Hawking Radiation - WordPress.com · 2018. 7. 26. · Hawking Radiation Shihao Wu Math 6130...

Hawking Radiation

Shihao WuMath 6130

Memorial University of Newfoundland

July 24, 2018

Shihao Wu ( Memorial University of Newfoundland) July 24, 2018 1 / 24

Introduction

Outline

Origin of Hawking radiation,

The particle perspective of Hawking radiation,

The temperature of the black holes.


Origin of Hawking radiation

Introduction

The first publication of the Hawking radiation is the article Black hole explosions?on Nature Vol. 248, 1974.

Stephen Hawking discovered that black holes have certain temperature, whichresults in certain mass loss or radiation from black holes.

Such discovery was very controversial. Even Hawking himself had doubts aboutthe physical cause of the radiation. He thought of the possibility that theradiation come from artifact of the mathematical structures.



Black Hole and Thermaldynamics

The four laws of the black hole mechanics and the four laws of thermaldynamicsare connected in forms.

The first law of black hole mechanics is

δM =1

8πκδA + ΩδJ + ΦδQ, (1)

where δM is the change in mass, κ is surface gravity, δA is change in area of theevent horizon, Ω is angular velocity, δJ is change in angular momentum, Φ iselectrostatic potential and δQ is change in charge. The expression is similar to thefirst law of thermal dynamics

δU = Q −W , (2)



Second Law

Black hole mechanics: the event horizon area A cannot be reduced.

Thermodynamics : the entropy of a closed system cannot be reduced.



Third Law

Black hole mechanics: the surface gravity κ cannot be reduced to zero by finiteamount of operations.

Thermodynamics : Nernst form is that the temperature of a system cannot bereduced to absolute zero in finite amount of operations.



Forth (or Zeroth) Law

Black hole mechanics: the surface gravity κ of a stationary black hole is constanton the entire surface of the event horizon.

Thermodynamics : if two thermodynamic systems are each in thermal equilibriumwith a third, then they are in thermal equilibrium with each other.



Problems with Hawking radiation

1. The generalized second law issue: GSL stated that the total entropy of ablack hole and its surroundings cannot decrease. This law can be violated byan object with minimal energy near a black hole, so that the increase of blackhole entropy would not make up for its own entropy lose.

2. Trans-Planck issue: The particles emitted by Hawking radiation near theevent horizon can have a wavelength shorter than Planck scale, which meansthe classical prospective of space time no longer apply to them.

3. Unitary issue: When a emitted photon fall into the black hole, which willrise a density matrix description of the radiation. This is a non-unitarydescription of the evolution, which is not a typical quantum mechanic process.



“Black hole explosion?”

Consider a massless Hermitean scalar field phi in an asymptotially flat space timecontaining a star which is collapsing into a black hole.

φ =∑i

(fiai + fia+i ), (3)

where fi are a complete orthornalmal family of wave equation solutions and ai anda+i are the annihilation and creation operator. φ can also be written as

φ =∑i

(pibi + pib+i + qici + qici ), (4)

where pi are the solutions of the wave equations which have positive frequencyand are zero on the event horizon and outgoing and qi are the solutions which arezero on the future null infinity I+.




pi and qi can be further expressed in terms of fi and fi , such as

pi =∑j

(αij fi + βij fi )

By comparing Eqn 3 and 4, one can write

bi =∑j

(αijaj − βija+j ) (5)

When there is no incoming particles in the expectation value, the outgoing stateof b+i bi is

< 0−|b+i bi |0− >=∑j

|βij |2




The outgoing solution plmw can be express as (with fixed lm)

pω =

∫(αωω′ fω′ + βωω′ fω′)dω′ (6)

To solve αωω′ and βωω′ , imagine a wave with positive frequency ω on I+ andpropagating backwards via spacetime in solitude through event horizon and someof it will be scattered by the event horizon and escape to I− with the samefrequency ω.




The other part of the wave will propagate backwards into the collapsing star andeventually out onto I− with significant blue shift with the form

Cω−1/2 exp(−iωκ2 log(v0 − v) + iωv) for v < v0

and0 for v ≥ v0,

where v0 is the last advanced time for a particle leave I−, return to the origin andmove to I+.




Eventually, one can solve αωω′ and βωω′ and find

|αωω′ | = exp(πω/κ)|βωω′ |, (7)

which indicates that the wave pocket mode would emit different amount ofparticles than it would absorb during a similar event on the black hole from I−,with the ratio of

1

exp(2πω/κ)− 1.

