# Brownian Motion in Robertson-Walker Space-Times from Electromagnetic Vacuum Fluctuations

###### Abstract

We consider classical particles coupled to the quantized electromagnetic field in the background of a spatially flat Robertson-Walker universe. We find that these particles typically undergo Brownian motion and acquire a non-zero mean squared velocity which depends upon the scale factor of the universe. This Brownian motion can be interpreted as due to non-cancellation of anti-correlated vacuum fluctuations in the time dependent background space-time. We consider several types of coupling to the electromagnetic field, including particles with net electric charge, a magnetic dipole moment, and electric polarizability. We also investigate several different model scale factors.

###### pacs:

04.62.+v,05.40.Jc,12.20.-m## I Introduction

Brownian motion of a particle in a thermal bath is a well-known phenomenon. (See, for example, Ref. Pathria .) In this case, the particle’s mean squared velocity grows linearly in time until dissipation effects become important, after which it approaches a non-zero equilibrium value. The linear growth phase is characteristic of any random walk process, in which each fluctuation is independent of previous fluctuations. Quantum fluctuations are quite different from thermal ones in that the former are strongly correlated or anti-correlated. This does not, however, prevent quantum Brownian motion, which will be the topic of this paper. The existence of Brownian motion in the Minkowski vacuum state is controversial. Although conventional quantum electrodynamics suggests that the only effect will be an unobservable mass renormalization, Gour and Sriramkumar gs99 have argued that there could be an observable effect on charged particles coupled to the fluctuating electromagnetic field. Brownian motion in the presence of boundaries is less controversial, and has been studied by several authors b9191 ; jr92 ; wkf02 ; yf04 ; wl05 ; sw07 . Barton b9191 was the first to examine fluctuations of the Casimir force. Wu et al wkf02 calculated the Brownian motion of an atom near a perfectly reflecting plate due to fluctuations in the retarded van der Waals force. The mean force here is the Casimir-Polder force cp48 . The analogous Brownian motion of a charged particle near a reflecting plate was treated by Yu and Ford yf04 . In all of these cases, the mean squared velocity of the particle approaches a constant even in the absence of dissipation. This is required by energy conservation, as there is no energy source in these static configurations. The fact that the late time mean squared velocity is non-zero can be attributed to the effects of switching when the interaction is turned on. Switching effects were recently discussed by Seriu and Wu sw07 .

The mechanism which enforces the lack of growth of the mean squared velocity can be understood as anti-correlated fluctuations. A charged or polarizable particle in a Casimir vacuum can acquire an energy from a fluctuation. However, that energy is typically surrendered on a time scale of order to an anti-correlated fluctuation. The correlation functions of the quantized electromagnetic field automatically enforces the required anti-correlations ford07 . The quantum fluctuations of the stress tensor in flat spacetime also exhibits subtle correlations and anti-correlations, as is discussed in Ref. fr05 .

The Brownian motion of test particles is an operational means to describe a fluctuating quantum field. This approach can be used to treat the quantum fluctuations of the gravitational field, which has been a topic of much interest in recent years f82 ; kf93 ; f05 ; fs96 ; fs97 ; nd00 ; c98 ; ch97 ; ccv97 ; nbm1998 ; mv99 ; hs98 ; tf06 ; wnf07 .

In the present paper, we will investigate the Brownian motion of various types of particles coupled to the quantized electromagnetic field in the background of a Robertson-Walker spacetime. Here the time-dependent background geometry can act as an energy source, so the particles can acquire a net kinetic energy.

In Sect. II we develop the basic Langevin equation formalism for calculating the mean squared velocity of classical particles coupled to a fluctuating force in a spatially flat Robertson-Walker background. The formalism is applied to a several specific choices for the scale factor of the universe in Sect. III. Our results are summarized and discussed in Sect. IV.

Unless otherwise noted, we work in Lorentz-Heaviside units with .

## Ii Basic Formalism

The equation of motion of a classical point particle moving in a curved spacetime with a four-force is

(1) |

where is the 4-velocity of the particle, is its mass and the proper time. The operator on the right-hand side of Eq. (1) is the covariant derivative given by

(2) |

Here we take the space-time geometry to be that of a spatially flat Robertson-Walker universe with metric

(3) |

where is the scale factor. We will restrict our attention to the case where the particles are moving slowly with respect to these coordinates, in which case the particle’s proper time becomes the coordinate time . Because of spatial isotropy, we can consider a particular direction, the -direction, and write the equation of motion as

