Time dilation?

[halfeye]

I'm having a problem thinking about time dilation in relation to black holes.

The gravitation at the event horizon of black holes is variable. Time dilation at black holes is due to the gravitational field, however it is assumed that it approaches infinity as it approaches the event horizon, even though the gravity at the event horizon is variable.

[Lvl 2 Expert]

even though the gravity at the event horizon is variable.

Is it? The event Horizon is the point where gravity is strong enough to counteract light escaping right? Now if there's a lot of gas and stuff light speed isn't an exact constant, but it's pretty close. Or they could mean "where the event horizon would be assuming no gas or stuff, which is very close to the real event horizon, even closer if you're falling into a black hole really fast".

Unless I'm missing something, in which case: gimme an infodump!

[Anymage]

C is always constant. In fact, its universal invariance is one of the core tenets of relativity. The speed of light through different mediums can have some interesting effects (https://en.wikipedia.org/wiki/Cherenkov_radiation), but is only tangentially related in this case; you can have trouble getting a message or object out, but you don't get the over-the-top weirdness that you do at and behind an event horizon.

As for the gravitational strength at an event horizon, gravitational strength is how it's defined. Tidal forces depend on the distance from the center of mass, which for sufficiently large black holes can result in negligible tidal differences near the event horizon. That doesn't negate the actual strength of the inwards force. And if my educated layman's guess counts for anything, I'd assume that the time dilation gradient is similarly more gradual; around a true supermassive monster, a clock one kilometer away and a clock two kilometers away should only report the a difference in scale similar to the difference in reported forces.

(This ignores how closer matter will be moving faster, and thus have that whole bundle of relativistic effects. I wouldn't know where to begin mathing that part out.)

[bbgun06]

How would gravity at the event horizon be variable?

[bbgun06]

MinutePhysics on YouTube has an excellent series explaining relativistic effects.

[halfeye]

Is it? The event Horizon is the point where gravity is strong enough to counteract light escaping right?

Unless I'm missing something, in which case: gimme an infodump!

The event horizon of a black hole is of variable size depending on the mass of the hole.

Light cannot escape from the event horizon. Light does not escape because it's path is bent by gravity.


As predicted by general relativity, the presence of a mass deforms spacetime in such a way that the paths taken by particles bend towards the mass.[66] At the event horizon of a black hole, this deformation becomes so strong that there are no paths that lead away from the black hole.[67]

https://en.wikipedia.org/wiki/Black_hole#Event_horizon

When the event horizon is small, gravity is bending light into a tighter curve than when the event horizon is large. Therefore the gravity at the event horizon is less for large event horizons.


Supermassive black holes have properties that distinguish them from lower-mass classifications. First, the average density of a SMBH (defined as the mass of the black hole divided by the volume within its Schwarzschild radius) can be less than the density of water in the case of some SMBHs.[6] This is because the Schwarzschild radius is directly proportional to mass, while density is inversely proportional to the volume. Since the volume of a spherical object (such as the event horizon of a non-rotating black hole) is directly proportional to the cube of the radius, the density of a black hole is inversely proportional to the square of the mass, and thus higher mass black holes have lower average density. In addition, the tidal forces in the vicinity of the event horizon are significantly weaker for massive black holes. As with density, the tidal force on a body at the event horizon is inversely proportional to the square of the mass: a person on the surface of the Earth and one at the event horizon of a 10 million M☉ black hole experience about the same tidal force between their head and feet. Unlike with stellar mass black holes, one would not experience significant tidal force until very deep into the black hole.

https://en.wikipedia.org/wiki/Supermassive_black_hole

[BaronOfHell]

The way I understood it, looking at it as space curves by gravity or light itself gets slowed down by gravity is mathematical equivalent.

Also my apologies that this post doesn't address your original question.

Edit: Also I might be way off here in regard to what you want to be discussed, but e.g. the Earth would also be a black hole if enough of its mass was focused into a sufficient low amount of volume, while the mass would still be that of the Earth. The black hole would of course be smaller than the volume of our planet.

