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 8G 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
8G 1
c
K
4
is
T00
8G
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 8G 8G 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 ]
8G
K
g T
1 2
g [R g R ]
8G 4
T
8G 1
c
4
K
g T
8G 1
c
4
K
T
c 1 8G 1 8G 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
8G 1
c
K
4
(T
1 8G 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
8G
c
4
(T
,
h 1
1 g T ) 2
(17)
According to Eq(15),
g h , h 1
R
8G
c
4
(T
1 g T 2
)
(18)
The tensor of weak gravity field is
R
8G
c
4
S , R
8G
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 Kc 4 2 dr 2 Kd 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 Kc 4 2
2
2(2c
2
K a sin2 K MG K r kGKQ ) cd K td Kc 4 2
Kc 4 d ( K r )2 Kd 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 Kc 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)
8GMkB is in Data General Relativity theory.
c 3 8GM k B
c 3 8G K MkB
T K
(34)
The black hole (PMBH)’s entropy
dS 8Gk B MdM / c
8GMkB 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 8M dM GkB / c
8GkB M 2 dQ c dM 3 T c
8Gk B K M 2 c d( K M ) c 3 8Gk 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 4rS
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 4rS K 4rS 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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