Elementary Question, Black Holes

[Sparki]

Okay, as a heads up, any answer you give are probably going to have to be given in the same manner as you would explain it to a rock.
That said, I've been thinking (I know, famous last words) about black holes. Mass of the black hole is increased when matter is "ingested". I'm aware that at some point past the event horizon space actually begins speeding up. I'm aware, even though I can't read mathematics, referencing relativity equations that black holes end in infinity. Quantum level analysis also yields infinity, albeit a lot more.
Switching gears, it's my understanding that faster than light travel theoretically isn't possible because as anything with mass approaches the speed of light, it's mass increases to the point of infinity. These are likely very simplified approaches to the topics, hence the need for simplified answers
Now to my specific question: If something with mass falls into a black hole and due to the pull of gravity approaches the speed of light, why does matter upon reaching infinite mass or light speed, not create another black hole within the black hole? Links or simplified answer are appreciated and thanks ahead of time

[Noclevername]

Welcome, Sparki!

Now to my specific question: If something with mass falls into a black hole and due to the pull of gravity approaches the speed of light, why does matter upon reaching infinite mass or light speed, not create another black hole within the black hole? Links or simplified answer are appreciated and thanks ahead of time

Well, several things come to my amateur mind; first, where does it say that something falling into a BH gets to lightspeed? For a supermassive black hole, IIRC, something can drift in relatively slowly. It's only the escape velocity of a BH that is lightspeed.

Second, it's only the relative or apparent mass of an object that changes as it approaches lightspeed, not the rest mass.

Third, BHs do not have infinite mass, as we've been able to measure their mass in terms of multiples of Solar mass.

Fourth, we have no idea what actually happens inside an event horizon, as it's impossible to observe the inside of one, but multiple black holes too close together are thought to merge-- violently. As most BHs we've observed absorb matter yet do not constantly display signs of mergence, there probably are not other black holes hiding inside the event horizon.

[Jeff Root]

I'm aware that at some point past the event horizon space
actually begins speeding up.

I don't know where you heard that or exactly what you
mean by "space begins speeding up", but... you're right.
Space is so strongly "curved" or "warped" in and around a
black hole that it acts as if it were flowing into the black
hole. The event horizon is located at the Schwarzschild
radius (R). That is the location where space is flowing
inward at the speed of light, so light at that location,
trying to escape the black hole by moving straight away
from it, is not able to make any progress. It is stuck at
the horizon. If the light is not moving straight away from
the black hole but is instead moving at a lower angle, it
will fall in.

Outside the event horizon, at disances greater than the
Schwarzschild radius, light can escape if it is moving close
enough to vertical. That is, straight away from the black
hole. But if it is moving at too low an angle, again it will
be pulled in. The event horizon is just the place beyond
which *everything* *always* falls in, nomatter what.
Above the event horizon things can escape if they are
moving at the right speed and in the right direction.

At 1.5 times the Schwarzschild radius (1.5 R), a really
interesting thing can theoretically happen with light.
Light which is emitted horizontally at that location could
"orbit" the black hole in a perfect circle. This location is
called "the photon sphere". If light moves the slightest
bit downward from the photon sphere, it is immediately
doomed to fall in. If it moves the slightest bit upward,
it will escape. So it would be very unlikely for any light
to actually orbit in the photon sphere for any length of
time. The "orbit" is just too unstable.

Space is "flowing" inward at the photon sphere just the
right amount so that photons "trying" to travel in straight
lines instead travel in circles.

Farther than three times the Schwarzschild radius (3 R),
any massive object can orbit a black hole just like it can
orbit any other massive body. But closer than 3 R, any
massive object will be pulled in by the "flowing" space.

The location of 3 R is called "the last stable orbit",
because anything closer to the black hole is not in a
stable orbit. Just inside 3 R, the effect is very slight,
so an object would very slowly spiral in. But by the
time it reached the photon sphere at 1.5 R, it would
be falling very rapidly.

A massive object dropped straight down into a black
hole from infinite distance will accelerate all the way,
reaching the speed of light as it crosses the event horizon.
Because light from the object trying to escape the black
hole is slowed by the flowing space, observers watching
the falling object from a distance will see the opposite:
As the object nears the event horizon it will appear to
slow down. The light from it will also redshift. When the
object crosses the event horizon the light will be infinitely
redshifted, so it can no longer reach distant observers.

