1 Introduction

In this paper, we study the kinematical quantities of accelerated motion in Minkowski spacetime, in which, as well known, one of the coordinates is the laboratory time. In the lowest dimensional (one time component and one spatial coordinate) Minkowski spacetime, the relativistic motion of constant proper acceleration (the hyperbolic motion) has been throughly discussed over more than a century, starting with Born ^[7]. Even when only considering uniform proper acceleration, intriguing quantum field effects, like the famous Unruh effect, show up ^[1,2].

In the recent literature on this topic, there have been discussions related to particular features of uniform and non-uniform relativistic acceleration ^[3], in which case, higher order time derivatives of the four-velocity, like the Jerk, J(s), Snap, S(s), Crackle, C(s), and beyond should be considered. Acceleration and further higher-order proper time derivatives become important and non-trivial quantities in generalizations related to intrinsic differential geometric parameters of the curves (worldlines) like in ^[4], or related to curved spacetimes as in ^[5,6].

The organization of the bulk of this paper is the following. In Sec. 2, we briefly present the kinematics of the hyperbolic motion in the 1 +1 Minkowski case with an iterative extension to higher order proper time derivatives. In Sec. 3, we consider the hyperbolic motion in four-dimensional Minkowski spacetime. In this case, a matrix representation of Lorentz transformations between a traveler proper frame and a laboratory frame can be used, see e.g. ^[8]. In this matrix approach, one obtains a nonlinear second order differential equation which is solved based on the standard parametrization of the four-velocity in terms of hyperbolic functions. The four-acceleration is calculated, together with its modulus. Similar calculations are performed for the Jerk and Snap. In Sec. 4, the general solution of the differential equation for hyperbolic parametrization is obtained and the same kinematical quantities are expressed in term of the general solution. In Sec. 5, the worldlines (Rindler-type hyperbolas) based on the general solution are obtained and displayed graphically. Section 6 contains the conclusions of this work.

2 Hyperbolic motion in the 1+1 dimensional Minkowski spacetime

The relativistic hyperbolic motion happens when a particle moves in a Minkowski spacetime with constant proper acceleration. It has been studied in detail by Born, already in 1909, who called it “hyperbolic motion" since the equation of the trajectory in the x, t plane (spacetime) is a hyperbola ^[7,9].

The basic physical concept in a hyperbolic frame is the hyperbolic velocity, also known as the rapidity, given by

where we have considered the speed of light c = 1and where v = r’, namely, the derivative with respect to laboratory time.

The hyperbolic acceleration, α, is defined as the derivative of the hyperbolic velocity V with respect to the proper time 𝑠. Note that

so the hyperbolic acceleration, α, is the proper acceleration, related to the laboratory acceleration by α=γ3(v)(dv/dt).

The relativistic trajectory can be found from

With the initial condition v = 0 at t = 0, one obtains

and considering that v=r'=x'=dx/dt, by one more integration one can obtain the 𝑥 coordinate

The latter equation can be written as the equation of a hyperbola

where b = 1/α. We set the initial condition to x₀ = 0 and use the hyperbolic parametrization

so that

which allows to identify u with the rapidity V, because

Furthermore

so that

Equation (7) contains the components of the position 1 + 1-vector

but to generate the iterative sequence of the higher order kinematical quantities, we find that the shifted 1 +1 position vector X-

is more appropiate. In terms of the shifted 1 + 1 position vector, the 1 + 1-velocity is

where u˙=α from (11) has been used. For the 1 + 1-acceleration, one obtains

Furthermore, the 1 + 1-Jerk and the 1 + 1-Snap are

The higher order derivatives can be written iteratively as follows:

for even derivatives, and

for odd derivatives. For constant proper acceleration, the higher order even derivatives (1 +1-acceleration, 1 + 1-Snap, etc.) are given in terms of rescaled 1 + 1-position vectors while the odd derivatives are in terms of rescaled 1 + 1-velocities. The moduli of both types of derivatives are succesive powers of fourth order in α. The even derivatives are spacelike 1 + 1-vectors, whereas the odd ones are timelike. There is not much functional change when passing from one kinematic derivative to the next one, which consists only in a permutation of the hyperbolic functions from the time component to the space component at each order and a power-law modification of the square modulus.

3 Hyperbolic motion in the 1+3 dimensional Minkowski spacetime

When considering relativistic hyperbolic motion in more dimensions it is suitable to use the matrix formalism since Lorentz transformations, such as general Lorentz boosts, have well defined matrix representations. In the case of proper acceleration and proper velocity, one can use the following matrix relationship ^[8]

where the laboratory velocity is defined as

and is related to proper velocity by the boost transformation

(the traveler is at rest in the proper frame) and B(s) is a matrix of the family of Lorentz transformations

where t = t(s) and the prime indicates derivative with respect to the laboratory time, t. From (19) we obtain

where γ'=(1/2)γ3d(r→ ')2/dt and γ=dt/ds. Here, r→ is the radius vector and the point indicates derivative with respect to proper time. One can notice that if one takes γ'=0 in (23) then one has d(r→ ')2/dt=0 and then (23) turns into a→=0, which tells us that the motion is a simple translation of constant velocity. Also, we get that the square modulus of the four-acceleration is

where the minus sign comes from the chosen signature (+ — — — ). We study the case with constant proper acceleration, a→, of constant square modulus a→ 2=α2. Considering γ=t˙, (23) becomes

We parameterize the four-velocity as

where n^ is a unitary vector, whose derivative is

Because n^˙⋅n^=0, we obtain

which leads to the second order nonlinear equation

A particular solution of this equation corresponding to the initial velocity UT(0)=(1,0) is

with n^˙p=0, a→⋅a→=α2 and for this particular case, n^p=a→/α. However, we move to calculate the 1 + 3-acceleration, Jerk, and Snap, mantaining an unspecified functional form of f(s) only constrained by (29) to allow for the possibility of using the general solution instead of the particular one (30).

