GENERALIZED SECOND LAW LIMITS ON THE VARIATION OF FUNDAMENTAL CONSTANTS PRL 99, 061301 (2007)

Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 12 - 18 2009

GENERALIZED SECOND LAW LIMITS ON THE VARIATION OF FUNDAMENTAL

CONSTANTS

PRL 99, 061301 (2007)

JANE H MACGIBBON

UNIVERSITY OF NORTH FLORIDA

MOTIVATION

Is the Fine Structure Constant constant?

e = the charge of the electron

ħ = Planck‘s constant

c = speed of light

2 /e c

MEASUREMENTSWebb et al.

M.T. Murphy, J.K. Webb & V.V. Flambaum M.N.R.A.S.. 345, 609 (2003)

Δα/α =(-0.543 ± 0.116) x10-5 over redshift range 0.2<z<3.7

2 independent samples, Keck/HIRES spectra, 78 absorption systems

greatest deviation seen at highest z (data marginally prefer a linear increase in α with time)

J.K. Webb et al. Phys.Rev.Lett. 87, 091301 (2001)

Δα/α = -0.72 ± 0.18 x10-5 over redshift range 0.5 < z < 3.5

3 large optical datasets and two 21cm/mm absorption systems provide four independent samples (each set shows same variation to 4 σ)

J.K. Webb et al. Phys.Rev.Lett. 82, 884-887 (1999)

Δα/α = -1.9 ± 0.5 x10-5 over redshift range 1.0 < z< 1.6 Δα/α = -0.2 ± 0.4 x10-5 over redshift range 0.5 < z < 1

MEASUREMENTSChand et al. VLT

H. Chand et al. astro-ph/0601194

Δα/α=(+0.05 ± 0.24)x10-5 at redshift z = 1.1508

H. Chand et al. astro-ph/0408200

Δα/α = (+0.15 ± 0.43) x10-5 over redshift range 1.59< z < 2.92

H. Chand et al. Astron.Astrophys. 417, 853 (2004)

R. Srianand et al. Phys.Rev.Lett. 92, 121302 (2004)

Δα/α = (-0.06 ± 0.06) x10-5 over redshift range 0.4<z<2.3

THEORETICAL LIMITS

P.C.W. Davies, T.M. Davis, & C.H. Lineweaver Nature 418, 602-603 (2002)

• If change in α is due solely to change in e , then Black Hole Entropy Law will be violated

But Davies, Davis & Lineweaver looked at

entropy change due to change in e

at fixed time


GENERALIZED SECOND LAW OF THERMODYNAMICS

over any time interval

0BH R MS S S


BLACK HOLE ENTROPY

Consider entropy change due to change in e of per second

over any time interval

23/ / 2 ~ 10e e

GENERALIZED SECOND LAW OF THERMODYNAMICS

over time interval Δt≥0

Entropy of Black Hole

Area of Charged Non-Rotating Black Hole

Temperature

0BH R MS S S 3

4BH BH

kcS A

G

2 2

2 24

4/BH

GA M M Q G

c

2 2 2 23

22 2

/ /2

2 /BH

BH

M Q G M Q GG cT

kc A kG M M Q G

ΔSBH

First Term contains Hawking Flux

Second Term

,

3

4BH BH BH

BH

dS A Akc deS t t

dt G t e dt

2 2

22 2

4 2 2

/8/

/BH H H

M M Q GA dM QG QM M Q G

t c dt G tM Q G

2 22

4 2 2

/8

/BH

M M Q GA G Q Q

e c G eM Q G

/ /Q e Q e

CASE I : Net radiation loss from black hole into environment

CASE (IA)

is not affected by Hawking radiation

So

TERM 1 TERM 2

BH R MT T

Q / 0HQ t

12 2 2 22

1 1 / /BH HdS dMkG Q Q deQ GM M M Q G

dt c dt G e dt

17, M 10 gBH e BHT m

CASE IA : Mass Loss due to Hawking Radiation

and (D.N. Page)

So ΔS≥0 until TERM 1 ≈ TERM 2 . This happens when black hole

charge satisfies

R M BH eT T m

44

2 2, 3x10 HdM c

dt G M

increase 1.62 ( decrease due to Hawking radiation)R M BHS S

1/ 2 1/ 24 4

1 2 1 2 3 12

/ /

c cQ

GM e de dt G M e de dt

1/ 23

1/ 223 -1 201 2 16

(If / ~ 10 s , 6x10 / gm 2 esu)1.8x10 g

Mde dt e Q M

CASE IA :

