PMBH Theory of Representation of Gravity Field

International Journal of Advanced Research in Physical Science (IJARPS) Volume 5, Issue 9, 2018, PP 36-45 ISSN No. (Online) 2349-7882 www.arcjournals.org

PMBH Theory of Representation of Gravity Field Equation and Solutions, Hawking Radiation in Data General Relativity Theory Sangwha-Yi Department of Math, Taejon University 300-716 *Corresponding Author: Sangwha-Yi, Department of Math, Taejon University 300-716

Abstract: In the general relativity theory, we find the representation of the gravity field equation and solutions. We treat the representation of Schwarzschild solution, Reissner-Nodstrom solution, Kerr-Newman solution, Robertson -Walker solution. We found new general relativity theory (we call it Data General Relativity Theory; DGRT). We treat the data of Hawking radiation by Data general relativity theory. This theory has to apply black hole (specially, Primordial Massive Black Hole; PMBH) because black hole(PMBH) is an idealistic structure. Keywords: General relativity theory, the other solution, Schwarzschild solution, Reissner- Nodstrom solution, Kerr-Newman solution, Robertson-Walker solution, Data General relativity theory, Hawking radiation, Primordial Massive Black Hole PACS Number: 04, 04.90. +e

1. INTRODUCTION In the general relativity theory, our article’s aim is that we find the representation of the gravity field equation and solutions. We found new general relativity theory (we can call it Data General relativity theory). We more obtain data of Hawking radiation by Data general relativity theory. First, the gravity potential g   is

ds 2  g  dx dx 

(1)

In gravity potential g   , we introduce tensor f  and scalar K . In this time, the condition is

K 0 x  i)

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ii)

If g 00  1, f   g   , K  1

(1-i)

If g 00  1 , f   Kg

(1-ii)



Hence, 

ds '  f dx dx 2





 g  d x d x



 g 

x  x  d x ' d x '   x ' x '

 g ' d x ' d x '  f' dx ' dx '

g '  g  

(2)

x  x  , g    g   , g '  g ' x ' x '

d x   K dx  , d x   K dx  , International Journal of Advanced Research in Physical Science (IJARPS)

Page | 36

PMBH Theory of Representation of Gravity Field Equation and Solutions, Hawking Radiation in Data General Relativity Theory

d x '  K dx ' , d x '  K dx ' In Christoffel symbol 

   

1 g 2





,

g   g g   1  (     )    x x x K

Therefore, in the curvature tensor R

R 



 

(3)



(4)

  ,

                           x x



1         1     (    R         )  K x  K x 

In Ricci tensor R

R   R



 



 

(5)

 ,

1



K

R



 



1

K

R 

(6)

R

(7)

In curvature scalar R

R g

1



R  

K

g  R   

1

K

Hence, in the gravity field equation of Einstein,

R   

1 1 1 g  R  (R    g  R ) 2 K 2

8G 1

c4 K

T 

(8)

In Newtonian approximation, Energy-momentum tensor T   is

 2 g 00 

1

K

c 2  T 00 ,

 2 g 00  

1

K

8G 1

c

4

K

T 00  

8G

c4

T 00

c 2  T 00

(9)

(10)

Hence,

1

K

T  T 

(11)

Therefore, revised Einstein’s gravity field equation is

R  

1 1 1 1 8G 8G g  R  (R    g  R )   T    4 T  4 2 K 2 K c c

(12)

Therefore, revised gravity field equation of tensor g   is able to reduce Einstein’s gravity field equation.

International Journal of Advanced Research in Physical Science (IJARPS)

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Therefore, 

[R   

g



8G

c

4

1 1 1 8G 1   8G 1 g  R ]  g  [R    g  R ]   4 g T    4 T 2 K 2 c K c K

g  T     1

 R  

K

R 

8G

c

8G 1

c

K

4



T

4







T







8G

c

4

T



(13)



Hence,

1

K

T





T



(14)



Ricci tensor is

R  

1

K

R   

8G 1 4

c

K

(T   

1 g  T 2





) 

8G

c

4

(T   

1 g  T 2





)

(15)

The proper distance is

ds 2  g  dx dx  , ds '2  f dx dx   g  d x d x 

(16)

2. WEAK GRAVITY FIELD APPROXIMATION Weak gravity field approximation is

g        h   , h    1 R   

8G

c

4

1 g  T 2

(T   





)

(17)

)

(18)

