What is the 'Unruh Effect'?

 Hi Russell and Friends,       I just ran across the following post and thought that you might find it interesting. Any comments?   Onward!   Stephen     On Thu, 11 Aug 2005 10:32:00 +0000 (UTC), in sci.physics.research [EMAIL PROTECTED] wrote:   The "Minkowski" or "inertial" vacuum state seen in an "acceleratingframe" is a thermal state at a temperature proportional to the"acceleration"; i.e., an heat bath containing an infinite number ofparticles (finite density) distributed in a fashion consistent with agas at a particular temperature.   The words in quotes are very misleading however, and require a largeamount of clarification, because the effect has little or nothing perse to do with acceleration or being inertial; but rather with theoccurrrence of a causal horizon.   A quantum field theory requires you first to define a "frame", the wordwhich -- unlike in Relativity -- does NOT a coordinate system; but a"flow of time".  Quantum theory, is you recall, treats time as aprocess, not a dimension.   The flow of time is represented by a vector field which is timelike.   The Minkowski or inertial frame has associated with it a constanttime-like field which (by suitable Lorentz transformation) can berepresented as T = d/dt -- i.e. the 4-vector T = T^{mu} d/dx^{mu} whoseonly non-zero component is T^0 (with x^0 = t).   The Unruh frame uses a time-like field which does NOT cover all ofspace.  The flow lines are all hyperbolic, each naturally associatedwith an observer at a given acceleration.  The hyperbolas all have, asasymptotes, the 2+1 boundary given by an equation of the form x = c|t|;the region associated with the field being x > c|t|.   The "acceleration" a is normally defined as that associated with one ofthe worldlines in the Unruh frame.  Different worldlines have differentaccelerations associated with them.   At this boundary, the timelike field T becomes null.  A second, mirrorregion, x < -c|t| has the boundary x = -c|t|.  Both boundaries meet att = 0.  In this region, the field T "flows" in the opposite direction.   The boundary x = c|t| is the causal horizon mentioned before.   A field is uniquely determined by its values at t = 0, and the space ofall states of a system is generally always associated with the initialvalues of whatever system is in question.  Here, that means, there is anatural split of the underlying state space H into H1 + H2, with H1being the state space associated with the region x > c|t|, and H2 beingthat associated with the region x < -c|t|.   (Solving the field equation by taking its initial values (and theinitial values of its time derivative) comprises what's called a Cauchyproblem.  For the Klein-Gordon field, the initial values play theanalogous role of coordinates, the initial time derivative thecongugate momenta.  The state space is then a Hilbert space in whichthese quantities act as operators satisfying the usual Heisenbergrelations).   H1, here, is the only one of physical relevance.  But a fulldescription of the Minkowski frame requires both H1 and H2.  Inparticular, the vacuum state |0> of a Klein Gordon field -- as seen inthe Minkowski frame -- when expressed in terms of the H1 & H2 states--becomes:                 |0> = sum |n>_1 |n>_2 exp(-pi a).   This is readily identifiable in the language of finite-temperaturequantum field theory.  The states |n>_1 can be thought of as particlestates, those |n>_2 can be thought of as states associated with vacuumfluctuations of the corresponding heat bath (i.e. "holes").  So, thesuperposition |n>_1 & |n>_2 has total energy 0, since |n>_2 reflects|n>_1.  All the states |n>_2 are negative energy since the time flow inregion 2, x < -c|t|, goes the other way.   Since only region 1, x > c|t|, is physically relevant (you can't seepast the boundary, the causal horizon), then the actual quantum stateassociated with it is arrived at by phase-averaging over the states ofregion 2.  This turns the Minkowski state into the region 1 state:               |0><0| --> Trace_2(|0><0|) = V_1with        |0><0| = sum |n>_1 _2 <0|) = sum |n>_1 c|t|.   The general lesson is that what appears as quantum noise in one framebecomes thermal noise seen from another frame; and there is no longerany covariant distinction between quantum and thermal noise.   There's an unlimited number of ways to define time-like fields in aregion of spacetime, and you can always have time-like fields definedin such a way that they have causal horizons somewhere.  A moredramatic example of this is where the region in question is actuallyfinite in size: |r| < 1 - |t| (using units where c = 1).  Then definingcoordinates (R, T) by                   r = R (1 - T^2)/(1 - (RT)^2)                   t = T (1 - R^2)/(1 - (RT)^2)this produces a metric            ds^2 = dT^2 ((1 - R^2)/(1 - (RT)^2))^2                 - ds_3^2 ((1 - T^2)/(1 - (RT)^2)^2where            ds_3 = dR^2 + R^2 ((d theta)^2 + (sin theta d psi)^2)   The worldlines associated with (R, theta, psi) = constant arehyperbolic worldlines that meet at (t,r) = (-1,0), (t,r) = (+1,0), andcross the t = 0 hyperplane at r = R; and the timelike field is justd/dT, that associated with the coordinate "T", itself.   Here, the causal horizon is |r| = 1 - |t| and the region enclosed isfinite, which means that the modes associated with the Klein-Gordonfield will form a discretely spaced set, rather than continuous (justas if you were to quantize "in a box").  So, there is a HUGE cut-off offield modes here, since the globally defined quantum field has acontinuously distributed set of modes.   As of yet, I still don't know what the Minkowski vacuum looks like inthis frame.  The wave equation has a particularly bizarre feature thathalf of the initial values become redundant, when the initial valuesare taken on the 1/2 causal horizon |r| = 1 - t, 0 < t < 1; or theother 1/2; |r| = 1 + t, -1 < t < 0; and the Cauchy problem for thefield becomes a Dirichlet problem (in part because there are notime-derivatives to define, since "time is frozen" on the causalhorizon).