[Anymage]

Tidal force != gravity. It's the difference between how strongly pulled the top vs. the bottom is. If the whole system felt a strong but near uniform pull, the tidal forces would be near zero.

The gravity difference at two points a meter apart near the horizon of a supermassive black hole is likely to be small, because the center of mass there is a good ways off. Your head and feet are unlikely to feel much time difference, the same way that they won't feel enough force differential to spaghettify you. You'll still be pulled in really fast.

[factotum]

Tidal force != gravity. It's the difference between how strongly pulled the top vs. the bottom is. If the whole system felt a strong but near uniform pull, the tidal forces would be near zero.

This. The actual gravitational pull at the event horizon will be the same for all black holes, no matter their mass. The *tidal* forces will vary, but that has nothing to do with time dilation.

[gomipile]

I'm having a problem thinking about time dilation in relation to black holes.

The gravitation at the event horizon of black holes is variable. Time dilation at black holes is due to the gravitational field, however it is assumed that it approaches infinity as it approaches the event horizon, even though the gravity at the event horizon is variable.

You seem to be conflating two different concepts here. One is the renormalized surface gravity, which is a scalar that numerically "acts like" and has units of acceleration. The other is the gravitational field, which is the tensorial differential geometry at a point on a spacetime manifold.

While one can define a renormalized surface gravity for a black hole, it doesn't physically act like an acceleration of a finite value at and within the event horizon. The event horizon is the border within which the geometry of the gravitational field is such that the radial direction is timelike instead of spacelike.

Now that that's out of the way, did you have a question? I didn't see a question in your OP.

[halfeye]

The way I understood it, looking at it as space curves by gravity or light itself gets slowed down by gravity is mathematical equivalent.

I don't think that really works, light has (allegedly) a constant speed in a vacuum (which the area around a black hole should be because any matter there was "eaten"), what happens with gravity is a doppler shift in freqency, but the speed stays constant.


Also my apologies that this post doesn't address your original question.

Don't worry, the thread as a whole hasn't been anywhere near there, you're not alone.


Edit: Also I might be way off here in regard to what you want to be discussed, but e.g. the Earth would also be a black hole if enough of its mass was focused into a sufficient low amount of volume, while the mass would still be that of the Earth. The black hole would of course be smaller than the volume of our planet.

Yeah, much smaller. However, if a crust could be built at the current Earth radius from the hole, we probably couldn't tell the difference without seismology and vulcanology.


Tidal force != gravity. It's the difference between how strongly pulled the top vs. the bottom is. If the whole system felt a strong but near uniform pull, the tidal forces would be near zero.

The gravity at two points a meter apart near the horizon of a supermassive black hole is likely to be small, because the center of mass there is a good ways off. Your head and feet are unlikely to feel much time difference, the same way that they won't feel enough force differential to spaghettify you. You'll still be pulled in really fast.

This idea is a problem in my view. If the force at the event horizon is the same, why are some event horizons bigger than others?

That might seem to be explained by the different masses, but if it depends on that, why does light escape near (1.5 event horizon radii) the event horizons of big holes? if the gravitational force was the same as for small holes, why isn't light captured much further out for bigger holes?


You seem to be conflating two different concepts here. One is the renormalized surface gravity, which is a scalar that numerically "acts like" and has units of acceleration. The other is the gravitational field, which is the tensorial differential geometry at a point on a spacetime manifold.

While one can define a renormalized surface gravity for a black hole, it doesn't physically act like an acceleration of a finite value at and within the event horizon. The event horizon is the border within which the geometry of the gravitational field is such that the radial direction is timelike instead of spacelike.

Now that that's out of the way, did you have a question? I didn't see a question in your OP.

Ah, someone who sounds as if the actually understand this stuff, thanks. What I have is confusion, and your explanation doesn't really help much, which is my failing not yours, probably.

A question: Why does the time dilation at the event horizon of black holes not vary like the gravitational attraction does?

[Kato]

I'm also having a hard time getting at what exactly your question is, since your initial post doesn't contain one and none of the other posts helps.

