not as general form of space-time line elements is [7–11] high as it is today, and 2.7 billion years ago, the air pressure 2 μ ] 2 on the Earth was only half of todayʼs. ,ese seem to be purely ds � g dx dx � B(r, t)dt − Q(r, t)dtdr μ] (2) geophysical problems, which can only be reasonably 2 2 2 2 − A(r, t)dr − D(r, t)􏼐dθ + sin θdφ 􏼑. explained from the perspective of cosmology because the evolution of the Earth is an epitome of the evolution of the ,e condition of this formula is only spherical symmetry, universe and the Earth must be reflected by cosmological which is applicable to the gravitational field of both static and events. On the contrary, the phenomena on the Earth can be oscillating gravitational sources. In this paper, we will just deal used to test the cosmological theory more accurately and with the static gravitational field, which is what Newtonian people do not have to go far to test the theory of cosmology. gravity describes. For the static case, g no longer contains μ] In a word, we are faced with some new problems that cannot time. Besides, the static case requires time version to be be avoided. We will see that when the speed limit of light, that symmetric, so g � Q(r) � 0. ,erefore, for the static case of is, the speed of light always 1 (in natural units), is still satisfied spherical symmetry, the space-time line element is in the gravitational field, the above problems can be solved in a 2 μ ] 2 2 2 2 2 package. ,e author thinks that it is a great mistake of general ds � g dx dx � B(r)dt − A(r)dr − D(r)􏼐dθ + sin θdφ 􏼑. μ] relativity that the invariance of the speed of light in the (3) gravitational field is not emphasized in the past, and it is this fault that leads to a series of misconceptions and absurd results; We just have to solve for three functions B(r), A(r), and for example, it is necessary to admit singularity as physical D(r). In order to ensure that the meaning of coordinates is reality, which will never be allowed in other parts of physics. In always clear and unchanged, this paper will not continue to a word, it is shameless to tie the correctness of general relativity simplify (3) into the so-called standard form through co- with some wrong conclusions such as big bang and black holes, ordinate transformation but directly solve with the gravi- and it is shameless to praise mistakes as successes. Leading to tational field equation. Firstly, determine the external the big bang, black holes and all kinds of other singularities are solution that satisfies the vacuum field equation R � 0, and μ] not the success of general relativity, but its failure. ,e reason is then the source internal solution is determined. simple: there is no singularity in real nature. No matter how In order to reflect the invariance of light speed, we re- much you boast big bang and black holes, they cannot be true. quire A(r) � B(r). From the following solving process, we ,e author thinks that if these absurd things are not stripped can see that such a solution not only exists but also is unique. away from general relativity, there will be no real progress in Equation (3) provides general relativity, the field of astrophysics will be dominated by g � B(r), all kinds of idealism, and more and more young students will be misled into the wrong way. In order to deal with these g � − A(r), problems systematically, to get to the bottom and bring order out from chaos, this paper begins with the most basic problem, g � 0(μ≠ ]), μ] that is, solving the metric of the spherically symmetric grav- itational field represented by coordinates in the usual sense. g � − D(r), 2. Spherically Symmetric Static Metric g � − D(r)sin θ, Represented in Usual Coordinates g � − , We just have to solve for the metric form in the usual (4) D(r) spherical coordinates; the form in other coordinates can be obtained by coordinate transformation. Indices g � − , μ 0 1 μ, ], λ, α, β � 0, 1, 2, 3. Space-time coordinates x � (x , x , D(r)sin θ 2 3 0 1 2 3 x , x ) � (t, r, θ, φ) and x � t, x � r, x � θ, x � φ repre- sent the usual time, radius, and pole angles, respectively. g � , ,ey have the same meaning as in quantum mechanics or B(r) electrodynamics. In the language of the observational theory of general relativity, t is the time recorded by a stationary g � − , A(r) observer at infinite distance, r is the distance the observer measures from the origin to another point, and θ, φ are the μ] g � 0(μ≠ ]). polar angles measured by the observer. Advances in Astronomy 3 According to the definition of connection, infinity, there must be C � 1/4, namely, AB � D /4 D. λ λα ] μ α Γ � (1/2)g (zg /zx + tzg /zx n − qzg /zx ), the And inserting A � D /4B D into equation (8) gets μ] αμ α] μ] repeating indices up and down means summing from 0 to 3, ′ ′ D D and it is not hard to figure out all of its nonzero connections ′ (10) B + B − � 0, 2 D 2 D as follows [2–7]: which is a differential equation with respect to B. Writing 1 zD 1 2 Γ � − sin θ, D � l , the general solution of (10) is given by B � 1 + C /l. 2A zr C is an integral constant. Because we must return to Newton gravitation in the distance, we have C � − 2GM. G cos θ 3 2 Γ � , Newtonʼs gravitational constant M is the mass of the source. sin θ It is important to insert B � 1 − 2GM/l and 1 zA ′ ′ 1 A � D /4B D � l /(1 − 2GM/l) into any one of (6)–(8). Γ � , You can obtain an identity with respect to l � l(r); namely, 2 zr no matter what l � l(r) is, this identity can be tenable, so we 1 zB can pick an appropriate l(r) so that A � B. And letting Γ � , 2B zr (5) A � B, namely, 1 − 2GM/l � dl/dr, we obtain 1 zB r � C + l + 2GM ln(l − 2GM), (11) Γ � , 2A zr where C is the integral constant and can be decided by the 2 3 continuity of l(r) on the surface of the source. In Section 4 of Γ � Γ 12 13 this paper, C will be calculated. So far, we obtain the ex- terior line element: 1 zD Γ � − , 2A zr 2GM 2GM 2 2 2 2 2 2 2 ds � 1 − dt − 1 − dr − l dθ + sin θdφ , 􏼒 􏼓 􏼒 􏼓 􏼐 􏼑 l l Γ � − sin θ cos θ. (12) According to the definition of curvature tensor, where l � l(r) can be inversely solved from (11). It can be λ ] λ λ α λ α λ R � zΓ /zx − zΓ /zx + Γ Γ − Γ Γ ; for μ≠ ], we μ] μλ μ] μλ α] μ] αλ seen from (11) that when l � 2GM, r becomes negative, have R � 0, which means that the vacuum field equation is μ] which means l − 2GM is always greater than 0 and so there is automatically satisfied. It is not hard to figure out all of the no horizon and no black hole. And considering nonzero components of R . Write A � dA/dr, μ] lim (ln x/x) � 0, we have r � l for l ⟶ ∞. 