4 The general solution to the nonlinear equation for f(s)

Equation (29) is vital to obtain the expressions for acceleration, Jerk, and Snap in their simplified form containing only terms of f and f.˙ Besides, the derivatives of their corresponding moduli vanish only when we consider (29). To find a solution to (29), we note that this is a non-homogeneous differential equation of second order where the sinh(f(s)) factor complicates the equation. This can be avoided by the following change of dependent variable

which turns (29) into

With a second change of the dependent variable

in (43), one obtains

The general solution of the latter equation is

where s₀ and β are integration constants that can be determined through initial conditions. Futhermore, we will use the shorthand notations sαβ=(1/2)α2-β2(s-s0) and Th2(sαβ)=(α2-β2)tanh2(sαβ); replacing (46) in (44) gives g(s) in the form

which implies

or alternatively

If β = 0, then the general solution (49) reduces to the particular one, fps=αs-s0.

Considering (49) in (36), the square modulus of the Jerk is obtained in terms of α and the initial condition β

which shows that the square modulus of the Jerk is a constant quantity less than α⁴. Regarding the square modulus of the four-Snap in (41), one obtains

by using the general solution (49), which shows that the four-Snap is spacelike of square modulus less than α⁶. One can also write the general solution (49) in terms of the Jerk modulus (50) as

Formulas of the general solution in terms of the moduli of higher order derivatives can be easily written down. For instance, using (51) one can substitute J2 in (52) by the modulus of the Snap.

Due to the performed changes of variable of f(s) and g(s), there are some restrictions we should take into account. Equation (42) implies that g(s) ≥ 1, while (44) requires that hs∈[0,1). Since, 0≤tanh2(sαβ)≤1 we make sure that hs∈[0,1) only if |β|<|α|, and the limiting case h(s)→1 happens when s→∞ or |α|=β.

5 Modified Rindler hyperbolas

If one uses the particular solution f_p (s), it is easy to obtain the standard geometric hyperbolic behavior in the plane defined by the coordinates t(s) and x(s) by integrating the corresponding components of the parametrized velocity (26). Instead of this, we use the general arctanh solution to see graphically the kind of geometric behavior of the worldlines. Using (26) and (49) we compute the coordinate time as

Similarly, by integrating the space component of the velocity along n^, which is sinh(f), leads to the projection rn

The plot of r_n(s) as a function of s is displayed in Fig. 1, for different values of β, whereas in the t vs r_n coordinates as frequently presented in the literature ^[7,18] are displayed in Fig. 2. The value β = 0 corresponds to the particular solution f_p(s) = αs.

Figure 1 Plots of r_n vs s, with α = 0.5, for the standard Rindler hyperbola (β = 0) and the deformed ‘hyperbolas’ corresponding to β = 0.1 and β = 0.2. 

Figure 2 For fixed α = 0.5, original Rindler hyperbola (β = 0) and modified hyperbolas for to β = 0.1 and β = 0.2. respectively. 

We notice that one can also work with the hyperbolic tangent of the equivalent of the hyperbolic velocity (rapidity) as defined in Eq. (1), where we recall that the four-velocity can be represented as UT=γ(1,r→ '), and we have for the hyperbolic parametrization that

then

where we can obtain

the modulus of the three-velocity. Considering equation (49), and considering that

we can write (60) as

The results are equivalent as |r'|=tanh(f)=r˙/γ=r˙/cosh(f), so r˙s=sinhf. However, it is of physical interest because |r'| is the magnitude of the three-velocity, the velocity that the laboratory frame measures for the moving observer.

6 Conclusions

We have studied the kinematical quantities of relativistic hyperbolic motion, i.e., of constant proper acceleration α, in 1 + 1- and 1 + 3-dimensional Minkowski spacetime. The standard hyperbolic parametrization of the spacetime coordinates has been used in the literature to obtain a nonlinear differential equation for the argument of the hyperbolic functions. In this paper, we have worked both with the particular linear solution as in the recent literature ^[8], but also with the general a𝑟𝑐𝑡𝑎𝑛ℎ solution to evaluate the kinematical higher quantities. All of the higher order derivatives beyond the acceleration depend only on the proper constant acceleration when the particular solution is employed, but if the general solution is used, they depend also on an integration constant corresponding to a nonzero initial condition. In the physics context, the effect of this nonzero initial condition is to produce deformed Rindler hyperbolas which still belong to the class of relativistic hyperbolic motion since all the proper time derivatives are of constant square modulus, although of smaller weight than in the case of the particular solution.