But if I were a guessing man, I'd agree with gomipile's assumption, if I understand it correctly.
Gravity / relativistic effects at the event horizon are identical for all black holes, since it is defined as the point where gravity is just strong enough to 'keep light from getting away' / the curvature is of spacetime is bent so far light cannot escape / similar phrasing of the same effect. Therefore equations for time dilation also result in infinity, but just outside give more 'normal' results depending on distance and size.

So, I'm not sure if that helps any but possibly it could help if you clarify your problem.


Edit: well, your edit helps. But I guess I must be misunderstanding something since it seems you assume black holes of differing mass have the same size :smallconfused:

[halfeye]

I'm also having a hard time getting at what exactly your question is, since your initial post doesn't contain one and none of the other posts helps.

But if I were a guessing man, I'd agree with gomipile's assumption, if I understand it correctly.
Gravity / relativistic effects at the event horizon are identical for all black holes, since it is defined as the point where gravity is just strong enough to 'keep light from getting away' / the curvature is of spacetime is bent so far light cannot escape / similar phrasing of the same effect. Therefore equations for time dilation also result in infinity, but just outside give more 'normal' results depending on distance and size.

So, I'm not sure if that helps any but possibly it could help if you clarify your problem.


Edit: well, your edit helps. But I guess I must be misunderstanding something since it seems you assume black holes of differing mass have the same size :smallconfused:

Which edit? I did a bunch!

I don't assume that at all, it seemed to me to follow from what someone was saying, I was pointing that out since it seems to show their argument to be mistaken.

Obviously this wasn't what anyone meant, but the singularities are all supposed to be infinitely small, so the size of those doesn't vary.

[danzibr]

What blows my mind is mass “increases” the greater the relative speed to whatever’s doing the measuring.

[factotum]

This idea is a problem in my view. If the force at the event horizon is the same, why are some event horizons bigger than others?


Because the mass is different. Let's put it this way: the acceleration due to gravity at Earth's surface is the conventional 9.81 m/s^2, right? Now, let's imagine you somehow replace the Earth with a black hole of the same mass. The acceleration due to gravity at the radius where the surface used to be (e.g. about 6,371km from the singularity) is still 9.81m/s^2 because the same mass exists inside; the new tiny black hole would continue to orbit the Sun as normal, and the Moon would continue to orbit as normal as well, because as far as it's concerned nothing much has changed. This is despite the massive difference in physical size between the two situations (the new Earthly black hole would be the size of a marble).

[halfeye]

Because the mass is different.

So it is. However, that's not IMO helpful to your case. There is a thing called a photon sphere at 1.5 times the radius of the event horizon. That is where light can orbit the black hole, so at that height above the black hole, the orbital speed is exactly c. This height also varies with the mass of the black hole, however its relation to the event horizon is static.

My point is that as a black hole gets bigger, the time it takes light to orbit the hole becomes longer, because the speed of light is constant, and it goes further, because the hole got bigger. So, from that, we can deduce that the curvature of space (i.e. gravity) around the black hole, becomes less at the photon sphere and event horizon, as the mass of the hole increases.


Let's put it this way: the acceleration due to gravity at Earth's surface is the conventional 9.81 m/s^2, right? Now, let's imagine you somehow replace the Earth with a black hole of the same mass. The acceleration due to gravity at the radius where the surface used to be (e.g. about 6,371km from the singularity) is still 9.81m/s^2 because the same mass exists inside; the new tiny black hole would continue to orbit the Sun as normal, and the Moon would continue to orbit as normal as well, because as far as it's concerned nothing much has changed. This is despite the massive difference in physical size between the two situations (the new Earthly black hole would be the size of a marble).

I'm not arguing against any of that, but I really don't think it helps your argument at all.

[Anymage]

So it is. However, that's not IMO helpful to your case. There is a thing called a photon sphere at 1.5 times the radius of the event horizon. That is where light can orbit the black hole, so at that height above the black hole, the orbital speed is exactly c. This height also varies with the mass of the black hole, however its relation to the event horizon is static.

