Is gravitation faster than light?

[CarstenDierks]

As far as I read, gravitation or better: gravitational waves expand with c.

However, particles like photons which are moving along with c are subject to the curvature of space. Regarding a black hole, light cannot escape with c but gravitation can.

Is gravitation thus faster than c?

Carsten

 

[dextercioby]

As far as I read, gravitation or better: gravitational waves expand with c.

However, particels like photons which are moving along with c are subject to the curvature of space. Regarding a black hole, light cannot escape with c but gravitation can.

Is gravitation thus faster than c?

Carsten

The short answer is:NO.What do you mean,"gravitation can (escape a black hole)"...?Gravitation IS a black hole (too)...A singularity in the gravitational field,that is...If u want to,a particular solution to the Einstein equations...

Daniel.

 

[Hurkyl]

The gravitational field outside the black hole is already there; it doesn't need to "escape".

Of course, anything that happens inside the black hole cannot propagate to the outside... so any gravitational waves that are inside the hole never make it out.

The same, incidentally, is true of the electromagnetic field of the hole.

 

[Problem+Solve=Reason]

Hasn't science, so far, proven it be impossible to move faster than c?

------ Life is a Problem... SOLVE IT!

 

[CarstenDierks]

Hi Problem+Solve,

Hasn't science, so far, proven it be impossible to move faster than c?

Not quite. Theoretical physics do not rule out speeds faster than c (see also: Dirac theory). Mass cannot move as fast or faster than c.

Particles faster than c are called tachyons. However, no one has found tachyons so far.

Of course, anything that happens inside the black hole cannot propagate to the outside... so any gravitational waves that are inside the hole never make it out.

Hm, Hurkyl, is it really true that gravitational waves cannot leave a black hole?

Changes in a gravitational field are supposed to be "communicated" by gravitational waves. If gravitational waves (which have not been detected so far) were not able to leave a black hole, nothing outside a black hole would notice changes in the gravitation of the black hole.

Since black holes of different masses are observable, are we not able to conduct that they can send out gravitational waves with this information?

Otherwise, black holes should all show the same mass (= gravitation): the mass which is necessary to initially form a black hole. Afterwards, information on the increase of its mass would not be able to be "communicated" outside the black hole...

Moreover, would there be only a 0-or-1 approach to the escape of gravitational waves from a black hole? A gradual influence on the gravitational waves seems more likely, since gravitational fields and the curvature of space also gradually influence c.

So far I read that the existing theories of gravitation (including SRT, GRT) and EM show explanatory gaps in situations of singularity (big bang, black holes). Is this one of the gaps or which are they?

Carsten

 

[Problem+Solve=Reason]

Alright, thank you... Do you know a good referece page on the Dirac Theory? I understand the basics of it, but want to know more. Again, thankyou...

---- Life is a Problem... SOLVE IT!

 

[CarstenDierks]

Alright, thank you... Do you know a good referece page on the Dirac Theory?

Well, I do not know the profound reference pages but here are some links:

A simple introduction:
http://encyclopedia.thefreedictionary.com/Tachyon

A simple introduction into the equation:
http://www.iscmns.org/iccf11/ppt/FilippovDtachyon.pdf

Not bad at all:
http://www.cbloom.com/physics/2d_dirac.html

More detailed, with coupling of branes:
http://www.iop.org/EJ/abstract/1126-6708/2001/02/002
http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:hep-th/0003122

Have fun reading!

Carsten

 

[Hurkyl]

Hm, Hurkyl, is it really true that gravitational waves cannot leave a black hole?

In GR, I'm quite sure -- otherwise we'd be able to get information from inside the black hole.

The resolution is this: the increase of mass doesn't occur inside the hole -- it occurs on its boundary as massive objects fall from the outside to the inside.

 

[CarstenDierks]

The resolution is this: the increase of mass doesn't occur inside the hole -- it occurs on its boundary as massive objects fall from the outside to the inside.