Maximal Possible Charge on Black Hole

So

BUT (Gibbons and Zaumen)

A black hole quickly discharges by superradiant Schwinger-type

e+e- pair-production around black hole if is greater than

R M BH eT T m

1/ 2MAXQ G M

Q

2 2 2 /PP eQ G m M ce

161 2 for 1.8 10 gMAXQ Q M

CASE IA : So for all lighter than

Superradiant discharge rate (Gibbons)

is greater than TERM 2 for all lighter than

M

1/53 6 2

5 4 1

2

/PP

e

c eM

G m e de dt

25 23 -13x10 gm for / 10 s PPM de dt e

3 2 23 4

2 2 23 2

exp , / /ePP c m rdQ Q er G M M Q G c

dt c r Qe

( / )( / ) /Q G Q e de dt M

1/ 25/ 2 2 5/ 2 14 4

2 3/ 2 3 1/ 22 2 2 1

4 /exp

2/

eG m M e de dtc e

c eG M e de dt

25 23 -17.0x10 gm for / 10 s PP EM de dt e

1 2PPQ Q

R M BH eT T m

CASE IA :

So if then ΔS≥0 for all lighter than

M25

, 3 7x10 gmPP PP EM

23 -1/ 10 sde dt e

R M BH eT T m


CASE IA :

So if then ΔS≥0 for all lighter than

Mass of black hole whose temperature is 2.73K

(cosmic microwave background temperature):

Coincidence?

M25

, 3 7x10 gmPP PP EM

254.5x10 gmCMBM

23 -1/ 10 sde dt e

R M BH eT T m

CASE IB : Charge Loss due to Hawking Radiation and so

For it is straightforward to show net entropy increase from

Hawking emission TERM 1 dominates TERM 2 so ΔS≥0 if

For higher , use work of Carter to show high temperature chargedblack hole discharges (via thermal Hawking and superradiant regimes)quickly over the lifetime of the Universe so ΔS≥0

SUMMARY OF CASE I:

If , ΔS≥0 for black holes emitting in the present Universe

H HQQ dN

et Q dt

2H H

av

Q dMc eQ Q

t E dt

MAXQ Q

Q

23 -1/ 10 sde dt e

23 -1/ 10 sde dt e

, BH R M e BHT T m T

CASE II:

Net accretion (which lowers and leads to more accretion)

Thermodynamics

With each accretion, increase = Environment Energy decrease

and so

increase due to accretion > decrease due to accretion

Also

increase due to accretion > decrease due to Hawking radiation

So compare effect of with increase due to accretion

BH R MT T

1 1 2

fixed

, R M BHR M BH

Q

S ST T c

E M

BHT

R MS BHS

M

BHS BHS

/de dt BHS

CASE II:

Cold Black Hole in Warm Thermal Bath

absorbs (and radiates )

per particle freedom Geometrical Optics Xstn

So

= mass of black hole whose temperature equals ambient temperature

(Note is max for thermal bath so this gives strictest constraint on ΔS)

R MS

2 44

3 260 S BH

dE kT

dt c

2 44

3 260 S R M

dE kT

dt c

2 2 427 /S G M c

244

2 2, ~ 10R M

R MR M R M

cdM M

dt G M M

R MM

BH R MT T

CASE II:So ΔS≥0 until TERM 1 ≈ TERM 2. This happens when black holecharge satisfies

If then only if

and only if Problem?

1/ 2 1/ 23/ 24 4

1 2 1 2 2 1' 2

/ /R M R M

R MR M R M

c cMQ

MGM e de dt G M M e de dt

BH R MT T

54 10 gM 1 2'MAXQ Q

1 2'PPQ Q 24 10 gM

23 -1/ 10 sde dt e

CASE II:So ΔS≥0 until TERM 1 ≈ TERM 2. This happens when black holecharge satisfies

If then only if

and only if Problem?

If at (mass at which )

require ( << )

1/ 2 1/ 23/ 24 4

1 2 1 2 2 1' 2

/ /R M R M

R MR M R M

c cMQ

MGM e de dt G M M e de dt

BH R MT T

54 10 gM 1 2'MAXQ Q

1 2'PPQ Q 24 10 gM

1 2'PPQ Q PP MAXQ Q39 10 gM

1 38 1 / 10 se de dt

23 -1/ 10 sde dt e

1 23 1 / 10 se de dt


CASE II:Resolution: Can charged black hole accreted opposite charge fast enough to avoid reaching ?