My point is that as a black hole gets bigger, the time it takes light to orbit the hole becomes longer, because the speed of light is constant, and it goes further, because the hole got bigger. So, from that, we can deduce that the curvature of space (i.e. gravity) around the black hole, becomes less at the photon sphere and event horizon, as the mass of the hole increases.

I'm really not following you, because both of these are basic newtonian mechanics. Escape velocity for a given mass at a given distance can be computed, and sometimes it spits out a value bigger than C. Required orbital velocities can be likewise computed, and again these can wind up being C or above. Relativity keeps us from reaching these speeds, but even the simplest number crunching can show times when what's allowed in our universe is only just good enough, or sometimes not even that.

As for photon spheres, the photon sphere of a larger black hole will be farther away from the event horizon in any absolute sense. An earth mass black hole will have a radius of ~1cm, and a photon sphere only about .5cm away from that. A supermassive black hole could have a radius of 1 AU, with the photon sphere being half an AU away from that. .5 AU is much bigger than .5cm, so I don't see how this is any problem.

[gomipile]

A question: Why does the time dilation at the event horizon of black holes not vary like the gravitational attraction does?

You're still conflating different mathematical objects and trying to call them the same thing. In a Newtonian two-body problem with a central mass much much larger than the other, yes, you can say that the "gravitational attraction" is a single real number. In general relativity, that is not possible.

In the sense that the relativistic effects on a test mass' trajectory in the direction towards the black hole's center could be called the "gravitational attraction," the "gravitational attraction" is the same at the event horizon for all Schwarzschild black holes.

Some components of the Riemann curvature tensor in Schwarzschild coordinates in the Schwarzschild metric approach zero or infinity as the radial position approaches the Schwarzschild radius. Some other components that don't do that are dependent on angular coordinates, so their effect is "stretched out" along paths that circle around the black hole.

You seem to have also conflated the idea of classical flat-space curvature of the circle of a circular orbit with the idea of the curvature tensor of spacetime at a point on a relativistic orbit. They are not the same thing.

[halfeye]

As for photon spheres, the photon sphere of a larger black hole will be farther away from the event horizon in any absolute sense. An earth mass black hole will have a radius of ~1cm, and a photon sphere only about .5cm away from that. A supermassive black hole could have a radius of 1 AU, with the photon sphere being half an AU away from that. .5 AU is much bigger than .5cm, so I don't see how this is any problem.

It is a problem for the idea that gravity is the same at the event horizon of all black holes, because gravity is an inverse square force.

https://en.wikipedia.org/wiki/Inverse-square_law

If the photon sphere is 1.5 times the distance from the singularity that the event horizon is, then the gravity at the photon sphere is 1/2.25 the strength that gravity is at the event horizon. So if the gravity at the photon sphere varies, the gravity at the event horizon does too.


You're still conflating different mathematical objects and trying to call them the same thing. In a Newtonian two-body problem with a central mass much much larger than the other, yes, you can say that the "gravitational attraction" is a single real number. In general relativity, that is not possible.

In the sense that the relativistic effects on a test mass' trajectory in the direction towards the black hole's center could be called the "gravitational attraction," the "gravitational attraction" is the same at the event horizon for all Schwarzschild black holes.

Some components of the Riemann curvature tensor in Schwarzschild coordinates in the Schwarzschild metric approach zero or infinity as the radial position approaches the Schwarzschild radius. Some other components that don't do that are dependent on angular coordinates, so their effect is "stretched out" along paths that circle around the black hole.

You seem to have also conflated the idea of classical flat-space curvature of the circle of a circular orbit with the idea of the curvature tensor of spacetime at a point on a relativistic orbit. They are not the same thing.

I understand some things, other things I don't, I don't feel I'm getting this, yet at least. I'm sort of happy with 3D Newtonian, going beyond that is difficult.

[gomipile]

It is a problem for the idea that gravity is the same at the event horizon of all black holes, because gravity is an inverse square force.

https://en.wikipedia.org/wiki/Inverse-square_law

If the photon sphere is 1.5 times the distance from the singularity that the event horizon is, then the gravity at the photon sphere is 1/2.25 the strength that gravity is at the event horizon. So if the gravity at the photon sphere varies, the gravity at the event horizon does too.

