theoretical physics - gravity

1. Ferdinand Joseph P. Roaa, Alwielland Q. Bello b,
Engr. Leo Cipriano L. Urbiztondo Jr.c

2. In this elementary exercise we consider the
Klein-Gordon field in the background of
Schwarzschild space-time metric. Very near the
event horizon the radial equation of motion is
approximated in form and we obtain oscillatory
solution in the Regge-Wheeler coordinate. The
time and radial solutions are then recast in the
outgoing and ingoing coordinates that
consequently lead to the outgoing and ingoing
waves that have respectively dissimilar
(distinct) analytic properties in the future and
past event horizons.

3. 1) Introduction
-Background of the present paper
-Related subject matter
-scope or limitation of the present
paper

4. - Scalar action, Lagrangian, equation
of motion
- Schwarzschild spacetime metric
- component equations of motion

5. - Recast of the radial component of
the equation of motion using
Regge-Wheeler coordinate

6. - Approximate radial equation of
motion, oscillatory solution
- Eddington-Finkelstein
coordinates
- out-going, in-going wave
solutions

7. - out-going, in-going null paths,
Carter-Penrose (CP) diagram
- Future and past event horizons

8. [1]Townsend, P. K., Blackholes – Lecture
Notes, http://xxx.lanl.gov/abs/gr-
qc/9707012
[2]Carroll, S. M., Lecture Notes On
General Relativity, arXiv:gr-qc/9712019
[3]S. W. Hawking, Particle Creation by
Black Holes, Commun. math. Phys. 43,
199—220 (1975)

9. [4]Ohanian, H. C. Gravitation and
Spacetime, New York:W. W. Norton &
Company Inc. Copyright 1976
[5]Bedient, P. E., Rainville, E. D.,
Elementary Differential Equations,
seventh edition, Macmillan Publishing
Company, 1989, New York, New York,
USA

10. - based on our answers to an exercise
presented on page 142 of [1]
- topic related to Hawking radiation
- this paper only covers the important
details in our solutions to Klein-
Gordon field equation against the
background of Schwarzschild space-
time metric

11. -Hawking radiation was explored in the middle
of 1970’s in Stephen Hawking’s paper [3]
-- quantum mechanics of pair production in
extreme proximity to a very strong
gravitational field of a blackhole : result led
to Planck distribution for black body
radiation at a given Hawking temperature –
BHs radiate

12. Scalar action
𝑆 = 𝑑4
𝑥 ℒ (1)

13. Lagrangian (density) for scalar field
ℒ = −𝑔
1
2
𝑔 𝜇𝜐
(𝜕 𝜇 𝜑) 𝜕𝜐 𝜑 + 2𝑉(𝜑)
(2)
scalar potential metric signature +2
𝑉 𝜑 =
1
2
𝑀2
𝜑2
(3)

14. equation of motion for the scalar field in
curved spacetime
1
−𝑔
𝜕𝜇 −𝑔𝑔 𝜇𝜐
(𝜕 𝜐 𝜑) − 𝑀2
𝜑 = 0 (4)
covariant four-divergence
1
−𝑔
𝜕𝜇 −𝑔𝑔 𝜇𝜐
(𝜕 𝜐 𝜑) = 𝛻𝜇 𝑔 𝜇𝜐
(𝜕 𝜐 𝜑)
(5)

15. Schwarzschild spacetime metric
𝑑𝑆2
= −𝜂𝑑𝑡2
+ 𝜀𝑑𝑟2
+ 𝑟2
𝑑𝜃2
+ 𝑟2
𝑠𝑖𝑛2
𝜃𝑑𝜙2
𝜂 = 𝜀−1
= 1 −
2𝐺𝑀 𝑞
𝑟
(6)
𝑐2
= 1 Heaviside units
𝑀 𝑞 is the mass of the gravitational body

16. spacetime coordinates
𝑥 𝜇
= (𝑥0
= 𝑡; 𝑥1
= 𝑟; 𝑥2
= 𝜃; 𝑥3
= 𝜙 )
product solution
𝜑 𝑥0
, 𝑟, 𝜃, 𝜙 = 𝑇 𝑡 𝑅(𝑟)Θ(𝜃)𝜓(𝜙)
(7)

17. component equations of motion
1
𝜓
𝜕 𝜙
2
𝜓 = −𝜇 𝜙
2
(8.1)
1
Θ
1
𝑠𝑖𝑛𝜃
𝜕 𝜃 𝑠𝑖𝑛𝜃(𝜕 𝜃Θ) −
𝜇 𝜙
2
𝑠𝑖𝑛2 𝜃
= −𝜇 𝜃( 𝜇 𝜃 + 1)
(8.2)

18. 1
𝑇
𝜕0
2
𝑇 = −𝜔2
(8.3)
1
𝑅
1
𝑟2
𝑑
𝑑𝑟
𝜂𝑟2
𝑑𝑅
𝑑𝑟
−
𝜇 𝜃 𝜇 𝜃 + 1
𝑟2
= 𝑀2
−
𝜔2
𝜂
(8.4)

