Constant Jerk Relativistic Rocket

Question

A quick google search will give many helpful pages and calculators regarding constant acceleration relativistic rockets, but my question is somewhat different.

What if jerk is the parameter to be held constant in a relativistic rocket?

Two versions of such a rocket: a simple flyby scenario, with beginning and ending accelerations specified, and a destination scenario, that will involve a flip turn at some point (but unlike the constant acceleration situation, not at halfway!). Initial velocities (and final for the destination scenario) should be zero.

I have just enough background physics to be interested in this, but my college courses never included the relativistic background to work it out easily for myself. I have a degree in maths so integrals etc. are fine, but space-time metrics may need some explaining if they come up.

A full-bodied relativistic rocket calculator offers distance, acceleration, max velocity (max lorentz factor), ship/earth times of journey, fuel/payload ratio. For constant jerk on a destination journey, I would also like to know when/where the flip would take place. I want to write such a calculator myself, I just need the equations, or a method of deriving them myself. (Of course I would not say no if someone pointed an existing one out to me.)

I also have the loftier goal of writing (or finding) a calculator that could give the above factors for any well-defined function of acceleration over time (or distance), if anyone knows of one or has been hoarding one on their hard drives.

[For the curious, there are various reasons why one might want to vary the initial and final accelerations that I can think of some offhand. For the flyby constant jerk rocket, it could set out to rendezvous with another ship having a different acceleration profile and wants its profile to smoothly match upon arrival, or it could depart with inhabitants temporarily acclimatized to one acceleration but with a preference for a "cruising acceleration" at a different level. For the destination constant jerk rocket, imagine flying from Earth to a Mars-gravity planet 10 light-years away, and having the entire flight as a slow adjustment period; alternatively, it seems to me that a constant acceleration rocket has to be working its engines a lot more in the beginning (when they have much more fuel mass to accelerate) than at the end; by keeping the engines on at full burn for constant thrust, acceleration will increase linearly throughout the flight.]

Answer 1

I'm going to use the same reasoning as in my answer here for the constant acceleration relativistic rocket.

Let $t$ be the accelerated observer's clock time. The accelerated observer feels constant proper jerk $\mathscr{J}$, meaning that at time $t$ and over time step $\Delta t$, they will add a velocity $\Delta v = \mathscr{J}\,t\,\Delta t$ to their current motion; as I reason in the other answer, this means that the rapidity change is $\Delta\eta = \mathscr{J}\,t\,\Delta t/c$ and so, adding up all these rapidity changes, this means that at accelerated observer time $t$ the total rapidity of their motion relative to the stay-at-home observer is

$$\eta(t) = \frac{1}{2\,c} \mathscr{J} \,t^2\tag{1}$$

and of course the relative velocity is

$$v(t) = c\,\tanh\eta = c\,\tanh\left(\frac{1}{2\,c} \mathscr{J} \,t^2\right)\tag{2}$$

Now we need to relate the two observers' clocks. Each $\Delta t$ of the accelerated observer's time is $\Delta t \gamma=\Delta t \cosh(\eta)$ of the stay-at-home observer's time. Thus the total time $T$ elapsed by the stay-at-home observer's clock when the accelerated observer's clock measures $t$ is:

$$T(t) = \int_{0}^t\,\cosh(\eta(u))\,\mathrm{d}\,u = \int_{0}^t\,\cosh\left(\frac{1}{2\,c} \mathscr{J} \,u^2\right)\,\mathrm{d}\,u\tag{3}$$

and the distance travelled when the clocks measure $t$ and $T$ is:

$$s(t) = \int_{0}^t\,v(u)\,\cosh(\eta(u))\,\mathrm{d}u = c\,\int_{0}^t\,\sinh(\eta(u))\,\mathrm{d}u= c\,\int_{0}^t\,\sinh\left(\frac{1}{2\,c} \mathscr{J} \,u^2\right)\,\mathrm{d}u\tag{4}$$

In this equation I have added the time dilation factor $\cosh\eta$ to relate the accelerated observer's $\mathrm{d}\,t$ to the stay-at-home observers $\mathrm{d} T$; the distance travelled in this interval is then $v(t)\,\mathrm{d} T = v(t)\,\cosh\eta\,\mathrm{d}t$. A general relationship relating (3) and (4) is:

$$\left(\frac{\mathrm{d}T}{\mathrm{d}t}\right)^2 - \frac{1}{c^2}\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^2 = 1\;\Leftrightarrow\;c^2\,\mathrm{d} T^2 - \mathrm{d} s^2=c^2\,\mathrm{d} t^2\tag{5}$$

which is simply the statement of the invariance of the proper time interval.

Given you have a mathematics background, these equations should get you going.