Number of thermally accreted positronswhere is positron fraction of background

So and

(taking which gives the strictest limit) provided

(ie for )

BH R MT T

1 2'Q 2 3 3

3 260e S R M

e

dN k T

dt c

23 3 310 R Me

MAX R M R MACC

c eTdQ M

dt GM M

1 2'BHQ Q

BH

MAXACC

QdQ de

dt e dt

1 9 2 1 / 10 se

e de dt 175 10

e

e

1 23 1 / 10 se de dt

CASE II:Resolution: Can charged black hole accreted opposite charge fast enough to avoid reaching ?

Number of thermally accreted positronswhere is positron fraction of background

So and

(taking which gives the strictest limit) provided

(ie for )

Also (Gibbons) BH can only gravitationally accrete particle of like charge if particle is projected at BH with initialvelocity and large BH is more likely to lose net charge by accretingparticle of opposite charge

BH R MT T

1 2'Q 2 3 3

3 260e S R M

e

dN k T

dt c

23 3 310 R Me

MAX R M R MACC

c eTdQ M

dt GM M

1 2'BHQ Q

BH

MAXACC

QdQ de

dt e dt

225 10ACC MAXQ Q Q

1 9 2 1 / 10 se

e de dt 175 10

e

e

1 23 1 / 10 se de dt


CASE II:

Special Case

ΔS due to absorption > ΔS due to emission

and from Case I for

ΔS due to emission ≥ ΔS decrease from

SO SUMMARY FOR CASE II:

For all ΔS ≥ 0 if

R M BH CMBT T T

BH CMBT T23 -1/ 10 sde dt e

BH R MT T 23 -1/ 10 sde dt e

BH R MT T


NOTES

Rotation

Above results also apply for charged rotating black holes

Get strictest constraints for charged non-rotating black hole

Second Order Effects

Changes in Hawking rate and pair production discharge rate

due to are second order effects/de dt


SUMMARY OF CASE I & II

is not ruled out by the GSL

is the maximum variation in allowed by the Generalized Second Law of Thermodynamics for black holes in the present Universe

23 -1/ 10 sde dt e

e23 -1/ 10 sde dt e


IMPLICATIONS Above only uses standard General Relativity

and standard QED (No Extensions)

• Use same methodology to find constraints

on independent and dependent variation in

, and (and )

For G see arXiv: 0706.2821

• Use same methodology for Extension Models by including extra terms in

cG s

S


IMPLICATIONS If Webb et al. measurements are correct

• Is varying at the maximal rate allowed by the GSL?

• Our constraint predicts the rate of increase in

and should weaken as the Universe ages now

• Are the other constants of Nature and/or coupling constants varying at the maximal rate allowed by the GSL?

• What is the physical mechanism for the change in ?

ee

e


IMPLICATIONS

Our constraint predicts the rate of increase in

and should weaken as the Universe ages now

• Extrapolating above constraint equations leads to

at about z ~ 40 BUT extrapolating

back in time may require inclusions of other effects

(eg how does accretion constraint change in pre-

re-ionization era?)

e

0/ 50%


POSSIBLE MECHANISM?

Our derivation suggests look for a coupling

between the electron and the cosmic photon

background (in standard QED)

Note: Schwinger effect is non-linear effect in

standard QED; know from accelerator

experiments that varies with energy scalee


POSSIBLE MECHANISM?

Our derivation suggests look for a coupling

between the electron and the cosmic photon

background (in standard QED)

Scattering of vacuum polarization e+e-

around bare electron off the cosmic photon

background?


GSL LIMITS ON VARIATION IN GDepends on how : • if n > -1/2 (including n = 0 ), GSL does not constrain

an increase in G but any decrease must be less than |G-1dG/dt|≈10-52 s-1

• if n < -1/2, GSL does not constrain an decrease in G but any increase must be less than |G-1dG/dt|≈10-52 s-1

• if n = -1/2, the GSL does not constrain a decrease but any increase must be less than |G-1dG/dt|≈ t -1

If restrict to only astronomically observed stellar-mass black holes, n > -1/2 and n < -1/2 limits are only weakened by 108 and n = -1/2 limit is unchanged

nM G

GENERALIZED SECOND LAW LIMITS ON THE VARIATION OF FUNDAMENTAL CONSTANTS PRL 99, 061301 (2007)

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