Intragalactic velocity past the central black hole

I don't have a solution; what I do have is a possible path to a numerical solution.

For the sake of simplicity and sanity, I will consider the special case of a non-rotating, chargeless, spherically symmetric black hole. This black hole causes space to take a shape described by the Schwarzschild metric. A small test particle - which in this case can be our ship - obeys the following general relativistic equations of motion: $$\ddot{r}=-\frac{G\mathcal{M}}{r^2}+r\dot{\theta}^2-\frac{2G\mathcal{M}}{c^2}\dot{\theta}^2,\quad \ddot{\theta}=-\frac{2}{r}\dot{r}\dot{\theta}$$ where $r$ and $\theta$ are polar coordinates centered on the black hole and $\mathcal{M}$ is its mass. In terms of $t$, we're working in the reference frame of the ship, so when we differentiate with respect to time, we're talking about the ship's proper time, rather than the coordinate time measured by an observer infinitely far away from the black hole.

These two equations are all we need to determine the path of the ship, given the initial conditions of the system (namely, the mass of the black hole and the initial velocity vector of the ship). We can vary those initial conditions until we get the result that we want; given that the ship can get arbitrarily close to the speed of light, there's no limit on the time dilation it will experience. It's going to take a lot of trial and error to get it right, but we can get there.

All that said, here are some more details.

How do we solve the equations of motion?

There's no analytical solution to these equations, so we have to resort to numerical methods. Fortunately, the equations of motion are a pair of coupled, second-order, nonlinear differential equations. Being general relativistic doesn't make them inherently more difficult to solve than the Newtonian versions (though they're slightly different). We can solve them via any method you please. Here are some you might consider:

• Euler's method: This is something you'd only want to do as a sanity check, although it's quite simple to apply in, say, Python, in a few minutes. Say your ship's position, velocity and acceleration at some proper time $t$ are $\mathbf{r}(t)$, $\mathbf{v}(t)$ and $\mathbf{a}(\mathbf{r}(t))$. Then the same variables, at a time $t+\Delta t$, are $$\mathbf{r}(t+\Delta t)=\mathbf{r}(t)+\mathbf{v}(t)\Delta t$$ $$\mathbf{v}(t+\Delta t)=\mathbf{v}(t)+\mathbf{a}(\mathbf{r}(t))$$ $$\mathbf{a}(t+\Delta t)=\mathbf{a}(\mathbf{r}(t+\Delta t))$$ Euler's method is relatively inaccurate, though, compared to the host of numerical methods at your disposal.
• Runge-Kutta methods: Runge-Kutta methods are a broader set of techniques that have much greater accuracy and are substantially more widely employed than Euler's method (though Euler's method is a special case). The fourth-order Runge Kutta (RK4) is the most common, and also not too hard to apply, but higher-order RK methods involve smaller rounding errors.
• The leapfrog method: This method computes temporary values of position, velocity, etc. in between timesteps and uses them for calculations of the next step, leading to better accuracy than the Euler scheme. Furthermore, it conserves energy.

A note on time

It's simple to calculate the proper time $d\tau$ measured by a ship traveling at a speed $v$ in some other reference frame measuring coordinate time $dt$, ignoring gravity: $$d\tau=dt\sqrt{1-\frac{v^2}{c^2}}$$ Similarly, the proper time measured by a ship a distance $r$ from the black hole is, ignoring motion: $$d\tau=dt\sqrt{1-\frac{2G\mathcal{M}}{rc^2}}$$ To put these together, you have to do a little bit of thinking to find that $$d\tau=dt\sqrt{1-\frac{v^2}{c^2}-\frac{2G\mathcal{M}}{rc^2}}$$ It should be pointed out, though, that we're solving the equations of motion in only one reference frame - that of the ship - and so we likely don't care much about doing this conversion at every point during the integration.

An idea of the on initial conditions

To get an idea of where we'll need to start looking, consider the case of a ship in traveling 66 million light-years without any sources of gravity nearby. This is an excellent approximation to most of the ship's journey, where it will be far from the black hole and thus essentially not affected by it. For the ship's proper time to be 2,000 years while traveling 66 million light-years, we can see that $dt$ should be close to 66 million years, so $$\sqrt{1-\frac{v^2}{c^2}}\approx\frac{2000}{66000000}\approx3.03\times10^{-5}$$ This implies that $v/c\approx0.9999999995408633$, a difference from the speed of light of less than one part in a billion.

posted almost 5 years ago

HDE 226868‭

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