gravitational casimir effect...b. s. dewitt, “superconductors and gravitational drag”, phys....
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Gravitational Casimir EffectJAMES Q. QUACH
Quach, PRL 114, 081104 (2015)
Hendrik Casimir (1948)
𝑃𝐼 𝑎 = −𝜋2
240ℏ𝑐𝑎4
𝐸𝑛 = ℏ𝜔(𝑛 +12)
𝐸0 =ℏ𝜔2
Casimir Effect
�Motivation�Gravitational Casimir Effect
�Ordinary Materials�Superconductors
Outline
QM & GR Interface
Hawking Radiation
Detectors
BA
DC
T
g
COW Experiment
QM & GR InterfaceDetectors
BA
DC
T
g
COW Experiment
A. Overhauser, R. Colella, S. Werner
QM & GR InterfaceDetectors
BA
DC
T
g
Colella, Overhauser, Werner, PRL 1975
COW Experiment
QM & GR InterfaceDetectors
BA
DC
T
g
Colella, Overhauser, Werner, PRL 1975
𝐻 =𝒑2
2𝑚𝑖+ 𝑚𝑔𝒈𝒓 − 𝝎𝑳
Δ𝛽𝑔 ≈𝑚𝑔𝑚𝑖𝒈𝑲𝑇𝑇′
COW Experiment
Lammerzahl, Gen. Rel. Grav.1996
WEP violation in QMExample: Dirac Equation
𝑑𝑠2 = 𝑉2 𝑑𝑥0 2 −𝑊2(𝑑𝒙 ∙ 𝑑𝒙)
𝑖ℏ𝛾𝛼𝐷𝛼 −𝑚𝑐 𝜓 = 0
𝐷𝛼 = ℎ𝛼𝑖 𝐷𝑖, 𝐷𝑖 = 𝜕𝑖 +𝑖4𝜎𝛼𝛽𝜔𝑖
𝛼𝛽
Obukhov, PRL, 2001
There is no experiment that can be done, in a smallconfined space, which can detect the differencebetween a uniform gravitational field and anequivalent uniform acceleration.
WEP violation in QMExample: Dirac Equation
𝑑𝑠2 = 𝑉2 𝑑𝑥0 2 −𝑊2(𝑑𝒙 ∙ 𝑑𝒙)
𝑖ℏ𝛾𝛼𝐷𝛼 −𝑚𝑐 𝜓 = 0
𝐷𝛼 = ℎ𝛼𝑖 𝐷𝑖, 𝐷𝑖 = 𝜕𝑖 +𝑖4𝜎𝛼𝛽𝜔𝑖
𝛼𝛽
𝑉 = 1 +𝒂 ∙ 𝒙𝑐2, 𝑊 = 1
𝐻𝑎 = 𝛽𝑚𝑐2 + 𝛽𝑝2
2𝑚 + 𝛽𝑚𝒂 ∙ 𝒙 +ℏ2𝑐 𝚺 ∙ 𝒂 +
ℏ2𝑚𝑐2 𝛽𝚺 ∙ (𝒂 × 𝒑)
(ii) Accelerated frame
Obukhov, PRL, 2001
ℏ𝑔𝑐= 2 × 10−23eV
𝑉 = 1 −𝐺𝑀2𝑐2𝑟
1 +𝐺𝑀2𝑐2𝑟
−1
, 𝑊 = 1 +𝐺𝑀2𝑐2𝑟
2
𝐻𝑎 = 𝛽𝑚𝑐2 + 𝛽𝑝2
2𝑚− 𝛽𝑚𝒈 ∙ 𝒙 −
ℏ2𝑐𝚺 ∙ 𝒈 −
ℏ𝑚𝑐2𝛽𝚺 ∙ (𝒈 × 𝒑)