According to Eq (15),

g        h   , h    1 R   

8G

c

4

1 g  T 2

(T   





The tensor of weak gravity field is

R   

8G

c

4

S  , R   

1 2

S    T      T



1 2

c4

S 

,



S    T      T

8G

(19)



The solution is



h  (t,x ) 

4G

c2







S  (t | x  x '|,x ') ,  d x ' | x  x'| 4


Page | 38


d

3

xT 00  M , otherwise T    0 

d

3



4G

h  (t,x ) 

d

c2

4

1





S  (t  | x  x '|,x ')

x'

xT 00   K K d 3 x

otherwise

(20)

  | x  x '|

1

K

,

T 00  K M  M

, T 00  K T 00

T  T   0

K

(21)

As



4G

h 00 (x )  

h ij(x ) 

rc 4G

rc

d

2

2

d

3

3

2 K GM

1 2

x '[T 00  T 00 ] 

2 K GM

1 2

x '[  ijT 00 ] 

,

rc 2 rc 2

 ij,

(22)

The proper distance is

 ds 2  c 2d  2  g  dx dx   (1 

2GM

rc

2

)c 2dt 2  (1 

2GM

rc

2

) ijdx idx

j

(23)

The proper distance is in this theory

 ds '2  Kds  K (1   (1   (1 

2GM

rc

2

2G M

rc

2

 Kc 2d  2  Kg

)c 2dt 2  K (1 

2 K GM

rc

2

2

)c 2d t 2  (1 

)c 2d t 2  (1 

2GM

rc

) ijdx idx

2 K GM

rc

2G M

rc

2

dx dx 



2

2

j

) ijd x id x j

) ijd x id x j

 g  d x d x 

K t  t , K xi  x i, K x j  x j , K r  r , K M  M

(24)

3. THE OTHER REPRESENTATION IN SCHWARZSCHILD SOLUTION, REISSNER-NODSTROM SOLUTION, KERR-NEWMAN SOLUTION AND ROBERTSON-WALKER SOLUTION Schwarzschild solution (vacuum solution) is

R   0 ds

2

dr 2  c (1  )dt   r 2d  2  r 2 sin2 d  2 2 2GM rc 1 rc 2 2

2GM

2

(25)

The other representation of Schwarzschild solution is International Journal of Advanced Research in Physical Science (IJARPS)

Page | 39


ds '2  f dx dx   Kg  dx dx   K  ds 2  c 2K (1 

 c 2 (1 

 c 2 (1 

2GM

)dt 2 

rc 2

2 K GM

rc

Kdr 2  Kr 2d  2  Kr 2 sin2 d  2 2GM 1 rc 2 dr2

)d t 2 

2

1

2G M

)d t 2 

rc 2

 r 2d  2  r 2 sin2  d  2

2 K GM

rc 2

dr2  r 2d  2  r 2 sin2  d  2  g  d x d x  2G M 1 rc 2

K t  t , K r  r ,    ,   , K M  M

(26)

Reissner-Nodstrom solution is

ds 2  g  dx dx   c 2 (1 

2GM

rc



2

kGQ 2 )dt 2  2 4 rc

dr 2 1

2GM

kGQ  2 4 rc

rc 2

 r 2d  2  r 2 sin2 d  2

2

(27)

The other representation of Reissner-Nodstrom solution is

ds '2  f dx dx   Kg  dx dx   K  ds 2  Kc 2 (1 

 c 2 (1 

 c 2 (1 

2GM



rc 2

2 K GM

rc 2G M

rc

2

2



kGQ 2 )dt 2  2 4 rc



Kdr 2  Kr 2d  2  Kr 2 sin2 d  2 2 2GM kGQ 1  2 4 2 rc rc

KkGQ 2 )d t 2  2 4 rc 2

kG Q )d t 2  2 4 rc

dr2 1

2 K GM

rc

2



KkGQ r 2c 4

2

 r 2d  2  r 2 sin2  d  2

2

dr  r 2d  2  r 2 sin2  d  2 2 2G M kG Q 1  2 4 2 rc rc

 g  d x d x 

K t  t , K r  r ,    ,   , K M  M , KQ Kerr-Newman solution is ds

 c 2 (1 



2c 2GMr  kGQ c 4

2

Q

2

(28)

 g  dx dx 

2

2

)dt 2  2(2c 2MGr  kGQ

c 4 r 2  c 2 2GMr  a 2  kGQ

2

2

a sin2  cdtd  c 4

)

dr 2  d  2


Page | 40


 sin2 [r 2  a 2  (2c 2GMr  kGQ

2

a 2 sin2  ]d  2 c 4

)

  r 2  a 2 cos 2 

(29)