That is a good point!

The boundary must be the Schwarzschild radius, right?

 

[Hurkyl]

The boundary is the event horizon. It doesn't have to be spherical... (but maybe it quickly changes into a spherical shape? I don't recall for sure)

 

[CarstenDierks]

As far as I know the Schwarzschild radius is the event horizon since the escape speed at this point becomes c.

 

[caribou]

Something I realized a day or two back is that in the sum-over-histories version of quantum theory, if a particle is near a black hole then some of the paths being summed up must cross over the event horizon and come back out again!

These are obviously faster-than-light histories which form part of the sum-over-histories that result in the light-speed-or-less particles that make up the universe.

I knew there were faster-than-light histories but this crossing over the event horizon was just an amusing thing I realized. Thought I'd mention it.

 

[CarstenDierks]

Thanks, Caribou, that is another interesting point. Maybe the particle will not be able to escape the black hole once it has passed the Schwarzschild radius - even if its probability inside the radius was just a fraction above zero. If it interacts with the black hole inside the radius its probability becoms 100% (like interaction with a measuring device).


But still my initial question remains. Maybe I should state it differently:

In the theory of gravitational waves: Are these waves influenced by relativity and the curvature of space (as photons and other particles are) or not?

If yes, I would consider it paradox because gravitation would influence itself.

If no, I would consider time dilation, length contraction and the like to behave differently with gravitational waves as they behave with other particles like photons. (But this would also imply that c is not the proper speed for gravitational waves.)

Does anybody know the answer?

Carsten

 

[caribou]

I just looked up Jim Hartle's book on gravity and noticed something about a definition.

Gravitational waves are "ripples in spacetime curvature" caused by "any mass in nonspherical, nonlinear motion". But a mass in spherical and/or linear motion must still affect other masses.

So I assume this means gravitational waves and gravitons are different things, with the former emitted in many situations and being made up of the latter.

I think gravitational waves from a black hole are like ripples when a stone is moved back and forth in water. The stone doesn't really change in any significant way.

Gravitons would be smeared out quantum particles with wave properties that can make up these ripples when they occur. So a graviton is a gravitational wave in a different sense.

My guess is that with gravitons we'd be talking about them in smeared out orbits outside and inside the black hole's horizon with them being in increasing and increasing numbers closer to the center. And this is where the gravity comes from, with these orbiting gravitons affecting the paths of other gravitons and anything else.

A black hole is a big cloud of gravitions going out from the singularity to the event horizon and beyond? And something which disturbs a black hole causes a whole lot of gravitons to be knocked out of orbit and form a gravitational wave?

I'm speculating well beyond my own level of knowledge now.

 

[selfAdjoint]

If you accept a graviton theory of gravity, a gravity wave would be a coherent stream of gravitons, i.e. their wave functions would be in macroscopic synch.

 

[Leo32]

gravity is curvature

As far as I read, gravitation or better: gravitational waves expand with c.

However, particles like photons which are moving along with c are subject to the curvature of space. Regarding a black hole, light cannot escape with c but gravitation can.

Is gravitation thus faster than c?

Carsten

According to the GRT, curvature is gravity and vice versa.
Gravitational waves should not be imagined like electro magnetic waves, but rather as curvature ripples which move across the 4 dimensional space.

As such, gravity does not escape from a black hole. It is created by the black hole, in this sense that the black hole curves the space around it.

Greetz,
Leo

 

[CarstenDierks]

Thank you caribou, selfAdjoint and Leo!

But I believe I still do not have an answer to my question. I am sorry for bugging you, but maybe I have missed the clue.

Let me put the question differently:

If we think of gravitons instead of gravitational waves, do they move with the speed of c?

And are they influenced by relativity as well as the curvature of space?

Example: Are gravitons from a mass m deflected if they pass nearby the space curvature of another mass M?

