If you think that's gross, the equation describing a rotating black hole is like a summoning spell for Cthulhu.   

But anyway yes, you're thinking about it correctly.   If you fall into a black hole inside a box, physics in your box behaves just like physics anywhere else (in zero gee), at least as long as the black hole is large enough, and your box is small enough, and until you get too close to the singularity (or the inner event horizon if it's a spinning black hole).  In either case it's a pretty surreal experience if you look outside that box, as you say.

[youtube]HuCJ8s_xMnI[/youtube]

^This video isn't fully accurate in terms of portraying what this would actually look like, but it's still pretty good and better than almost anything else that I'm aware of.  It's also using the equations for an electrically charged black hole rather than a spinning one, because the math is easier, but it has many of the same features for what the interior is like.

If you think that's gross, the equation describing a rotating black hole is like a summoning spell for Cthulhu.   

Well, I got one that definitely beats that one for length. I tried to factor the ellipse equation. 

This is more of a math question than an astronomy question, but whatever:
Does anyone know how to calculate the position in an orbit per time for a body in an elliptical orbit? I've been trying to figure this out for a while, and my efforts to find the answer have led to ↑that↑ ginormous equation up there.

I have no idea how to, but these online calculators look useful.
http://www.1728.org/ellipse.htm

Calculates the ecliptic longitude and latitude, right ascension, celestial declination, and distance of the planets from Earth.
http://keisan.casio.com/exec/system/1224748262

I'm not hopeful for those calculators, because neither seem to be in depth on the body's position in space and time. But I might be on to something, because if I can figure out how to make the radius of a circular orbit oscillate in the right way, it'll be elliptical. This is good, because I have a good grip on circular motion.

Nah, we can get very far below the event horizon before our knowledge of physics starts breaking. In terms of space-time curvature, things break at about 10^13 per centimeter, which is *very strong* and occurs extremely close to the singularity -- like within the radius of an atom!  For comparison, the space-time curvature at the surface of the Earth is about 10^-18 per centimeter, or a radius of curvature of about a light year.

A caveat to that however is that this assumes an idealized, non-rotating black hole.  In nature, black holes rotate, and that really messes things up in the deep interior, around the inner event horizon.  So for real black holes our knowledge of physics is still good within the outer event horizon, but breaks down farther out from the singularity than it does for a non-rotating black hole.

That's why I simply adore ring singularities  It would be amazing if our universe was actually inside a rotating black hole with a ring singularity.

They're very neat as mathematical entities, but that's all they are.  Ring singularities do not exist in nature, but are simply consequences of assuming the mass of a rotating black hole exists at a point to begin with.  That assumption leads to predictions that contradicts itself, and the true nature of them requires understanding the details of the collapse.

Mr. Missed Her, I'm afraid this may be disappointing, but simply using the equation for an ellipse will not help you determine the position as a function of time for an orbit, because the speed is not constant but rather must obey the vis viva equation.  In fact, there is no algebraic expression which will do it.  The solutions to the equation of motion for a particle in a gravitational well are transcendental, meaning they cannot be written as a finite combination of elementary functions.  So this is a fairly advanced problem, and a topic of quite some length in celestial mechanics.

To give a hint to the scope of it, I'll briefly overview a standard approach, which is to move to polar coordinates (problems involving radial forces become much easier to analyze in polar coordinates), and solve for the radius as a function of polar angle.  The relation between radius and time for a particle in a gravitational field is given by a differential equation:

, 

where the first (negative) term is the gravitational force, the second term is the centrifugal force, l is the angular momentum and [tex]\mu[/tex] is the reduced mass.  Using the substitution u=1/r and transforming time into polar angle (and skipping all the steps therein), we obtain



which is a differential equation which is much more readily solvable.  Transforming the solution back to r(ϕ),



I won't prove this, but it turns out that this ε is in fact the eccentricity of an ellipse, or a conic section in general.  So the orbit is a conic section, and we can determine its exact shape in terms of the masses, gravitational constant, eccentricity, and angular momentum.

Finally, to bring time back into the picture, we relate the angle (or "mean anomaly") with respect to perihelion, back to the other parameters and initial conditions, and essentially are solving r(ϕ(t)).

So, hopefully that gives some intuition to the problem and how to approach it, though my explanation certainly is not thorough enough to be workable.  If this is something you'd like to pursue more deeply or apply to a project, there are some good resources available out there which can help guide you through it.

They're very neat as mathematical entities, but that's all they are.  Ring singularities do not exist in nature, but are simply consequences of assuming the mass of a rotating black hole exists at a point to begin with.  That assumption leads to predictions that contradicts itself, and the true nature of them requires understanding the details of the collapse.

Yes, I love math too!  Which is why I was so puzzled by it being called "gross."  I mean raw sewage is gross (or sewage of any kind)- but math?!  No!
How long will it be before we understand the details of the collapse and rotating black holes do you think? Do we need a workable theory of quantum gravity first (perhaps that will also settle the question of black hole cosmology too.)

I saw you were using transcendental math up there, what's your opinion of Stephen Hawking using Imaginary Time to eliminate the Big Bang singularity problem?  This would also be workable for black holes.

Mr. Missed Her, I'm afraid this may be disappointing, but simply using the equation for an ellipse will not help you determine the position as a function of time for an orbit, because the speed is not constant but rather must obey the vis viva equation.

Yeah, I figured that out. My main problem is figuring out how to differentiate between the focus with the influencing body and the other focus.