2 x⟶∞ 2 2 2 ″ ′ A � d A/dr , A � (dA/dr) , and note that R � R sin θ; we are left with the following three equa- 33 22 3. Link with the Mechanics of Special Relativity tions about B(r), A(r), and D(r): ,e invariance of the speed of light described by (11) is easy ′ ′ ′ ′ ′ ′ B ′ B B A D B to see. Suppose that the photon moves in the radial direction, (6) R � − 􏼠 􏼡 + − 􏼠 + + 􏼡 � 0, 2A 2AB 2A 2A D 2B dφ � dθ � 0, ds � 0, and from (11), we can get 2 2 dr /dt � B(r)/A(r) � 1, namely, dr/dt � ± 1, which shows 2 2 that radial speed of light is constant. And now we look at the ′ ′ ′ ′ ′ ′ ′ ′ D B D B A D B ⎝ ⎠ ⎛ ⎞ R � 􏼠 + 􏼡 + + − 􏼠 + 􏼡 � 0, 2 2 light moving tangentially and set θ � π/2, dr � 0, ds � 0; D 2B 2A D 2B 2D 4B the tangential speed of light is given from (12) √��������� (7) byrdφ/dt � (r/l) 1 − 2GM/l. And (11) tells us that the √��������� smaller 1 − 2GM/l than 1, the larger r/l than 1, so the √��������� ″ ′ ′ ′ D D A B deviation of (r/l) 1 − 2GM/l from 1 is actually very small, R � + 􏼠− + 􏼡 − 1 � 0. (8) and you can still think of it as 1. It is in this sense that we say 2A 4A A B that (11) describes the invariant speed of light, not strictly From R × (1/B) + R × (1/A) � 0 we obtain constant. ,is slight change of the tangential speed of light 00 11 causes light to bend near a celestial body; otherwise, it travels ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ AB + A B D D D (AB) D D in straight lines. ,e previous result is − + + � − + 􏼠 􏼡 􏼠 􏼡 􏼠 􏼡 √��������� 2AB D D 2AB D D 2D rdφ/dt � 1 − 2GM/r which can be obtained from (28), and it is not hard to find that when r � 2GM, the tangential D speed of light is zero and the deviation from 1 is severe; + � 0. although it means also that light can bend, it does not reflect 2D the invariance of the speed of light. (9) ,e correctness of (12) is not only in the invariance of the Equation (9) is a differential equation with respect toAB, speed of light but also in the natural connection with rel- its general solution is AB � C D /D, and C is the integral ativistic mechanics under weak field approximation. 1 1 ′ ″ constant. Since A � B � 1, D � r , D � 2r, D � 2 at Equation (12) provides 4 Advances in Astronomy dv Γ � 0, 2 € _ _ _ € � 􏼐r − rφ 􏼑e + (2rφ + rφ)e , r φ dt GM dl GM 2 1 0 dv Γ � Γ � � , 2 2 11 01 2 2 (13) � 2r _r € + 2r φ _ φ € + 2rr_φ _ , dr (21) (1 − 2GM/l)l l dt GM d(mv) dv dm dv dv Γ � . � m + v � m + mv . 2 2 dt dt dt dt l 2 1 − v dt 􏼐 􏼑 ,e dynamic equation describing the motion of free Using (20), we have particles is the geodesic equation, the proper time must be 2 2 2 2 2 eliminated when solving for acceleration, and the geodesic 1 − r φ _ r r _φ _ φ € φ _ rr _ GM (22) € _ r + + − rφ � − , 2 2 2 2 equation after the elimination of proper time is 1 − v 1 − v 1 − v r 2 μ ] λ ] λ μ d x dx dx dx dx dx μ 2 2 3 (14) + Γ · − Γ · · � 0, r − rr_ rr _φ _ r € r φ _ r _ ]λ ]λ (23) dt dt dt dt dt φ € + + + 2r _φ _ � 0, dt 2 2 2 1 − v 1 − v 1 − v which is derived in detail in post-Newtonian mechanics [9]. eliminating φ € in the use (22) ×(1 − r_ )− (23) ×rφ _ r _ gets Let the particle move on the plan θ � π/2, set μ � 1, 3, and write dr/dt � v , dφ/dt � v /r; we have r φ GM GM (24) r − + − v � 0, 2 2 2 2 2 lv r r d r GM GMv 1 1 2 0 2 1 2 � − Γ − Γ r _ + 2Γ r_ + Γ φ _ � − + + , 2 00 11 01 33 2 2 2 dt l l r which is exactly equation (17), and inserting it into (23), we (15) get 2 GM € (25) ′ φ + v v − v v � 0, d φ 2l 1 2GM r φ r φ 3 0 2 3 � − 2Γ r _φ _ + 2Γ r_φ _ � − v v + v v . r r r φ r φ 2 13 01 2 rl 1 − 2GM/l dt l r which is (19) omitting the higher-order small GMv v /r . So r φ (16) far, it can be seen that (12) in the weak field approximation In the weak field or in the distance, 2GM/l≪ 1, l ⟶ r, can link well with the special relativistic mechanics. 2 2 l ⟶ r , l ⟶ 1, (15) and (16) become, respectively, And again, when the particle moves along the radial direction, if v � 1, the acceleration of the particle is equal to 2 2 d r GM GMv zero, which means that the invariance of light speed and the (17) + − − � 0, 2 2 2 dt r r light speed limit are unified. d φ 2 2GM (18) + v v − v v � 0. 4. The Light Speed Invariant Solution within a r φ r φ 2 2 3 dt r r Spherically Symmetric Static Gravity Source Ignoring the higher-order small quantity 2GM/r , (17) Now, we solve the interior A(r), B(r), and D(r). In order to becomes keep the constant speed of light, we still require A(r) � B(r) d φ 2 inside the source. From the following solution’s process, we (19) + v v � 0. r φ 2 2 know such a solution not only exists but also is unique. Note dt r that the constant speed of light means that the speed of light It is not difficult to prove that (17) and (19) are just the passing through the cavity in the source is 1, but that passing relativistic Newton equations of gravity: through the medium is 1. ,e equation of gravitational field in the source is d(mv) GMm � − e , (20) dt (26) R � c􏼒T − Tg 􏼓, μ] μ] μ] √����� where m � m / 1 − v is the motion mass of the particle, 2 2 2 _ _ v � r_e + rφe , v � r _ + r φ , and e , e are the base vectors. where c is the coupling constant, and that we do not write its r φ r φ Prove as follows: from theoretical mechanics, we know specific value here is to lay a hint for the following Advances in Astronomy 5 modification of the constant. And T � (ρ + p)u u − pg the identity with respect to l � l(r); that is, this is true re- μ] μ ] μ] is the energy-momentum tensor of the source. Note that gardless of the function of l(r), regardless of the value of c. μ μ ] μ μ] μ] u � dx /ds, u � g u , u u � 1, g g � 4, T � g T � So, we can pick a function l(r) by the following equation (41) μ μ] μ μ] μ] ρ − 3p. to make A(r) � B(r): i i And for the static source, u � dx /ds � 0, i � 1, 2, 3. ] 2 0 2 l l ,en, u � g u � 0, 1 � B(dt/ds) � B(u ) , 2GM i i] − 2 2 √� � 􏼠1 − 􏼡exp 􏽚 − cA 􏼠pl + l 􏽚 l ρdl􏼡dl 0 2 u � g u � B, T � pA, T � p D, T � p D sin θ, 0 00 11 22 33 l l 0 e e T � ρB; from (26), we have (35) − 1 ′ ′ ′ ′ ′ ′ − 1 2 B ′ B B A D B c � 1 + l c 􏽚 ρl dl l . 􏼠 􏼡 R � − 􏼠 􏼡 + − 􏼠 + + 􏼡 � (ρ + 3p)B, 2A 2AB 2A 2A D 2B 2 (27) On the other hand, we know T � 0, for the static μ;] source; it is [2–7] 2 2 ′ ′ ′ ′ ′ ′ ′ ′ D B D B A D B ⎛ ⎝ ⎞ ⎠ √� � zp z R � 􏼠 + 􏼡 + + − 􏼠 + 􏼡 2 2 (36) D 2B 2A D 2B � − (ρ + p) ln B. 2D 4B μ μ (28) zx zx � (ρ − p)A, 5. Modification of the Coupling Constant and ″ ′ ′ ′ D D A B c the Internal D(r) R � + 􏼠− + 􏼡 − 1 � (ρ − p)D. (29) 2A 4A A B 2 When the pressure at the surface of the gravitational source In the use of (R /2B) + (R /2A) + (R /D) � ρc, we is set as zero, it can be seen from (39) that once the geodesic 00 11 22 get equation is required to return to Newtonian gravity under the weak field approximation, the coupling constant c must ″ ′ 1 2D D 1 2 D 􏼒 􏼓 + 􏼠 − 􏼡 − − 2cρ � 0, (30) be − 8πG, which is the previous result. But this result should ′ ′ ′ A D 2 D A D D be considered as a mistake because it leads to a lot of sin- gularities that should not occur. For example, when the ratio which can be looked as a differential equation with respect to of the mass to the radius of an object is 2GM/R> 8/9, the 1/A. And writing D � l , in the requirement of ensuring D � pressure inside the object becomes infinite [7–9], which is 0 and A being limited at origin, the solution of (31) is given obviously absurd [10–16]. ,e root of all kinds of singularity by is the improper selection of − 8πG. As we will see, when the − 1 c pressure is taken negative, the coupling constant is identified ′ (31) A � 􏼠1 + 􏽚 ρl dl􏼡 l , as 4πG, which not only avoid singularity of Schwarzschild l 0 metric but also remove in a package cosmological problems. 