Gravity waves finally detected, Einstein proven right

[Vorpal]

Hum. (36^3+29^3)^0,333333≃41,4 min solar masses.

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Area scales with the square of the mass, so the naive lower limit would be 46 solar masses or so for the merged black hole. But angular momentum can shrink the area of a black hole by up to a factor of two, so the areas of the initial and final black holes can deviate significantly from such a straight mass comparison.

As for the non-spinning naked singularities and cosmic censorship goes, I'm reminded of the xkcd what-if of turning moon into electrons.
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1​For a sphere of electrons with electrons gathered on rQ, this can probably be relatively trivially calculated (I'd guess n*(q*n*q/r) off the top of my head).

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Yeah, ignoring general relativity, it's basically charge (Ne) separated by r from charge (Ne), i.e. itself, so (Ne)²/r up to Coulomb's constant and some factor that depends on the specifics of the distribution in question: 1/2 for a uniform sphere, 3/5 for a uniform ball, etc.

It turns out that if r is the areal radial coordinate, this is OK in GTR as well: the external field of a spherically symmetric charge distribution is Reissner–Nordström, and a Reissner–Nordström spacetime of parameters (M,Q) has enclosed mass at r as M - Q²/(2r), where the last term can be interpreted as the mass-energy due to the electric field outside the enclosing sphere. There are two horizons, at r±​ = M±(M²-Q²)½​, so the extremal case is M = Q, i.e. the degenerate case of a single horizon at r = M. (I assume natural units of G = c = 4πε₀ = 1.)

For the extremal RN black hole, half of the total mass due to the electric field outside the black hole, and the other half could be interpreted as a 'bare' mass independent of the electric field contribution. However, this is not the same thing as the sum of the masses of the constituent particles, because that would ignore their gravitational interaction. A Moon's-mass-worth of electrons distantly separated from each other would have a total charge of Q = 11.77 ly, so in principle one could assemble this into a near-extremal RN black hole of total mass M = 23.5 ly (3.00E44 kg). You can't do better because pushing more charge into it would also add more mass.

The distantly-separated requirement underscores an issue of interpretation of this scenario: why did Munroe take N = (moon mass)/(electron mass) ∼ 1053​ electrons in the first place? We're talking about a 'Moon made of electrons', not a diffuse collection of very distant electrons. Well, it's not really wrong because if we're talking about blatant violations of conservation of charge, why not violate conservation of (ADM) mass too, so ok, whatever. But it is somewhat arbitrary what a 'Moon made of electrons' even means. For example, we could alternatively interpret this phrase as a spherically symmetric charged object with the same charge-to-mass ratio as an electron. Or, say, a Moon-sized charged sphere or ball such that there is a Moon's mass of total energy including the electric field (basically the idea behind 'classical electron radius', up to a factor of order 1). To my intuition, that's the most straightforward interpretation of 'Moon made of electrons', in as much as I would guess the intent of the question to try to keep both the mass and size of the Moon the same. But YMMV, whatever.

Then, since rs​ = 2 G(M0​+ (Q^2/rQ​)/c^2)/c^2 and rQ​ = (G/piε0​)0,5​*Q/2c^2, when looking at specific case of rs​=2rQ​, it becomes G(M0​+ 2Q*(piε0​/G)^0,5)/c^2 = (G/piε0​)0,5​*Q/2c^2 →[crat=(G/piε[sub]0[/sub])0,5​]→ GM0​+2*Q/crat=Q*crat/2 → M0​ = Q*(crat/2+2/crat).

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The rs​=2rQ​ condition is for extremal RN black hole, so if you're using rs​ for Schwarzschild radius and rQ​ for charge radius, then in natural units, the critical charge-to-mass ratio is just Q/M = 1. In more common units, it's sqrt(4πε₀G) = 4.900E-22 e/me​ = 8.617E-11 C/kg.

 

[tzar1990]

While we're talking about charged black holes anyways, I was wondering: Would said charge change where the black hole's event horizon was from the perspective of a charged particle? That is, assuming we have a black hole with meaningful net charge, would the point of no return be different for a proton and an electron?

