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Watsisname
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09 Oct 2017 17:35

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

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.



^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.
 
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Mr. Missed Her
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11 Oct 2017 03:56

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

Well, I got one that definitely beats that one for length.
factoring sucks.png

I tried to factor the ellipse equation. 
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11 Oct 2017 16:41

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.
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11 Oct 2017 18:01

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
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12 Oct 2017 06:58

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.
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A-L-E-X
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12 Oct 2017 15:08

Watsisname wrote:
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.
 
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12 Oct 2017 15:37

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.
 
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12 Oct 2017 19:16

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:

Image

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

Image

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

Image

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(ф(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. :)
 
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13 Oct 2017 00:47

Watsisname wrote:
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.
 
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13 Oct 2017 09:57

Watsisname wrote:
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.
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13 Oct 2017 13:17

Yes! Yes! YES! I HAVE FOUND IT!!!! I HAVE IT YES AHAH HAHAHA [maniacal laughing]
[math]

I realized that my point for circular rotation was a polar coordinate in disguise. With this point: [math], 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
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15 Oct 2017 20:45

A-L-E-X wrote:
Source of the post 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!

;)

Image



A-L-E-X wrote:
Source of the post 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?

Image

A-L-E-X wrote:
Source of the post 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.  
 
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15 Oct 2017 23:34

Watsisname wrote:
Source of the post 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.
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19 Oct 2017 16:32

Gravitational waves were detected from a neutron star merger. 11 hours afterward, a nova was seen in visible-light from a location in the sky consistent with the source of the g-waves. X-ray and radio emission were then seen over the next few weeks, and SWIFT detected a gamma-ray burst from it. These observations are the first time we have actually *seen* the source of detected gravitational waves, and confirm the source of fast gamma ray bursts.

Multi-messenger Observations of a Binary Neutron Star Merger

Section 4 deals with the 1.74±0.05 second difference between the detected gravitational waves and the GRB. I'm not really sure I understand exactly what they're talking about but it looks like there are understood mechanisms that would produce the offset, with some arguing for lag times out to even a thousand years by previous models. But ultimately, "Future joint GW-GRB detection should allow disentangling the emission time difference from the relative propagation time, as only the latter is expected to depend on distance."

Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A

My question is, why do g-waves reach us before the visible light is detected? What is this lag time and what causes it?
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Watsisname
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19 Oct 2017 23:09

Gnargenox wrote:
Source of the post My question is, why do g-waves reach us before the visible light is detected? What is this lag time and what causes it?

They were not emitted at the same time.

It's a bit similar to how we receive neutrinos from a core collapse supernova before we see that the star explodes and brightens.  The neutrinos did not travel faster than light, but rather the light was delayed because it takes more time (hours!) for the shockwave to reach the surface of the star and cause it to visibly brighten, whereas the neutrinos from the core could flood straight through.

In the case of the neutron star merger, the gravitational waves were emitted before the GRB occurred.  Gravitational waves are produced by the orbital motion of the neutron stars around each other.  The waves increase in amplitude and frequency as they get closer together, peaking as their surfaces make contact.  The resulting GRB however will not occur at that same instant -- its emission depends on the physics of how the neutron stars actually merge together following that point of contact, and we would expect there to be some delay.  

The peak (electromagnetic) emission also happens first in gamma rays, and then shifts down through the spectrum through x-ray, UV, optical, IR, and finally radio wavelengths.  It's truly a multi-spectrum event, with gravitational waves serving as the first indicator. :)




Dynamical Mass Ejection from Binary Neutron Star Mergers

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