2 l 2 − 1 − 1 2 where l � (dl/dr) . Writing A � (1 + l c 􏽒 ρl dl) , we I need to say a few words about the negative pressure. have Einstein did not refuse the negative pressure. In the book %e Meaning of Relativity, Princeton University Press Published, ′ ′ ′ ′ ″ A � A l + 2A l l , 1 1 1922, (page 117), for T � ρu u , Einstein said, we “shall μ] μ ] add a pressure term that may be physically established as ′′ ′ ″ 􏼐l 􏼑 � 2l + 2ll , follows. Matter consists of electrically charged particles. On (32) the basis of Maxwellʼs theory these cannot be conceived of as l dB electromagnetic fields free from singularities. In order to be B � . consistent with the facts, it is necessary to introduce energy dl terms, not contained in Maxwellʼs theory, so that the single Insert these into (30), we obtain electric particles may hold together in spite of the mutual repulsion between their elements, charged with electricity of dB − 2 2 � − cA 􏼠pl + l 􏽚 l ρdl􏼡. (33) one sign. For the sake of consistency with this fact, Poincare Bdl 0 has assumed a pressure to exist inside these particles which balances the electrostatic repulsion. It cannot, however, be To make B(r) continuous on the boundary of source, the asserted that this pressure vanishes outside the particles. We solution of (33) is shall be consistent with this circumstance if, in our phe- l l nomenological presentation, we add a pressure term. ,is 2GM − 2 2 B � 􏼠1 − 􏼡exp 􏽚 − cA 􏼠pl + l 􏽚 l ρdl􏼡dl, (34) must not, however, be confused with a hydrodynamical l 0 pressure, as it serves only for the energetic presentation of where l � l(r ) is the value on the boundary and r is the the dynamical relations inside matter.” From this statement, e e e radius of the source. it can be seen that Einstein did not equate pressure as a It is important that, similar to the external solution, source of gravity with the dynamic pressure of a fluid but substituting (32) and (34) into any of (28)–(30), you can get regarded it as a phenomenological representation of all the 6 Advances in Astronomy action within matter, including the electromagnetic force. It 14 l � r − πGρr , e e is not surprising that a negative value is taken. Now solve for D(r) in the source with negative pressure and in the meantime determine the coupling constant. For 3 r � l + πGρl , (42) e e e the convenience of calculation, we use the average density 9 instead of the density of each point that is to take the interior 3 3 5 ρ � const. ,e density itself is a statistical average, such l � r − πGρr . e e e treatment is equivalent to treating the whole celestial body as a statistical volume element, so it is suitable for some not too Inserting (42) into (40), we obtain large celestial bodies as such the sun, and it is also an ap- 2 2 4πρ 80π ρ G proximation for larger celestial bodies. 3 5 (43) M � r − r + · · · . e e When ρ is regarded as a constant, p � − ρ is the solution 3 9 of (36). Since the geodesic under the weak field approxi- Writing 4πρr /3 � M , we have e i mation must return to Newtonian gravity, there must be 5G Γ ⟶ GM/r on the surface of the source, and we conclude 2 M � M − M + · · · , that, from (33), the coupling constantc � 4πG. Notice that r under the weak field approximation l ⟶ r, A ⟶ 1, (44) 2 3 5G 4π 􏽒 ρl dl ⟶ M, and p � − ρ ⟶ − 3M/4πr . 2 M � M + M + · · · , With p � − ρ, c � 4πG, and l as the fixed value, the r e e integral of both sides of (35) is easy, and we get where M is the inertial mass and M is the gravitational mass. 􏽳���������� � dl 1 − 2GM/l 4πGρl ,e reason why M is called inertial mass is that ρ represents � 􏼠1 + 􏼡. (37) the inertial density of matter measured in comoving coordinate dr 3 1 + 4πGρl /3 system, while M is introduced from the perspective of gravity, ′ so it is natural to call it gravitational mass. Ensuring B � 0 for r � 0, which is because the force on the particle at the origin is zero, the solution of (37) is Equation (44) distinguishes gravitational mass from 􏽳���� � 􏽳����������� 􏽲���� � inertial mass, which is the result of an in-depth discussion in 3 4πGρ 1 − 2GM/l this paper, and for high-density celestial bodies, the dif- arctg􏼠l 􏼡 � r . (38) 4πGρ 3 1 + 4πGρl /3 ference is obvious. We see l � r − 7GM/6 + · · ·, and so far, the integral e e On the other hand, we require Γ to be also continuous constant C in (11) can be decided according to the con- on the boundary, which is a necessary condition to ensure tinuity on the boundary; that is (11) is applied to the surface the continuity of gravity on the boundary. So, there exists on of the source the boundary C � r − l − 2GM ln l − 2GM � r − l 􏽳���������� � 3 e e e e e zB zl 8πGρl 1 − 2GM/l 2GM 1 1 e e (45) Γ � � � Γ � . 00 2 00 2 7GM B zl zr 3 1 + 4πGρl /3 l e e − 2GM ln r − 2GM􏼁 ≈ − 2GM ln r . e e (39) 􏽰�������� � Inserting C into (11), we can complete the calculation of 2 2 Solve (39), M � G k + kl − Gk, in which Mercury precession and ray bending, and the calculated 2 2 5 2 k � 16π ρ l /(9 + 12πGρl ). e e result is that the difference between the new results and the ,e explicit form of M can be solved under the weak field original ones is very small and completely consistent with approximation, and expanding the square root in (39) by the observation. ,e concrete calculation is not written here, Taylor, we obtain readers can do it by themselves. In a word, with the new coupling constant 4πG, the 4πρl 8 e 2 2 5 (40) M � − π ρ Gl + · · · . gravitational field equation is now modified as 3 3 (46) Applying (39) on the boundary and taking the ap- R � 4πG􏼒T − Tg 􏼓, μ] μ] μ] proximation M � 4πρl /3 into, we have and correspondingly the pressure p as the source of grav- 􏽶���������� � 􏽳���� � 􏽲���� � itation takes negative. Multiplying the two sides of (46) with 3 4πGρ 1 − 8πGρl /3 μ] g , we have R � − 4πGT, so the equivalent form of (46) is (41) arctg􏼠l 􏼡 � r . e e 4πGρ 3 1 + 4πGρl /3 R − (1/2)Rg � 4πGT . μ] μ] μ] Obviously, for 1 − 8πGρl /3 � 0, r ⟶ ∞; that is to say, 