Here’s a great question I saw recently. I’m wondering if anyone has a good answer for this:
I just watched a video of two black holes merging. I read about how time slows down near a black holes surface. If that’s true, shouldn’t the black holes slow down as they get close to each other, so that they never actually merge?
I guess this question is not completely well posed as it does not tell in what frame time is being measured. Where is the observer ?
@ecm that’s a great point!
Since we can only observe black hole mergers from earth, it probably makes sense to think about an observer that is very far (millions of light years away!) from the merger.
We know that a test particle falling into a black hole, as observed by a distant observer, would appear to “freeze” near the event horizon, as time seems to slow down neat the event horizon from the perspective of the distant observer.
However, with black hole mergers, this doesn’t seem to happen. Instead, LIGO and Virgo observe black hole mergers, and they merge in finite time as observed from earth. As best I can tell, this means that the “test particle” approximation for time dilation doesn’t work for two black holes merging, and instead we need to understand them in a more holistic context.
I meant to reply to this earlier, but didn’t have the time. Even with a point particle falling into a black hole, the ringdown signal comes primarily from when the particle passes the light ring, not when it goes all the way to the horizon (see, e.g., Cardoso, Franzin, and Pani), so there are no concerns with the merger signal being infinitely redshifted.
Hello fom Monterrey, México. Is my first day here, tks a lot.
What if we take tripartite aproach to the configuration.
First:
m[quote=“jonah, post:1, topic:23, full:true”]
Here’s a great question I saw recently. I’m wondering if anyone has a good answer for this:
I just watched a video of two black holes merging. I read about how time slows down near a black holes surface. If that’s true, shouldn’t the black holes slow down as they get close to each other, so that they never actually merge?
]
A
Del Bel, J. (2025). Hypatia’s Math. Zenodo. Hypatia's Math
For two Schwarzschild black holes with masses M_1, M_2, their merger trajectory is modeled through Kruskal-Szekeres coordinates (u,v,\theta,\phi) with recursive phase cancellation:
\tau_{\text{merge}} = \int \frac{dt}{\sqrt{1 - \frac{2G(M_1 + M_2)}{c^2 r(t)}}}
where r(t) follows null geodesics in the combined spacetime. The apparent time dilation paradox resolves through coordinate singularity cancellation at r = 2G(M_1+M_2)/c^2 using Morse theory on the event horizon cobordism.
Measurement outcomes derive from parallel transport in SU(2) principal bundles over the experimental configuration space \mathcal{C}. For spin-½ systems:
\psi(\theta) = \text{Hol}_{\gamma}(A) \psi_0,\quad A = -i\frac{\sigma_z}{2}d\theta
where \text{Hol}_\gamma(A) is the holonomy along path \gamma in the fiber bundle
Contextuality arises from non-flat connections (F_A = dA + A \wedge A \neq 0).
Open systems evolve on contact manifolds (M,\alpha) with Liouville form \alpha = dS - \sum p_idq^i. The dissipative term becomes:
\iota_X d\alpha = \epsilon \xi(t)
where X is the Reeb vector field and \xi(t) white noise
Attractor basins correspond to Legendrian submanifolds.
The Newtonian phase space extends via:
0 \to \mathbb{R} \to \mathfrak{g}^* \to \text{ham}(T^*Q) \to 0
where \mathfrak{g}^* is the thermodynamic state space with coordinates (S,T,\mu). Entropy S emerges as the Casimir function under Kirillov-Kostant-Souriau bracket
Each Einstein metric g_{\mu\nu} corresponds to a Hopf fibration \mathbb{S}^3 \to \mathbb{S}^2 through Petrov classification:
g = -\frac{\Delta}{r^2}dt^2 + \frac{r^2}{\Delta}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)
with \Delta = r^2 - 2GMr + a^2 encoding the fibration’s twist
Koch-Cantor Information Manifold
Memory evolution occurs on the product space K \times C with Hausdorff measure:
\mu_H(E) = \lim_{\delta \to 0} \inf\left\{\sum (\text{diam } U_i)^D \right\}
where D = \log 4/\log 3. Noether charges become:
Q = \int_{K \times C} j^\mu d\Sigma_\mu^{(D)}
maintained under Hutchinson iterated function systems
The time dilation paradox resolves through:
showing finite energy radiation
The adelic spacetime combines Archimedean and p-adic structures:
\mathbb{A}{\text{ST}} = \prod{p\leq\infty}'\mathbb{Q}_p^{3,1}
with Dirac quantization:
\oint_{C_p} g_{\mu\nu}dx^\mu \otimes dx^\nu \in \mathbb{Z}_p
for all prime loops C_p
Space-time intervals become:
ds^2 = \sum_{i=1}^D \left(\frac{dx_i}{\phi^{n_i}}\right)^2 - c^2\left(\frac{dt}{\phi^{n_t}}\right)^2
where \phi = (1+\sqrt{5})/2 ensures self-similarity under n \to n+1
Translation: Yes, but only from the perspective of an external observer. Time dilation near the event horizon of a black hole is a well-established consequence of general relativity. As two black holes approach each other and merge, their gravitational fields intensify, causing extreme spacetime curvature. For an external observer, the apparent time slows down as matter approaches the event horizons due to this curvature. In theory, it would seem that the black holes never fully merge because the process asymptotically approaches completion as seen from outside.
This is not the case for observers closer to the merging black holes or for the dynamics of spacetime itself. From the perspective of an infalling observer or in terms of spacetime evolution, the merger occurs within finite proper time. Gravitational waves emitted during the merger—detected by instruments like LIGO—confirm that black holes do indeed merge and form a single entity in finite time. These waves provide direct evidence of the merger process and its completion.
While time dilation is a significant factor near black holes, it does not prevent their actual merger; it only affects how this process is perceived externally.
Perceived.
thanks for this perspective, but there are still somethings that i can not fully understand, and i put them here:
first, one event occur or not should have only one definite answer for any observer with proper coordinate, but here in this question, i don’t know how to understand the process as an observe in the earth: the black holes seem never fully merge and suddenly we get signal from the GW.
second, when we choose the observer near the black hole system, we may find that any thing can happen within finite time, so it’s not strange that LIGO can detect this signal, but for the coordinate/metric for the distant observers, there seem to be some odd point in the black hole(maybe something similar to the event horizon in the Schwarzschild metric), the point is whether this odd point is from the chosen metric, or the physical process?
i am new to this filed, and not very well at GR, hope there can be some suggestions for me to understand this field better, thx! 
The key is that gravitational waves are produced throughout the merger process, not just at the final moment of horizon merger. As the black holes spiral inward, they emit gravitational radiation that propagates outward at the speed of light. While the merger itself appears to slow down asymptotically when viewed in Schwarzschild coordinates, the gravitational waves generated during earlier phases of the inspiral reach us without being affected by the extreme time dilation at the horizons.
The crux of the issue: the “odd point” you mention is indeed analogous to the coordinate singularity at the Schwarzschild radius. This is a mathematical artifact rather than a physical reality. In Schwarzschild coordinates, the time component of the metric becomes singular at the event horizon, making it appear as though processes there take infinite time. However, this is purely a coordinate effect.
We can use alternative coordinate systems (like Kruskal-Szekeres or Eddington-Finkelstein coordinates) that remove this coordinate singularity, revealing that the merger completes in finite proper time. The mathematics confirms that infalling observers experience nothing unusual when crossing event horizons (other than death) - spacetime remains smooth there despite the coordinate breakdown.