PMBH theory of representation of gravity field

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data of Hawking radiation by Data general relativity theory. This theory has ..... Stephen Hawking found black –hole's thermodynamics. .... [4]S. Hawking and G. Ellis,The Large Scale Structure of Space-Time(Cam-bridge University Press,1973).
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

ABSTRACT

In the general relativity theory, we find the representation of the gravity field equation and solutions. We treat the representation of Schwarzschild solution, ReissnerNodstrom 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.

PACS Number:04,04.90.+e Key words: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 e-mail address:[email protected] Tel:010-2496-3953

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  In gravity potential

g  , we introduce tensor f 

(1)

and scalar

K

.

In this time, the condition is

K 0 x  i)

If

g00  1, f   g  , K  1

(1-i)

ii)

If

g 00  1, f   Kg 

(1-ii)

Hence,

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

x  x  dx ' dx '   x ' x '

 g ' dx ' dx '  f ' dx ' dx '

x  x  , g   g  x ' x ' dx   K dx  , dx   K dx  , dx '  K dx' , dx '  K dx'  In Christoffel symbol   ,

g '  g 

   





,

g'  g '

(3)

1  g  g g  1  g (    )     2 x x x K

Therefore, in the curvature tensor

R

(2)



(4)

R   ,

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

(5)

R  ,

In Ricci tensor

1

R   R    In curvature scalar

K

R   

1

K

R 

(6)

R

(7)

R

R  g R  

1

K

g R  

1

K

Hence, in the gravity field equation of Einstein,

1 2

1 g R ) K 2 8G 1  4 T  c K

R   g R 

1

(R  

(8)

In Newtonian approximation, Energy-momentum tensor

 2g 00 

1

K

2g 00  

1

c 2  T00 ,

K

T 

8G 1

c

K

4

is

T00  

8G

c4

T00

(9)

c 2  T00

(10)

Hence,

1

K

T   T 

(11)

Therefore, revised Einstein’s gravity field equation is

1 2

R   g R 

1

(R  

K

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

Therefore, revised gravity field equation of tensor

g 

(12)

is able to reduce Einstein’s gravity field

equation. Therefore,

1 2

g [R   g R ]  

8G

K

g T   

1 2

g [R   g R ]  

8G 4

T



8G 1

c

4

K

g T   

8G 1

c

4

K

T 



c 1 8G 1  8G T   4 T  R   R   4 K c K c Hence,

c

4

1





(13)

1

K

T   T





(14)

Ricci tensor is

R  

1

K

R   

8G 1

c

K

4

(T  

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





)

(15)

The proper distance is

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

(16)

2. Weak gravity field approximation. Weak gravity field approximation is

g      h 

R   

8G

c

4

(T  

,

h   1

1 g T   ) 2

(17)

According to Eq(15),

g      h  , h   1

R   

8G

c

4

(T  

1 g T 2





)

(18)

The tensor of weak gravity field is

R   

8G

c

4

S  , R   

8G

c4

S 

1 2

S   T    T   , 1 2

S   T    T





(19)

The solution is

   S  (t  | x  x '|, x ' )  4G 4 h (t , x )  2  d x ' ,   | x  x '| c

d

3

xT00  M , otherwise T   0 (20)



h (t , x ) 

4G

c2



d

4

x'





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

,

d

3

xT00   K K d 3 x

1

K

otherwise

T00  K M  M 1

K

,

T00  KT00

T   T   0 (21)

As



4G

h00(x ) 

d

3

1 2

x '[T00  T00] 

2 K GM

rc 2 rc  2 K GM 4G 1  ij hij (x )  2  d 3 x '[  ijT00]  2 rc 2 rc 2

,

,

(22)

The proper distance is

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

2GM

rc

2

)c 2dt 2  (1

2GM

rc

2

) ij dx i dx j (23)