(4) |

Here we have used , where . We will take the four-force to be of the form

(5) |

where is a non-fluctuating external force, and is a fluctuating force produced by the electromagnetic vacuum fluctuations whose mean value vanishes:

(6) |

First, let us consider the case of free particles, which corresponds to the case when . Thus Eq. (4) can be written as,

(7) |

which after integration reduces to

(8) |

Assuming that these particles are initially at rest (), we find that the velocity-velocity correlation function is given by

(9) |

Another case of interest is when there is an external force which cancels the effect of the cosmological expansion:

(10) |

This is the case for any particles in bound systems such as galaxies or molecules. Such particles do not participate in the cosmological expansion and in this case two such particles do not move apart on the average. We will refer to these as bound particles. In this case,

(11) |

and the velocity correlation functions for particles which start at rest at is

(12) |

Note that the above expression is a coordinate velocity correlation function. In a Robertson-Walker space-time, proper distance between particles, , is related to the coordinate separation at by . Thus the proper velocity correlation function is given by

(13) |

### ii.1 Charged Particles

In this section, we will consider electrically charged particles with charge coupled to a fluctuating electromagnetic field. In this case, the four-force is

(14) |

For the case of free particles, Eq. (9) yields

(15) |

where the sub-indexes 1 and 2 refer to the coordinates and , respectively, and the subscript denotes a vacuum correlation function in Robertson-Walker spacetime.

This correlation function is obtained from the corresponding correlation function in flat spacetime by a conformal transformation. First write the Robertson-Walker metric in its conformal form

(16) |

with . The field strength tensor in these coordinates is given by

(17) |

where the subscript refers to the Minkowski space field strength. This may be seen, for example, from the fact that the Lagrangian density, is invariant under the conformal transformation. From this and Eq. (15), we find

(18) |

Here the appropriate component of the Minkowski space correlation function is given by Eq. (126). The key feature of this result is that the scale factor does not appear inside the integrand. Thus the cosmological expansion has no effect on the Brownian motion, and hence we do not find an interesting result in this case.

### ii.2 Magnetic Dipoles

In this section we will consider particles with a magnetic dipole moment. In flat spacetime, such particles experience a force when there is a non-zero magnetic field gradient:

(20) |

where , is the magnetic potential energy, is the magnetic moment and is the magnetic field. Writing this force in covariant form, we have for the -component

(21) |

where, and , and such that .

For free particles, the velocity-velocity correlation function, Eq. (9), becomes

(22) |

Again, we may use Eq. (17) to write the above expression in terms of , which may be evaluated to write

(23) | |||

In Eq. (23), we have used the coincident limit only in the numerator, in order to simplify our expressions. This will not alter our final results, because we will take this limit after the integrations.

For bound magnetic dipoles, we may start with Eq. (12) and follow the same procedure to find

where again the coincidence limit in the spatial coordinate was taken in the numerator of the integrand.

### ii.3 Polarizable Particle

We will consider in this section a polarizable particle, described as a point particle with a static polarizability . In an inhomogeneous electric field, such a particle experiences a force

(25) |

which in covariant notation becomes,

(26) |

where the low velocity limit was taken with , . For a free particle, Eqs. (9) and (26) lead to

where we used the fact that and , in conformal coordinates.

Here we use the Wick theorem to calculate the two-point function: , finding

(28) |

or, from the procedure outlined in Appendix A,

(29) |

To simplify our expression we will consider the coincident limit in the spatial coordinate , only in the numerator of all factors, as in the previous cases. Then the velocity-velocity correlation function is,

(30) | |||

## Iii Specific Universe Models

In this section we will apply the basic formulas obtained in Sect. II to investigate the influence of different scale factors on the Brownian motion of particles induced by quantum vacuum fluctuations of the electromagnetic field.

### iii.1 Asymptotically Static Bouncing Universe

The study of bouncing universe was considered by some authors in the past as78 . Here we will study a special case, which is asymptotically static in the past and future. We will take the scale factor to have the form

(31) |

where and are constants, and is a positive integer. Note that when , the universe is asymptotically flat and goes to unity. It will be convenient to consider different choices of for different types of particles in order to simplify the corresponding integrals.

#### iii.1.1 Bound Charged Particles

In this case, we set , so that Eq. (19) becomes

(32) | |||

where we used Eq. (19). This integral is evaluated in Appendix B, with the result being a rather complicated expression, Eq. (B.1). This result simplifies considerably in the limit that and to

(33) |

Note that because , this expression also gives the mean squared proper velocity .