6. Further Interpretation of the Physical e e no matter how big the radius of the celestial body, there is Meaning of the Negative Pressure always l < 3/8πGρ; we do not have to worry about whether l e e has a solution. And expanding the two sides of (41) by Taylor Einstein did not interpret the pressure term in a gravitational and taking second-order approximation, we obtain source as the dynamic pressure of a fluid, but as a Advances in Astronomy 7 phenomenological representation of the pressure within a usually. So, we say that dark energy is just the binding energy matter to balance the electromagnetic force and prevent of matter, rather than an independent existence. charged particles from being disintegrated by electrical re- pulsion, which we should accept. Einstein did not point out 7. The Problems of Schwarzschild Metric that pressure is produced by which power. Today, it is easy to infer and prevent the disintegration of the protons and Schwarzschild metric is neutrons are strong; preventing electronics is the disinte- − 1 2GM 2GM 2 2 2 2 2 2 2 gration of the weak force. ,erefore, the pressure term ds � 􏼒1 − 􏼓dt − 􏼒1 − 􏼓 dr − r 􏼐dθ + sin θdφ 􏼑. r r should be understood as a phenomenological representation (47) of the combined effects of the strong, weak, electromagnetic, gravitational, and all other forms of action within a matter, Because the precession angle of Mercury orbit predicted representing the total binding energy that holds the matter by (47) is consistent with the observation, it is generally together, represented by the potential energy of the system. believed that it is correct. However, there are also some In other words, if you divide the matter infinitely, and you serious problems with the metric; for example, it exposes the move each part to infinity, the work done is the volume incompatibility of electromagnetic theory and gravitational integral of the pressure, which is negative, and the absolute theory: when a charged particle moves in the radial direction value is equal to the mass of the gravitational source, namely, at a speed dr/dt ≈ 1, near the singularity r � 2GM, 􏽒 pdxdydz � − M, which is like adding a physical condition − 1 2 2 2 ds /dt � (1 − 2GM/r) − (1 − 2GM/r) (dr/dt) < 0, to make the solution of the pressure definite. In fact, the which is ridiculous because there is no reason to think that form of the gravitational source already determines the the electromagnetic equipment in the gravitational field interpretation of p. As gravitational source cannot make the speed of a charged particle close to 1. T � (ρ + p)u u − pg , if p is still understood as the μ] μ ] μ] In order to avoid the defect with (47), textbooks do not common dynamic pressure, then the field equation can only be interpret r in (47) as the usual radius but instead refer to it as used to solve for the metric of an ideal fluid, which is obviously the radial parameter with fuzzy meaning [7, 8]. But this is not hoped by general relativity, and the dynamic pressure in a unhelpful and leads only to conceptual confusion since it has solid is generally considered to be zero. let alone (36). Can p be been used as the usual radius when calculating the pre- understood as a thermal pressure? No, because the thermal cession angle and the bending angle of light; there should be motion is absorbed by ρ in the form of thermal kinetic energy no other explanation. and cannot be repeated to appear. p is called pressure only Now, we calculate the ordinary pressure given by the because it appears in the equation of motion in the form of Schwarzschild interior metric, from which we can see the pressure; of course, p includes the effect of common pressure. defects of the Schwarzschild metric and the necessity of μ] ,e equation of motion refers to T � 0. ;] modifying the field equation. According to the definite of When the field equation with the coupling constant 4πG pressure, it refers to the stress per unit area. It may as well let is applied to the universe, T � (ρ + p)u u − pg repre- μ] μ ] μ] the celestial body be a fluid with ρ � const, P denotes the sents the energy-momentum tensor of space of the universe, common stress, and the common pressure given by the and taking the statistical average of ρ and p, we have p � − ρ, interior solution of Schwarzschild is which is just the equation of state of the dark energy said r r r e e e zB 1 P � ρ 􏽚 Γ dr � ρ 􏽚 dr � ρ 􏽚 dB c 00 A zr A r r r 􏽱����������� 􏽱����������� ρ 8πGρ 2 2 � 􏽚 1 − r d 3 1 − 8πGρr /3 − 1 − 8πGρr /3 􏼒 􏼓 􏼒 􏼓 (48) 4 3 􏽳��������� � 2 3/2 2 2 2 2 3ρ 8πGρr ρ 8πGρr 8πGρr ρ 8πGρr e e � − 􏼠1 − 􏼡 + 1 − 􏼠1 − 􏼡 − 􏼠1 − 􏼡 . 8 3 2 3 3 8 3 In weak field approximation, P � ρ(GM/r n − q But, the common pressure given by (39) is c e 2 3 GMr /r ), which is Newton’s result. And at the center, (48) r r r r e e e e zB zB gives 􏽳��������� � P � ρ 􏽚 gdr � ρ 􏽚 Γ dr � ρ 􏽚 dr � ρ 􏽚 dl c 00 2 B zr B zl 2 r r r r 3ρ 8πGρr ρ 8πGρr ρ e e (49) P � − 􏼠1 − 􏼡 + 1 − − . 8 3 2 3 8 8πGl 1 + 4πGρl /3 � ρ 􏽚 dl � ρ ln , 2 2 Obviously, for 1 − 8πGρr /3 � 0, P < 0, and for e c 1 + 4πGρl /3 1 + 4πGρl /3 1 − 8πGρr /3 � 1/9, P � 0, which are abnormal because the e c (50) pressure should not be zero anyway. 8 Advances in Astronomy − 1 − 1 in which l � l(r) and l � l(r ) satisfy (38). ,ere is no parameter of today H � H(t ) � 70 km · s · Mpc into e e 0 0 singularity in (50). Under weak field approximation, l ⟶ r, the above equation, we get t � 1.37 × 10 years, that is, 13.7 2 3 P � ρ(GM/r n − qGMr /r ), which is just the result of billion years, the same as the previous theoretical results. c e e Newton. It can be seen from (54) that the cyclical period of expansion 􏽰����� � and contraction of the universe is 2π/ω � 3π/Gρ � 2 × 10 years, namely, 200 billion years, so the universe is currently in 8. The Application of (46) in Cosmology the expansion stage and will begin to contract in 36.3 billion years. Contraction is the reverse course of expansion. In the comove coordinate system (t, l, θ, φ), the metric that Now, we derive the new relation between distance and describes cosmic space is the Robertson–Walker metric: redshift given by (46). Similar to the previous operation, 2 2 2 2 2 2 2 2 2 letting the light given out by distant galaxy at the time t in ds � dt − R (t)􏼠 dl + l dθ + l sin θdφ 􏼡, 1 − kl past and reach the Earth at the time t of today, its redshift z � (λ − λ /λ ) � (R(t )/R(t)) − 1. λ is wavelength. We (51) 0 e e 0 may as well put today’s R(t ) � 1. Note that the subscript 0 where R(t) is the cosmic scale factor and k is a constant. l is represents today. And differentiating 1 + z � 1/R(t), we get the radial coordinate, and in other books it is denoted by r, dR dR dt just to distinguish it from the usual radius, here l instead of r. dz � − � − . (57) Rdt R When the new field equation (46) is applied to the universe, R (t) that is, combined with the Robertson–Walker metric, the And the derivative of equation (52) gives following two equations are given: 2 2 € _ 4πGρ/3H � − RR/3H , where H � dR/Rdt, R � dR/dt. dR 4πG Writing today’s 4πGρ/3H � q , H(t ) � H , and ap- 2 0 0 0 0 (52) 􏼠 􏼡 + k � − ρR , plying (52) to today, we have dt 3 k � − H 1 + q 􏼁 , 0 0 dρ dR (53) R + 3 (p + ρ) � 0. (58) dt dt 4πGρ k 2 2 2 H � − − � H 􏽨 1 + q 􏼁 (1 + z) − q 􏽩. 