The proper distance is in this theory

 ds'2  Kds 2  Kc 2d 2  Kg dx dx   K (1

2GM 2

)c 2dt 2  K (1

2GM 2

) ij dx i dx j

rc rc 2 K GM 2 2 2 K GM  (1 )c dt  (1 ) ij dx i dx j 2 2 rc rc 2GM 2 2 2GM  (1  )c dt  (1  ) ij dx i dx j 2 2 rc rc

 g dx dx 

Kt  t

,

K xi  x i ,

Kxj  x j

,

Kr  r

,

KM  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  c 2 (1

2GM

rc 2

)dt 2 

dr 2  r 2d 2  r 2 sin2 d 2 2GM 1 rc 2

The other representation of Schwarzschild solution is

(25)

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

 c (1  2

 c 2 (1

Kt  t

2GM

rc 2

)dt 2 

2 K GM

rc 2 2GM

rc 2

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

dr 2 )dt   r 2d 2  r 2 sin2  d 2 2 K GM 1 rc 2

)dt

2

2



dr 2  r 2d 2  r 2 sin2  d 2  g dx dx  2GM 1 rc 2

K r  r ,    ,  

,

,

KM  M

(26)

Reissner-Nodstrom solution is

ds 2  g dx dx   c 2 (1

2GM

rc 2



kGQ 2 2 )dt  r 2c 4

dr 2  r 2d 2  r 2 sin2 d 2 2 2GM kGQ 1  2 4 rc 2 r c

(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 2 2GM

rc 2

 g dx dx 



kGQ 2 2 )dt  r 2c 4



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

KkGQ 2 )dt 2  2 4 r c

kGQ 2 )dt 2  r 2c 4

dr 2  r 2d 2  r 2 sin2  d 2 2 2 K GM KkGQ 1  2 4 rc 2 r c

dr 2  r 2d 2  r 2 sin2  d 2 2GM kGQ 2 1  2 4 rc 2 r c

Kt  t

,

K r  r ,    ,   , K M  M , KQ 2  Q 2

(28)

Kerr-Newman solution is

ds 2  g dx dx  2 2c 2GMr  kGQ 2 2 2 2 a sin  ) dt  2 ( 2 c MGr  kGQ ) cdtd c 4 c 4 c 4  2 dr 2  d 2 2 2 2 r  c 2GMr  a  kGQ a 2 s i n2   s i n2 [r 2  a 2  (2c 2G M r k G Q2 ) ]d 2 4 c  2 2 2   r  a cos 

 c 2 (1

(29)

The other representation of Kerr-Newman solution is

ds'2  f dx dx   Kg dx dx   K  ds 2 2 2c 2GMr  kGQ 2 2 2 2 a sin  ) dt  2 K ( 2 c MGr  kGQ ) cdtd c 4 c 4 Kc 4  2 dr 2  Kd 2 2 2 2 r  2c GMr  a  kGQ a 2 s i n2   K s i n2 [r 2  a 2  (2c 2G M r k G Q2 ) ]d 2 4 c 

 Kc 2 (1 

2c 2G K M K r  kGKQ 2  c (1 )dt Kc 4 2

2

 2(2c

2

K a sin2  K MG K r  kGKQ ) cd K td Kc 4 2

Kc 4  d ( K r )2  Kd 2 2 2 2 2 Kr  2c G K M K r  Ka  k G K Q Ka 2 s i n2   s i n2 [Kr 2  Ka 2  (2c 2G K M K r  kGKQ 2 ) ]d 2 4 Kc 2 2c 2GM r  kGQ 2 2 2 2 a sin   c 2 (1  ) d t  2 ( 2 c M G r  kG Q ) cdt d c 4 c 4 c 4  2 dr 2  d 2 2 2 2 r  2c GM r  a  kGQ a 2 s i n2   s i n2  [r 2  a 2  (2c 2GM r  kGQ 2 ) ]d 2 4 c   g dx dx 