The speed of the falling object is always less than the
speed of light relative to other objects nearby. It can
be more than the speed of light relative to objects which
are far away from the black hole, but observers there
will never see the object moving so fast. It is the flow
of space, drawing the falling object down, which makes
superluminal speeds possible. But that same flow of
space ensures that the superluminal speeds can never
be observed. Any observer who falls in along with an
object in order to see what happens to it will see it
moving at ordinary speeds relative to him.

I'm aware, even though I can't read mathematics,
referencing relativity equations that black holes end
in infinity.

Again, I don't know where you heard that or what you
mean by it, but you're right.

First, though... "Infinity" is not a thing. It is a property
of something. Redshift of light can be "infinite". Some
other properties might be infinite.

General relativity provides the math which predicts that
all matter falling into a black hole falls in forever, so
that the matter becomes more and more dense, without
limit. You could say that it is tending toward infinite
density. The curvature of space at the center of a black
hole increases without limit, so that you could say it
is tending toward infinite curvature.

Quantum level analysis also yields infinity, albeit a
lot more.

That I'm not so sure about.

Quantum mechanics suggests that particles cannot be
squished to infinite density, or even very close to infinite
density. That is a conflict with what general relativity
predicts. The two theories are each very well proven
over a wide range of energies and densities, but neither
theory can be tested at the energies and densities near
the center of a black hole. The two theories appear to
say opposite things. At least one of them must be
incomplete, or wrong, at extreme energies and densities.
There's no way to say which. Maybe they're both wrong.

So although general relativity predicts a singularity of
infinite density and infinite curvature at the center of a
black hole, we just don't know, and might never know.

Switching gears, it's my understanding that faster than
light travel theoretically isn't possible because as anything
with mass approaches the speed of light, it's mass increases
to the point of infinity.

You can look at it that way. A technically correct description
of why FTL is impossible is beyond me (though it has been
explained here in this forum many times by others). Just
remember that increasing mass is not the actual reason,
but it's a good enough explanation for most purposes.

A few years after Einstein introduced the idea of relativistic
mass, it was realized that the idea is unnecessary and
confusing. It has generally been dropped in favor of other
descriptions of what is observed at high relative speeds.
Essentially, the kinetic energy and momentum of a mass
account for everything that needs to be accounted for.
When a physicist says "mass" nowadays, he or she almost
always means "proper mass", the mass of the object as
measured in its own rest frame, not relativistic mass.

If something with mass falls into a black hole and due
to the pull of gravity approaches the speed of light, why
does matter upon reaching infinite mass or light speed,
not create another black hole within the black hole?

Relativistic mass is merely relative. It is what an
observer, moving relative to the mass, calculates for
the object's mass given the relative speed. If the
observer changes his speed to match the object's speed,
he will measure the object's proper mass, and the extra
apparent mass due to the relative speed vanishes.

Energy is what actually causes the curvature of space,
and therefore causes gravity. Mass is by far the biggest
contributor to the curvature because a little bit of mass
has a huge amount of energy, as shown by that formula
everyone knows. Mass is the most compact form of
energy.

The kinetic energy of an object free-falling into a black
hole is provided by the gravity of the black hole itself,
so the total amount of energy added to the black hole
by the object falling into it is pretty much just that of
the object's proper mass. So the mass of the black
hole only increases by the object's proper mass.

-- Jeff, in Minneapolis

[Sparki]

Thank you for the quick replies. I'm digesting and already have follow ups forming so bear with me

[Disinfo Agent]

Hi. Another amateur here, but perhaps I can add to the other replies. There are several things in your post that are not quite right as far as I know:

Okay, as a heads up, any answer you give are probably going to have to be given in the same manner as you would explain it to a rock.
That said, I've been thinking (I know, famous last words) about black holes. Mass of the black hole is increased when matter is "ingested". I'm aware that at some point past the event horizon space actually begins speeding up. I'm aware, even though I can't read mathematics, referencing relativity equations that black holes end in infinity.

While "space speeding up" may be right, that does not mean that a massive body will reach an infinite velocity as it falls into a black hole. As I understand it, any mass that falls into a black hole will reach the centre in a finite amount of time, from its own point of view.

Quantum level analysis also yields infinity, albeit a lot more.

