Your spacecraft (mass m) has to travel to a far away (distance d) planet. Your engine can deliver a useful energy E. You must start and finish at rest.
What is the minimum travel time given classical mechanics?
What is the minimum travel time (proper and earth) given relativistic mechanics?
I think that the time-optimal approach is to accelerate till the maximum speed (given by the condition T = E/2) as soon as possible, then keep the speed constant until we are close enough to start decelerating in a similar manner.
If we assume that the distance travelled during the first and last stage is negligible to d, the optimal time needed is
t = d / v_max.
The two descriptions of mechanics differ in the expression of v_max.
In classical mechanics,
1/2*m*v_max^2 = E/2
v_max = sqrt(E/m)
t1 = d*sqrt(m/E).
In special relativity,
m / sqrt(1-v_max^2) = m + E/2
1 – v_max^2 = m^2 / (m + E/2)^2
v_max^2 = ((E/2)^2 + mE) / (m + E/2)^2
t2 = d * (m + E/2) / sqrt((E/2)^2 + mE)
(this is the Earth time),
for the proper time,
t3 = τ = ∫ (dτ/dt) dt
However, dτ/dt = sqrt(1-v^2) is constant for most of the interval, leaving a limit of
t3 = sqrt(1-v_max^2) * t2 = m / (m + E/2) * t2 = d * m / sqrt((E/2)^2 + mE).
It can be quickly seen that
t3/t1 = sqrt(mE / (mE + (E/2)^2)) < 1,
from which it follows that
t3 < t1.
Similarly, expanding t2^2 – t1^2, we find that
t1 < t2.
In summary, the special relativity causes a longer travelling time when observed from Earth but shorter proper time for the travellers.
Appendix: the derivation of the condition used in the beginning
For the time-wise optimal scenario, we will use variational calculus. The function to be optimized will be the dependency of consumed energy on the position, E(x). For symmetry reasons, we will study only the acceleration part (we would need some absolute values for the second part, the power is apparently not <0 when decelerating).
Our fixed boundary conditions will be E(0) = 0 and E(d/2) = E/2 and we will restrict the solution on nondecreasing functions.
The cost function is defined as the total time,
Φ[E(x)] = ∫ dt = ∫ (dt/dx) dx = ∫ [0 to d/2] 1/v dx,
where v is expressed in terms of E(x). Apparently, there is no dependence on E’(x), so the Euler-Lagrange equations are unapplicable and there is no regular optimum. However, the message is clear: when there is no restriction on the behaviour of E’(x), the closer we get v(E(x)) to its potential maximum, the better will be the cost. For both mechanics, v is monotoneously increasing in E, so the solution is the better the closer is E(x) to the constant function E(d/2) = E/2 in L1.