The other representation of Kerr-Newman solution is

ds '2  f dx dx   Kg  dx dx   K  ds 2  Kc 2 (1  

2c 2GMr  kGQ c 4

2

)dt 2  2K (2c 2MGr  kGQ

K c 4 r 2  2c 2GMr  a 2  kGQ

2

 c (1 



2c 2G

K M K r  kGKQ K c 4

Kr  2c G K M

 c 2 (1 

K r  Ka

2

 sin2 [Kr 2  Ka



2

2

a 2 sin2  ]d  2 4 c 

)

)d t  2(2c 2

K c 4 2

2

 (2c 2G

2c 2G M r  kG Q c 4

c 4 r 2  2c 2G M r  a

 sin2  [r 2  a

2

2

2

2

 kGKQ

KM

2

2

2

K r  kGKQ

K MG

2

K a sin2  ) cd K td K c 4

d ( K r)2  K d  2

K r  kGKQ

2

Ka 2 sin2  ]d  2 4 K c

)

)d t 2  2(2c 2M G r  kG Q

 kG Q

a sin2  cdtd  c 4

)

dr 2  K d  2

 K sin2 [r 2  a 2  (2c 2GMr  kGQ 2

2

2

a sin2  cd td  c 4

)

d r 2  d  2

 (2c 2G M r  kG Q

2

a 2 sin2  ]d  2 4 c 

)

 g  d x d x 

  K   Kr 2  Ka 2 cos 2   r 2  a 2 cos 2 

K t  t , K r  r ,    ,   , K M  M , KQ

2

Q

2

, K a a ,

J  c M a  KcMa  KJ

(30)

In this time, we obtain the data of the time t , the distance r , the mass M , the charge Q and the angular momentum J . Robertson-Walker solution is

ds 2  g  dx dx  dr 2

 c 2dt 2   2 (t)[  r 2d  2  r 2 sin2 d  2 ] 2 1  kr International Journal of Advanced Research in Physical Science (IJARPS)

Page | 41


k  (1,0,1)

(31)

The other representation of Robertson-Walker solution is by Eq(1-i)

ds 2  f dx dx   g  dx dx  dr2

 c 2d t 2   2 (t )[  r 2d  2  r 2 sin2  d  2 ] 2 1  kr

 g  d x d x 

k  (1,0,1), t  t , (t)  (t ), r  r ,    ,  

(32)

4. OBTAINING PROCESS INFORMATION OF HAWKING RADIATION Stephen Hawking found black –hole’s thermodynamics. By Hawking Radiation, we obtain the new data from formulas of black-hole’s thermodynamics. We start the obtaining process informations of Hawking Radiation. In Wikipedia (Hawking Radiation), we know formulas of Hawking Radiation. The radiation temperature T of Schwarzschild black hole (In this theory, PMBH)

T 

c 3

(33)

8GMk

B

The radiation temperature T is in Data General Relativity theory.

T 

c 3

c 3



8G M k B

8G

K Mk

 B

T

(34)

K

The black hole (PMBH)’s entropy

dS  8Gk B MdM

/ c 

8GMk c

B

3

c 2dM 

dQ , T

dQ  c 2dM

(35)

The black-hole (PMBH)’s entropy S is in Data General Relativity theory.

d S  8M d M Gk 

8Gk

K

K

KM

B

c

3

8Gk B M c

3

B

/ c 

8Gk B M c

3

c 2d M 

dQ T

c 2d ( K M ) c 2dM

dQ  KdS , dQ  c 2dM T

d Q  c 2d M  K c 2dM  K dQ

S  KS , Q  K Q

(36)

Black-hole (PMBH) radiation’s power P ev is International Journal of Advanced Research in Physical Science (IJARPS)

Page | 42

PMBH Theory of Representation of Gravity Field Equation and Solutions, Hawking Radiation in Data General Relativity Theory 2

P ev  A S  T 4 , A s  4rS , rS 

2GM

c2

,  , is constant

(37)

Black-hole (PMBH) radiation’s power P ev is in Data General Relativity theory. 2

2

A s  4rS  K 4rS  KA s , rS  P ev  A S  T

4

 KA s 

2G M

c

2



2G

KM c2

 K rS

T 4 P ev  K K2 4

 2k B Stefan-Boltzmann constant   , 60  3c 2 Black-hole (PMBH) is a perfect black body (   1)

(38)

The evaporation time tev of a black hole (PMBH) is

tev 

5120 G 2M c 4

3

(39)

The evaporation time tev

tev 

5120 G 2M c 4

3



of a black hole (PMBH) is in Data General Relativity theory.