Actually, we don't know what the right quantum analysis would be for black holes. This remains an open question in physics! (The LHC experiments are an attempt to find some answers to this question, among many others.) The whole classical concept of "black hole" is derived from general relativity only, neglecting quantum effects, and likely just a first approximation to reality.

Switching gears, it's my understanding that faster than light travel theoretically isn't possible because as anything with mass approaches the speed of light, it's mass increases to the point of infinity.

There are two kinds of mass in relativity: the inertial mass increases to infinity, but the rest mass of the object — which more closely matches our everyday understanding of the word "mass" — remains finite. The whole black hole with everything inside it has a finite mass which can be measured. For instance, the supermassive black hole at the centre of the Milky Way has a mass on the order of a million solar masses. But ordinary stellar black holes have only up to a dozen solar masses.

Now to my specific question: If something with mass falls into a black hole and due to the pull of gravity approaches the speed of light, why does matter upon reaching infinite mass or light speed, not create another black hole within the black hole? Links or simplified answer are appreciated and thanks ahead of time

I think you can see by now that your question started from wrong premises, but here are some links I hope you'll find enlightening:

What would happen to me if I fell into a black hole?
Can black holes ever really form?
If you go too fast, do you become a black hole?

[onomatomanic]

[...] here are some links I hope you'll find enlightening:

What would happen to me if I fell into a black hole?
Can black holes ever really form?
If you go too fast, do you become a black hole?

48. H.C. Says:
May 2nd, 2008 at 3:19 am

Still can’t understand how you could add mass to a black hole in finite (external) time. Say a black hole existed 10 billion y ago. Something started falling into it at that time. At this moment, because to me as an external observer it is slowing down, it still hasn’t gone through the event horizon, it is still falling and is not a part of the black hole.

And this will go on forever.

As far as I understood that was the question, and it isn’t answered.

What is the answer to that question? Does a black hole (in the sense of mass enclosed by an event horizon, not necessarily in the sense of having a singularity at its core) "really exist", or is it just the theoretical end-product of a formation process that will take an infinitely long time to run its course, from an outside observer's point of view?

Observationally, the two cases should be indistinguishable, as matter will both contribute to the compact object's gravitational mass and no longer be able to emit energy, regardless of whether it has *actually* crossed an *actual* event horizon or is merely frozen in time just above a not-quite-event-horizon, I'm thinking.

I think I may know the answer, and that it is that, yes, black holes can genuinely and fully form in a finite amount of time, and that there's a flaw in the scenario outlined in the comment above. I'm not going to give my reasoning for the time being, if that's alright, as I'd be more interested to see if others arrive at the same answer and how they make their case.

[Jeff Root]

onomatomanic,

You appear to have the essential answer: observationally,
the effect of the apparent slowing of the fall of an object
free-falling into a black hole, as seen by distant observers,
is tiny. Light coming from the object is redshifted more
and more rapidly as the object nears the event horizon,
and it nears the event horizon more and more rapidly
because it is still accelerating downward. So it redshifts
to invisibility in perhaps a few microseconds. Since light
is composed of photons, a finite number of photons are
emitted or reflected by the object on outward trajectories
before the object has passed the event horizon. So there
will be a final such photon, after which no more photons
are able to escape. A distant observer will see that last
photon, greatly redshifted, something like a hundredth
of a second after the object is seen to get close to the
event horizon. Mathematically, you could calculate that
idealized light waves of unlimited length would still be
reaching you after an unlimited amount of time, but the
actual photons would stop reaching you very quickly.
The time delay does make a difference, but it wouldn't
be much of a difference. You'd need a high-speed
camera to see it.

While the object actually does cross the event horizon,
even if it is seemingly, from the viewpoints of outside
observers, plastered onto the event horizon forever, my
understanding is that it acts to add to the mass of the
black hole uniformly. As the falling object approaches
the singularity, outside observers see the object spread
across the surface of the event horizon so that the
gravitational effect is indistinguishable: The added
mass acts as if it were at the center.

The problem here, as I understand it, is to say when
events happen. I don't think there can be any general
agreement on when events near a black hole happen.
There can be no generally agreed-on "now". Certainly
every event observed happened at some earlier time,
but different observers will always disagreee on how
much earlier that was.

-- Jeff, in Minneapolis

[Disinfo Agent]

What is the answer to that question? Does a black hole (in the sense of mass enclosed by an event horizon, not necessarily in the sense of having a singularity at its core) "really exist", or is it just the theoretical end-product of a formation process that will take an infinitely long time to run its course, from an outside observer's point of view?