There is no physical "distance to the singularity." Because of that and the fact that space is not at all flat near a black hole, the inverse square law can not be assumed to be true here. The inverse square law is a feature of flat three dimensional space.

[halfeye]

There is no physical "distance to the singularity." Because of that and the fact that space is not at all flat near a black hole, the inverse square law can not be assumed to be true here. The inverse square law is a feature of flat three dimensional space.

Are you saying that the gravity at the event horizon is the same for all black holes? That is definitely not what I'm understanding from Wikipedia and the like.

[gomipile]

Are you saying that the gravity at the event horizon is the same for all black holes? That is definitely not what I'm understanding from Wikipedia and the like.

I'll give your question a qualified "no," because I answered it in more nuanced detail up-thread. Your asking that question in response to the part of mine you just quoted is a non-sequitur, though. I'm just saying that your use of the inverse square law specifically, and your continued use of Newtonian intuition in general is useless for general relativity problems.

[Anymage]

Are you saying that the gravity at the event horizon is the same for all black holes? That is definitely not what I'm understanding from Wikipedia and the like.

Let's step back a second.

There's a type of black hole, called a schwarzchild black hole, that comes from the same place as the frictionless planes and point masses you see in physics exams. It isn't rotating, isn't charged, isn't near any other significant sources of gravity, and is basically idealized to help you get the basics down. For these you can approximate them, at least from the outside, by assuming a newtonian point mass in the center and everything inside the event horizon is "here there be dragons". Gravity at the event horizon of a schwarzchild black hole is the same no matter the size, because the event horizon is defined by a certain gravitational strength.

Throwing any wrinkles into that complicates the matter. And of course, if you want more details and specifics, you're asking for people to explain special cases which may require them to catch you up to speed on how the special cases work. In extremis, understanding the bleeding edge of black hole physics will require finding a very good school, signing up for physics courses, and then devoting the rest of your life to being an astrophysicist yourself.

So I guess the biggest question is, how much abstraction and handwaving are you okay with in the name of this just being people putzing around on the internet? If you want links to help refine your intuitions I'm sure people can find them, but that'll take a lot of study time. (And might be tricky finding the right learning curve so that you can go from where you are now to understanding some of the deeper equations.) If you want a physicist to answer a specific question, articulate your specific question and accept that the underlying details could very well be "...and if you want to understand fully why, here are the courses you'll need to take first". And if you just want a "works in the most basic case but ignores all the specialist complications that can get thrown in" answer, understand that you'll wind up with a highly abstracted answer too.

[keybounce]

So a question on black holes, time dialation, and the event horizon.

From what I understand (and looking at the post right above me, I might be woefully out of my depth here), time slows down as you approach the horizon, and becomes infinite at the horizon.

But if it actually goes to infinite, how can you "Cross" the horizon and wind up inside? At least, it seems to me that you'd actually wait just outside the horizon until the BH evaporates over "forever".

So what am I misunderstanding?

[Anymage]

It's a perspective thing.

From the outside, infalling matter does seem to get stretched infinitely thin and frozen in time at the event horizon. That's what you'd see if you were floating a safe distance away and saw something else fall in.

From the perspective of something falling in, the path looks more or less normal. There's a lot of wibbly wobbly, timey-wimey relativistic stuff going on in there I'm not sure about after it passes the event horizon and then as it approaches the singularity (I'm pretty content seeing the whole thing as just "here there be dragons"), especially if you try to look at it as a collection of particles instead of a point particle. There is no hard stop before the singularity, though.

[factotum]

I think that the outside observer would see the infalling person frozen at the event horizon, but the image would fade away and disappear fairly rapidly due to doppler effects (the image getting red-shifted due to the black hole's gravity). VSauce did a video years ago about what things look like falling into a black hole, I assume it's still relevant today:

https://www.youtube.com/watch?v=3pAnRKD4raY

[Anymage]

Since this topic is up, might as well ask something I've been wondering myself.

