Mathematical Principles of Theoretical Physics

7.5. THE UNIVERSE 467 Rc semi-sphere (a) Rc upper semi-sphere (b) lower semi-sphere Figure 7.14: (a) A 3D hemisphere, and (b) a ...

468 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY whereMis the mass of the flat space. By the invariance of density, M/ 4 π 3 R^3 c=Mtot ...

7.5. THE UNIVERSE 469 whereMis the cosmic mass of hemisphere, and ̃ris the distance from the light source to the opposite radial ...

470 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY By (7.5.56), we have Mr= 4 π 3 r^3 ρ= r^3 R^3 M. Then it follows from (7.5.57) that (7 ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 471 1) The cosmic observable mass M and the total mass Mtotal, which includes both M ...

472 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY Dark matter was first postulated in 1932 by Holland astronomer Jan Oort, who noted tha ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 473 Dark energy. Dark energy was first proposed in 1990’s, which was based on the hy ...

474 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY and byTμ ν=diag(c^2 ρ,g 11 p,g 22 p,g 33 p), we have T 00 − 1 2 g 00 T= c^2 2 ( ρ+ 3 p ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 475 then, from (7.6.5), (7.6.7) and (7.6.9)-(7.6.11) we can deduce that (7.6.12) (R′) ...

476 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY Here, we remark that in the classical Einstein field equations whereφ=0, the relation ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 477 we obtain from (7.6.22) that (7.6.24) 3 α 2 = 24 kπ α 1 +k+ 1. By the relation (7 ...

478 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY Now we are ready to deduce from (7.6.27) the PID gravitational interaction formula as ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 479 To this end, we need to make the following transformation to convert the field eq ...

480 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY In fact, by (7.6.31) and (7.6.33) we can see that all Schwarzschild solutions lie on t ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 481 Inserting (7.6.41)-(7.6.42) into the first two equations of (7.6.34), we deduce t ...

482 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY It is easy to show that the orbits inΩ 1 withx 1 >0 will eventually cross thex 2 −a ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 483 We remark that Theorem7.30is valid provided the initial valueαis small because th ...

484 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY In view of (7.6.52), we obtain that φ′′=−evR+ 2 r φ′+ 1 2 (u′−v′)φ′ Again by the Schwa ...

7.6. THEORY OF DARK MATTER AND DARK ENERGY 485 wherek 0 andk 1 will be determined by the Rubin rotational curve and the astronom ...

486 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY It is known that VΩ^1 = 3 π 4 |Ω|. ForVΩ^2 , we propose that VΩ^2 =π^2 r^30 , r 0 the ...