  K  Kr 2  Ka 2 cos2   r 2  a 2 cos2 

Kt  t

K r  r ,    ,   , K M  M , KQ 2  Q 2 , K a  a

,

In this time, we obtain the angular momentum

J  cM a  KcMa  KJ data of the time t , the distance r ,

, (30)

the mass

M , the charge Q and

the

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 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   c 2dt

2

dr 2

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

 g dx dx 

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 T 

The radiation temperature

T 

T

of Schwarzschild black hole (In this theory, PMBH)

c 3

(33)

8GMkB is in Data General Relativity theory.

c 3 8GM k B



c 3 8G K MkB



T K

(34)

The black hole (PMBH)’s entropy

dS  8Gk B MdM / c 

8GMkB 2 dQ c dM  , 3 T c

dQ  c 2dM

(35)

The black-hole (PMBH)’s entropy

S

is in Data General Relativity theory.

dS  8M dM GkB / c 

8GkB M 2 dQ c dM  3 T c

8Gk B K M 2 c d( K M ) c 3 8Gk B M 2 K c dM c 3 

dQ  KdS , dQ  c 2dM T 2 dQ  c dM  K c 2dM  K dQ S  KS , Q  K Q K

Pev

Black-hole (PMBH) radiation’s power

Pev  AS T 4 , As  4rS

2

Black-hole (PMBH) radiation’s power 2

,

is

rS  Pev

2

4

2GM

,  ,  is constant

c2

(37)

is in Data General Relativity theory.

As  4rS  K 4rS  KAs , Pev  AS T

(36)

rS 

 KAs 

2GM

c

2



2G K M

c2

 K rS

T 4 Pev  K K2

4

Stefan-Boltzmann constant

 2k B   60  3c 2

,

Black-hole (PMBH) is a perfect black body( 

 1) (38)

The evaporation time

t ev 

of a black hole (PMBH) is

5120 G 2M 3 c 4

The evaporation time

t ev 

t ev

t ev

(39)

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

5120 G 2M 3 5120 G 2K K M 3   K K t ev c 4 c 4

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

(40)

Pev  

dE ev dt ev

(41)

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

Pev  

dE ev d (E ev K ) Pev   dt ev K K K dt ev

,

Mevc 2  E ev  K E ev  K Mevc 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

Table 1. Astrophysical obect

PIMBH1

Mass solar masses

Period

20MΘ

τ seconds

Angular momentum ㎏㎢/s

0.013s

3.0ⅹ1037

PIMBH2

100MΘ

0.063s

7.2ⅹ1038

PIMBH3

1000MΘ

0.63s

7.2x1040

PIMBH4

104 MΘ

6.3s

7.2ⅹ1042

PIMBH5

105MΘ

63s

7.2ⅹ1044

PSMBH6(M87)

6ⅹ109MΘ

3.8x106s

2.6x1054

According to DGRT, BH (PMBH)’s mass

M

is to be

KM ,

time

 K

is to be

 K

,Angular

is 5, PIMBH3’s (from J is to be KJ .Hence, PIMBH2’s (from PIMBH1) PIMBH 2) K is 10,PIMBH4’s(from PIMBH3) K is 10, PIMBH5’s (from PIMBH4) K is 4 10, PSMBH6’s (from PIMBH5) K is 6  10 . momentum

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”,

20M   M PIMBH  100 ,000 M  10 5 M   M PSMBH  10 17M  Therefore, calculated data is in DGRT, Table 2. Astrophysical obect

Mass solar

Period

τ seconds

Angular momentum ㎏㎢/s

masses PIMBH1

20MΘ

0.013s

3.0ⅹ1037

PIMBH2

100MΘ

0.065s

7.5ⅹ1038

PIMBH3

1000MΘ

0.65s

7.5x1040

PIMBH4

104 MΘ

6.5s

7.5ⅹ1042

PIMBH5

105MΘ

65s

7.5ⅹ1044

PSMBH6(M87)

6ⅹ109MΘ

3.9x106s

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  4J . 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.

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