This means the particles are emitted from the black hole in total.


Particle perspective of Hawking radiation

Cross section and Decay rate

The total decay rate

Γtot =n∑

i=1

Γi ,

The expression for the decay rate Γi is

Γi =

∫|M |2(2π)4δ4(p1 − p2 − · · · − pn)

n∏j=2

d3pj(2π)32Ej

, (8)

which is known as the Fermi’s golden rule. It consists of 3 maincomponents-amplitude, phase space factor and the delta function.



Cross section Example

the amplitude expression of the electron-muon scattering can be constructed as

Figure: Feynman diagram for electron-muon scattering.

M =(2π)4∫

[ue(p3, s3)igeγµue(p1, s1)]

−igµνq2

[umuon(p4, s4)igeγνumuon(p2, s2)]

× δ4(p1 − p3 − q)δ4(p2 − p4 + q)d4q(9)




The Hawking radiation can be considered as a black hole decay. The black holeloses a massless quantum, such as photon, and become a new black hole.

(black hole)i → (black hole)f + γ (10)




The estimated expression for the decay rate of Hawking radiation is

~Γ(bi → bf + γ) = ε

(~ωmP

)2(AH

`2p

)(~ω)

(2

2Jf + 1

2Ji + 1

)Nf

Ni, (11)

with the units of c = 1 and G =`pmp

=`2p~ = ~

m2p.




~Γ(bi → bf + γ) = ε

(~ωmP

)2(AH

`2p

)(~ω)

(2

2Jf + 1

2Ji + 1

)Nf

Ni,

It consists of 6 terms and one assumption.

ε is a free scalar parameter for normalization.(~ωmP

)is the coupling constant of the decay process. To be more precise, it is

the coupling between matter and gravity,(AH

`2p

)is the ratio between the area of the event horizon and the areal of

Planck length square. It provides a scale for how many individual decay canoccur all over the event horizon, since each unit area of `2P can emit radiationseparately. It works as the sum of all Γi




~Γ(bi → bf + γ) = ε

(~ωmP

)2(AH

`2p

)(~ω)

(2

2Jf + 1

2Ji + 1

)Nf

Ni,

~ω is the energy of the radiation, which is proportional to the phase spacecomponent of a typical decay process.(

2 2Jf +12Ji+1

)is the spin factors. This term comes from the sum over the squares

of Clebsch-Gordon coefficients. Clebsch-Gordon coefficients are related to theangular momentum components of the amplitude. i.e. the si terms in Eqn 1.For the most simple version of the decay, one can set Ji = Jf = 0.




~Γ(bi → bf + γ) = ε

(~ωmP

)2(AH

`2p

)(~ω)

(2

2Jf + 1

2Ji + 1

)Nf

Ni,

The last term Nf

Niis the statistical factors. It averages over the initial stats Ni

and sums over the final states Nf .

The one assumption it has was the conservation law holds, which serves thesame purpose as the delta function terms.




Nf

Ni= exp(Sf − Si ) ≈ exp

(∂S

∂M[Mf −Mi ]

)≈ exp

(− ∂S∂M

~ω)

= exp(−~ω/T )

(12)

Now, we can rewrite Eqn 11 as

Γ(bi → bf + γ) = ε

(~ωmP

)2(AH

`2p

)ω

(2

2Jf + 1

2Ji + 1

)exp(−~ω/T ), (13)

with the dω increment at ω

dΓ(bi → bf + γ) = ε

(~ωmP

)2(AH

`2p

)(2

2Jf + 1

2Ji + 1

)exp(−~ω/T )dω




Therefore the power from the decay/radiation is

dP(ω) = ~ωdΓ(bi → bf + γ) = 2ε (~ω)3 AH

(2Jf + 1

2Ji + 1

)exp(−~ω/T )dω

Let Jf = Ji = 0 and compare it to the original paper by Hawking, we have ε = 13π

and T = κ.


Conclusion

Conclusion

It is an interesting combination of both quantum mechanics and general relativity.However, Hawking radiation only produce a limited effect on black holes to beobserved astronomically. On the other hand, it is impossible to simulate blackhole effects experimentally in the near future.

With more developments on the gravitational detection field, the chance ofdiscovering Hawking radiation related data could be increasing. The gravitationalcoupling constant obtained from other sources can provide more precise model forHawking radiation. In return, a better Harking radiation model can helpintroducing additional massless quanta, such as graviton to the Standard model.


Questions

Questions?


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