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111

Nice discovery! The formula for time dilation outside a spherical body is $$\tau = t\sqrt{1-\frac{2GM}{c^2r}}$$ where $\tau$ is the proper time as measured by your object at coordinate radius $r$, $t$ is the time as measured by an observer at infinity, $M$ the mass of the spherical body, and $G$ and $c$ the gravitational constant and the speed of light. ...


17

There seem to be several confusions here. Massive and massless particles behave qualitatively differently, even if the massive particle is traveling very fast. The minimum radius for a stable orbit for a massive particle is $3 r_s$. Circular orbits above this radius are all stable. Massless particles only have circular orbits at the photon sphere, $(3/2) ...


10

Note that this is an incomplete answer. Imagine an object of mass $m$ at a distance $r$ from the centre of a black hole of mass $M$. The gravitational potential energy is $$ U(r)=-G\frac{Mm}{r} $$ This has its highest value when $r=\infty$ and its lowest value when $r$ is at the event horizon of the black hole, i.e. the Schwarzschild radius $$ R = ...


7

Thanks to the hint given by knzhou I figured out that if one wants to give the particle a proper initial velocity of $v_0$, the initial velocity in terms of Schwarzschild coordinates $v_i$ would then be $$\dot{\theta}(0)\cdot r(0) = \frac{{v\perp}_0}{ \color{green}{\sqrt{ 1-v_0^2/c^2}}}$$ for the transversal component, and $$\dot{r}(0) = ...


4

The "theory" describing the black hole interior (in the classical approximation) is the same theory that implies the existence of the black holes, namely the general theory of relativity. As the OP correctly said, the singularity at the event horizon is a coordinate singularity – one that is an artifact of a bad choice of coordinates. When a coordinate ...


3

When black holes form from the collapse of stars, we think they usually produce bright supernovae explosions which release tremendous amounts of energy. Some models suggest that a non-negligible fraction of stars which produce black holes may not produce normal supernovae, these are often called 'failed supernovae', see for example astrobites: Gone Without ...


3

If you believe classical parameters for black holes down to a neutron's scale, then it turns out that a neutron has too much angular momentum to form a black hole, and would be best interpreted as a naked singularity. The reason why is that the radius of a black hole's event horizon is predicted to be: $$M \pm \sqrt{M^{2} - a^{2}}$$ where $M$ is the mass ...


3

Yes. You're exactly right, deviations from no-hair do occur for example after BH mergers --- and hints of the "quasi-normal" mode ("ringdown") were observed in the LIGO detection. The no-hair theorem is constructed for a static, stationary BH (i.e. fully settled). In general, deviations from no-hair (magnetic fields, asymmetry, etc) will be radiated away ...


3

The simplest answer is that you integrate the emission from annuli radiating at various temperatures. Explicitly, the luminosity per wavelength in this approximation is $$L_\lambda = 2 \int_{r_{\rm in}}^{r_{\rm out}} 2 \pi r [\pi B_\lambda(r)] dr$$ where the overall factor of 2 is for the two sides of the disk, $2 \pi r dr$ is the area of each annulus, ...


3

Any observer outside the Schwarzschild radius sees the same thing: matter approaching the Schwarzschild radius at slower and slower (asymptotically zero) speed, forming a thin shell around the event horizon. The matter takes an apparently infinite time to collapse, and infinity is infinitely larger than a large finite the same way it's infinitely larger than ...


3

In addition to what @Jack Mazy said about force due to gravitational attraction, there are also charged (Reissner–Nordström) black holes. I would imagine in theory you could take another charged object and place it near the black hole which would cause Coulomb force on black hole putting it in motion.


2

Hawking, I believe, is referring to a more metaphorical 'hovering'. As light, or anything, approaches the event horizon, it becomes more and more redshifted---it's motion appearing to go slower and slower and slower, approaching zero apparent velocity to an outside observer (approximately) infinitely far away. Anything falling into a BH, thus appears to ...


2

Well, plenty of black holes show jets of highly energetic gas or plasma coming out perpendicular to their rotation plane, seemingly out of the hole. It's just accreting highly energetic gas that's spiraling into the black hole but can not make it in, much of it does but much of it is expelled out. It is not from the hole, just the energetics works out that ...


2

That's just a very poorly written article. Nothing was observed to exit from inside the event horizon. There was a high energy event that launched a flare from near the BH. The event is unexplained, but doesn't obviously violate any known laws of physics.


2

The general relativity model for an electron is the Kerr-Newman solution, that is a charged rotating black holes. Unfortunately the radius of the exterior horizon in geometric units is given by: $$r_{+}=M + \sqrt{M^2-Q^2-P^2-\left(\frac{J}{M}\right)^2}$$ Here P is the magnetic charge, zero in this case. If you convert this relation to standard units ( for ...


2

Torsion is not frame dragging. Torsion is having an anti-symmetric spacetime connection. As you do parallel transport in general relativity (GR) you drag frames the frames roll as they move. With torsion they would twist. The connection is GR is the Christopher symbols, symmetric in the two bottom indices. The torsion is an anti-symmetric tensor. It will ...


1

There's nothing different about a supermassive black hole except that its mass is large. So, if things work for small black holes they work for large ones. It turns out that, by the measure of density given, the density decreases with the mass of the object. This follows immediately from the formula for the Schwarzschild radius, $r=2MG/c^2$: this goes as ...


1

A black hole $m\approx300,000M_{sun}$ at the center of galaxy $NGC 4845$ is thought to have a giant gas planet with the mass no larger than that of a brown star in orbit, though it is currently being drained into the black hole. (And by currently I mean what we are seeing right now, which happened a long time ago). The gas giant is thought to have been ...


1

A black hole in the Schwarzschild case has a radius $$ r=\frac{2GM}{c^2} $$ Let us solve for the mass with a radius of about Fermi or $10^{-15}$m, or the radius of a baryon such as a neutron, $$ M = \frac{rc^2}{2G}=\frac{10^{-15}m\times 9\times 10^{16}m^2/s^2}{2\times 6.67\times 10^{-11}Nm^2/kg^2} $$ $$ =6.7\times 10^{11}kg. $$ This is close to a billion ...


1

This is indeed possible! With a risk of overstatement, this fact is extremely important for astrophysics because it turns out that by dipping in and out of the ergogregion (the region between the surface of infinite redshift and the event horizon) one may extract quite a bit of energy and angular momentum from a rotating black hole in a process known as the ...


1

Traversable - Overlapping (actually intersecting) region would not be Traversable even if the gravity at some parts of the region may be zero. For exampple, between earth and moon, gravity will be zero at some point. That does not mean something in that region can go out of earth/moon system. As soon as an observer leaves that region, it either falls towards ...


1

Yes. Black Holes (BH) can grow from accreting anything with energy --- including dark matter (DM). I'm not entirely clear on the second part of your question, but probably the most important thing to keep in mind is that the black hole information paradox is still unresolved. Answering how information is not lost for any type of particle, including DM, ...


1

You state that: there is literally no way to squeeze more information (entropy) into a given volume than that in a black hole occupying that volume But you must keep in mind that the volume occupied by the radiation+BH system is larger than the volume of the black hole by itself. When the black hole initially forms the horizon has a radius $r$ which ...


1

There is a very general theorem that there is no stable configuration of matter that can exist above a threshhold that fundamentally just depends on the mass of the constituent particles that make up the matter (and the interatomic forces in the matter). There is some debate about the exact limit for neutron stars, but it is something like 2.5 - 3 solar ...



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