(i) Gravitational field 𝒈 = −𝐺𝑀𝒓𝑟3
(Foldy–Wouthuysen transformation)
WEP violation in QMExample: Dirac Equation
𝑑𝑠2 = 𝑉2 𝑑𝑥0 2 −𝑊2(𝑑𝒙 ∙ 𝑑𝒙)
𝑖ℏ𝛾𝛼𝐷𝛼 −𝑚𝑐 𝜓 = 0
𝐷𝛼 = ℎ𝛼𝑖 𝐷𝑖, 𝐷𝑖 = 𝜕𝑖 +𝑖4𝜎𝛼𝛽𝜔𝑖
𝛼𝛽
𝑉 = 1 +𝒂 ∙ 𝒙𝑐2, 𝑊 = 1
𝐻𝑎 = 𝛽𝑚𝑐2 + 𝛽𝑝2
2𝑚 + 𝛽𝑚𝒂 ∙ 𝒙 +ℏ2𝑐 𝚺 ∙ 𝒂 +
ℏ2𝑚𝑐2 𝛽𝚺 ∙ (𝒂 × 𝒑)
(ii) Accelerated frame
Obukhov, PRL, 2001
ℏ𝑔𝑐= 2 × 10−23eV
𝑉 = 1 −𝐺𝑀2𝑐2𝑟
1 +𝐺𝑀2𝑐2𝑟
−1
, 𝑊 = 1 +𝐺𝑀2𝑐2𝑟
2
𝐻𝑎 = 𝛽𝑚𝑐2 + 𝛽𝑝2
2𝑚− 𝛽𝑚𝒈 ∙ 𝒙 −
ℏ2𝑐𝚺 ∙ 𝒈 −
ℏ𝑚𝑐2𝛽𝚺 ∙ (𝒈 × 𝒑)
(i) Gravitational field 𝒈 = −𝐺𝑀𝒓𝑟3
(inertial spin-orbit coupling)
(gravitational spin-orbit coupling)
𝑉 = 1 +𝒂 ∙ 𝒙𝑐2 , 𝑊 = 1
𝐻𝑎 = 𝛽𝑚𝑐2 + 𝛽𝑚𝒂 ∙ 𝒙 + 𝛽𝑝2
2𝑚+ℏ2𝑐𝚺 ∙ 𝒂 +
ℏ2𝑚𝑐2𝛽𝚺 ∙ (𝒂 × 𝒑)
(i) Accelerated frame
𝑉 = 1 −𝐺𝑀2𝑐2𝑟
1 +𝐺𝑀2𝑐2𝑟
−1
, 𝑊 = 1 +𝐺𝑀2𝑐2𝑟
2
𝐻𝑎 = 𝛽𝑚𝑐2 − 𝛽𝑚𝒈 ∙ 𝒙 + 𝛽𝑝2
2𝑚−ℏ2𝑐𝚺 ∙ 𝒈 −
ℏ𝑚𝑐2𝛽𝚺 ∙ (𝒈 × 𝒑)
(ii) Gravitational field
ℏ𝑔𝑐= 2 × 10−23eV
Altschul, et al., Advances in Space Research (2015) 𝜂 = 2𝑎𝐴 − 𝑎𝐵𝑎𝐴 + 𝑎𝐵
Eötvös ratio:
𝒈 = −𝐺𝑀𝒓𝑟3
WEP violation in QM
B. S. DeWitt, “Superconductors and Gravitational Drag”, Phys. Rev. Lett. 16 , 1092 (1966).
G. Papini, “London moment of rotating superconductors and lense-thirring fields of general relativity”, Nuovo Cimento B 45 , 66 (1966).
G. Papini, “Particle Wave Functions in Weak Gravitational Fields”, Nuovo Cimento B, 52, 136–41 (1967).
G. Papini, “Detection of inertial effects with superconducting interferometers”, Phys. Lett. A 24, 32–3 (1967).
G. Papini, “Superconducting and Normal Metals as Detectors of Gravitational Waves”, Lettere Al Nuovo Cimento 4, 22 1027 (1970).
