Mass-energy, so General Relativity tells us, puts dents in Space-time. Most concentrations of mass-energy, like stars, planets and galaxies, form shallow dents. Black Holes go deeper, forming an inescapable waterfall of space-time inwards to their centres. The closest that light can approach and still escape is the so-called Event Horizon, which forms a spherical boundary at the distance of the Schwarzschild Radius. But that’s light, which travels at the maximum possible speed – light-speed. Light follows the curvature in space-time, traveling the shortest pathways (geodesics). For particles with rest mass, like atoms, dust and space-ships, geodesics can’t be followed, merely approached.
Instead of flying towards a Black Hole, let’s ponder merely orbiting it. For most orbital distances from a Black Hole any small mass in orbit will experience nothing different to orbiting around any other large mass. Too close and you’ll experience extreme tidal forces – but that’s also true of neutron stars. To avoid being torn to shreds when approaching really close to a Black Hole we need a really big Black Hole. So we’ll consider the Milky Way’s Super-Massive Black Hole (SMBH) which masses 4.3 million solar masses and so has a Schwarzschild radius of 12.9 million kilometres. The Tidal Force at the Event Horizon is very mild for small objects, though significant for things as big as stars and planets.
Most orbital distances are fine – we have multi-year ‘movies’ of stars orbiting around our SMBH. Close orbits get measured in multiples of M – which is half the Schwarzschild radius. At a radius of 3M space-time is so curved that the geodesics form a circle around the Black Hole. Light can orbit indefinitely, building up to potentially extraordinary energy densities if nothing else gets in its way. But the centre of the Galaxy is full of dust and gas, so something is always getting in the way.
For things that don’t follow geodesics, merely approximate them, the Innermost Stable Circular Orbit (ISCO) is further out, at 6M. Objects here travel at half the speed of light. Other shapes of orbits can dip a bit closer in, down to 4M. Deeper in and motion near the Black Hole is no longer “orbital”. You must point away from the Black Hole and apply thrust or in-fall is inevitable.
The equations of motion from just inside the ISCO all the way to the centre of the Black Hole were only recently worked out in closed form. Previously numerical Relativity methods were used. The Equation of Motion of a non-rotating Black Hole, which our SMBH approximates, is quite simple:
$$\frac{r}{6M} = \left(\frac{1}{1+\frac{\phi^2}{12}}\right)$$
Which I’ve plotted here:
The Red Circle is 3M and the Yellow is the Schwarzschild Radius at 2M. In this case the plot starts at r = 5.95 M with the test particle circling the Black Hole 6 times before hitting the central point.
The proper time experience by an observer infalling on the spiral is a bit more complicated. Let \(x = (\frac{r}{r_I})\), where \(r_I\) is the ISCO radius.
We parameterise \(x\) as follows:
$$x = \frac{1}{2}(1-cos\psi) = sin^2(\frac{\psi}{2})$$
with \(\psi\) running from \(\pi\) to 0. Then the Proper-Time \(\tau\) of the inspiral is:
$$\tau = \sqrt{\frac{3r_I^3}{2M}}\left[\frac{1}{2}(3\psi-sin\psi)-2tan(\frac{\psi}{2})\right]$$
In the case of the non-rotating Black Hole ISCO is at 6M, so the equation simplifies to:
$$\tau = 18M\left[\frac{1}{2}(3\psi-sin\psi)-2tan(\frac{\psi}{2})\right]$$
But what is M? It’s the “geometrised” mass of the Black Hole, which is derived by muliplying the mass by \(\frac{G}{c^2}\). Similarly the proper time is in units of “geometrised” time, so it needs to be divided by the speed of light, \(c\), to convert to seconds.
In the case of the fall from r = 5.95M to 0 (\(\psi\) = (5.95/6)\(\pi\) to \(\psi\) = 0) the total time is 18 * 71.73 * M = 1291.14 M. In the case of our Galaxy’s SMBH a proper time of M is 21.5 seconds. So the inspiral time is about 27,760 seconds or 7.71 hours. The point of no return, the Event Horizon, is reached with 217 seconds to go, 7.65 hours later.
Beneath the Event Horizon we’ve explored in a previous post: Surviving the Plunge.
Reference:
Mummery, A. & Balbus, S. “Inspirals from the innermost stable circular orbit of Kerr black holes: Exact solutions and universal radial flow” (2022) https://doi.org/10.48550/arXiv.2209.03579
[accepted to Physical Review Letters: Phys. Rev. Lett. 129, 161101 – Published 12 October 2022]