We can gain some insight into this result by writing it in terms of a characteristic measure of the maximum curvature. Consider the scalar curvature and evaluate it using the scale factor Eq. (31) with . We find the following result,

(34) |

where is the Ricci scalar when . If we consider the limit , is negative and

(35) |

Then in terms of is

(36) |

We can also write Eq. (36) in terms of the redshift, defined by: , with , where is the minimum scale factor. Considering , we get

(37) |

The mean squared velocity is proportional both to the squared redshift and to the maximum curvature. This can be associated with an effective temperature using the non-relativistic equation: . Where is the effective temperature and is the Boltzmann constant in Lorentz-Heaviside units . Then, is,

(38) |

where is the lenght curvature and is the particle’s Compton wavelength.

#### iii.1.2 Free Magnetic Dipoles

Again take the scale factor to be Eq. (31) with . Then Eq. (23) for the mean squared velocity becomes,

Using the same procedure as before, we find the following result in the coincident limit when

(40) |

In terms of the redshift (), we have

(41) |

In contrast to the result for bound charges, the effect decreases with . For the case of electrons, it is convenient to write the magnetic moment as

(42) |

The effective temperature in terms of curvature length and Compton wavelength is,

(43) |

#### iii.1.3 Bound Magnetic Dipoles

Here we choose the scale factor to be of the form of Eq. (31) with . In this case, the mean squared velocity from Eq. (II.2) is,

We may evaluate this integral using the same technique as before, with the result in the coincident limit

(45) |

The physical velocity when is,

(46) |

where now the scalar curvature at is

(47) |

In terms of the redshift, given by , we can write

(48) |

which shows that the effect of quantum fluctuations grows with . Associating an effective temperature we have,

(49) |

Here the temperature grows with as in the bounded electric particle due the extra force that acts on the magnetic dipole. Indeed, we see that the effect here is smaller than that one indicated by Eq. (38).

#### iii.1.4 Free Polarizable Particle

Again we take the scale factor to be of the form of Eq. (31) with . Equation (II.3) for the mean squared velocity can be written as

(50) | |||

Following the procedure previously used, we find, in the coincident limit,

(51) |

This physical (comoving) velocity can be expressed in terms of given in Eq. (47) as

(52) |

We can also write Eq. (52) in terms of the redshift as

(53) |

The mean squared velocity decreases with the redshift in contrast with the bounded particle cases investigated in the previous section. This is due to the fact that the atoms are free of external forces. This effect can be associated with an effective temperature using the non-relativistic equation: . Thus, we obtain

(54) |

This result shows that the temperature decreases with the redshift because the particles are free of external forces.

### iii.2 Asymptotically Bounded Expansion

A universe with asymptotically bounded expansion was studied in Ref. bd77 , where the production of massive particles were considered. Here we will investigate the Brownian motion effects in scale factors of the form

(55) |

where is a positive integer, and are dimensionless constants and is a constant with dimension of time. We note that when . Then, this universe is asymptotically flat in past and future, but it is not symmetric and exhibits only expansion.

#### iii.2.1 Bound Charged Particles

Here we take the scale factor to be given by in Eq. (55). The mean squared coordinate velocity is then given by

(56) | |||

This integral is calculated in Appendix B, with the result given by Eq. (134).

In this model, the physical velocity is related to the coordinate velocity by , where . If we will make a Taylor expansion in up to the zeroth order term, we find

(57) |

where . Here the expression in parenthesis is a negative constant and is the Riemann zeta function. We can write Eq. (57) in terms of the scalar curvature at , given by

(58) |

It is also interesting write the mean squared velocity in terms of the redshift defined here as ). Then, in terms of and the redshift is given by

(59) |

when . The effective temperature is now

(60) |

#### iii.2.2 Free Magnetic Dipole

Now take the scale factor to be Eq. (55) with . Then the mean squared velocity is given by

(61) | |||

Following the same procedure as before we find that in the coincidence limit ,

(62) |

As before, . The physical velocity is

(63) |

where now

(64) |

The temperature in terms of and is,

(65) |

#### iii.2.3 Bound Magnetic Dipole

In this case, let in Eq. (55). The mean squared velocity becomes

(66) | |||

In the coincidence limit ,

(67) |

where

(68) |

The effective temperature is given by

(69) |

#### iii.2.4 Free Polarizable Particle

The scale factor is again given by Eq. (55) with and the mean squared coordinate velocity is given by

(70) | |||

Following the method used previously, we find the result

(71) | |||

in the limit . Here is again the zeta function, and the expression in brackets is positive. The physical speed satisfies the relation

(72) |

We can write Eq. (72) in terms of the scalar curvature , and of the redshift as

(73) |

when . The corresponding effective temperature is