5120 G 2K K M c 4

3

 K K tev

(40)

The power of evaporation energy of the black-hole (PMBH) is

P ev  

dE ev dtev

(41)

The power of evaporation energy of the black-hole (PMBH) is in Data General Relativity theory.

P ev  

d E ev d (E ev K ) P ev ,   d tev K K K dtev

M ev c 2  E ev  K E ev  K M ev c 2

(42)

5. PMBH DATA AND DGRT DATA In DGRT, we have to apply only black-holes (PMBHs). According to [27] Paul H. Frampton, Physical Letter B (2017), if the mass of sun is M

,

data is

Table1. Astrophysical obect

Mass solar masses

Period τ seconds

Angular momentum ㎏㎢/s

PIMBH1

20MΘ

0.013s

3.0ⅹ1037

PIMBH2

100MΘ

0.063s

PIMBH3 PIMBH4

1000MΘ 104 MΘ

0.63s 6.3s

7.2ⅹ1038 7.2x1040

PIMBH5

105MΘ

63s

PSMBH6(M87)

6ⅹ109MΘ

3.8x106s


7.2ⅹ1042 7.2ⅹ1044 2.6x1054 Page | 43


K M , time  is to be  K ,Angular momentum J is to be KJ .Hence, PIMBH2’s (from PIMBH1) K is 5, PIMBH3’s (from PIMBH 2) K is 10,PIMBH4’s(from PIMBH3) K is 10, PIMBH5’s (from PIMBH4) K is 10, PSMBH6’s (from PIMBH5) K is 6  10 4 . According to DGRT, BH (PMBH)’s mass M

is to be

In this time, PIMBH is Primordial Intermediate Massive Black hole, PSMBH is Primordial Super Massive Black Hole. According to [27]“Angular momentum of dark matter black holes”,

20 M



10 5 M

M



PIMBH

M

PSMBH

 100 ,000 M

 10 17 M





Therefore, calculated data is in DGRT, Table2. Astrophysical obect

Mass solar masses

Period τ seconds

PIMBH1

20MΘ

0.013s

3.0ⅹ1037

PIMBH2

100MΘ

0.065s

PIMBH3 PIMBH4

1000MΘ 104 MΘ

0.65s 6.5s

7.5ⅹ1038 7.5x1040

5

PIMBH5

10 MΘ

65s

PSMBH6(M87)

6ⅹ109MΘ

3.9x106s

Angular momentum ㎏㎢/s

7.5ⅹ1042 7.5ⅹ1044 2.7x1054

If we compare Table 1. and Table 2., we know DGRT has to apply PMBHs. 6. CONCLUSION We find the other representation of solutions in the General relativity theory. We more obtain the information of black-hole thermodynamics in Data General Relativity theory. If we use variable A instead of A , Data General Relativity theory is reduced to normal general

K  2 and black hole (PMBH)’s mass M is M K  2M , black hole (PMBH)’s distant of gravitation r is r K  2r , black hole (PMBH)’s proper time  is  K  2 .If rotating black hole (PMBH)’s mass M is to be M K  2M , we predict the angular momentum J of the black-hole (PMBH) is to be KJ  4 J . relativity theory. This theory’s remarkable thing is if

In this time, we have to apply only black-holes (PMBHs) because black hole (PMBH) is an idealistic structure. BH is Black hole. REFERENCES [1] [2] [3] [4] [5] [6] [7]

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PMBH Theory of Representation of Gravity Field Equation and Solutions, Hawking Radiation in Data General Relativity Theory [12] [13] [14] [15] [16] [17] [18] [19]

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Citation: Sangwha-Yi, " PMBH Theory of Representation of Gravity Field Equation and Solutions, Hawking Radiation in Data General Relativity Theory", International Journal of Advanced Research in Physical Science (IJARPS), vol. 5, no. 9, pp. 36-45, 2018.

Copyright:© 2018 Authors, This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. International Journal of Advanced Research in Physical Science (IJARPS)

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