If so, would it be theoretically possible to observe the inside of an effective event horizon by observing the behaviours of infalling particles with a charge opposite to that of the black hole (since the event horizon for them would be further out than the event horizon for uncharged particles)?

 

['Lement]

For your first question, I believe that's indeed why there's the ± term in the radius. + for positively charged particles, - for negatively.

As for your second question....I doubt it? Don't think the dual horizons applies to something electrically neutral (like light, maybe?) from the charged particle.

Well, it could be more interesting in case of non-fundamental particle like proton, given it has 1 gluon-​ and 2 gluon+​s, but I'd expect it to behave like proton behaves in strong enough electric field to split in twain: split into two new protons.

 

[Vorpal]

The true event horizon is defined in terms of ideal light rays that can escape to infinity, or more formally in terms of causal (timelike or lightlike) signals—the idea here is that if you can communicate in any way whatsoever with the region far away from the black hole, you're not inside the horizon. So the question of different charges or masses doesn't really apply, or one could say it's vacuously the same for everything because it's defined in this highly idealised way in terms of all possible causal signals.

This definition is completely non-local, and in a dynamic situation, you can't know where the event horizon is exactly unless you know the future, e.g. where the horizon is now depends on not only on what has already fallen into the black hole, but also on what will fall in eventually. As a black hole is fed, the horizon expands to meet the infalling matter.

For stationary spacetimes, we don't have to worry about that, though. In the Reissner–Nordström case, the event horizon is at r = r+​, where r±​ = M±(M²-Q²)½​. The inner horizon at r = r-​ is not, properly speaking, an event horizon (but is not infrequently called such anyway**), but rather a Cauchy horizon. This has a rather technical definition, but one can imagine the particular case like so: suppose you have a collapsing charged spherical shell. Then as it collapses, the external geometry is provably Reissner–Nordström, but when it hits the inner horizon, what happens after that is no longer well-defined by the initial conditions (the charged shell) and the Einstein field equation.

The inner structure of the Reissner–Nordström spacetime is physically quite ill. If you treat it seriously, you'll find that a test particle going inside the inner horizon will be repelled back outside—not the region it came from, but out of a white hole. The spherically symmetric maximal extension of this geometry has an infinite chain of universes (asymptotically flat regions). But since any light rays impinging on the inner horizon will be infinitely blueshifted, we can expect any perturbation whatsoever to chaotically change the inner geometry, so that the innermost part of the conventional Reissner–Nordström geometry has no physical relevance. It's a mathematical artefact of over-idealisation.

** It's ok to call it that, as long as one keeps in mind that when talking about the black hole surface or its area, it's the other sense that's intended.

 

[Vorpal]

Turns out that Stephen Colbert had a pretty cool segment on this:

which included a demonstration of the idea of interferometry.

By the way, a side point from the Einstein-was-right-again perspective that the video also echoes: this is a confirmation of Einstein's general theory of relativity, and though Einstein made a prediction about the gravitational waves in 1916 (which the earlier Hulse–Taylor binary system indirectly confirmed, netting them a Nobel prize in 1993), in 1936 Einstein argued that gravitational waves do not exist. But that paper was rejected as a result of peer review, and sometime after withdrawing it and re-submitting it it a different journal, Einstein realised his mistake. But hey, even Einstein's missteps are usually very interesting.

 

[serra2]

Been a long time, but this is back in the news because several follow-up independent studies on the data are concluding, with at least two confirming.

Article:

Just last month, we told you about a small group of Danish physicists who were casting doubt on the original gravitational wave signal detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO), saying it was an "illusion." The researchers alleged that the collaboration mistook patterns in the noise for a signal. Now Quanta is reporting that two independent analyses have been completed that confirm that detection. This should lay any doubts about the momentous discovery to rest.

"We see no justification for lingering doubts about the discovery of gravitational waves," the authors of one of the papers, Martin Green and John Moffat of the Perimeter Institute for Theoretical Physics, told Quanta. That paper appeared in Physics Letters B in September. A second paper by Alex Nielsen of the Max Planck Institute for Gravitational Physics in Hannover, Germany, and three coauthors, was posted to the physics preprint site arXiv.org last month and is under review by the Journal of Cosmology and Astroparticle Physics.