We consider the relativistic generalization of the problem of the “least uncomfortable” linear trajectory from point A to point B. The traditional problem minimizes the time-integrated squared acceleration (termed the “discomfort”), and there is a universal solution for all distances and durations. This universality fails when the maximum speed of the trajectory becomes relativistic, and we consider the more general case of minimizing the squared proper acceleration over a proper time. The least uncomfortable relativistic trajectory has a rapidity that evolves like the motion of a particle in a hyperbolic sine potential, agreeing with the classical solution at low velocities. We consider the special case of a hypothetical trip to Alpha Centauri and compare the minimal-discomfort trajectory to the one with uniform Earth-like acceleration.

In a recent paper in this journal, Anderson, Desaix, and Nyqvist1 (ADN) considered the “least uncomfortable” trajectory of linear motion from point A to point B, defined as the path covering a distance X in time T for which the integrated squared acceleration (the “discomfort”) is minimized. These authors derived an elegant solution, showing that the jerk (third derivative of position) is constant on such a trip, and compared their solution with those found by variational approximation methods. The least uncomfortable solution, as presented by ADN, is universal in that the velocity relative to its maximum as a function of distance or time relative to the total distance or duration is independent of X and T.

While the ADN solution nominally applies to any trip of distance X over time T, if T becomes short enough and X large enough, the maximum speed reached during the trip can approach the speed of light. There is in fact an unstated assumption in the ADN solution that X/Tc. In this note, we generalize the solution and consider the least uncomfortable relativistic journey from A to B.

The magnitude of acceleration depends on the reference frame in which it is measured, be it the “lab” inertial frame fixed to the start and end points or the “ship” accelerating frame that moves between them. In the lab frame, as the ship asymptotically approaches the speed of light, its coordinate acceleration d2x/dt2 approaches zero. In the ship frame, however, the proper acceleration dictates the inertial forces experienced onboard. To minimize acceleration-induced discomfort, we wish to minimize the total proper acceleration experienced. This quantity can be minimized by considering the total duration of the trip in the lab frame, but because observers in the ship frame are experiencing discomfort, we wish to minimize the squared proper acceleration integrated over a proper time.

In their paper, ADN considered a one-dimensional trip over distance X and time T, beginning and ending at rest with velocity v = 0. Their solution minimizes the discomfort integral

F=0Ta2dt,
(1)

subject to the constraint

T=0Xdxv.
(2)

The solution, as derived by ADN, is a cubic function of time

xX=3(tT)22(tT)3,
(3)

which implies that velocity is quadratic in time [with vmax = (3/2) X/T], that acceleration is linear, and that jerk is constant.

In the non-relativistic limit, both proper time and acceleration are equivalent to the coordinate time and coordinate acceleration, and so the relativistic disagreement between lab and ship clocks need not be considered. However, when considering the least uncomfortable relativistic journey, it is desirable to minimize the “proper discomfort” as experienced in the frame of the traveller. In the following, units in which c = 1 are implicitly used. We define the discomfort functional F as the squared proper acceleration integrated over proper time τ, for a total proper duration T, to be

F=0Ta2dτ.
(4)

Since the functional to be minimized depends on the second time derivative of the position, it is convenient to phrase the question in terms of a minimization over velocity profiles β(τ), so that F is first order in derivatives of β and the problem can be solved with the usual Lagrangian formalism. In light of the dimensionality of the problem, we choose to consider the rapidity r(τ)=tanh1β(τ) as the dynamical variable; F is rewritten as

F=0TL0(r,ṙ)dτ=0Tṙ2dτ,
(5)

where we have identified the Lagrangian L0(r,ṙ), and the overdot denotes a derivative with respect to proper time τ.

We take boundary conditions such that the velocity is zero at τ = 0 and τ=T:

r(τ=0)=0,r(τ=T)=0,
(6)

and impose that the total distance travelled (in the lab frame) is X as a constraint:

0Xdx=0Tβdt=0Tβγdτ=0Tsinhrdτ=X.
(7)

This formulation presents the question as a standard constrained optimization problem that can be solved2 through the introduction of a Lagrangian multiplier λ for the constraint. In practice, we switch to the minimization of the extended functional

F[r(τ)]λ0Tsinhr(τ)dτ,
(8)

where the minus sign has been introduced for later convenience, which implies an extended Lagrangian

L=(drdτ)2λsinhr.
(9)

It is immediately recognized that Eq. (9) is the Lagrangian for a particle moving in a potential

U(r)=λ2sinhr,
(10)

after we have identified the particle's position with r and its velocity with dr/. The resulting Euler-Lagrange equation is

r¨=λ2coshr.
(11)

The (conserved) energy of the particle is given by

E=12ṙ2+λ2sinhr=λ2sinhrmax,
(12)

with rmax being the maximum rapidity reached at τ=T/2, which means

ṙ=±λsinhrmaxλsinhr,
(13)

and therefore

±dr1sinhr/sinhrmax=λsinhrmaxdτ.
(14)

The two signs of the radical are for τ<T/2 and τ>T/2, respectively. We now define the function

L(r;rmax):=0rds1sinhs/sinhrmax,
(15)

so that Eq. (14) can be integrated to give

λsinh(rmax)τ=L(r;rmax).
(16)

Equation (16) is an expression for the proper time elapsed τ as a function of the current rapidity and the two parameters λ and rmax. This expression only holds for 0<τ<T/2, until r = rmax at τ=T/2; the remaining section is determined by symmetry.