0 0 2 0 R (t) Equation (52) shows that k must be negative, which proves that space-time is infinite. Equation (52) is similar to On the other hand, for the motion of light, the original Friedman equation; just replace G there with − (G/2). Equation (53) is the so-called energy equation, dl dt dz dl 􏽱������� � 􏽰����� � − � � − 􏽚 which is in the same form as the original. Now putting p � R(t) H 􏼐1 − kl 􏼑 1 − kl − ρ in (52), we obtain p � − ρ � const, which means that the (59) density and pressure remain the same while the universe 1 dz expands or contracts, and new matter must be created 􏽱����������������� � 􏽚 . continuously in the universe. ,e solution of (52) is 0 1 + q 􏼁 (1 + z) − q 0 0 􏽲��� � πGρ Note that l as superscript of the integral sign refers to the R(t) � k sin􏼠2t + k 􏼡, (54) 1 2 galaxy’s unchanged comoving coordinate. Using the relation between luminosity distance and redshift √����� � which shows that the universe expands and contracts in d � (1 + z)R(t ) 􏽒 dl/ 1 − kl and completing the inte- L 0 cycles. Here, k and k are two integral constants. Since time 1 2 gration of the right of (65), we get has no beginning and no end, the moment of R(t) � 0 has 􏽱����������������� 􏽰����� occurred countless times. Let us define the nearest moment (z + 1) q + 1 + q + 1 􏼁 (z + 1) − q z + 1 0 0 0 of R(t) � 0 as zero; that is to say, at the moment t � 0, then 􏽰����� 􏽰����� H d � ln . 0 L q + 1 1 + q + 1 0 0 k � 0. Hubble parameter: 􏽲���� � 􏽲���� � (60) dR 4πGρ 4πGρ � (55) H(t) ≡ cot􏼠t 􏼡. Here, d is Luminosity distance and q is the deceleration Rdt 3 3 L 0 parameter today. As z ⟶ 0, expanding it, we have Since everything disappears at R(t) � 0 [17], including light, the universe in the last cycle is unobservable and no 1 − q 3q − 2q − 1 0 2 0 3 (61) H d � z + z + z + · · · , 0 L concern to us. What we care about is the age of our universe, 2 6 which is the time from the beginning (t � 0) of the most recent which is classical Hubble law after omitting high-order cycle to today, and using (55), we obtain our universe’s age: terms. ,e conclusion of (60) is in good agreement with the 􏽳���� � 1 4πGρ observed distance and redshift data [9–19], which strongly 􏽰������ � t � arctan . (56) indicates that the modified field equation (46) is correct, the 4πGρ/3 3H so-called dark energy does not have to exist, and the ex- pansion of the universe is still decelerating. ,e curve in By substituting the observed density − 28 3 Figure 1 is the simulation of (60) with q � 0.14 and ρ � 3.1 × 10 kg/m of the universe and the Hubble 0 Advances in Astronomy 9 50 expanding, and new matter is continuously generating in galaxies. In a word, just like the night sky we see with a magnifying glass, everything is expanding but the periods of various rotations and revolutions are unchanged. ,e essence of cosmic expansion is the simultaneous gener- ation of space and matter. Many people have recognized the fractal structure of the 40 universe, but they are unwilling to explain the formation of galaxies as the growth of fractals [17, 18]; the obstacle is obviously that people do not know the mechanism of the generation of new matter. Now, the generation of matter is Red shi (z) no longer a problem. Figure 2 is a step-by-step magnification of the Solar System. It represents the actual growth process of the Solar Figure 1: ,e recent Hubble diagram of 69 GRBs and 192 SNe Ia. System. With the universe expanding, the Solar System is becoming bigger and bigger; not only do its size and mass − 1 − 1 H � 70 km · s · Mpc , H is Hubble parameter of today, 0 0 increase, but also brightness increases. For example, the distance-modulus is equal to 5lgd + 25, and unit of d is L L Earth is moving away from the Sun at a speed of Mpc. v � H s � 9m/year, namely, following Hubble expansion. 0 0 0 Note that according to the observation of ρ, people Since the Hubble expansion does not change the revolution deduce q � 4πGρ/3H � 0.1 ± 0.05. 0 0 period, the revolution speed of the Earth increases today at a ,e redshift-distance relation derived from the original rate of H v � 61m/year , and, accordingly, the mass of the 0 0 field equation cannot explain the observation, in order to be Sun increases at a rate of 3H M � 4 × 10 kg/year. Here, 0 0 consistent with the observation, dark matter and dark energy s � 1.49 × 10 km is the distance between the sun and the must be introduced temporarily, but such an operation has Earth today, M � 2 × 10 kg is the mass of the Sun, and no scientific value because dark matter and dark energy are v � 30000m/s is the revolution speed of the Earth today. no different from the copy of ether, and in essence, they Again, in addition to the tide, the expansion of Hubble belong to the pseudoscientific concept that can never be recedes the Moon 2.7 cm away from the Earth every year, the verified by experiments. ,e accelerating expansion of the tide only recedes the Moon 1.1 cm away from the Earth, and, universe advocated by some people cannot be consistent meanwhile, the radius of the Earth increases at a speed of with the facts. ,ough the data of the distance and redshift v � H r � 0.5mm/year, the Earthʼs mass increases at a rate 0 0 they measured are right, the theoretical basis for analyzing of 3H m � 1.2 billion tons per year, r � 6400 km is the 0 0 0 these data is wrong; that is to say, the middle derivation from radius of today’s Earth, and m � 5.96 × 10 kg is the data to conclusion is wrong. Earth’s mass of today. ,e Earthʼs rotation is slowing down at a rate of 3.8 cm/year, just because of the tide and not the Hubble expansion. If the 3.8 cm/year is all the effect of tides, 9. Galaxies and Celestial Bodies are Formed by the result calculated according to the theory of tidal damping Gradual Growth rather than by the is that the rotation period of the Earth increases by 1.7 Convergence of Existing Matter millisecond every year, which is inconsistent with the ob- servation, and if the tides make the Moon only 1.1cm away Since the negative pressure is confined to the inner part of from the earth every year, the calculated result is that the the celestial body, the new matter can