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I have read this question:

What is the general relativity explanation for why objects at the center of the Earth are weightless?

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?
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  • $\begingroup$ What is the centre of mass? I mean what is the definition? $\endgroup$ – MBN Jan 6 at 10:12
  • $\begingroup$ @MBN arxiv.org/pdf/1807.11489.pdf this defines COM for BH binaries. $\endgroup$ – Árpád Szendrei Jan 6 at 17:19
  • $\begingroup$ Can you say which page/paragraph? $\endgroup$ – MBN Jan 7 at 9:42
  • $\begingroup$ @MBN 2.1 paragraph $\endgroup$ – Árpád Szendrei Jan 7 at 15:46
  • $\begingroup$ @safesphere thank you I edited to make it clear, do you think it is OK now? $\endgroup$ – Árpád Szendrei Jan 7 at 17:57
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You can be weightless at any point in spacetime. A free-falling observer always experiences weightlessness. This is one way of stating the equivalence principle.

If you want to say this in terms of the Christoffel symbols, then you can always pick coordinates such that the Christoffel symbols all vanish at a given point.

The equivalence principle is a local thing. The inability to escape from a black hole is a global thing. We can see this in Newtonian gravity as well. Escape velocity has to do with one's gravitational potential relative to the potential at a distant point.

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  • $\begingroup$ thank you do you think that at the center of mass of the binary $\Gamma_{tt}^r$=0? $\endgroup$ – Árpád Szendrei Jan 5 at 23:42
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    $\begingroup$ @Árpád As Ben said, "you can always pick coordinates such that the Christoffel symbols all vanish at a given point". $\endgroup$ – PM 2Ring Jan 6 at 0:48
  • $\begingroup$ @PM2Ring thank you, is it the answer the same as if I ask would you feel weightless at the center of mass? $\endgroup$ – Árpád Szendrei Jan 6 at 2:03
  • $\begingroup$ @Árpád You always feel weightless in freefall. And you don't feel gravity standing on the Earth, you feel the normal force from the ground which is preventing you from falling towards the centre of the Earth. OTOH, you can feel tidal forces if they're strong enough, eg near the event horizon of a stellar black hole. $\endgroup$ – PM 2Ring Jan 6 at 2:25
  • $\begingroup$ @PM2Ring thank you so basically at the center of mass you would be in freefall right? $\endgroup$ – Árpád Szendrei Jan 6 at 2:56

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