I'm not a point particle, but rather a complicated, interconnected bunch of particles. If I were to kill the thrusters on my spaceship and fall into a supermassive black hole (so that tidal forces near the event horizon would be negligible), would either the strong absolute forces or the oddities of physics inside do bad things to the chemistry/biology keeping me alive, or would everything feel locally fine until I reached the point that tidal forces became significant and I became spaghetti?

I know the vsauce link says "fine until spaghetti", but the vsauce hosts all strike me as big picture types instead of hardcore physicists.

[factotum]

I think the problem there is...we don't know. What happens inside the event horizon is eternally hidden from us for obvious reasons. However, while I'm not an astrophysicist nor do I play one on TV, the only force that's really strong inside a black hole is gravity, and gravity is overall a very weak force compared to even electromagnetic forces, much less the super-strong forces that hold atoms together. I can't see any reason in ths case why the VSauce video would not be correct and that you would quite happily fall inward until the point where tidal forces become strong enough for spaghettification to take place.

[Anymage]

I guess my thinking is that, if the event horizon is defined as the point at which anything inside is causally disconnected from the outside universe, it stands to reason that anything closer to the singularity is likewise causally disconnected from anything farther out. If my heart is a little closer than my brain, blood can't be pumped up and I'm pretty much screwed.

I could very well be wrong. But that's why I'm curious to hear from someone who knows the deeper nitty gritty.

[factotum]

I guess my thinking is that, if the event horizon is defined as the point at which anything inside is causally disconnected from the outside universe, it stands to reason that anything closer to the singularity is likewise causally disconnected from anything farther out.

I'm not really seeing the logic there. "Causally disconnected" happens because the light cones of anything inside the horizon point toward the singularity, so there's no future in which you can somehow get out. It's not some sort of mystical field that disconnects everything higher up from everything lower down, it's an either-or thing--either you're doomed to fall into the centre, or you're not.

[Douglas]

I'm not really seeing the logic there. "Causally disconnected" happens because the light cones of anything inside the horizon point toward the singularity, so there's no future in which you can somehow get out. It's not some sort of mystical field that disconnects everything higher up from everything lower down, it's an either-or thing--either you're doomed to fall into the centre, or you're not.
It's not even that, really, it's more that the basic principle of relativity still applies. If your blood is inside the event horizon, it's true that it can't go outward. It can, however, go inward less quickly than your arteries do, which would work out the same as far as your internal biology is concerned.

[BaronOfHell]

From what I understand (and looking at the post right above me, I might be woefully out of my depth here), time slows down as you approach the horizon, and becomes infinite at the horizon.

But if it actually goes to infinite, how can you "Cross" the horizon and wind up inside? At least, it seems to me that you'd actually wait just outside the horizon until the BH evaporates over "forever".


To continue on this tangent, if it's alright, something I wonder about is if the gravity from an object formed as a flat ring is the same at all points outside the ring as it would be for all points outside a massive disk of the same mass?

If so, wouldn't there be nothing inside a black hole?

If not, how did "the dragons" get inside the hole (from our perspective) if it requires infinite time from our perspective for this to happen?

And when a black hole increases in size (radius and mass), does the density of the matter at the disk increase as more stuff is pushed into a larger circumference away from the black hole center, or does stuff somehow fall beyond the event horizon as the event horizon is pushed further away from the center of the black hole due to the increase in mass?

[gomipile]

To continue on this tangent, if it's alright, something I wonder about is if the gravity from an object formed as a flat ring is the same at all points outside the ring as it would be for all points outside a massive disk of the same mass?


This seems like a Newtonian question. The answer (in Newtonian physics) is no. Rings and discs produce different gravitational fields. Deriving a similar result for general relativity with enough mass to make it clearly non-Newtonian would be pretty difficult, and I don't know what the result would look like.



If so, wouldn't there be nothing inside a black hole?



I feel like you left out some reasoning after your question and before this. I have no idea what you think the connection could be between rings and discs and black holes.

[BaronOfHell]

Thank you for your reply.