19. - gravity takes effect through the
metric tensor component, 𝜂

20. Regge-Wheeler coordinate,
𝑟∗
= 𝑟 + 2𝐺𝑀 𝑞 𝑙𝑛
𝑟
2𝐺𝑀 𝑞
− 1
∀𝑟 > 𝑟 𝐻 (= 2𝐺𝑀 𝑞 )
𝜕𝑟
𝜕𝑟∗
=
𝜕𝑟∗
𝜕𝑟
−1
= 𝜂
(9.1)

21. 1
𝑅
𝑑2
𝑅
𝑑𝑟∗2
+
1
𝑅
2(𝑟 − 2𝐺𝑀 𝑞)
𝑟2
𝑑𝑅
𝑑𝑟∗
+ 𝜔2
=
𝜇 𝜃 𝜇 𝜃 + 1
𝑟2
+ 𝑀2
𝜂
(9.2)

22. 𝑟 ≈ 𝑟 𝐻 for approximate radial eq’n of
motion
𝑑2 𝑅
𝑑𝑟∗2 + 𝜔2
𝑅 = 0 (9.3)
oscillatory solution
𝑅 𝑟∗
= 𝑅01 𝑒𝑥𝑝 −𝑖𝜔𝑟∗
+ 𝑅02 𝑒𝑥𝑝 𝑖𝜔𝑟∗
(9.4)

23. For (8.3) oscillatory solution
𝑇 𝑡 = 𝑇01 𝑒𝑥𝑝 −𝑖𝜔𝑡 + 𝑇02 𝑒𝑥𝑝 𝑖𝜔𝑡
(9.5)
Eddington-Finkelstein coordinates
𝑢 = 𝑡 + 𝑟∗
ingoing (9.6.1)
𝑣 = 𝑡 − 𝑟∗
outgoing (9.6.2)

24. Outgoing wave
Φ(𝑟∗ 𝑡)
+
= 𝐴0
+
𝑒𝑥𝑝 −𝑖𝜔 𝑣 (9.7.1)
Ingoing wave
Φ(𝑟∗ 𝑡)
−
= 𝐴0
−
𝑒𝑥𝑝 −𝑖𝜔 𝑢 (9.7.2)

25. contrasting case, limit as 𝑟 → ∞ very
far from the event horizons
𝑑2 𝑅
𝑑𝑟∗2 + (𝜔2
−𝑀2
)𝑅 = 0
(9.8.1)
massless 𝑀 = 0
𝑑2 𝑅
𝑑𝑟∗2 + 𝜔2
𝑅 = 0 (9.8.2)

26. Note:
- case for waves very near event
horizons mass term in (9.2) drops off,
vanishing 𝜂 – effectively massless
scalar that corresponds to a massless
scalar field very far from horizons

27. - very near the event horizon the scalar
field is effectively massless, very far
from the horizon, there corresponds
the same radial equation of motion for
a massless scalar field

28. - the crude approximation:
-- same out-going solution (9.7.1) for
the two cases of waves very near the
horizon and waves very far from the
horizon
-- assume that the same out-going
waves very near the horizon that
reached very far from the horizon

29. waves massless
𝛾+
: 𝜒 − 𝜂 = −𝑎+
(10.1.1)
out-going null path
𝛾−
: 𝜒 + 𝜂 = 𝑎−
(10.1.2)
in-falling null path

30. 𝑟 𝐻 < 𝑟 < ∞
region I Carter-Penrose diagram

31. 𝐻+
: 𝜒 − 𝜂 = −𝜋 (10.1.3)
future event horizon
𝐻−
: 𝜒 + 𝜂 = −𝜋 (10.1.4)
past event horizon

32. ℑ+
: 𝜒 + 𝜂 = 𝜋 (10.1.5)
future null infinity
ℑ−
: 𝜒 − 𝜂 = 𝜋 (10.1.6)
past null infinity

33. given changes of coordinates
𝜒 + 𝜂 = 2 𝑢′, tan 𝑢′ = 𝑢 (10.2.1)
𝜒 − 𝜂 = −2 𝑣 ′ , tan 𝑣 ′ = 𝑣 (10.2.2)

34. very near the horizon, 𝑟 ≈ 𝑟 𝐻
𝑟∗
→ −∞
Future event horizon,
𝐻+
: 𝑟∗
→ −∞ and 𝑡 → ∞

35. - along 𝑢 = 𝑐𝑜𝑛𝑠𝑡, infalling wave hits 𝐻+
in an infinite coordinate future
𝑟∗
→ −∞, 𝑡 → ∞, 𝑣 = ∞
outgoing wave not defined on the
future event horizon

36. past event horizon,
𝐻−
: 𝑟∗
→ −∞, 𝑡 → − ∞
- along 𝑣 = 𝑐𝑜𝑛𝑠𝑡, the out-going wave
hits 𝐻−
in an infinite coordinate
past

37. 𝑟∗
→ −∞, 𝑡 → − ∞, 𝑢 = −∞
infalling wave not defined on past
event horizon

38. Future continuing discussions:
-parametrized forms of the wave
solutions Φ(𝑟∗ 𝑡)
+
= 𝐴0
+
𝑒𝑥𝑝 −𝑖𝜔 𝑣 𝑢
𝑣 𝑢 = −2𝐺𝑀𝑙𝑛 − 𝑢 , −∞ < 𝑢 < 0
- Fourier components, scalar field
operators, Bogoliubov coefficients
- Planck distribution for black body
radiation, Hawking temperature

39. - Planck distribution for black body
radiation(pp. 129, [1])

40. - late time particle flux through ℑ+
given a vacuum on ℑ−

41. - Hawking temperature

42. Surface gravity
𝜅 =
1
4𝐺𝑀 𝑞
𝑐2
= 1 Heaviside units