In principle, Eq. (16) can be inverted to obtain r(τ; λ, rmax), the general solution to the constrained minimization problem. In practice, for any given values of the parameters λ > 0 and rmax > 0, the function r(τ; λ, rmax) minimizes the functional F for some values of the total distance X and proper time T. To obtain the solution relative to a given X and T, this relationship must also be inverted to reconstruct the corresponding parameters λ and rmax. To find T as a function of (λ, rmax), the total proper time T can be found as the solution to r(T/2)=rmax. In accordance with Eqs. (15) and (16), we have

T=2λsinhrmaxL(rmax;rmax)=2λsinhrmax0rmaxds1sinhs/sinhrmax.
(17)

Thus, T can be expressed numerically as a function of (λ, rmax) in one integral. Making use of Eq. (7), a similar computation yields X also as an integral

X=0Tsinhrdτ=20T/2sinhrdτ,
(18)

and using Eq. (14), we then have

X=2λsinhrmax0rmaxsinhsds1sinhs/sinhrmax.
(19)

Let us verify that the results of ADN are recovered in the classical limit. If β ≪ 1, proper acceleration and proper time are equivalent to acceleration and time, r ∼ β, and γ ∼ 1. Thus, the Lagrangian equation (9) reduces to

L=v̇2λv,
(20)

whose equation of motion is v¨=λ/2, also obtainable from a Taylor expansion of Eq. (11). With boundary conditions v(0) = v(T) = 0, the solution for v(t) is

v(t)=λ4t2+λ4Tt.
(21)

With the constraint X=0Tvdt=(λ/24)T3, the Lagrange multiplier is determined to be λ = 24 X/T3, so that the velocity profile v(t; X, T) is

vX/T=6(tT)2+6tT.
(22)

The corresponding trajectory is then found by direct integration to be

xX=2(tT)3+3(tT)2+constant,
(23)

which matches the result determined in ADN.

While we regrettably lack a closed-form expression for the solution, for a given distance and a desired proper time, the minimal-discomfort trajectory can be determined through the constraints of Eqs. (17) and (19). These equations contain two unknowns, the maximum rapidity and the Lagrange multiplier. It is possible to solve these two equations simultaneously for the two unknowns, but it is difficult. Rather, we focus on the ratio of X and T, a characteristic “average” velocity (that may be superluminal). Dividing Eq. (17) by Eq. (19) gives

XT=0rmaxsinhsds1sinhs/sinhrmax0rmaxds1sinhs/sinhrmax,
(24)

which fixes the relationship between X/T and rmax, eliminates the dependence on λ, and can be more easily found numerically, being a one-dimensional relationship. Then, the desired proper time can be used to calculate the Lagrange multiplier using the known maximum rapidity. From this, the initial acceleration can be found iteratively.

Having derived the least uncomfortable trajectory for relativistic travel, we can calculate the rapidity as a function of proper time by solving Eq. (11) numerically using the Runge-Kutta algorithm and then use the integral equations Eqs. (19) and (17) to constrain the Lagrange multiplier and the initial acceleration to constrain the total distance and proper time.

We examine the trajectories in terms of the fractional velocity (v/vmax) as a function of relative proper time (τ/T) as shown in Fig. 1. While the trajectory of the classical solution is universal, the relativistic solution is characterized by a single free parameter, the maximum speed (which, in effect, determines the degree of deviance from the classical solution), or equivalently the maximum rapidity or Lorentz factor. We classify solutions in terms of the maximum Lorentz factor, and our solutions reduce to the classical solution in the limit of γ = 1. Because vmax plateaus at c, the faster trajectories spend comparatively more of the trip near their maximum velocity. The behavior of the trajectories as the speed increases is similiar to that of solutions of the non-linear Schrodinger equation for repelling particles in a box, whose wavefunction changes from cosine-like to a more uniform distribution as the density increases.3 

Fig. 1.

Velocity profiles of the least uncomfortable trips for varying maximum speeds from solutions to Eq. (11). The inner solid curve denoted by the car is the classical solution, Eq. (3), with trips of increasing maximum speed characterized by maximum Lorentz factors γmax = 1.4, 2.5, 4.1, 9.1, 25, and finally γmax = 100 denoted by the outer solid curve and the spaceship.

Fig. 1.

Velocity profiles of the least uncomfortable trips for varying maximum speeds from solutions to Eq. (11). The inner solid curve denoted by the car is the classical solution, Eq. (3), with trips of increasing maximum speed characterized by maximum Lorentz factors γmax = 1.4, 2.5, 4.1, 9.1, 25, and finally γmax = 100 denoted by the outer solid curve and the spaceship.