My thoughts were that since nothing can ever pass the event horizon (from our perspective), all the mass of the black hole might be at the event horizon.
Then I imagined a black hole as a ring and wondered if it would be possible based on the gravitational pull from a black hole to determine if it is a ring or a disk.

After asking the question I later speculated that my assumption of a black hole being either a disk or a ring could be incorrect, which I take from your reply to be the case (that it was an incorrect assumption).

[Bohandas]

I'm having a problem thinking about time dilation in relation to black holes.

The gravitation at the event horizon of black holes is variable. Time dilation at black holes is due to the gravitational field, however it is assumed that it approaches infinity as it approaches the event horizon, even though the gravity at the event horizon is variable.

It becomes infinitely concentrated at the singularity, not the event horizon. The event horizon has a variable distance from this depending solely on how much matter ypu're infinitely compressing

[Anymage]

It's not even that, really, it's more that the basic principle of relativity still applies. If your blood is inside the event horizon, it's true that it can't go outward. It can, however, go inward less quickly than your arteries do, which would work out the same as far as your internal biology is concerned.

Fair. That solves the immediate question, and I'm sure that others boil down to nobody knowing the exact details because they're untestable.


To continue on this tangent, if it's alright, something I wonder about is if the gravity from an object formed as a flat ring is the same at all points outside the ring as it would be for all points outside a massive disk of the same mass?

If so, wouldn't there be nothing inside a black hole?

If not, how did "the dragons" get inside the hole (from our perspective) if it requires infinite time from our perspective for this to happen?

And when a black hole increases in size (radius and mass), does the density of the matter at the disk increase as more stuff is pushed into a larger circumference away from the black hole center, or does stuff somehow fall beyond the event horizon as the event horizon is pushed further away from the center of the black hole due to the increase in mass?

Moving up from 2D versions to 3D, it's basic high school physics that a spherical shell can be treated as a point mass at the center for anybody outside, and is gravitationally null for anybody inside. Since falling inside means crossing the horizon yourself and all that entails, for a very basic approximation you could treat it as a big spherical shell.

There are relativistic and quantum effects that happen nearby that may or may not be compatible with treating it as the empty spherical shell model, I wouldn't even know where to begin checking those.

[halfeye]

I'll give your question a qualified "no," because I answered it in more nuanced detail up-thread.

I didn't understand that answer or notice the nuance, sorry.


Your asking that question in response to the part of mine you just quoted is a non-sequitur, though. I'm just saying that your use of the inverse square law specifically, and your continued use of Newtonian intuition in general is useless for general relativity problems.

It's all I have to use *shrug*, if I don't understand something, I don't/can't remember it.


Let's step back a second.

There's a type of black hole, called a schwarzchild black hole, that comes from the same place as the frictionless planes and point masses you see in physics exams. It isn't rotating, isn't charged, isn't near any other significant sources of gravity, and is basically idealized to help you get the basics down. For these you can approximate them, at least from the outside, by assuming a newtonian point mass in the center and everything inside the event horizon is "here there be dragons". Gravity at the event horizon of a schwarzchild black hole is the same no matter the size, because the event horizon is defined by a certain gravitational strength.

@gomipile: So, if my previous question was limited to Schwarzchild black holes, would that change your reply?

I appreciate that they probably don't actually exist, but supposing they did, would the gravity at the event horizon be constant despite changes in mass/volume/density?

[gomipile]

@gomipile: So, if my previous question was limited to Schwarzchild black holes, would that change your reply?


No. The Schwarzschild metric is still full-on general relativity. It's just one of the simplest cases. The Schwarzschild solution predicts precession of the perihelion of Mercury, for instance. So it still has curved, non-Euclidean space.

The Schwarzschild solution is the general relativity version of a point mass. It's what you get from Einstein's equation if you just account for a point mass, with no angular momentum or anything else.


Basically, the equivalent in Newtonian gravity would be calculating the orbit of a satellite as though the Earth was a simple point mass. In the real world you see effects from the rotation of the Earth, local variations in the Earth's gravity, the Moon and Sun, etc. But you're still using full-on Newtonian physics for calculating the motion of the satellite. Likewise, the Schwarzschild solution still uses the full field equations of general relativity, it just has fewer initial conditions than other, more complex solutions.