Watsisname wrote:
In fact, there is no algebraic expression which will do it.  The solutions to the equation of motion for a particle in a gravitational well are transcendental, meaning they cannot be written as a finite combination of elementary functions.  So this is a fairly advanced problem, and a topic of quite some length in celestial mechanics.

Dang. I suppose it can still be written finitely with non-elementary functions? (Whatever the distinction is.)

Edit: Also, it would be useful to figure out how to convert (r,θ) points into (x,y) points. The graphing program I use doesn't have a way to directly input polar coordinates.

Yes! Yes! YES! I HAVE FOUND IT!!!! I HAVE IT YES AHAH HAHAHA [maniacal laughing]
[tex](\frac{r}{1+ε \cos (2 \pi ft + 2 \pi fa)} \sin (2 \pi ft)+h, \frac{r}{1+ε \cos (2 \pi ft + 2 \pi fa)+k)} \cos(2 \pi ft)[/tex]

I realized that my point for circular rotation was a polar coordinate in disguise. With this point: [tex](r \cos t,r \sin t)[/tex], input r to r and Θ to t, you've just converted your polar point to a normal point. So I replaced r with that equation at the end of Watsisname's post, and I had an elliptical orbit. I'll have to refine that big equation* up there, but it should be able to create an elliptical orbit with any position, size, and speed. And, as a bonus, any para- and hyperbolic orbits.


*HELPHELP ITS AN EQUATION

Yes, I love math too!  Which is why I was so puzzled by it being called "gross."  I mean raw sewage is gross (or sewage of any kind)- but math?!  No!

How long will it be before we understand the details of the collapse and rotating black holes do you think? Do we need a workable theory of quantum gravity first (perhaps that will also settle the question of black hole cosmology too.)

I think theorists are very slowly getting there, but it's an exceptionally difficult mathematics problem made more difficult by the weird property of rotating black holes which causes their interior geometry to depend on what happens in the future as well as in the past.  

You can get a sense for why this is the case just by looking at the Kerr metric as is -- things that fall into it never hit the singularity, but rather cross relativistically with more matter falling in -- including itself.  You end up with three singularites -- the initial one at the center, plus a "past" singularity which fell in after the hole formed but before you, and a "future" singularity which is coming in after you.  Thus the interior of the black hole depends on what happens in the future!  

That property still becomes relevant when studying what should actually happen in a real rotating black hole, not just the Kerr vacuum solution. 
 
Rotating black holes are weird!

A further difficulty is, as you said, what actually happens to the matter distribution as it approaches singularity conditions, or when the space-time curvature becomes very large.  A huge hurdle here is that different researchers have all sorts of ideas on how to approach the problem (how to bridge toward a theory of quantum gravitation), but they are enormously difficult to test experimentally.

So for the time being, when asked what really happens in an astrophysical, rotating black hole?

I saw you were using transcendental math up there, what's your opinion of Stephen Hawking using Imaginary Time to eliminate the Big Bang singularity problem?  This would also be workable for black holes.

I'm not too knowledgeable on this, and I'm definitely not as smart as Hawking, so I wouldn't know what to conclude about it.  Sometimes very complex problems may have elegant solutions, but I have a difficult time knowing how that approach would be helpful for this one.  

So for the time being, when asked what really happens in an astrophysical, rotating black hole?

That picture nicely sums up how every discussion on black holes has ended in Discord.

The other day gnargenox asked a great question about why with the recent binary neutron star merger, the GRB was observed after the gravitational waves.  What causes the delay?  Unfortunately the forum ate the question and reply, so I'm reposting it.

In short, the GRB and the gravitational waves were not emitted at the same time.  A good analogy is how we observe neutrinos from a core collapse supernova before we observe the visible supernova.  The neutrinos did not go faster than light, but rather it took some extra time for the shockwave to reach the star's surface and cause it to brighten, whereas most of the neutrinos could stream right through.

With a neutron star merger, gravitational waves are emitted as the stars orbit around each other.  Their amplitude and frequency increase as they get closer together, peaking at the moment their surfaces collide.  But the GRB occurs a bit later.  There is some very interesting physics behind the delay, described in more detail in section 5 of this paper.  

The first factor is that after the neutron stars merge, the remnant oscillates for a while before finally collapsing into a black hole.  We can see this in the following simulation which shows the density of the material.  In this simulation it takes just 15 milliseconds to start the collapse, but the timescale depends very sensitively on the dynamics of the merger and the equation of state of the stars.

[youtube]KgUKAbX3x0w[/youtube]

Then it takes some more time for the black hole's jet to blast through the surroundings, and longer still for the resulting fireball to become transparent to gamma rays.  A similar effect occurs in nuclear explosions, where the surrounding air rapidly becomes ionized and opaque to light, thus momentarily shielding the observer from the much hotter conditions inside the fireball.  Visually this appears as a double flash at detonation:

[youtube]C4LqqA4GdsY[/youtube]

Only after the fireball becomes transparent to gamma rays is the GRB finally released, which in this merger happened a little over one second after the peak gravitational wave signal.  The electromagnetic emission also shifts downward in frequency as the nova evolves, from gamma rays to x-ray, UV, optical, IR, and finally radio.  It's truly a multi-spectrum event, and the gravitational waves act as the first indicator.

1. Is it conceivable for a universe to have different rules regarding entropy? I'm trying to imagine one where, instead of energy slowly running down, it stays exactly the same or builds up. What sort of adjustments would be necessary to make this happen?

2. What are some ideas of how sleep would work for organisms on tidally locked planets? I thought about looking at Earth animals that live underground... but the whole 'being underground' thing creates a systemic bias.