Observationally, the two cases should be indistinguishable, as matter will both contribute to the compact object's gravitational mass and no longer be able to emit energy, regardless of whether it has *actually* crossed an *actual* event horizon or is merely frozen in time just above a not-quite-event-horizon, I'm thinking.

I would say that the answer has two parts:

The first is that, as we've all learned to recite, but often forget, in relativity time is relative. The very question "When, if ever, did the black hole actually form?" assumes that there could be a unique, universal answer. But there isn't! The answer depends on the observer's reference frame. Essentially, if you are part of the matter falling into the black hole, then you will get there in finite time; but if you are looking at it from far away - causally disconnected from the black hole - then for you the matter just gets indefinitely close to falling in (indefinitely redshifted, to the point where it fades to black for all intents and purposes).

The second part of the answer is that "if it walks like a duck and quacks like a duck..." From afar, something indefinitely close to forming a black hole is observationally indistinguishable from an actual black hole already formed.

[onomatomanic]

You both make good points, but the answer I was thinking of is somewhat more clear-cut than that: If I'm not mistaken, the perceived paradox is almost solely the result of a flaw in the way in which people imagine the process by which a black hole grows, likely helped along by the vocabulary that's commonly used to describe that process.

Naively, it should work like this: First, a particle physically passes through a static event horizon, and then, that event horizon grows by a miniscule amount. This is often framed in terms of food consumption (the black hole "swallows" the particle, et cetera), and since it does work that way there (we need to move food physically inside our body to grow fatter), it's naturally to imagine it working that way here as well.

But, of course, it doesn't, because the horizon is not actually static. Instead of the particle crossing the horizon, the horizon can move to engulf the particle. And since the horizon isn't a physical object subject to relativistic effects, it's neither frozen in time nor required to move smoothly. As a very simple thought experiment, consider a non-rotating solar-mass black hole. The event horizon is at the Schwarzschild radius of ~3 km. Now take another solar mass of some substance, preferrably something dense and inert like neutronium, arrange it into a spherical shell with a radius of, say, ~10 km, and let the shell contract under the influence of the self-gravity of the combined mass distribution. The naive view says that our shell will asymptotically approach the ~3 km event horizon but never quite get there, according to an observer who remains at the initial distance of ~10 km, because yadda yadda yadda. However, the interesting stage of the contraction actually occurs long before that point. As soon as the shell has contracted to ~6 km, the event horizon will jump outwards to that radius, since the mass distribution as a whole now satisfies the Schwarzschild criterion. So there's a two solar-mass black hole, formed in what appears to be very much finite time to our outside observer, considering that the time dilation due to the black hole is tiny at ~6 km.

Similarly, consider the black hole equivalent of a gun-assembly nuclear weapon. You have a big ball of neutronium sitting there, just short of the Schwarzschild criterion, and you are at a comfortable distance holding a little pellet of neutronium. You let it drop. As it impacts the ball, which will happen with some non-negligible but finite relativistic time-delay due to the not-quite-black-hole nature of the ball, it brings the combined mass to "super-criticality" and voila, an entirely proper event horizon suddenly forms.

That being said, these thought experiments have their own flavour of naivete: The closer the pellet gets to the ball, the closer the combination is to being a black hole, and the larger the time dilation. So in actuality, this process is one of competing divergences: As the distance between the two objects approaches zero, their approach speed, as seen by the outside observer, also approaches zero. However, unlike in the original naive model with the static horizon, this is now a situation in which it is not a priori clear whether that means a finite or an infinite time to contact. To illustrate that, consider the distance x(t) to vary as (t0-t)^3 and as e^(-t) where t is the time as measured by the outside observer. Both cases can be trivially scaled such that x(0)=x0 is our initial distance. In both cases, v(t)=dx/dt->0 as x(t)->0 where v(t) is the approach speed. Nevertheless, in the former case, there is a finite time t0 at which x(t) smoothly attains (and then passes) zero, whereas in the latter case, this only happens asymptotically as t approaches infinity.

One would have to do the maths rigorously to figure out which type of description applies here. I suspect, though, that it is the finite-time one, based on the counter-intuitive fact that a black hole's "density" (mass divided by the volume of a Euclidean sphere with Schwarzschild radius) decreases as mass increases. This would seem to imply that the bigger the black hole gets, the easier it becomes for the event horizon to migrate outwards to encompass infalling material.

