# Black hole dilation for space exploration

Just curious, if you could contain a small black hole, such as has been theorized possibly occurring after the period of rapid expansion, curious if the local time dilation from the proximity of mass would allow journey to distant stars in something less than a lifetime.

• That awesome, thanks mate.
– Luke
Feb 3 '20 at 10:46

It's a nice idea, but I'm afraid it won't work, even if you do manage to find a primordial black hole.

To get substantial gravitational time dilation you need to get very close to the event horizon, but that's an extremely dangerous activity around small black holes, due to the tidal effects. It's bad enough near a 3 solar mass black hole, which is about the smallest black hole that can be formed in the modern universe (through a supernova explosion). A black hole of that mass has a Schwarzschild radius of about 8.862 km, and of course, carting around something like that in your spaceship isn't exactly easy. ;)

If you captured an Earth mass primordial black hole, which has a Schwarzschild radius of about 8.87 mm, not only do you have the problem of carting around that much mass in your spaceship, it's too small for you to get your whole body inside the desired time dilation "zone". And anything that gets too close to it will get ripped to shreds by the intense tidal force.

According to the Hawking radiation calculator, an Earth mass black hole has surface tides of $$1.14249 \times 10^{21} m/s^2/m$$.

From Wikipedia, the formula for the time dilation of a stationary observer at a (Schwarzschild) distance of $$r$$ from the centre of a non-rotating black hole of Schwarzschild radius $$r_s$$, relative to the time of a distant observer in a negligible gravitational potential is:

$$t_0 = t_f\sqrt{1 - \frac{r_s}{r}}$$

Eg, if $$r = 4r_s$$ the time dilation factor is $$\sqrt{3/4} \approx 0.866$$, which isn't very useful.

A more practical way to get time dilation for a long space journey is to travel at high speed. This requires a huge amount of energy, and we don't currently have the required technology to boost a spacecraft to such speeds. But if you accelerate at 1g for 5 years ship time, in Earth's reference frame your flight will last for 83.7 years and you'll cover a distance of 82.7 lightyears. Please see the classic Usenet Physics article, The Relativistic Rocket for further details.

• Not to mention that if you had a black hole small enough to somehow carry around in a spaceship, it would be very, very hot. Feb 2 '20 at 9:32
• @Bob Indeed! Even a 1 million metric ton BH (which has a Schwarzschild radius about a thousand times smaller than a proton) radiates with a luminosity of 356 trillion watts at a rather toasty 123 trillion kelvin. ;) An Earth mass BH doesn't have that problem, though, it's only 0.02 kelvin. Feb 2 '20 at 10:08
• You've only addressed the 1st part of the question: "if you could contain a small black hole". What about the time-dilation effects of traveling with a heavy mass? Feb 2 '20 at 14:01