only generate in the μ Earthʼs rotation period slows down by 0.6 millisecond per celestial body not wherever. Applying T � 0 to a celestial ];μ year, which is in consistence with observation. Of course, body’s interior, we obtain dm � d(ρV) � − pdV � ρdV. these data belong to today and do not represent the past, and Here, V is the volume of the object, m is its mass. if you want to infer the past or future situation, you need to In order to keep the density of the universe unchanged do a similar derivation; I will not discuss it here. during the expansion process, the celestial body must grow Figure 3 is a step-by-step magnification of the Milky with time and its volume satisfies V∝ R (t). From Way. It represents the actual growth process of the Milky dm � ρdV � (m/V)dV, we know m � CV and C is integral 3 Way. With universe expansion, not only do its size and mass constant, so we obtain m∝ R (t); that is, for any two increase, but also its brightness increases. ,at is to say, all moments t and t , 1 2 parts of it have been expanding according to Hubble; at the m t 􏼁 R t 􏼁 same time, new matter is continuously generated in the 1 1 � . (62) celestial bodies. For example, the radius of the galactic disk m t R t 􏼁 (refers to the luminous part) is expanding at a rate of Further, dm � 3Hm. Of course, (62) is also suitable for v � H d � 600m/s; d � 30, 000 light-years is the radius of 0 0 0 0 describing the mass change of a galaxy. So, we get a new the luminous part of the galactic disk today. ,e Solar picture of the evolution of the universe: everything is System is moving away from the center at a rate of expanding in Hubble; not only is the space between v � H r � 450m/s; r � 8.5 kpc is the distance of the Sun 0 0 0 0 galaxies expanding, but the galaxies themselves are to the galactic center. Distance-modulus 10 Advances in Astronomy Figure 2: Schematic diagram of the growth process of the solar system with time. Figure 3: Schematic diagram of Gradual growth of the Milky Way. It is because the Milky Way is formed by gradual growth, L M (63) not by the accumulation of existing matter that its spiral � 􏼠 􏼡 , L M ⊙ ⊙ arms are not getting tighter and tighter; otherwise, they would have been destroyed. L is the luminosity of the star and L and M are, re- ⊙ ⊙ Figure 4 is a step-by-step magnification of a piece of spectively, luminosity and mass of the sun. ,e brightness cosmic space, which represents the actual expansion and temperature of celestial bodies have the following process of cosmic space. ,e white spot in the figure relations: represents galaxies, not only is the space between galaxies 2 4 2 2 4 2 L � 4πr · σT � 4πr · l � 4πd · σT � 4πd · l , (64) expanding, but also the galaxies themselves are expand- e e e e p p p p ing. It tells that the more backward we look, the more where l is absolute brightness of the star, l is its vision evenly matter is distributed, which is just reflected by the e p brightness, d is the distance from the star to us, σ is Ste- microwave background radiation. ,erefore, we say that fan–Boltzmann constant, and T and T are the temperature the microwave background radiation is the comprehen- e p of surface and the vision temperature, respectively. sive effect of redshifted photons emitted by the matter at a Now we treat M as a variable, namely, M∝ R (t). Since distant and indistinguishable distance on our instrument, r ∝ R(t), d ∝ R(t), for the same star, at any two moments e p and these photons have a blackbody spectrum because t and t , we have following relations: they come from different stars. ,is is a simple and re- 1 2 alistic explanation, but it is like a myth to describe it as a 4 10 l t 􏼁 T t 􏼁 l t 􏼁 T t 􏼁 R t 􏼁 p 1 p 1 e 1 e 1 1 relic or sound of the big bang. ,e distant sky we see with � � � � . (65) 4 4 10 l t 􏼁 l t 􏼁 T t 􏼁 T t 􏼁 R t 􏼁 e 2 p 2 the naked eye is uniform, and, similarly, the distant sky we 2 2 2 e p see with the telescope should be also uniform. It is And assume t � t � 1.37 × 10 years, which is our 2 0 shameless to deliberately tie microwave background ra- universe age, then 1 billion years ago t � 1.27 × 10 years, diation with the big bang. and using (52) and the approximated formula x ≈ sin x for It should be noted that the inverse process of the ex- x ⟶ 0, we have pansion of the universe is its contraction, and in the con- traction process all galaxies and space atrophy reversibly. l t 􏼁 l t 􏼁 1.27 p 1 e 1 � � 􏼒 􏼓 � 0.46, For a more detailed discussion of the expansion process l t l t 1.37 􏼁 􏼁 p 2 e 2 of the universe and the fractal structure of galaxies, see the (66) authorʼs paper and related papers [17–22]. 2.5 T t 􏼁 T t 􏼁 1.27 p 1 e 1 � � 􏼒 􏼓 � 0.82, T t 􏼁 T t 􏼁 1.37 p 2 e 2 10. TheTemperatureandBrightnessofCelestial which means that the Sunʼs brightness was less than half of Bodies Are Increasing todayʼs and the temperature of the solar light is 82% of today It is found that the mass of a celestial body is related to its 1 billion ago. For the change of temperature of the surface of luminosity, generally speaking, the greater the mass, the the Earth, we can also roughly estimate to use (15), if the greater the luminosity. For a main sequence star, we have the Earthʼs surface temperature is 25 C (298 K) today, 1 billion following empirical formula: ° years ago the temperature was 246 k (− 27 C), and in 30 Advances in Astronomy 11 Figure 4: Schematic diagram of the generation process of cosmic space. billion year its temperature will reach 6000 k (5727 C), derive the planetary orbit equation from the well-known which is equal to the surface temperature of the Sun today. Schwarzschild metric and, by the way, point out the And as the universe will contract in 3.6 billion years, it can shortcoming in the previous calculation. ,e orbital equa- become reality for the Earth to shine like the Sun today. tion described by the Schwarzschild metric is Similarly, the evolution of gravity acceleration on the 2 2 du a − 1 2GM 2 3 surface of the Earth can be deduced; 1 billion years ago the (69) 􏼠 􏼡 � + u − u + 2GMu , 2 2 dφ acceleration of gravity on the surface was h h R t 􏼁 1.27 2 2 removing the final term, which is Newtonʼs ellipse orbit g t 􏼁 � g t 􏼁 � 10m/s × � 9.2m/s . (67) 1 2 equation. Here, u � 1/r, and h � r dφ/ds � const and a � R t 􏼁 1.37 (1 − 2GM/r)dt/ds � const are two integral constants. ,e Todayʼs atmospheric pressure is 101 kPa, since density derivation of (69) can be found in any textbook and I will not does not change and the height of the atmosphere increases repeat it here. As an initial condition, we can let the peri- following Hubble expansion; then, 1 billion years ago, the helion on the x-axis; then, atmospheric pressure was φ u 