Carsten

 

[MiGUi]

As far as I remember, fields are considered to be established when you do a problem. They travel at the speed of light, so if you don't have mass or charge in the infinite, then the fields in the infinite are zero always.

 

[DB]

Leo said it well,

As such, gravity does not escape from a black hole. It is created by the black hole, in this sense that the black hole curves the space around it.

Light cannot escape a black hole because the force of gravity i.e gravitons, are forcing it them back. Gravity is the black hole (though the singularity is mass).

When we say gravity waves travel at c, a good example of this is such: If the sun were to disapear right now, it would take ~8mins for Earth's orbit to be affected (aswell for us to see it). So picture a wave through 4 dimensional spacetime traveling at c. So if a black hole was to disapear, let's say a black hole powering a galaxie, the gravitons would travel at c, so each star would lose its orbit at different moments.

 

[CarstenDierks]

Hi DB,

that is correct. By theory, the observer on such a star (or an observer orbiting it) should notice ("watch") the black hole vanish at the same time his star loses the orbit around it.

As such, gravity does not escape from a black hole. It is created by the black hole, in this sense that the black hole curves the space around it.

Also correct. But are gravitons or gravitational waves the same as curvature of space? The curvature itself is static in the sense of resting in space. The wave (or gravitons) propagate with a speed (c?). So it must be two different effects even if they have the same origin.

But this would imply, that

a) gravitons are either able to propagate at a higher escape speed from the black hole than photons. (However it would mean our observer´s star would lose orbit before the observer notices the black hole vanish.)

or

b) gravitons propagate at c but we do not know how they escape a black hole. (And does the relativity of c apply to gravitons as well: speed, time, length contraction?)


I do not want to be impolite. My insisting on this point might seem like it but it is truly not and I sure do not want it to look like it. I am just a little curious.

Of course, anything that happens inside the black hole cannot propagate to the outside... so any gravitational waves that are inside the hole never make it out.

Hurkyl, is this an accepted theory or rather a personal assumption?

If gravitons or gravitational waves occur from the boundary of a black hole (Schwarzschild radius), why wouldn´t photons do the same? But wouldn´t we see a white hole instead of a black one?


Your curious

Carsten

 

[Problem+Solve=Reason]

Hurkyl, is this an accepted theory or rather a personal assumption?

If gravitons or gravitational waves occur from the boundary of a black hole (Schwarzschild radius), why wouldn´t photons do the same? But wouldn´t we see a white hole instead of a black one?


Your curious

Carsten

Are you purposing that photons would originate from the boundary of the black hole, and then be sucked into the middle therefor making the black hole bright? Plus, I am wondering if the Schwarzschild Radius is the same thing as an event horizon. I am currently reading up on Schwarzschild Radius and should know more soon...
Also, why would gravitons have to escape the black hole when EVERTHING is being sucked into it. Wouldn't the photons and gravitons be travaling "side by side" at the same rate of speed, going towards the black hole. I'm not sure where I am going with that one but hopefully ill find something interesting when I get where I am going.

---- Life is a Problem... SOLVE IT!

 

[CarstenDierks]

Are you purposing that photons would originate from the boundary of the black hole, and then be sucked into the middle therefor making the black hole bright?

Hi Problem+Solve,
No, I was referring to an earlier post by Hurkyl (#8). His suggestion was that gravitons might originate from the boundaries to be able to escape the black hole.

So my reasoning was: Why should the same not apply for photons?

Or the other way around: Black holes are black because they do not emit photons at their boundary. So why should they emit gravitons at that point when capturing additonal mass without emitting photons at the same time?

Carsten

 

[DB]

But are gravitons or gravitational waves the same as curvature of space?

A gravitational wave is packets of gravitons, so we consider gravitational waves to be made up of gravitons. When a star is born, it's mass and density will curve space, by doing so it automatically sends gravitational waves traveling at c. Like throwing a rock in a pond, the second the rock hits the water it "curves' the body of water, with ripples traveling at a certain speed.

