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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July GENERALIZED SECOND LAW LIMITS ON THE VARIATION OF FUNDAMENTAL CONSTANTS PRL 99, (2007) JANE H MACGIBBON UNIVERSITY OF NORTH FLORIDA

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MOTIVATION Is the Fine Structure Constant constant? e = the charge of the electron ħ = Planck‘s constant c = speed of light

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MEASUREMENTS Webb et al. M.T. Murphy, J.K. Webb & V.V. Flambaum M.N.R.A.S.. 345, 609 (2003) Δα/α =( ± 0.116) x10 -5 over redshift range 0.2

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MEASUREMENTS Chand et al. VLT H. Chand et al. astro-ph/ Δα/α=(+0.05 ± 0.24)x10 -5 at redshift z = H. Chand et al. astro-ph/ 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, (2004) Δα/α = (-0.06 ± 0.06) x10 -5 over redshift range 0.4

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THEORETICAL LIMITS P.C.W. Davies, T.M. Davis, & C.H. Lineweaver Nature 418, (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

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July GENERALIZED SECOND LAW OF THERMODYNAMICS over any time interval

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July BLACK HOLE ENTROPY Consider entropy change due to change in e of per second over any time interval

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GENERALIZED SECOND LAW OF THERMODYNAMICS over time interval Δt≥0 Entropy of Black Hole Area of Charged Non-Rotating Black Hole Temperature

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ΔS BH First Term contains Hawking Flux Second Term,

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CASE I : Net radiation loss from black hole into environment CASE (IA) is not affected by Hawking radiation So TERM 1 TERM 2

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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

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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

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CASE IA : So for all lighter than Superradiant discharge rate (Gibbons) is greater than TERM 2 for all lighter than

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CASE IA : So if then ΔS≥0 for all lighter than

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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?

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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 charged black 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

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July

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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

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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)

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CASE II: So ΔS≥0 until TERM 1 ≈ TERM 2. This happens when black hole charge satisfies If then only if and only if Problem?

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July

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CASE II: So ΔS≥0 until TERM 1 ≈ TERM 2. This happens when black hole charge satisfies If then only if and only if Problem? If at (mass at which ) require ( << )

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July CASE II: Resolution: Can charged black hole accreted opposite charge fast enough to avoid reaching ? Number of thermally accreted positrons where is positron fraction of background So and (taking which gives the strictest limit) provided (ie for )

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CASE II: Resolution: Can charged black hole accreted opposite charge fast enough to avoid reaching ? Number of thermally accreted positrons where 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 initial velocity and large BH is more likely to lose net charge by accreting particle of opposite charge

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July

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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

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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: Use same methodology for Extension Models by including extra terms in

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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 ?

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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?)

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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 scale

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July 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?

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Jane H MacGibbon "GSL Limits on Varn of Constants" MG12 Paris July GSL LIMITS ON VARIATION IN G Depends 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 -1 dG/dt|≈ s -1 if n < -1/2, GSL does not constrain an decrease in G but any increase must be less than |G -1 dG/dt|≈ s -1 if n = -1/2, the GSL does not constrain a decrease but any increase must be less than |G -1 dG/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 10 8 and n = -1/2 limit is unchanged

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