# Weightless at the center of mass of two merging black holes?

When $$r = 0$$ the Christoffel symbol $$\Gamma_{tt}^r$$ is zero and that means the radial four-acceleration is zero and that means you're weightless.

This example is for a spherically symmetric system, but it is basically saying that because the gravitational attraction cancels out from all directions, you would feel weightless. As an analogy, if we could assume that the mass is distributed so along the binary BH system, that there is a point (center of mass) where the gravitational attraction cancels out from both sides, then at this center of mass point we could calculate the same Christoffel symbol.

Now at the center of mass of the black hole binary system, based on this, if this calculation showed 0 (I have not found such calculation), you could feel weightless. So far so good.

What gives me doubt, is that in a merging binary system, initially the center of mass point could be outside the event horizons.

But eventually as the event horizons join, the center of mass point is incorporated by the joining event horizons, and thus the center of mass point moves inside the event horizon (of the now joint system). At that moment, in the process, there are two singularities, and if I understand correctly, being inside the horizon means having the spatial and temporal dimensions oddly interchanging roles.

One of the consequences of this is a inevitable move of any object toward the singularity, the singularity becomes the future. Could this in any way change the fact of weightlessness at the center of mass of the binary system?

As per the comments, I am asking about whether we could feel weightless at the COM of the BH binary (not moving through that point but being held there with no relative motion to the COM).

Question:

1. Could you feel weightless (is $$\Gamma_{tt}^r$$=0) at the center of mass of the BH binary?
• What is the centre of mass? I mean what is the definition?
– MBN
Jan 6, 2020 at 10:12
• @MBN arxiv.org/pdf/1807.11489.pdf this defines COM for BH binaries. Jan 6, 2020 at 17:19
• Can you say which page/paragraph?
– MBN
Jan 7, 2020 at 9:42
• @MBN 2.1 paragraph Jan 7, 2020 at 15:46
• @safesphere thank you I edited to make it clear, do you think it is OK now? Jan 7, 2020 at 17:57

• thank you do you think that at the center of mass of the binary $\Gamma_{tt}^r$=0? Jan 5, 2020 at 23:42