But this would imply, that

a) gravitons are either able to propagate at a higher escape speed from the black hole than photons. (However it would mean our observer´s star would lose orbit before the observer notices the black hole vanish.)

or

b) gravitons propagate at c but we do not know how they escape a black hole. (And does the relativity of c apply to gravitons as well: speed, time, length contraction?)

Remeber that a graviton is a very hypothetical particle, it is used mainly in string theory, helping to intertwine both quantum mechanics and relativity.
Gravitons have rest mass zero just as photons, so we now know that they will travel at c and can have similar properties.So don't get confused that gravitons are escaping a black hole, they don't need to, they are lost in space. What I mean is, that when a black hole is created it sends gravitational waves (gravitons) outward, through space, "forever". So it doesn't need to escape a black hole, the creation of a black hole itself sends them outward. You might ask then what is keeping photons from escaping if gravitons are gone. Spacetime curvature.

I've explained this in a different thread

A common anology of spacetime fabric is "space foam" (like those beds where you can leave a hand print) a rubberish fabric. If you were to place a bowling ball in the center of space foam is would sink, curving the fabric along it's circumference. Now if you took a ball with exactly the same mass but with 1/10th the circumference and radius it will sink to the same depth that the bowling ball did and curve the space foam around its circumference. But since the circumference is much smaller, it would lead to a greater curvature of space foam. It would look like a hole. I think you can picture it. So knowing this situation, general relativity tells us that: the greater the curvature of spacetime, the strong the gravitational force. So if we took a marble and placed it on the space foam for both situations (very important that we don't place it with a force or initial velocity because it would fall into "orbit" for a bit) Here's what would happen:

The bowling ball: the marble would roll towards the center of the bowling ball at a moderate accelerated speed.

The denser bowling ball of same mass: the marble would actually roll slower towards the "event horizon" of the hole, though once it passed it, it would fall towards the ball at much faster accelerated speed than our other situation.

Edit: I realize now surface area might be better to say than circumference.

I found this too: http://www.bun.kyoto-u.ac.jp/~suchii/embed.diag2.jpg
So mathematically Einstein proved (in short) that a photon in a black hole cannot escape because the hole is to deep. Though spacetime curvature doesn't exist as an example of a space fabric situation, it is just a way we use to see what going on, math is the true way of understanding gravity.

If gravitons or gravitational waves occur from the boundary of a black hole (Schwarzschild radius), why wouldn´t photons do the same? But wouldn´t we see a white hole instead of a black one?

A bondary of a black hole is the event horizon. Any photon (anything) passed the event horizon can shine to it's hearts content. Pass the event horizon black holes aren't so bad, they power our galaxies, they are key to life. Also whites holes have not been proven to exist, and are very complicated phenomena, breaking laws of thermodynamics.

If you have anymore questions or I didn't cover what you want to know just ask.

 

[DB]

Plus, I am wondering if the Schwarzschild Radius is the same thing as an event horizon. I am currently reading up on Schwarzschild Radius and should know more soon...

Picture a cone with the base upward. The tip of the cone is the singularity, the circumference of the circular base is the event horizon, the radius of that base is simply the schwarzschild radius.

 

[Hurkyl]

No, I was referring to an earlier post by Hurkyl (#8). His suggestion was that gravitons might originate from the boundaries to be able to escape the black hole.

I said nothing about gravitons! Nor about photons! GR is a classical theory, which appears to be inconsistent with current methods of quantization. Similarly, the EM field that one could describe in GR is a classical field, not a quantum field. AFAIK, any talk about photons in GR is really speculative -- "Once we get these things united, things probably should behave like this..."

Or the other way around: Black holes are black because they do not emit photons at their boundary.

No, they're called black because light cannot escape from the inside.

 

[CarstenDierks]

If you accept a graviton theory of gravity, a gravity wave would be a coherent stream of gravitons, i.e. their wave functions would be in macroscopic synch.

