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On the Unruh Effect (From a Textbook) On the Unruh effect (from a textbook) M. Besançon October 8th 2019 Unruh effect (W.G. Unruh, Notes on black-hole evaporation, PRD 14 (1976) 870 sect III) But start with textbook treatments heavily relying on : 1) V. F. Mukhanov and S. Winitzki : Introduction to Quantum Fields in Classical Backgrounds (Chapter 8 The Unruh effect) and a tiny bit on : 2) R. M. Wald : Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chapter 5 The Unruh effect) Chicago Lectures in Physics 3) R. M. Wald : The History and Present Status of Quantum Field Theory in Curved Spacetime gr-qc/0608018 4) K. Thorne, R.H. Price, D.A.MacDonald : Black holes the membrane paradigm, Yale University Press (Chap VIII.B6) Thanks to Jean François Glicenstein for pointing 1) and 4) to me Unruh effect summarized - in quantum field theory particles are excitations of quantum fields - Unruh effect predicts that particles will be detected in a vacuum by an accelerated observer - simplest case : observer moves with constant acceleration through Minkowski spacetime and measures the number of particles in a massless scalar field - even though the field is in the vacuum state → observer finds a distribution of particles characteristic of a thermal bath of blackbody radiation Unruh effect consider the trajectory of an observer moving with constant acceleration in Minkowski spacetime acceleration of observer in its own frame of reference (proper acceleration) is constant introduce several reference frames : - laboratory frame the usual inertial reference frame with the coordinates (t, x, y, z) - proper frame the accelerated system of reference that moves together with the observer also called the accelerated frame - comoving frame : defined at a time t 0 is the inertial frame in which the accelerated observer is instantaneously at rest at t = t 0 thus the term comoving frame actually refers to a different frame for each t 0 Unruh effect : trajectory of accelerated observer - consider a uniformly accelerated observer with a time-independent proper acceleration equal to a given 3-vector a - trajectory of such an observer may be described by a worldline xμ (τ) → τ proper time measured by the observer - 4-acceleration in the laboratory frame (see appendix) μ 2 μ μ μ du d x μ d x μ a ≡ = with u ≡ and u uμ = 1 d τ d τ2 d τ related to the three-dimensional proper acceleration a by μ 2 a aμ = − |a| Unruh effect : trajectory of accelerated observer - assume now that the acceleration is parallel to the x axis a ≡ (a, 0, 0) with a > 0 and that the observer moves only in the x direction → then (see appendix) : 1 1 x(τ) = x - + cosh a τ 0 a a 1 t (τ) = t + sinh a τ 0 a - the trajectory has a simpler form if we choose the initial conditions x(0) = a −1 and t(0) = 0 → then the worldline is a branch of the hyperbola x 2 − t 2 = a −2 - this trajectory has zero velocity at τ = 0 (which implies x = x , t = t ) 0 0 Unruh effect : trajectory of accelerated observer - the worldline of the uniformly accelerated observer in the Minkowski spacetime is a branch of the hyperbola x2 − t2 = a−2 - at large |t| the worldline approaches the lightcone - the observer comes in from x = +∞ decelerates and stops at x = a −1 and then accelerates back towards infinity - in the comoving frame of the observer, this motion takes infinite proper time, from τ = −∞ to τ = +∞. - dashed lines show the light-cone - observer cannot receive any signals from the events P , Q and cannot send signals to R Unruh effect : horizon - an accelerated observer cannot measure distances longer than a−1 in the direction opposite to acceleration : for instance the distances to the events P and Q one says that the accelerated observer perceives a horizon at proper distance a−1 Unruh effect : proper coordinates to describe quantum fields as seen by an accelerated observer → need to use the proper coordinates (τ, ξ) where τ is the proper time and ξ is the distance measured by the observer the proper coordinate system (τ, ξ) is related to the laboratory frame (t, x) by some transformation functions τ (t, x) and ξ (t, x) 1 + aξ t (τ ,ξ) = sinh a τ a 1 + aξ x(τ ,ξ) = cosh a τ a coordinates (τ, ξ) vary in the intervals −∞ < τ < +∞ and −a−1 < ξ < +∞ for ξ < −a−1 → ∂t/∂τ < 0 i.e. the direction of time is opposite to that of τ Unruh effect : horizon proper coordinate system of a uniformly accelerated observer in the Minkowski spacetime - solid hyperbolae are lines of constant proper distance ξ - hyperbola with arrows is the worldline of observer ξ = 0 or x2 − t2 = a−2 - lines of constant τ are dotted - dashed lines show lightcone which corresponds to ξ = −a−1 events P , Q , R are not covered by the proper coordinate system subdomain x > |t| → Minkowski wedge Unruh effect : Rindler spacetime Minkowski metric in the proper coordinates (τ, ξ) is : ds2 = dt 2 - dx2 = ( 1 + a ξ )2 d τ2 - d ξ2 spacetime with this metric is called Rindler spacetime curvature of Rindler spacetime is everywhere zero since it differs from Minkowski spacetime merely by a change of coordinates. to develop quantum field theory in