J. Anandan, “Gravitational and Inertial Effects in Quantum Fluids”, Phys. Rev. Lett. 47 , 463 (1981).
R. Y. Chiao, “Interference and inertia: A superfluid-helium interferometer using an
Enhanced gravitational interaction with quantum systems
Gravitational Casimir Effect
𝛻 ∙ 𝑬 = 𝜅𝝆(𝐸)𝛻 ∙ 𝑩 = 𝜅𝝆 𝑀
𝛻 × 𝑬 = −𝜕𝑩𝜕𝑡− 𝜅𝑱 𝑀
𝛻 × 𝑩 =𝜕𝑬𝜕𝑡+ 𝜅𝑱 𝐸
𝜅 =8𝜋𝐺𝑐4
𝐸𝑖𝑗 = 𝐶0𝑖0𝑗𝐵𝑖𝑗 = ⋆ 𝐶0𝑖0𝑗
𝐽𝜇𝜈𝜌 = −𝑇𝜌 𝜇,𝜈 +13𝜂𝜌[𝜇𝑇,𝜈]
𝜌𝑖𝐸 = −𝐽𝑖00
𝜌𝑖𝑀 = − ⋆ 𝐽𝑖00𝐽𝑖𝑗𝐸 = 𝐽𝑖0𝑗𝐽𝑖𝑗𝑀 = ⋆ 𝐽𝑖0𝑗
Gravitoelectromagnetism (GEM)
𝐸0 =ℏ4𝜋 0
∞𝑘∥𝑑𝑘∥
𝑛
𝜔𝑛+ + 𝜔𝑛× 𝜎
Δ 1 +𝜅𝜒2𝑬𝑇𝑇 = 0
Δ𝑩𝑇𝑇 = 0
𝑬𝑇𝑇 =𝛼(1 −
sin2𝛾2) 𝛽cos𝛾
𝛽cos𝛾 −𝛼(1 −sin2𝛾2)𝑒𝑖(𝒌∙𝒓−𝜔𝑡)
𝑩𝑇𝑇 =−𝛽(1 −
sin2𝛾2) 𝛼cos𝛾
𝛼cos𝛾 𝛽(1 −sin2𝛾2)𝑒𝑖(𝒌∙𝒓−𝜔𝑡)
Gravitational Casimir Effect
(smoothness principle)
Gravitational Casimir Effect
𝛼′ 1 +𝜅𝜒2 1 −
𝑆′
2 𝑒−𝑞′𝑎/2 = 𝛼 1 −
𝑆2
2 𝑒−𝑞𝑎/2 + 𝛼′′ 1 −
𝑆2
2 𝑒𝑞𝑎/2
𝑆 = sin𝛾𝐶 = cos𝛾
𝛼′′′ 1 +𝜅𝜒21 −𝑆′
2𝑒−𝑞′𝑎/2 = 𝛼 1 −
𝑆2
2𝑒𝑞𝑎/2 + 𝛼′′ 1 −
𝑆2
2𝑒−𝑞𝑎/2
𝛼′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−
𝑞𝑎2 − 𝛼′′𝐶𝑒𝑞𝑎/2
-𝛼′′′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−
𝑞𝑎2 − 𝛼′′𝐶𝑒−𝑞𝑎/2
𝑞2 = −𝑘𝑧2 = 𝑘∥2 −𝜔2
𝑐2
𝑞′2 = −𝑘𝑧′2 = 𝑘∥2 − (1 + 𝜅𝜒)
𝜔2
𝑐2
𝛼′ 1 +𝜅𝜒2 1 −
𝑆′
2 𝑒−𝑞′𝑎/2 = 𝛼 1 −
𝑆2
2 𝑒−𝑞𝑎/2 + 𝛼′′ 1 −
𝑆2
2 𝑒𝑞𝑎/2
𝑆 = sin𝛾𝐶 = cos𝛾
𝛼′′′ 1 +𝜅𝜒21 −𝑆′
2𝑒−𝑞′𝑎/2 = 𝛼 1 −
𝑆2
2𝑒𝑞𝑎/2 + 𝛼′′ 1 −
𝑆2
2𝑒−𝑞𝑎/2
𝛼′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−
𝑞𝑎2 − 𝛼′′𝐶𝑒𝑞𝑎/2
-𝛼′′′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−
𝑞𝑎2 − 𝛼′′𝐶𝑒−𝑞𝑎/2
Δ+ = 𝑒−𝑎𝑞′{ 𝐶′ 𝑆2 − 2 − 1 +𝜅𝜒2𝐶 𝑆′2 − 2
2𝑒−𝑎𝑞 − 𝐶′ 𝑆2 − 2 + 1 +
𝜅𝜒2𝐶 𝑆′2 − 2
2𝑒𝑎𝑞 = 0
Gravitational Casimir Effect
𝑛
𝜔𝑛(𝑞) =12𝜋𝑖 𝑖∞
−𝑖∞𝜔(𝑞) 𝑑lnΔ(q) +
𝐶+𝜔(𝑞)𝑑lnΔ(q)
𝐸0 =ℏ4𝜋 0
∞𝑘∥𝑑𝑘∥
𝑛
𝜔𝑛+ + 𝜔𝑛× 𝜎
𝐸 𝑎 =𝐸0𝜎− lim𝑎→∞
𝐸0𝑎
𝐸 𝑎 =ℏ4𝜋2 0
∞𝑘∥𝑑𝑘∥
0
∞ln 1 − 𝑟+2𝑒−2𝑞𝑎 + ln 1 − 𝑟×2𝑒−2𝑞𝑎 𝑑𝜉
𝑟+ =𝐶′ 𝑆2 − 2 − 1 + 𝜅𝜒2 𝐶(𝑆
′2 − 2)
𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)