Close modal

The classical solution has a cubic time evolution for the position, implying that jerk is constant. This is not the case for the relativistic version; as the velocity approaches that of light, it is nearly unchanging, and the trajectory approaches linearity (Fig. 2(a)). The coordinate acceleration thus approaches zero for most of the trip; however, the proper acceleration acquires a strong jerk as the maximum velocity increases (Fig. 2(b)).

Fig. 2.

Top: Fractional distance as a function of fractional proper time for non-relativistic (dashed), slightly relativistic (dotted), and highly relativistic (solid) solutions. The trajectory is cubic at low velocities, but with greater speed, the trajectory approaches a line as v ≈ c. Bottom: Proper acceleration relative to its initial value as a function of fractional time, for the same three cases.

Fig. 2.

Top: Fractional distance as a function of fractional proper time for non-relativistic (dashed), slightly relativistic (dotted), and highly relativistic (solid) solutions. The trajectory is cubic at low velocities, but with greater speed, the trajectory approaches a line as v ≈ c. Bottom: Proper acceleration relative to its initial value as a function of fractional time, for the same three cases.

Close modal

Analyses similar to ADN have been considered for the design of optimal train driving strategies.4 Trains and cars do not approach light speed and have a transverse source of terrestrial gravity, but in interstellar spaceflights for which this analysis may become relevant, a constant proper acceleration may be used as a source of artificial gravity. One could consider a trip to Alpha Centauri, for example, where +1 g of proper acceleration is applied for the first half of the 4.37 light-year trip and −1 g is applied over the second half (with a brief period of nonzero jerk in the middle), coming to rest at the destination while enjoying Earth-like gravity for (nearly) the entire trip. In a reference frame in which both Earth and Alpha Centauri are close to being at rest, such a trip would take almost exactly 6 years through a pleasant coincidence and a proper 3.6 years onboard, reaching 95% of the speed of light. We can compare this trip with the minimal-discomfort trajectory with the proper time over the same distance (Fig. 3). The minimal-discomfort trajectory only reaches 90% light speed but spends more time closer to its maximum velocity, experiencing greater acceleration (roughly 1.5 g) at the start and end of the trip.

Fig. 3.

Two possible velocity profiles on a hypothetical trip to Alpha Centauri. The trip maintaining constant Earth-like proper acceleration reaches 0.95c and has a kink discontinuity, while the minimal-discomfort scenario is smooth and reaches 0.9c.

Fig. 3.

Two possible velocity profiles on a hypothetical trip to Alpha Centauri. The trip maintaining constant Earth-like proper acceleration reaches 0.95c and has a kink discontinuity, while the minimal-discomfort scenario is smooth and reaches 0.9c.

Close modal

The constant-acceleration trip contains a kink discontinuity in the velocity corresponding to infinite jerk at the halfway point as well as at the start and end of the journey. In their earlier paper, ADN1 suggested an alternative comfort scheme in which the squared jerk is minimized rather than the squared acceleration. While this note was motivated by the desire to see a relativistic generalization of the least-discomfort path, the most comfortable trajectory may be the one that keeps gravity as close to g as possible while minimizing a higher-order kinematic derivative such as jerk. Considerations of jerk in special relativity have been explored previously,5,6 and minimizing the jerk for a relativistic journey is left as an exercise for the astute reader. We note that the minimal-discomfort path may be more suited to the transport of acceleration-sensitive equipment or self-replicating machines, rather than humans themselves.

We have extended the result of ADN1 for the least uncomfortable linear trajectory to incorporate special relativity and found a class of solutions that deviate from the classical solution as the maximum velocity approaches that of light. Although relativistic interstellar travel is likely many generations away, as physicists we believe it is not too early to consider the details of exotic-seeming transportation schemes.7 We hope that this work encourages students and researchers to consider the assumptions inherent in published results in physics and to examine the implications of breaking those assumptions.

A.R.K. was supported by an NSERC postdoctoral fellowship.

1.
D.
Anderson
,
Mats
Desaix
, and
R.
Nyqvist
, “
The least uncomfortable journey from A to B
,”
Am. J. Phys.
84
,
690
695
(
2016
).
2.
F.
Clarke
,
Functional Analysis, Calculus of Variations and Optimal Control
, Graduate Texts in Mathematics (
Springer
,
London
,
2013
).
3.
Lincoln D.
Carr
,
Charles W.
Clark
, and
Wilhelm P.
Reinhardt
, “
Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity
,”
Phys. Rev. A
62
,
063610-1
10
(
2000
).
4.
Peter
Pudney
and
Phil
Howlett
, “
Optimal driving strategies for a train journey with speed limits
,”
J. Aust. Math. Soc., Ser. B
36
,
38
49
(
1994
).
5.
T. R.
Sandin
, “
The jerk
,”
Phys. Teach.
28
,
36
40
(
1990
).
6.
Jorge G.
Russo
and
Paul K.
Townsend
, “
Relativistic kinematics and stationary motions
,”
J. Phys. A
42
,
445402
(
2009
).
7.
Alexander R.
Klotz
, “
The gravity tunnel in a non-uniform earth
,”
Am. J. Phys.
83
,
231
237
(
2015
).