If this is complete nonsense, please don't be shy about telling me so.

[noncryptic]

"...Say a black hole existed 10 billion y ago. Something started falling into it at that time. At this moment, because to me as an external observer it is slowing down..."

An object approaching c does not so much "slow down" (and eventually stop) as seen by an outside observer, rather its *acceleration* approaches zero as its speed approaches c. So from an outside perspective the object would be seen 'coasting' past the event horizon instead of "falling".

[Jeff Root]

noncryptic,

The *measured* speed of an object falling into a black
hole depends on what that speed is measured relative to,
and how it is measured. Relative to distant observers
far outside the event horizon, the speed reduces toward
zero as it approaches the event horizon, when measured
against a grid undistorted by the black hole's gravity.

That is...

Suppose there are several objects arranged in a circle
around you. Each object is 100 million kilometers away.
You have tape measures that extend that far and people
at the other ends of the tapes to postion the ends at the
same distance from you as the centers of the objects.
They don't actually place the ends of the tapes at the
centers of the objects, but slightly off to the side.

One of the objects is a black hole. You say that the
center of the black hole is 100 million km away, as
measured by your tape, just like all the other, more
ordinary objects. I'm already done with those other
objects. They were just for comparison with finding
the distance to the black hole.

Now you drop a probe into the black hole and measure
its distance from you against the same tape that you
used to measure the distance to the black hole. You
find that as the probe approaches the event horizon, it
slows down relative to the tape. Light from the probe
redshifts and fades dramatically just as the probe gets
to the event horizon, and the probe simultaneously
comes to a stop and disappears.

You have another assistant take the end of another
tape straight to the black hole. You say goodbye to
him and watch him fall in. He also slows down as he
nears the event horizon, relative to the first tape,
but the tape he is holding keeps reeling out faster
and faster. When you see him redshift and fade out
at the event horizon, the tape is reeling out at nearly
the speed of light. It continues to reel out at that
speed for as long as there is tape left.

By this measure, the speed of the infaller reaches the
speed of light at the event horizon, and the distance
to the event horizon is not measureable.

The black hole's gravity ensures that anything falling
into it keeps accelerating. According to special relativity,
nobody will see the speed reach or exceed the speed
of light. The tape measure you held while your assistant
fell in never moved faster than the speed of light because
it broke somewhere near the event horizon. The far end
was being pulled down faster than the electrical forces
between atoms in the metal of the tape could pull the
tape out of the case you were holding. Almost the whole
length of the tape is pulled apart this way as it falls in.
Only the last little bit survives a while longer because it
isn't pulling in more tape behind it, so it can fall some
distance inside the event horizon before it is pulled apart
by the steep gravitational gradient near the center.

The answer to the person who made the quoted comment
is that the falling object appears to splat against the
event horizon, redshift and fade to invisibility, all in a
fraction of a second. That is for two reasons: Only a
finite number of photons escape the falling object before
it crosses the event horizon, and a photon can only be
detected if the entire photon is detected. There is no
such thing as detection of part of a photon.

-- Jeff, in Minneapolis

[caveman1917]

If this is complete nonsense, please don't be shy about telling me so.

It's not complete nonsense and the jumping horizon has been my answer to this question as well. However upon closer examination this answer is flawed. It is most easily seen by dropping a spherical shell of mass into the black hole. Birkhoff's theorem then states that outside this shell the spacetime is schwarzschild with as the mass at infinity. So to the outside observer the shell of mass approaches zero speed as it goes towards the location of the new horizon, not the old one. The same argument still applies that he will never see it pass into the black hole, since the relevant quantity is the "new" black hole from the start.

Things get even weirder, and i mean seriously weirder . It turns out that there never even is a black hole to the schwarzschild observer. His slices of constant time "wrap around" in the time direction to before the black hole would have formed, so at no point is there a black hole to him. The event horizon as far as he is concerned is the limit of his infinite future, and the inside of the black hole is "beyond his infinite future" so to speak. An infalling particle quite literally in its own finite time travels to the infinite future of the schwarzschild observer and beyond.

