II. Plan de Investigación
11. Cronograma de actividades
A mirror accelerating uniformly emits quanta, but there is no radiation of energy. Davis and Fulling first discovered this seemingly paradoxical effect, see [5]. A hy- perbolic trajectory is:
z(t) =B−√B2−t2 , (3.28)
where a plot is given in Figure 3.4.
Using a splice between the static mirror and hyperbolic mirror, Davies and Fulling [5] found βωω′ = B πe i(ω′−ω)B K1(2B √ ωω′), (3.29)
(see the Hyperbolic plus Static mirror in Section 3.6), whereK1is the modified Bessel
function. This result was used with questionable assumptions, involving a static mirror spliced to a hyperbolic mirror. We present the results for the hyperbolic tra- jectory throughout all times.
t
-x
Figure 3.4: Hyperbolic,B = 12.
detector absorbing quanta and making a transition to an excited state in the absence of all field energy. A detector, as described by Walker[15] emits negative energy into the field to compensate. This strange result is not without precedent in quantum field theory. Similar situations are explored by Epstein, Glaser and Jaffe (1965)[57], who construct many-particle states with negative or zero fluxes even in the absence of mirrors or detectors.
Hyperbolic and Mobius Trajectories on Null Surfaces
Constant acceleration mirror trajectories have beta coefficients which can be solved straightforwardly using null hypersurfaces for the Cauchy surface in the scalar prod- uct. For two hyperbolic mirrors (Mobius trajectory), an arguably more natural man- ifestation of hyperbolic motion, there is no appearance of particles or energy. This is to be contrasted with the single hyperbolic mirror which has particle production, yet
hTuui= 0. The Mobius expression is more simple and both±branches are included. In addition there is no transformation shift to intersect zero.
Hyperbolic Mirror on Null Surface: Particles without Energy
The hyperbolic trajectory is zh(t) = B −
√
B2+t2. The modes solutions used that
satisfy the boundary condition and wave equation are, using the CW normalization convention:
φω′ =e−iω ′v
−e−iω′p(u)Θ(B+u), φω =e−iωf(v)Θ(B−v)−e−iωu , (3.30)
while the beta coefficient is βω′ω = −iB
πe i(ω−ω′)
K1(2B
√
ωω′). To solve for this beta coefficient, the conversions (usingu=t−z,v =t+z) begin via
u(t) = −B+t+√B2+t2 v(t) = B+t+√B2+t2 . (3.31) Inversion gives t(u) = u2((2BB++uu)) , t(v) = v2((2BB−−vv)) . (3.32)
In null coordinates, Eqn. (2.35), and the inverse, Eqn. (2.36), are
p(u) = Bu
B+u u >−B , f(v) = Bv
B−v v < B .
(3.33)
For solving for the beta coefficient on the past hypersurface, the mode solutions are therefore,φ∗ ω′ =eiω ′v ,φ∗ ω =eiω Bv
B−v and the integral
βω′ω = 1 4π√ωω′ Z ∞ −∞ dv eiω′veiωBBv−v(ω′−ω B 2 (B −v)2)Θ(B−v), (3.34)
βω′ω = 1 4π√ωω′ −4i√ωω′Bei(ω−ω′)B K1(2B √ ωω′) , (3.35) βω′ω =−i B πe i(ω−ω′)B K1(2B √ ωω′). (3.36)
Solving for the beta by using the future surface results in the integral
βω′ω = 1 4π√ωω′e i(ω−ω′)B Z ∞ −∞ da eiωae−iω′Ba2(ω′B 2 a2 −ω)Θ(a), (3.37)
which also gives Eqn. (3.36).
The energy flux is found by the expectation value of the stress-energy tensor, Eqn. (2.45) where the total energy is found by Equ (2.48),E =Rumax
umin hTuuidu. Herep(u) =
Bu
B+u. Using Eqn. (2.45), one finds zero flux,hTuui = 0. Therefore, incorporating Eqn. (3.36), the well known ‘particles without energy’ or ‘radiation without energy’ result is confirmed. Obadia and Parentani study how to switch off the interaction of the mirror-field coupling at large times. [58] They found that if the coupling is constant they were able to obtainhTuui= 0.
Two Hyperbolic Mirrors: Acceleration without Particles
Two hyperbolic mirrors (Mobius mirror) produce no energy or particles[23]. This double mirror trajectory is included in this constant acceleration section as the only demonstration of a radiationless, yet accelerated trajectory. It is a trivial example of preserving purity during the mode ‘evolution’ fromI−
R toIR+. Asymptotically iner- tial mirrors in Section 3.4 are unitary like the Mobius mirror. There is no particle production because the two constant acceleration mirrors cancel each other’s parti- cle creation effect. Using the CW normalization, the modes solutions that satisfy the boundary condition and wave equation are, Equ. (3.19). The Mobius trajectory has
been studied before[59] from a conformal field theory point of view in an attempt to illustrate the close relationship between the production of particles and energy. The only new aspect presented here is the use of the past and future hypersurfaces to derive the same results and confirm our technique. The trajectory is,
zmobius(t) =±√B2 +t2 . (3.38)
t
-x
Figure 3.5: Mobius Trajectory: two hyperbolic mirrors,B = 1 2.
The ray tracing formulae are solved by invertingu(t)andv(t):
u(t) = t±√B2+t2 ,
v(t) = t±√B2+t2 ,
(3.39)
with the result,
t(u) = u2 − B2u2 , t(v) = v2 − B2v2 .
. (3.40)
p(u) = −Bu2 u6= 0 , f(v) =−B2
v v 6= 0 .
(3.41)
The beta transformation coefficient integral is done using the past hypersurface by βω′ω = i
R∞
−∞dv φ∗ω′
↔ ∂v φ∗
ω such that the modes are, φ∗ω′ = eiω ′v andφ∗ ω = eiω −B2 v . The resulting integral is βω′ω = Z ∞ −∞ dv eiω′veiω−Bv2(ω′ −ωB 2 v2)≡ Z ∞ −∞ dv I(B, v, ω, ω′), (3.42) which must be done avoiding the pole atv = 0. Split into two, we obtain
βω′ω = Z 0 −∞ dv I(B, v, ω, ω′)Θ(−v) + Z ∞ 0 dv I(B, v, ω, ω′)Θ(v) = 0 . (3.43)
For integration using the future hypersurface, this result is obtained by a similar method which also gives zero. The energy flux, Eqn. (2.45) is found as was done for the hyperbolic mirror, however in this case, p(u) = −B2
u and the result is zero:
hTuui= 0.