I "Testing the black-hole area law with GW150914"

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(I haven't been actively following this line of research... but I think it is possibly interesting reading.
It's been in the science news today.)

"Black Hole Area Law Tested" (synopsis)
https://physics.aps.org/articles/v14/s87

"Testing the Black-Hole Area Law with GW150914"
Maximiliano Isi, Will M. Farr, Matthew Giesler, Mark A. Scheel, and Saul A. Teukolsky
Phys. Rev. Lett. 127, 011103 – Published 1 July 2021
http://dx.doi.org/10.1103/PhysRevLett.127.011103 (preprint: https://arxiv.org/abs/2012.04486 )
We present observational confirmation of Hawking’s black-hole area theorem based on data from GW150914, finding agreement with the prediction with 97% (95%) probability when we model the ringdown including (excluding) overtones of the quadrupolar mode. We obtain this result from a new time-domain analysis of the pre- and postmerger data. We also confirm that the inspiral and ringdown portions of the signal are consistent with the same remnant mass and spin, in agreement with general relativity.
 
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I asked a question here, probably over 15 years ago on entanglement and I appreciated the thoughtful answers I received back then. The intervening years haven't made me any more knowledgeable in physics, so forgive my naïveté ! If a have a piece of paper in an area of high gravity, lets say near a black hole, and I draw a triangle on this paper and 'measure' the angles of the triangle, will they add to 180 degrees? How about if I'm looking at this paper outside of the (reasonable)...
From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
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