Of course we cannot say for sure if graviton particles exist. For string theory and quantum mechanics it would be very important, as I understand.

But can you (or anybody else) tell me, whether in theory gravitational waves (= gravitons) and the curvature of space (= gravitation, gravitational force) are equal?

Carsten

 

[CarstenDierks]

Hi DB,

thanks for the substantial explanations. That helps already to increase my understanding.

If you have anymore questions or I didn't cover what you want to know just ask.

I would like to, if I may.

I am still a little confused when relating all facts.

For me it is difficult to understand why gravitational waves (gravitons) should propagate through spacetime with c and, thus, cause space curvature (if I have understood that correctly).

Difficult because of the following:

(1) If gravitons (gravitational waves) are not at rest in a gravitational field, this would imply to me that mass needs to constantly emit gravitational waves (gravitons) to "replace" those which are "gone".

(2) If mass curves spacetime: Is the curvature once "engraved" in spacetime and "rests" there until a new gravitational wave "updates the information"?

Or:
(3) Is the curvature of spacetime just "newly" evoked by every ripple of a gravitational wave passing by?

(4) Do gravitational waves interfere with each other? Probably yes because gravitational forces and the curvature of space of two objects do add up.

Arising questions:

(5) Is it allowed to conclude out of (4) and (1) that the path of gravitons is not straight but also influenced by the curvature of spacetime (of other objects)? Meaning: Gravitons (gravitational waves) have to travel along our (curved) cosmos as it exists?

(6) Out of (4): What about gravitational forces of 2 objects of identical mass on a 3rd object right in-between the two? Is the gravitational force for the 3rd object zero? But is spacetime not curved at that point due to the sum of the curvature of the first two objects?

(7) Out of (6): So are gravitons (gravitational waves) and the curvature of spacetime really equal?

(8) Out of (2): Is this true for black holes? Do they curve spacetime and the curvature "rests" there because the gravitons (gravitational waves) cannot escape from inside of the black hole? Is the gravitational field of black holes never "updated" by gravitons (gravitational waves)?

(9) Out of (1), (3), (4), (7) and (8): How can black holes capture gravitons (gravitational waves) inside the Schwarzschild radius and, at the same time, emit gravitons (gravitational waves) to curve space and exert gravitational force?

(10) Out of (9): How does quantum and/or string theory explain the speed and escape speed of gravitons (gravitational waves)?


I hope I was able to put everything into the context as it currently occurs to me - and to show where my gaps in understand (relating) it are situated.

Carsten

 

[Louis Cypher]

Gravitons

Vey interesting but arent you forgetting that the graviton has never been detected so all of this is just conjecture, personally I think gravity has no force carrier and we have already unified all the forces; gravity is an effect of mass, there is no mediator, thus it can escape a black hole, as we know even photons probably have mass, all be it extremely small, so gravitons ought not to be able to escape from a black hole either.

Thus my point mass is caused by the atoms themselves not by any mediating force carrier.

 

[CarstenDierks]

Thus my point mass is caused by the atoms themselves not by any mediating force carrier.

Hm. How should one mass particle get the information where the second mass particle is located at? And how is the (gravitational) force enacted which attracts both particles?

Since a theory on gravitational waves and gravitons does exist, it should also provide answers to the questions I am still having. That is what I am looking for.

Carsten

 

[DB]

I'm not an expert, but I'll try to anwser your questions with my best knowledge of general relativity. I hope you like reading...

For me it is difficult to understand why gravitational waves (gravitons) should propagate through spacetime with c and, thus, cause space curvature (if I have understood that correctly).

Yup your're gettin it. I guess you could say that gravitational waves cause space curvature, though you could say vice versa, and as well that gravitational waves are space curvature. They propagate at c because they have rest mass zero as does a photon.

Difficult because of the following:

(1) If gravitons (gravitational waves) are not at rest in a gravitational field, this would imply to me that mass needs to constantly emit gravitational waves (gravitons) to "replace" those which are "gone".