􏽱������������������������������� � 2 􏽚 dφ � 􏽚 du, R t 􏼁 2 2 2 3 2 1 0 u 􏼐a − 1􏼑/h + 2GMu/h + 2GMu − u P t 􏼁 � P t 􏼁 � 86kPa. (68) c 1 c 2 R t 􏼁 (70) Equation (66) tells us that planets can evolve into stars; where u denotes the reciprocal of the perihelion distance. this should be the main mechanism of star formation. We On the other hand, according to the theorem of factoriza- usually think that the objects that do not emit light are older tion, we have objects and the luminous objects are younger; this idea 􏽳������������������������ � should be changed. ,e reason why a celestial body does 2 a − 1 2GMu 3 2 not emit light is that its mass is not large enough, and the + + 2GMu − u 2 2 h h second reason is that the material that makes up the ce- (71) 􏽱������������������������ lestial body is too loose. ,e age of a celestial body refers to � 2GM u − ε u − ε u − ε , the time when the celestial body exists as an independent 􏼁 􏼁 􏼁 1 2 3 individual, not the time when the matter that makes up the where ε , ε , ε are the three roots of the cubic equation celestial body exists. ,e chemical composition of a ce- 1 2 3 2 2 2 3 2 (a − 1)/h + 2GMu/h + 2GMu − u � 0. And since lestial body should be determined by its temperature, not 2GMu is regarded as a perturbation, two of ε , ε , ε must be having a direct relationship with the existence time of the 1 2 3 very close to u and u . ,erefore, as an approximation, we celestial body. ,erefore, it may not be appropriate to use 1 2 may as well let ε � u and ε � u , where u and u are the the content of radioactive elements to infer the age of 1 1 2 2 1 2 two roots of the quadratic equation celestial bodies. I do not advocate talking about the concept 2 2 2 2 (a − 1)/h + 2GMu/h − u � 0, which corresponds to the of celestial age. perihelion and the aphelion. Note that there must be du/dφ � 0 at the extreme points. And according to Vedaʼs 11. More Reasonable Derivation of theorem, ε � 1/2GM − u − u ; then, 3 1 2 Orbit Precession 􏽱������������������������ 2GM u − ε u − ε u − ε 􏼁 􏼁 􏼁 1 2 3 In the case of weak field and low speed, the conclusion of (12) 􏽱���������������������������������� � (72) is almost the same as that of the Schwarzschild metric. As � − u − u 􏼁 u − u 􏼁 􏼂1 − 2GM u + u + u 􏼁 􏼃. 1 2 1 2 long as r in (69) is replaced by l(r), the orbit equation described as (12) can be obtained. ,erefore, it is advisable to Next, (70) becomes 12 Advances in Astronomy 1 + GMu + GM u + u 1 2 􏽱�������������� � φ � 􏽚 du 1 − u − u u − u 􏼁 􏼁 1 2 u u d􏽨− u + u + u 􏼁 u − u u 􏽩 GM 1 + 3GM u + u 􏼁 1 1 2 1 2 1 2 􏽱������������������� � 􏽱�������������� � (73) � − 􏽚 + 􏽚 du 2 u 2 u 1 1 − u − u 􏼁 u − u 􏼁 − u + u + u 􏼁 u − u u 1 2 1 2 1 2 􏽱�������������� � 3GM u + u 􏼁 2u − u − u 1 2 1 2 � GM − u − u 􏼁 u − u 􏼁 − 􏼢1 + 􏼣arccos . 1 2 2 u − u 1 2 Obviously, for u � u , φ � π[1 + 3GM(u + u )/2] � Our foothold is still the spherically symmetric metric 2 1 2 2 2 2 π + 3πG M /h , which implies that the processional angle is field. And for a spherically symmetric metric field, no matter 2 2 2 Δφ � 6πG M /h . its source is static, oscillatory, or variable-mass, as long as the And since u + u � 2GM/h , (u − u )/(u + u ) � e, spherical symmetry is kept, the exterior solution is still the 1 2 1 2 1 2 further we have same form, namely, − 1 u + u u − u 1 2 1 2 2Gk 2Gk 2 2 2 2 2 2 2 u � + ds � 􏼠1 − 􏼡dt − 􏼠1 − 􏼡 dλ − λ 􏼐dθ + sin θdφ 􏼑. 2 2 λ λ 􏽱�������������� � (76) 2GM − u − u u − u 􏼁 􏼁 1 2 ⎡ ⎢ × cos⎣ 2 + 3GM u + u 􏼁 ,at is, with t, λ, θ, φ as independent coordinate vari- 1 2 (74) ables, (76) is the solution of the vacuum field equation 3GM u + u R � 0. Do not consider the meaning of λ for the moment, 1 2 μ] − φ/ 1 + 􏼠 􏼡􏼣 and k is only thought of as a constant. ,e proof of (76) is similar to the proof of Birkhoff law; I will not repeat here. 2 2 GM 3G M Equation (76) offers the orbit equation of the planets: ≈ 􏼢1 + e cos􏼠1 − 􏼡φ􏼣, 2 2 h h du a − 1 2Gk 2 3 � + u − u + 2Gku , (77) 􏼠 􏼡 2 2 whose final step takes advantage of the formula cos(α − β) � dφ 􏽰��������������� h h cos α cos β + sin α sin β and2GM − (u − u )(u − u )≪ 1, 1 2 whose derivation is the same as (69). However, here u � 1/λ 1/[1 + 3GM(u + u )/2] ≈ 1 − 3GM(u + u )/2. 1 2 1 2 It should be pointed out that the second-order ap- and h � λ dφ/ds � const.a � (1 − 2Gk/λ)dt/ds � const. Similar to (74), we have proximate solution obtained by using 1/r � u � 2 2 2 (1 + e cos φ)G M /h as the first-order approximation is 2 2 Gk Gk 3G k wrong, that is, the following (75) is wrong: u � + e cos 1 − φ. (78) 􏼠 􏼡 2 2 2 h h h GM 3 3 3 u � (1 + e cos φ) + G M eφ · sin φ. (75) 2 4 ,e above is the result of the coordinate system h h (t, λ, θ, φ), and our purpose is to solve the orbit equation in ,e shortcoming of (75) is that when φ � 2nπ, the orbital the coordinate system (t, r, θ, φ). To this end, we introduce two crossover points with the x-axis are always invariant, so the coordinate transformation. λ � l/R(t) and meanwhile set the shape of the ellipse is not guaranteed when it rotates, and k � M/R (t), where R(t) is the cosmic scale factor, and M � 2 2 2 the precession angle Δφ � 6πG M /h cannot be obtained M(t) is the mass of the central celestial body and satisfies from (75) when φ is quite big; that is to say, the transition (62). And in the light of (11), l � l(r, t) satisfies − 2 2 2 − 2 from (75) to u � GMh [1 + e cos(1 − 3G M h )φ] can- not be realized. In short, using (75) to explain the precession 7GM(t) l � r − 2GM(t)ln[r − 2GM(t)] − + 2GM(t)ln r (t), of Mercury is not only grudging but also causing serious other problems. And again, Einsteinʼs original calculations (79) were also ambiguous and cannot obtain the correct pro- Of course, without considering the expansion of space- cessional angle according to Einsteinʼs calculation [14]. time, all equations must go back to the previous. Now (77) is transformed into 12. Planetary Orbit Equations of Giving 2 2 R(t) R(t) Gk Gk 3G k Consideration to the Expansion of Space- ≈ � + e cos􏼠1 − 􏼡φ, (80) 2 2 2 time: The Evolution of Planetary Orbit l(r, t) r h h h Now, let us look at the orbital equation of planets in which is just the orbital equation of planets and shows that expanding space-time, which is also the equation that de- while planets are moving around the center, they recede termines the formation and evolution of galaxies. from the center in Hubble. Advances in Astronomy 13 Figure 5: ,e schematic diagram of the speed distribution of matter in the Milky Way. 3 2 Besides, give consideration to Keplerʼs law a /T � GM, reached without being thrown out of the galaxy. Because the 3 3 3 since a ∝ � R (t) and M∝ R (t); then, T � const. ,at is thickness of the galactic disk decreases slowly, its speed does to say, the period of