Unruh temperature with maximal acceleration

Unruh temperature with maximal acceleration Elmo Benedettoa and Antonio Feoli a , Abstract In this paper, we modify the geometry of Rindler space so as to include an upper limit on the acceleration. Caianiello and his collaborators, in a series of papers, have analyzed the corrections to the classical spacetime metrics due to the existence of a maximal acceleration. Our goal is to derive, in this context, in a very simple way, the so called Unruh temperature. a Department of Engineering, University of Sannio, Piazza Roma 21, 82100– Benevento, Italy

1

Introduction

It is well known that uniformly accelerated observers detect a black-body radiation where an inertial observer would not. This e¤ect is a generalization of the famous Hawking e¤ect for black holes derived for the …rst time in [1] and was described by Stephen Fulling, Paul Davies and W. G. Unruh [2],[3],[4] and then called Unruh e¤ect. The equivalence principle guarantees indeed the similarity of the two e¤ects and recent modern interpretations are present in [5],[6]. The Unruh e¤ect has played a crucial role in understanding that the particle content of a …eld theory is observer dependent. In fact, the notion of vacuum depends on the path of the observer through spacetime and, from the point of view of an accelerating observer, empty space contains a gas of particles at a temperature proportional to the acceleration. Indeed Unruh e¤ect attests that a detector, moving with a uniform proper acceleration of magnitude a, sees the vacuum state of a quantum …eld as a thermal bath with temperature T =

~a 2 ckB

(1)

where kB is the Boltzmann constant, ~ is the reduced Planck constant and c is the speed of light. Therefore uniformly accelerated observers can see as real those particles which inertial observers claim to be virtual. Many rigorous demonstrations of the relation (1) exist (see for example [7]) Our aim is to estimate the Unruh temperature through a simpli…ed path and to show how the formula is modi…ed in the case of existence of a maximal acceleration. In this paper we start to consider the spacetime of observers moving with a constant acceleration of magnitude a and direction along the x - axis. This is a well known 1

spacetime called Rindler space [8]. However, we want to modify its geometry in such a way to include an upper limit on the acceleration. One possible model of a dynamics with a maximal acceleration was developed in a series of papers and assumes that an accelerated particle is embedded in a curved spacetime given by the metric [9] [10] de2 =

1

j• x x • j A2

d

2

=

1

a2 A2

d

2

(2)

related to the old standard metric of General Relativity d 2 by a conformal factor depending on the acceleration. In the equation (2) a dot means derivative respect to the proper time (that is x_ = dx =d ) and we denote with A the maximal acceleration (MA). This model is a consequence of a geometric approach to Quantum Mechanics [11],[12], and has been applied to di¤erent sectors of theoretical physics such as black hole physics, cosmology, the dynamics of accelerated strings, the energy spectrum of a uniformly accelerated particle, lamb shift, mass of Higgs Boson, neutrino oscillations, etc. [13],[14],[15],[16], [17],[18],[19],[20],[21],[22],[23],[24],[25]. In Caianiello’s proposal the maximal proper acceleration is a basic physical property of all massive particles and must therefore be included in the physical laws from the outset. But maximal acceleration can be also discovered at the end as a consequence of already existing classical or quantum theories. During the years, strong evidence has appeared that the acceleration of any physical object cannot be arbitrarily large, but it should be superiorly limited. For example in string theory, it was derived that string acceleration must be less than some critical value, determined by the string tension and its mass. Otherwise in string dynamics Jeans–like instabilities arise, leading to unlimited growth of the string length [26],[27],[28],[29]. MA also appears in the context of Weyl space and of a geometrical analogue of Vigier’s stochastic theory [30],[31]. Recently it has been found in loop quantum gravity as well [32] It is also invoked to regularize black hole entropy [33]. Other classical and quantum arguments were used to support the existence of MA [34],[35],[36]. From the classical point of view (as Wheeler suggested), if we consider an extended object in rotating motion, we have a = v 2 =R and it follows that a must be at least limited by c2 =R: In quantum mechanics, instead, Caianiello [37] [38] showed that the natural limit for the proper acceleration of any massive particle follows from Heisenberg’s uncertainty relations and is …xed by the particle rest mass itself according to the relation A = 2mc3 =~. On the other side, some authors regard A as a universal constant …xed by Planck’s mass [39]. The paper is organized as follows: In Sect.2 we introduce the modi…ed Rindler space, while in Sect.3 we analyze the time dependent Doppler e¤ect; in Sect.4 we give the conclusions.

