# Brownian Motion in Minkowski Space

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- A Brownian random walk in which a single time variable determines both the stopping times and the time between jumps. For example, one could adapt Keller’s analysis [14] of the model of a particle moving along the x axis such that during a time interval of duration Δτ = 1 from τ = i − 1 to τ = i, i = 1, 2,…, the particle moves with velocity ν = +1 or ν = −1, each with probability 1/2.
- A Brownian random walk in which two independent time variables are used, the stopping times and the the proper time difference between two consecutive stopping times, both of which are measured with respect to the universal time τ.

_{i}= cχ

_{i}define two random variables χ and s, that represent the times between successive impulses as recorded in the rest frame and the corresponding length between impulses respectively. Their variances, Var(χ) and Var(s) = c

^{2}V ar(χ), depend on properties of the bath, such as its density ϱ and temperature Θ, which we assume to be constant; hence, for simplicity, we omit these parameters in our notation. Also, we assume Var(χ) is positive and therefore that the trajectory segments between impulses are timelike.

_{i}is a proper time (recall the local time of the rest frame is equivalent to the proper time) it remains invariant in general relativity under coordinate transformations, which also includes the laboratory frame. This is the only information transmitted from frame to frame. Moreover, as the heat bath is held at a constant temperature, then it is reasonable to assume using the strong Markov property that the set ${\left\{{\chi}_{i}\right\}}_{i=1}^{\infty}$ defines a set of stationary independent increments [17]. This statement is independent of reference frames, and will be assumed throughout.

_{χ}of χ, and it contains most of the probability. Our approximation of the density function will be Lorentz invariant (see below) and therefore will allow us to calculate probabilities in any Lorentz frame.

## 2. Geodesics

^{a}in which its equation of motion is that of a straight line in space-time, that is

_{i}= s

_{i}will be invariant under parameter and coordinate transformations. In the case of a piecewise geodesic on the i-th component of the curve, we let

_{i}can be synchronized with the laboratory frame by means of the expression

_{i}, and the measure of a universal unit of time [34].

_{i}of the stopping times (in this paper we do not investigate how the distribution of the increments χ

_{i}depends on Θ, as we keep Θ fixed.)

## 3. Brownian Motion from the Perspective of the Laboratory Frame

_{j}= (cT

_{j}, iX

_{j}, iY

_{j}, iZ

_{j}) between two impulsive events, such that [36]:

_{x}, V

_{y}, V

_{z}), with ${V}_{x}^{2}+{V}_{y}^{2}+{V}_{z}^{2}<{c}^{2}$, we assume IE(X) = IE(Y ) = IE(Z) = 0, and IE(V

_{x}) = IE(V

_{y}) = IE(V

_{z}) = 0, where IE denotes expected value. The time elapsed between two impulsive events will be positive with mean µ

_{T}> 0 if we consider future events, but negative with mean μ

_{T}< 0 if we consider the past. Therefore, we may take μ

_{T}= 0, there is no loss of generality due to this assumption, as μ

_{T}≠ 0 merely implies a translation along the time axis. We further assume a positive standard deviation σ

_{cT}> 0 and σ

_{X}= σ

_{Y}= σ

_{Z}= σ > 0, where the equalities are due to the assumed isotropy (the isotropy condition can be easily relaxed, allowing different standard deviations for the different directions of space. However, we prefer to keep our notation simple). It follows from Equation (3) that the expected value given by

_{i}corresponding to the impulse at time ${\widehat{\tau}}_{i}$. As the only constraint that our variables have to obey is the time-like condition, we also assume the X

_{i}and their components are i.i.d. variables, in accord with the notion of the Brownian motion.

_{n}is approximated at large n by the normal distribution in Minkowski space, (See Appendix 2 for a justification of this form of CLT) with variance growing linearly with n, and density function given by

^{x}, ip

^{y}, ip

^{z}) replacing (t, ix, iy, iz) in (7). Note, the distribution for the 4-vector (ct, x, y, z) is compatible but obtained without reference to the distribution for the 4-momentum; it is only based on the validity of the CLT for (ct, x, y, z).

## 4. Concluding Remarks

_{n}= (ct, ix, iy, iz)

_{n}, with n ∈ ℕ a function of the proper laboratory time and not the local time variable t, which is part of the event. By doing this, we avoid the difficulties that arise (for example in the telegraph equation) from using only the single time variable t as an index for the 3-dimensional space events (x, y, z)

_{t}. Because of the second order time derivative, events described by the telegraph equation do not enjoy the Markov property, in accordance with the aforementioned theorem of Hakim [41]. Moreover the solutions of the telegraph equation do not necessarily remain normalized and positive at all times [5].

## Acknowledgments

## Appendix

## A. Properties of Curves

## B. Central Limit Theorem in Complex Space

_{1}, X

_{2},…, X

_{n}are independent and identically distributed with mean μ ∈ $\mathcal{R}$

^{k}and covariance Σ, where Σ has finite entries then

_{1}, x

_{2},…, x

_{k}) and each x

_{i}is independent with IE(x

_{i}) = 0 then

_{1}= ct, x

_{2}= ix, x

_{3}= iy and x

_{4}= iz we find using the standard 1-dimensional proof of the CLT that

## Author Contributions

## Conflicts of Interest

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O'Hara, P.; Rondoni, L.
Brownian Motion in Minkowski Space. *Entropy* **2015**, *17*, 3581-3594.
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Brownian Motion in Minkowski Space. *Entropy*. 2015; 17(6):3581-3594.
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