motion of planets does not change and not weaken, which is normal. We have no reason to deny − 24 3 the speed of planets increases gradually while they go away that the density of halo can reach 9 × 10 kg/m ; such from the center. density is extremely thin. In a word, there is no need to introduce dark matter; let alone a black hole. Figure 5 shows the speed distribution of matter in the 13. Modern Observations Do Not Confirm the Milky Way, the red line represents the result without Existence of Dark Matter considering halo mass, and the white line represents the observation result. ,e white line represents the observation Take the Milky Way as an example. It is composed of result and is also the result of the calculation of considering galactic ball, galactic disk, and galactic halo. Near the the halo mass. center, the material distribution is dense, so the galactic ball I do not think dark matter, dark energy, and black holes can be treated as a rotating rigid body, so that it is natural exist. Although peopleʼs observation technology is con- that the velocity of matter at the center is proportional to stantly improving and data is constantly accumulating, the radius, and there is no need to assume that dark matter peopleʼs interpretations of the observed phenomena and or black hole exists. ,e calculated velocity of the material data are basically wrong. ,e reason for this lies in the in the halo is lower than the measured velocity because the contradiction between these interpretations. Dark matter, mass of the halo itself is ignored; that is to say, once dark energy, and black holes have pushed science into considering the mass of the halo, there is no need to assume metaphysics, which is not progress but retrogression. It is dark matter or black hole, too. Here is a rough estimate of shameless for those who deliberately bind the correct the velocity of the material in the halo. Since the halo is conclusion of general relativity with the contemporary spherical with a radius of about 100,000 light-years, at etheric, namely, dark matter, dark energy, and black holes. rfrom the center, the acceleration of gravity provided by the It is imperative to separate general relativity from these halo itself is (let us not consider the effect of the mass of the absurd sermons. People attribute the incomprehensible dish): phenomena to that dark matter, dark energy, and black GM(r) 4πGρr v holes are the passivation of human intelligence. In a word, (81) g � � � , 3 r r all singularity physics is unreal, no matter how many halos 􏽰������ � it has. where ρ is the density of the halo, v � r 4πGρ/3 is the speed of a moving particle around the galactic center, and probably Data Availability as well setting v � 30km/s and r � 70, 000 light-years, we obtain ,e data used to support the findings of this study are available from the corresponding author upon request. 3v − 24 3 (82) ρ � � 9 × 10 kg/m . 4πGr Conflicts of Interest ,at is to say, as long as the halo density reaches − 24 3 9 × 10 kg/m , the particle can maintain the speed of ,e author declares that there are no conflicts of interest. 30 km/s to make a circular motion and not be thrown out of the galaxy. It is possible that the density of the halo can reach Acknowledgments − 24 3 9 × 10 kg/m , and considering that the galactic ball and disk also have mass, even if the mass density of the halo is ,e study was supported by the National Key Research and − 24 3 smaller than 9 × 10 kg/m , the speed 30km/s can still be Development Plan (973 Plan), no. A030101. 14 Advances in Astronomy References [1] L. Lorio, “Gravitational anomalies in the solar system,” In- ternational Journal of Modern Physics D, vol. 24, no. 6, pp. 1–37, 2015. [2] M. Ness and D. Lang, “,e X-shaped bulge of the Milky way revealed bywise,” %e Astronomical Journal, vol. 152, no. 1, p. 14, 2016. [3] C. Martinez-Lombilla and I. Trujillo, “Discovery of disc truncations above the galaies’smid-plane in Milky May-like galaxies,” Monthly Notice of the Royal Society, vol. 483, no. 1, pp. 664–691, 2019. [4] J. T. Nielsen, A. Guffanti, and S. Sarkar, “Marginal evidence for cosmic acceleration from type Ia supernovae,” Science Reports, vol. 6, pp. 1–8, 2016. [5] D. Herwartz, A. Pack, D. Krylov et al., “Revealing the climate of snowball Earth from Δ17O systematics of hydrothermal rocks,” in Proceedings of the National Academy of Sciences, vol. 112, no. 17, pp. 5337–5341, 2015. [6] S. M. Som, R. Buick, J. W. Hagadorn et al., “Earthʼs air pressure 2.7 billion years ago constrained to less than half of modern levels,” Nature Geoscience, vol. 9, no. 6, pp. 448–451, [7] S. Weinberg, Gravitation and Cosmology, Wiley, New York, NY, USA, 2013. [8] S. Carroll, Lecture Notes on General Relativity, Columbia University, New York, NY, USA, 2013. [9] L. D. Landau, %e Classical %eory of Fields, Pergmon Press, Oxford, UK, 1987. [10] Einstein, %e Meaning of Relativity, Princeton University Press, Princeton, NJ, USA, 1922. [11] B. C. Tolman, Relativity, %ermodynamics and Cosmology, Oxford Clarendon Press, Oxford, UK, 1934. [12] J. L. Yang, “Criticism to universal big bang,” Astrophys and Aerospace Technology, vol. 4, p. 1, 2016. [13] T. Felicead, “f (R) theories,” Living Reviewing in Relativity, vol. 13, no. 1, p. 3, 2012. [14] X. Mei and P. Yu, “Did LIGO really detect gravitational waves?” Journal of Modern Physics, vol. 7, pp. 1098–1104, [15] P. Bhar and N. Pant, “Relativistic anisotropic stellar models with Tolman VII spacetime,” Astrophysics and Space Science, vol. 359, no. 1, 2015. [16] M. Yang, “Modification of gravitational field equation and rational solution to cosmological puzzles,” International Journal of Physical Science, vol. 5, no. 2, 2010. [17] J. L. Yang, “Unavoidable correction to the coupling constant in Einstein field equation,” International Journal of Advanced Research in Physical Science, vol. 6, no. 11, pp. 4–30, 2019. [18] J. Gaite, “,e fractal geometry of the cosmic web and its formation,” Advance in Astronomy, vol. 1, p. 25, 2019. [19] J. de Haro, A. Paliathanasis, and R. J. Slagter, “Evolution and dynamics of a matter creation model,” Monthly Notices of the Royal Astronomical Society, vol. 460, no. 2, pp. 1445–1456, [20] S. N. Gurbatov and A. T. Saichev, “Large-scale structure of the universe,” Physics-Uspekhi, vol. 55, p. 3, 2012. [21] S. L. Blibbikov and A. D. Dolgov, “Cosmological accelera- tion,” Physics-Uspekhi, vol. 62, no. 6. [22] J. L. Yang, “Light speed invariant solution and its enlight- enment of field equation of general relativity,” Advances in Astronomy, vol. 2020, Article ID 3930947, 12 pages, 2020.

Journal

Advances in AstronomyHindawi Publishing Corporation

Published: Mar 20, 2021

References