Not exactly so. First let's consider gravitational waves as a force through spacetime. Let's almost forget about graviton particles for they are very hypothetical and are just what make up the gravitational wave. So we don't say gravitation is emited, it's exerted. We know that when space is curved by mass, it will "deforme" all of space, most anologically spacetime fabric.
Like the rock in the pond, when we throw the rock in, ripples will not be continuesly exerted forever, but the ripples that were exerted along the body of water will continue traveling as so until they barely have a wavelength (or are slowed or stopped by more ripples of another force). Once the rock has hit bottom, it's deformation towards the body of water is finished. Now the water will adapt to have this new rock in its pond. Translation: the second a star is formed in space it deformes (curves) spacetime fabric, and by doing so it effects spacetime by sending gravitational waves along space forever. Once the star is formed and balanced in space, it's job is done, and space has adapted to the deformation of this new star. So the star is the rock, the gravitational waves are the ripples and spacetime fabric is the water. So gravitational waves are not constantly emited, it happens once and travels as a wave at the speed of light. Those which are "gone" have already done their job of curving space. They will keeping doing their job forever, but always at further and futher parts of the cosmos. It is now space curvature (created by the wave) to do it's job of creating and exerting gravity. (Think of gravitational waves through the water anology, but don't think of it as what gravity is, because you can't sink to the bottom of spacetime. The bowling ball anology is good for gravity.)

(2) If mass curves spacetime: Is the curvature once "engraved" in spacetime and "rests" there until a new gravitational wave "updates the information"?

Yes. Engraved is a great word. But the altering of a star's gravitational field is usually very minimal because stars are far away from each other, so there is minimal effect. But it happens and I will give an example soon.

Or:
(3) Is the curvature of spacetime just "newly" evoked by every ripple of a gravitational wave passing by?

Ill say no, but I don't exactly understand the question. Once a gravitational wave has passed a certain spot, that spot is officially considered effected by the mass that sent the wave off. The only way that the certain spot can be effected again by the same mass is if the mass or density were to change (of the mass) or if the mass would disapear, "taking it's wave back". What I mean by this is that if the object were to disapear it would send of a wave that would un-deforme space.
And remember that all this doesn't happen at once, the waves have a finite speed "c", and each spot is affected at different times.

(4) Do gravitational waves interfere with each other? Probably yes because gravitational forces and the curvature of space of two objects do add up.

Right. Right. Here's what I said I would get to. In our solar system, planet's orbits are elliptical, they are eccentric. There are debatable reasons for why, some say it's because planets don't orbit the sun, they orbit the common center of mass between the sun and that planet, some say that other gravitational force of other planets effect orbits, mostly Jupiter. And Jupiter is a great example for more understanding. Because it's so big it has a much stronger force of gravity i.e it curves spacetime more. (this is why it has 63 moons and probably more. More objects are likely to fall in orbit around Jupiter because of its strong gravitational force.) So now picture the sun sending a gravitational wave and Jupiter doing the same. The sun and Jupiter (in astronomical terms) are very close to each other. So the two waves sent of by each mass will clash together. Once the clash is finished with, spacetime has adapted to the deformation. It's engraved. Now in the middle of this weird deformation of space (between the sun and Jupiter), we've got planets. The sun obviously has a stronger gravitational force, so the planets orbit the sun, but they are still effected by Jupiter's gravity. Making their orbits eccentric. (This is not completely proven, but today it is said that Jupiter probably has an effect in our solar system.) (And aswell, planet's past Jupiter still feel the effect of both the sun's and Jupiter's gravity) So yes. Gravitational waves interfere with each other.

*The reason planets orbit is around bigger mass is because they start off with velocity.Iif we were to slightly place a planet in space near a star in would be forced towards the star at an accelerated speed (dependent on the mass). "In would fall down the slope". In a black hole, "it would fall into the hole and be stuck".