And now for the "seriously weirder" part, no we weren't there yet . Since we have the particle travel beyond the infinite future of the schwarzschild observer, it basically travels to the limit of the infinite future as it approaches the event horizon, and then travels infinitely far back in time again as it plunges inside the black hole such that it does indeed increase the mass of the black hole in finite time for the schwarzschild observer. This means that as the particle comes close to the event horizon, it exists in two places at once, both inside the black hole (as it is travelling back from the infinite future) and just outside the black hole as it is travelling towards the infinite future. Since it is at some point both inside the black hole and outside the black hole, our schwarzschild observer can determine that it is inside by noting the gravitational effect of the increased mass of the black hole.

But this duplicate forwards and backwards in time travelling begs the question, can we not have some fun with causality that way? What if our schwarzschild observer waits until he notes the increased mass of the black hole, and then quickly "recalls" the particle before it has a chance to cross the event horizon (after all that will take infinite time to our schwarzschild observer)? It turns out, quite beautifully, that that is impossible. The point where our schwarzschild observer notes the increased mass is also the point at which a light ray sent out after the infalling particle will fail to reach the particle before crossing the event horizon. Or in terms of the schwarzschild observer, where the light ray will fail to catch up with the infaller. So basically the "travelling back from the infinite future" particle only appears at the moment that it becomes impossible to stop the particle from travelling forwards towards the infinite future in the first place, and causality is preserved.

Note that this gives us a way to calculate when (in finite time for the schwarzschild observer) the mass of the black hole increases because of an infalling particle, namely the moment that a photon will not catch up to it anymore.

So actually both points of view are correct here. Jeff is right that the outside observer will never see a particle reach the event horizon (or the "new" event horizon for the added mass), but you are correct that the mass of the black hole will increase in finite time for that observer. The thing here being that the mass increases not because of the particle actually crossing the horizon, but because of its "backwards in time travelling" twin appearing inside the black hole at some finite time.

[caveman1917]

"...Say a black hole existed 10 billion y ago. Something started falling into it at that time. At this moment, because to me as an external observer it is slowing down..."

An object approaching c does not so much "slow down" (and eventually stop) as seen by an outside observer, rather its *acceleration* approaches zero as its speed approaches c. So from an outside perspective the object would be seen 'coasting' past the event horizon instead of "falling".

Its speed goes to zero in schwarzschild coordinates. The "approaching c" business is in GP coordinates, which do not represent actual measurables for any observer.

[onomatomanic]

Many thanks, caveman1917 - that was the most cogent explanation of "serious weirdness" I've read in quite a while!

[noncryptic]

noncryptic,
...

Jeff, sorry but to me your explanation seems to be more an extensive reiteration than an explanation.

It left me wondering though: why is it that when a photon travels at the speed of light it does not appear to have stopped moving, but when something else travels (nearly) that fast it does appear to have (nearly) stopped moving?

Its speed goes to zero in schwarzschild coordinates. The "approaching c" business is in GP coordinates, which do not represent actual measurables for any observer.

Thanks, but i'm not sufficiently familiar with that jargon ("schwarzschild coordinates", "GP coordinates") for that to be a useful statement to me.
Does an object falling into a black hole appear to slow down (rather then appear to slow its acceleration) as it approaches c, or not (as seen by an observer at considerable distance from- and at rest wrt to the bh)?

[Jeff Root]

Jeff, sorry but to me your explanation seems to be
more an extensive reiteration than an explanation.

It seems adequate to me, but I don't know what you
want explained or what you would consider to be an
adequate explanation. I'd very much like to do better
if I can.

You said, in post #10:

An object approaching c does not so much "slow down"
(and eventually stop) as seen by an outside observer,
rather its *acceleration* approaches zero as its speed
approaches c. So from an outside perspective the object
would be seen 'coasting' past the event horizon instead
of "falling".

You appear to be conflating two different things:

- The appearance of an object accelerating to near c

- The appearance of an object falling into a black hole

It is reasonable that you might think the same description
applied to both situations since the object also accelerates
to near c in the second case. But the two very different
situations don't look anything alike.

You correctly described what happens when an object is
accelerated without limit: the speed relative to the observer
does not also increase without limit. Instead, the observed
acceleration (change in speed) decreases as the object's
speed approaches c, while the object's kinetic energy
and momentum balloon.

An object free-falling into a black hole is also accelerated
without limit, and will have a speed of c as it crosses the
event horizon. But it will appear to all outside observers
to slow to a stop at the event horizon.

