# How close would merging black holes have to be to feel gravitational waves? [duplicate]

Recently LIGO discovered gravitational waves caused by two black holes that were orbiting each other, and then collapsed into one black hole. A few months later, we find out that this actually happens quite often. However, these collisions are far enough away from Earth that no one can feel the gravitational waves, and it actually takes very precise measurement to even determine that they happen. So my question is, how close would a black hole collision have to be for us to feel the gravitational waves pass through Earth?

Secondly, is this even possible? As in, would the distance have to be so short that we'd get swallowed up before the black holes collided? And, if we were to feel such gravitational waves, what would they feel like?

EDIT: Let's say that the black holes have the same mass as Earth, giving their event horizons a radius of about 9 mm (the size of a dime). So two Earth-sized black holes that are stuck in an orbit finally collapse is the scenario here.

• What exactly do you mean by "feel"? Like your human senses would detect distortions in space-time? You also can't just arbitrarily pick a mass and size of a black hole. A $2\:M_\odot$ black hole would have a radius of about $6\:km$, significantly smaller than Earth. – zephyr Jun 24 '16 at 14:01
• Yes, I mean "feel" as in, you and I would be able to tell that gravity was being distorted. And thanks for pointing out that mass-radius inconsistency, I'll make an edit @zephyr – Cameron Payton Jun 24 '16 at 14:04
• Just for a reference, your change to require that the black holes be Earth-sized implies a mass of $\sim1000\:M_\odot$ for each black hole. – zephyr Jun 24 '16 at 14:11
• I realized that and made a second edit to give them the mass of the Earth, giving them the radius of a dime – Cameron Payton Jun 24 '16 at 14:14
• Cross-posted from Worldbuilding. Please don't do this. – HDE 226868 Jun 24 '16 at 21:53

It's a bit hard to answer this question as there are some ambiguities and certain parameters of the setup aren't defined, but I can try to give you a general idea.

If we were to feel such gravitational waves, what would they feel like?

I'll start here as we need to have some well established meaning for how we can feel a wave before calculating what conditions produce such a wave. However, this is the ambiguous part. A gravitational wave passing over us would effectively just be felt as a differential gravitational field, which can induce spaghettificiation as Andy points out. One side of your body might feel a stronger pull than the other being that one side is experiencing slightly different gravity than the other. But I have to say that you'd need extreme waves to have this effect. The frequency of the waves would have to be extremely high for the wavelength to be on the order of the human body's size. If the wavelengths are much longer than your body, the change, even from a powerful gravitational wave, would probably be too slow for you to detect.

I'll note that the frequency of the waves detected by LIGO were about $250\:Hz$ for the first detection (at the peak) and $450\:Hz$ for the second detection (again at the peak). That implies a wavelength of $\sim600\:km$ at the shortest. At these extreme wavelengths, the gravity differential is negligible. You'd need wavelengths on the order of a meter which implies frequencies of $10^9-10^{10}\:Hz$ which is right near the peak of astrophysical possibilities. So already your proposal of Earth-sized black holes is out. There's no way they can produce these extreme waves.

But now we run into another problem. In order to make the wavelengths on the order of the human body such that you can actually feel the differential, you've made the frequency extremely high. Could your body even notice gravitational changes happening in $10^{-10}\:sec$? I doubt it. That'd be like me blasting you with hot air, then cold, but flipping back and forth between the two at a rate of 10 billion times a second. I have a feeling everything would just cancel out and you wouldn't really feel anything, but it's hard to say.

How close would a black hole collision have to be for us to feel the gravitational waves pass through Earth?

Let's ignore the issue of whether you'd ever feel a wave and look at the main question. I'll just consider raw energy in the wave and hope that somehow equates to feeling it. The power output of two orbiting bodies of mass $M$ (assuming the same mass for simplicity), and orbiting at a radius of $r$ is given by

$$P_{grav} = \frac{32}{5}\frac{G^4}{c^5}\frac{2M^5}{r^5}$$

Note that $G=6.67\times10^{-11}\:m^3kg^{-1}s^{-2}$ is the gravitational constant and $c=3\times10^8\:m/s$ is the speed of light. Now $P_{grav}$ is the total power radiated out in energy per second in the form of gravitational waves. You of course would not be experiencing this entire power though because this is radiated in all directions. The power is not radiated spherically symmetrically, but for simplicity's sake, let's just assume it is. That means we can say the power you experience given your area $A$ and distance $d$ from the system is given by

$$P_{exp} = P_{grav}\frac{A}{4\pi d^2} = \frac{32}{5}\frac{G^4}{c^5}\frac{2M^5}{r^5}\frac{A}{4\pi d^2}$$

Play with some numbers for this and see what is necessary to have an appreciable power.

I will point out though that at the distance you'd have to be where this power became appreciable, it would be overwhelmed by the tidal forces involved.

So in short, no I don't think you'll ever be able to feel gravitational waves, no matter the circumstance.