Arising questions:

(5) Is it allowed to conclude out of (4) and (1) that the path of gravitons is not straight but also influenced by the curvature of spacetime (of other objects)? Meaning: Gravitons (gravitational waves) have to travel along our (curved) cosmos as it exists?

Ya, because all large object's are symmetric. They are all completely 3-Dimensional, and will exert gravitational waves (gravitons) in every direction of the cosmos.

(6) Out of (4): What about gravitational forces of 2 objects of identical mass on a 3rd object right in-between the two? Is the gravitational force for the 3rd object zero? But is spacetime not curved at that point due to the sum of the curvature of the first two objects?

Let's make sure that the objects have the same density also.
This is a complicated question and is being studied today by trying to understand what's going on in binary star systems and star clusters. But I'll say this, because the question is answerable. As long as the objects are sperated by inches, centimeters, millimeters, they will both exerted the same gravitational wave (because they are the same mass and density). But when these waves "clash" together they will have a different overall curvature of space. Possible equal to the sum of each of their gravitational forces, but not exactly sure. But if we put an object in the middle I'm stumped. One would have to study the motion of this 3rd object a lot to understand what kind of pattern (orbit) it would follow, and why is that so. It's being studied as we speak. There are a lot of experts on this site that might have a basic answer, but it is a complicated matter.

(7) Out of (6): So are gravitons (gravitational waves) and the curvature of spacetime really equal?

Let's put it this way. Gravitational waves (gravitons) curve spacetime at equal magnitudes.

(8) Out of (2): Is this true for black holes? Do they curve spacetime and the curvature "rests" there because the gravitons (gravitational waves) cannot escape from inside of the black hole? Is the gravitational field of black holes never "updated" by gravitons (gravitational waves)?

None of this happens. The curve rests with every object in space wherther it's a black hole or pluto. The same as any object in space, once it's there, it's there with the speed of light, then it's engraved in space. Your question does makes sense though. But the let's say a black hole gained mass. The second it did so it would curve space more, in all directions. The reason the gravitational waves can "escape" the "hole" (the strong force of gravity) is because they travel along space curvature itself. Gravitons aren't exerted by the mass, they are exerted by the change in mass. Gravitons don't come out of the singularity of a black hole. They come from around.

One slight change is space curvature will effect the next and so on. So picture a gravitational wave in a black hole, it will travel in all directions around the singularity. This wave will curve this spot, that spot will curve the spot right after it and so on. It's one big wave. Not a particle. (String theory is the real use of the graviton.) So as one spot in space is affected (more curved), at the speed of light the next will be affected (curved). Leading to a spacetime wave climbing up the hole, climbing up the space fabric. The spacetime (gravitational) wave created the curve in the first place. You can't say that once the hole has changed (is deformed) the hole itself won't let it. It's already been changed. (The change is due to the change in mass)

(9) Out of (1), (3), (4), (7) and (8): How can black holes capture gravitons (gravitational waves) inside the Schwarzschild radius and, at the same time, emit gravitons (gravitational waves) to curve space and exert gravitational force?

Think I covered this. I also think you mean the event horizon not the Schwarzschild radius, which is where nothing can escape. Gravitational waves (packets of gravitons) don't have to escape the event horizon, "they go around it". In other words they travel along the curve of a black hole as a wave. They are what put the event horizon there in the first place.

(10) Out of (9): How does quantum and/or string theory explain the speed and escape speed of gravitons (gravitational waves)?

The speed of a graviton is that of light: "c". Quantum and string theory, uses the graviton's properties and finds that they match those to supersymmetric strings. Example: they both have spin-2. The graviton is a way to incorperated gravity into particle physics and string theory.

Carsten I think your questions are very valid and I understand their meaning. I hope I was able to cover them, and it is a difficult task to both ask, answer and understand what is going on in spacetime curvature of a black hole. Hope I helped