, 𝑟× =1 + 𝜅𝜒2 𝐶
′ 𝑆2 − 2 − 𝐶(𝑆′2 − 2)
1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶(𝑆′2 − 2)
Gravitational Casimir Effect
𝐸 𝑎 =ℏ4𝜋2 0
∞𝑘∥𝑑𝑘∥
0
∞ln 1 − 𝑟+2𝑒−2𝑞𝑎 + ln 1 − 𝑟×2𝑒−2𝑞𝑎 𝑑𝜉
𝑟+ =𝐶′ 𝑆2 − 2 − 1 + 𝜅𝜒2 𝐶(𝑆
′2 − 2)
𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)
, 𝑟× =1 + 𝜅𝜒2 𝐶
′ 𝑆2 − 2 − 𝐶(𝑆′2 − 2)
1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶(𝑆′2 − 2)
𝛼′ 1 +𝜅𝜒21 −𝑆′
2𝑒−𝑞′𝑎/2 = 𝛼 1 −
𝑆2
2𝑒−𝑞𝑎/2 + 𝛼′′ 1 −
𝑆2
2𝑒𝑞𝑎/2
𝛼′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−
𝑞𝑎2 − 𝛼′′𝐶𝑒𝑞𝑎/2
𝛼′′
𝛼=−𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆
′2 − 2)
𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)
𝑒𝑞𝑎
Gravitational Casimir Effect
𝐸 𝑎 =ℏ4𝜋2 0
∞𝑘∥𝑑𝑘∥
0
∞ln 1 − 𝑟+2𝑒−2𝑞𝑎 + ln 1 − 𝑟×2𝑒−2𝑞𝑎 𝑑𝜉
𝑟+ =𝐶′ 𝑆2 − 2 − 1 + 𝜅𝜒2 𝐶(𝑆
′2 − 2)
𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)
, 𝑟× =1 + 𝜅𝜒2 𝐶
′ 𝑆2 − 2 − 𝐶(𝑆′2 − 2)
1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶(𝑆′2 − 2)
𝛼′′
𝛼=−𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶 𝑆
′2 − 2
𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶 𝑆′2 − 2
𝑒𝑞𝑎 = |𝑟+|
𝛽′′
𝛽=− 1 + 𝜅𝜒2 𝐶
′ 𝑆2 − 2 + 𝐶 𝑆′2 − 2
1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶 𝑆′2 − 2
𝑒𝑞𝑎 = r×
Gravitational Casimir Effect
Ordinary material
𝑛 = 1 +2𝜋𝐺𝜌𝜔2
𝜒 =1 − 𝑛 𝜔 2
𝜅𝑐2
𝐺 = 6.67 × 10−11m3kg−1s−2
Peters, PRD, 1974
𝑂 𝜌 = 𝑂[𝑐2
𝐺] ≈ 1027kg/m3
𝑂 𝜌 = 104kg m−3
𝑃 𝑎 = −𝜕𝐸 𝑎𝜕𝑎
𝑃(10−6) ≈ 10−21nPa
Superconductor𝐻 =𝒑 − 𝑞𝑨 −𝑚𝒉 2
2𝑚+ 𝑉
𝒋𝐺 =1𝑚Re 𝜓∗
ℏ𝑖𝛻 − 𝑞𝑨 −𝑚𝒉 𝜓
=1𝑚−𝑞𝑨 −𝑚𝒉 𝜓∗𝜓
𝒋𝐺 = 𝜎𝐺𝑬𝐺
𝑟𝐺 = 1 +2𝛿2
𝑐𝑑𝜉−1
𝑟𝐸 = 1 +2𝜆𝛿2
𝑐𝑑2𝜉−1
Minter, Wegter-McNelly, Chiao, Physica E, 2010
Heisenberg-Coulomb effect
𝐻 =𝒑 − 𝑞𝑨 −𝑚𝒉 2
2𝑚+ 𝑉
𝒋𝐺 =1𝑚Re 𝜓∗
ℏ𝑖𝛻 − 𝑞𝑨 −𝑚𝒉 𝜓
=1𝑚−𝑞𝑨 −𝑚𝒉 𝜓∗𝜓
𝒋𝐺 = 𝜎𝐺𝑬𝐺
𝑟𝐺 = 1 +2𝛿2
𝑐𝑑𝜉−1
𝑟𝐸 = 1 +2𝜆𝛿2
𝑐𝑑2𝜉−1𝑑 = 2 nm 𝛿 = 37 nm 𝜆 = 83 nm
Superconductor