Rindler spacetime, we first rewrite the metric in a conformally flat form i.e. choosing a new spatial coordinate : ~ ~ ξ ≡ ln ( 1 + aξ ) ξ is called the conformal distance so that we obtain a common factor in the metric ~ 2 2a ξ 2 ~2 ds = e (d τ - d ξ ) the proper distance ξ is constrained by ξ > −a −1 then the conformal distance varies in the interval [ −∞ , +∞ ] relation between the laboratory coordinates and the conformal coordinates is ~ ~ ~ −1 a ξ ~ −1 a ξ t (τ , ξ ) = a e sinh a τ x(τ , ξ ) = a e cosh a τ Unruh effect : quantization of a scalar field - quantize a scalar field in the proper reference frame of a uniformly accelerated observer simplify problem → consider a massless scalar field in 1+1-dimensional spacetime - procedure of quantization is formally the same in both coordinate systems i.e. laboratory and accelerated frames : - mode expansion in the laboratory frame annihilation operator creation operator +∞ ^ d k 1 −i|k|t + ikx - i|k|t − ikx + ϕ ( t , x ) = ∫ / [ e a^ k + e a^ k ] −∞ ( 2π )1 2 √2|k| vacuum state in the laboratory frame i.e. the Minkowski vacuum |0 > M is the zero eigenvector of all the annihilation operators â− â− |0 > = 0 for all k k k M - mode expansion in the accelerated frame +∞ ~ ~ ^ ~ d k 1 −i|k|τ + ik ξ ^ - i|k|τ − ik ξ ^ + ϕ ( τ , ξ ) = ∫ [ e bk + e bk ] −∞ ( 2π )1/ 2 √2|k| vacuum state in the accelerated frame i.e. the Rindler vacuum |0 > R is defined by b̂ − |0 > = 0 for all k k R Unruh effect : different vacuum - mode expansions are decompositions into linear combinations of 2 different ± ± sets of basis functions with operators â and b̂ k k - operators â and b̂ are different although they satisfy similar commutation k k relations → Rindler vacuum |0 > and Minkowski vacuum |0 > R M are 2 different quantum states of the field Φ Unruh effect : which is the correct vacuum ? which of the state |0 > or |0 > is the “correct” vacuum ? M R - observers accelerating would agree that the field in the state |0 > has R the lowest possible energy and the Minkowski state |0 > has a higher energy M → thus a particle detector at rest in the accelerated frame will register particles when the scalar field is in the state |0 > M - however in the laboratory frame the state with the lowest energy is |0 > M and the state |0 > has a higher energy R → therefore the Rindler vacuum state |0 > will appear to be an excited state R when examined by observers in the laboratory frame neither of the two vacuum states is “more correct” if considered by itself ultimately the choice of vacuum is determined by experiment: the correct vacuum state must be such that the theoretical predictions agree with the available experimental data Light cone mode expansion : density of particles and Unruh temperature - one can relate the two sets of operators â± and b̂ ± k k → Bogolyubov transformations (after some gymnastic rewritting the previous mode expansions in lightcone coordinates - see appendix) → transformations linking the vacuum states of the quantum field in the Rindler frame and the Minkowksi frame (i.e. accelerated frame and laboratory frame) - vacua |0 > and |0 > corresponding to operators â − and b̂ − are different M R → the a-vacuum is a state with b-particles and vice versa - one can compute the mean density of massless b-particle of energy E in the a-vacuum (from the Bogolyubov transformation coefficients - see appendix) : -1 acceleration E a n( E) = exp - 1 with T ≡ [ ( T ) ] 2π thermal spectrum Unruh temperature Unruh effect : one physical interpretation a physical interpretation of the Unruh effect as seen in the laboratory frame is the following : - the accelerated detector is coupled to the quantum fields and perturbs their quantum state around its trajectory - this perturbation is very small but as a result the detector registers particles although the fields were previously in the vacuum state - the detected particles are real and the energy for these particles comes from the agent that accelerates the detector finishing with an exercise Unruh effect : exercice a glass of water is moving with constant acceleration what is the smallest acceleration that would make the water boil due to the Unruh effect ? Unruh effect : exercice a glass of water is moving with constant acceleration what is the smallest acceleration that would make the water boil due to the Unruh effect ? expressing all quantities in SI units : a ℏ a T ≡ becomes k T ≡ 2π c 2π where k ≈ 1.38 10-23 J/K is the Boltzmann’s constant the boiling point of water is T = 373 K so the required acceleration is a ≈ 10 22 m/s2 the Unruh effect is quite difficult to use in practice because the acceleration required to produce a measurable temperature is enormous Possible next steps - Hawking like radiation from accelerated mirrors (from an analog/similar framework) - Hawking radiation from Schwarzschild BH - ‘t Hooft approach : scattering matrix approach, black hole unitarity, back reaction, antipodal entanglement - what about a possible role of BMS (Bondi, van der Burg, Metzner, Sachs) asymptotic symmetries, relation to soft hairs on BH Strominger at al.
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