2

2

The Modi…ed Rindler Space

In Minkowski Space the trajectories of uniformly accelerated particles are described by the equations c a sinh a c

=

x c a = cosh c a c

=

t=

where

= p

1 1 a2 =A2

c a sinh e a c

(3)

c a cosh e a c

(4)

We can obtain the corresponding Rindler space using the

parametrization:

c a a ; = e= ; a c c so that the equations (3) and (4) become =

=

(5)

t = sinh

(6)

x = cosh c

(7)

hence the standard metric is d

2

=

dx2 c2

2

dt2 =

d

2

+d

2

(8)

while the Rindler space with maximal acceleration can be calculated knowing that the components of velocity are t_ = cosh

(9)

x_ = sinh c and the components of acceleration are t• = x • = c

(10)

1

sinh

(11)

1

cosh

(12)

So we obtain de2 =

j• x x • j A2

d

2

that has an event horizon for rewritten in this way:

=

c A

de2 =

2

1

=

c2 A2

1

2

2

d

2

+d

2

(13)

that is for a ! A. The metric can be also c2 A2

3

d

2

+

d

2 2

(14)

3

Time Dependent Doppler E¤ect

In order to determine the form of the Unruh temperature taking into account the existence of a maximal acceleration, we consider the time dependent Doppler e¤ect and we follow the same path of [40] [41]. All that we want to do is to rewrite the standard relations obtained by Alsing and Milonni [40] introducing the maximal acceleration suitably using the term = p 1 2 2 1 a =A ! We consider a plane wave in the Minkowski (M ) frame with wave vector k along x axis and frequency ! k = kc : B(t; x) = B0 ei'

(t;x)

(15)

with ' (t; x) = kx ! k t The observer in the origin of the M frame sees the wave B(t) = B0 e i!k t , while the observer in the origin of the Rindler (R) frame moves along the trajectory (2,3) and sees the wave B(e) = B[t(e); x(e)] = B0 exp[i! k so

a e c i e=c :

c a

cosh

h

sinh

a e ] c

(16)

c i! k e a (17) a Therefore the R observer does not see a plane wave, but a superposition of plane waves (time dependent Doppler e¤ ect): B(e) = B0 exp

+1 Z e )e B(e) = d B( 1

i e

(18)

where is the frequency of R frame plane waves and (for waves moving toward x) e )= 1 B( 2

+1 Z h c 0 i a d 0 B0 exp i ! k e c ei a

Z1

e

ic a !k y

y

i ac

(19)

1

a

Introducing the new variable y = e c 0 d = dy=( ac y): 0 c a c 0 Moreover ei = e( c ) ( a ) i = y i a

e ) = B0 B( 2

0

c y a

1

0

we have dy =

e

a c

0

d 0 , and hence

and therefore

B0 c dy = 2 a

0

a c

Z1

c c c cos ! k y + sin ! k y y i a a a

1

0

(20) The integral can be calculated following note n. 8 of ref. [40]. It converges for 0 < Re(ic =a ) < 1 but can be regularized considering ! ia "=c and

4

dy

taking the limit for " ! 0 The result is: e ) = B0 c ( c ! k ) B( 2 a a

i ac

2

Using the relation j (ix)j = (i

c a

x sinh( x)

=

(i

c a

)

2 c a

(21)

we have

2

)

c 2 a

e

e

c a

e

c a

(22)

and the equation (21) becomes e ) B(

2

=

cB02 2 a e2

1 c a

1

Time dependent Doppler e¤ect therefore results in the Planck factor e~ typical of a Bose-Einsten distribution with temperature T = called Unruh temperature.

4

~a ~a p = 2 kB c 2 kB c 1 a2 =A2

(23) =kB T

(24)

Conclusion

We have modi…ed the Rindler space to include the existence of a maximal acceleration. The consequence was a change in the Unruh relation between acceleration and temperature. The standard relation (1) has been often used to show that if a maximal temperature exists, then also the acceleration must be upper limited and viceversa [39]. Taking into account the existence of a maximal acceleration, we considered the time dependent Doppler e¤ect and we derived in a simple way the new Unruh temperature. From our new equation (24), on the contrary, we argue that for a ! A we have T ! 1 that is the temperature could be unlimited even in the case of existence of a maximal acceleration. If there is a maximal temperature the acceleration cannot reach its maximal value A.

4.1

Acknowledgements

This research was partially supported by FAR fund of the University of Sannio

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