[Devils_Advocate]

So if the gravity at the photon sphere varies, the gravity at the event horizon does too.
I.e. (by modus tollens (https://en.wikipedia.org/wiki/Modus_tollens)), if the gravity at the event horizon doesn't vary, the gravity at the photon sphere doesn't either.

And I would expect the gravitational attraction towards the black hole to be the same everywhere on the photon sphere, if every point on the photon sphere is equidistant from the black hole's center of mass, which I assume to be the case. Just like I would expect the force of gravity to be the same everywhere on the event horizon, if every point on it is also the same distance from the black hole's center of mass. (That is to say, the same distance from the center of mass as every other point on the event horizon. Not the same distance from the center of mass as the points on the photon sphere. That's clear, right?)

With all of the relevant caveats that we're talking about an idealized case.

So it's still not clear where you think the problem is.

Do you think that some explanation of how something works is incompatible with some other explanation of how something works? If so, could you tell us which explanations you think contradict each other, and how?

[gomipile]

I.e. (by modus tollens (https://en.wikipedia.org/wiki/Modus_tollens)), if the gravity at the event horizon doesn't vary, the gravity at the photon sphere doesn't either.

And I would expect the gravitational attraction towards the black hole to be the same everywhere on the photon sphere, if every point on the photon sphere is equidistant from the black hole's center of mass, which I assume to be the case. Just like I would expect the force of gravity to be the same everywhere on the event horizon, if every point on it is also the same distance from the black hole's center of mass. (That is to say, the same distance from the center of mass as every other point on the event horizon. Not the same distance from the center of mass as the points on the photon sphere. That's clear, right?)

With all of the relevant caveats that we're talking about an idealized case.

So it's still not clear where you think the problem is.

Do you think that some explanation of how something works is incompatible with some other explanation of how something works? If so, could you tell us which explanations you think contradict each other, and how?

Halfeye has been using that phrase structure to mean "varies with the mass of the black hole" not "varies with position around a black hole of a given constant mass."

[halfeye]

I.e. (by modus tollens (https://en.wikipedia.org/wiki/Modus_tollens)), if the gravity at the event horizon doesn't vary, the gravity at the photon sphere doesn't either.

I did prove that the gravity at the photon sphere varies with the mass of the black hole. That's why I'm not getting the gravity at the event horizon being constant despite the variation in masses of various black holes.


And I would expect the gravitational attraction towards the black hole to be the same everywhere on the photon sphere, if every point on the photon sphere is equidistant from the black hole's center of mass, which I assume to be the case. Just like I would expect the force of gravity to be the same everywhere on the event horizon, if every point on it is also the same distance from the black hole's center of mass. (That is to say, the same distance from the center of mass as every other point on the event horizon. Not the same distance from the center of mass as the points on the photon sphere. That's clear, right?)

Any particular schwartchild black hole's photon sphere has a uniform radius yes. Different photon spheres have radii that will be different if the masses of the respective black holes are different.


With all of the relevant caveats that we're talking about an idealized case.

So it's still not clear where you think the problem is.

Do you think that some explanation of how something works is incompatible with some other explanation of how something works? If so, could you tell us which explanations you think contradict each other, and how?

Right, troll, I forgot for a minute. Thanks for resurrecting the thread before it died.

[Douglas]

I did prove that the gravity at the photon sphere varies with the mass of the black hole. That's why I'm not getting the gravity at the event horizon being constant despite the variation in masses of various black holes.
Yes, gravity at the photon sphere varies with the black hole's mass, and yes the ratio of the photon sphere's radius and the event horizon's radius is constant. And yes, if inverse square held universally true, that would imply that gravity at the event horizon varies with the black hole's mass.

However, inverse square is an approximation. For most objects, even up to stellar masses, it is a very very good approximation, but it is still an approximation. Close to a black hole is the one major area where it's a bad approximation. At the event horizon itself, inverse square outright breaks and is completely wrong.