It is the stretching -- or expansion -- of spacetime close to
the black hole that makes this happen. My description with
the measuring tapes was intended to make this clear to you.

The measuring tape positioned beside the black hole shows
a definite distance from you to the black hole's center. The
tape which goes straight into the black hole shows no definite
distance, and is pulled into the black hole at an accelerating
rate until it is pulled apart by the expansion of spacetime,
somewhere near the event horizon. Exactly where depends
on the strength of the tape, its length, and how it is allowed
to move as it is pulled into the black hole.

Earlier, in post #2, you said:

Well, several things come to my amateur mind; first,
where does it say that something falling into a BH gets
to lightspeed? For a supermassive black hole, IIRC,
something can drift in relatively slowly. It's only the
escape velocity of a BH that is lightspeed.

Just like with ordinary bodies, such as the Earth, an
object free-falling from infinitely far away has the same
speed at a given distance from the body as the escape
speed at that distance.

The only way an object could approach the event horizon
slowly would be to fire a rocket engine downward to slow
its descent. Even an infinitely-powerful rocket could not
hover at the event horizon because the engine's exhaust,
instead of pushing up on the inside of the combustion
chamber, would be pulled down by gravity.

What you correctly remember as being weak at the event
horizon of a supermassive black hole is the gravitational
gradient. An object free-falling into the black hole, not
trying to hover or greatly slow its fall, will not be stretched
or torn apart by tidal forces until it is far below the event
horizon.

The tape measure is stretched and torn apart because you
are holding onto one end of it far above the event horizon,
so there is a very large gravitational gradient between you
and the other end of the tape. A tape made of an ideally
strong material would not stretch or break until the far end
reaches the event horizon and the whole tape is falling at
nearly c relative to you.


According to special relativity, the speed difference
between any two objects cannot be as much as c.
This describes the limit on acceleration that you referred
to in post #10.

According to general relativity, spacetime is stretching
downward toward the center of a black hole. This is the
cause of the apparent freezing of an object at the event
horizon.

Two different phenomena.

Back to your most recent post:

It left me wondering though: why is it that when a photon
travels at the speed of light it does not appear to have
stopped moving, but when something else travels (nearly)
that fast it does appear to have (nearly) stopped moving?

You again appear to be conflating two different things.

The only time something travelling at or near the speed
of light appears to stop moving is when it approaches the
event horizon of a black hole. It appears to stop moving
because it is near the event horizon, not because it is
moving fast.

The locally-measured speed of light is always c.
If you could measure the speed of light falling into a
black hole from your position far above the event horizon,
you would see that it appears to slow to a stop just like
the massive objects falling in. It would not be a local
measurement.

Everything falling into a black hole appears to slow to
a stop at the event horizon, including light, when seen
from a distance, because expanding spacetime near the
event horizon is pulling light downward, inducing longer
and longer light-travel delay times, the closer the light
starts out from the horizon.

Remember that you can only see what light shows you,
and you can only see it when that light reaches you.

-- Jeff, in Minneapolis

[noncryptic]

Earlier, in post #2, you said:

Well, several things come to my amateur mind;...

Actually that was not me, that was Noclevername.
No offense taken (nor intended) but i do know that in its simplest form, 'impact' velocity = escape velocity, which at the event horizon of a bh is c.

You appear to be conflating two different things:

- The appearance of an object accelerating to near c
- The appearance of an object falling into a black hole

...

It is the stretching -- or expansion -- of spacetime close to
the black hole that makes this happen.

That clears it up, thank you.

[Cougar]

This means that as the particle comes close to the event horizon, it exists in two places at once, both inside the black hole (as it is travelling back from the infinite future) and just outside the black hole as it is travelling towards the infinite future.

Explanation appreciated, caveman. Is this a general result of GR or specifically a Penrose* interpretation?

___________________
* I'm reading Penrose's popular book Cycles in Time about his Conformal Cyclic Cosmology, which sounds very similar to this idea of "traveling back from the infinite future." He is applying it, or a similar idea, to the big bang itself. It's a little too dense for me, pun intended, but I don't feel too stupid since Sean Carroll even said, "I’ve been hesitant because, frankly, I don’t really get it." - Carroll on Penrose.

[caveman1917]

Does an object falling into a black hole appear to slow down (rather then appear to slow its acceleration) as it approaches c, or not (as seen by an observer at considerable distance from- and at rest wrt to the bh)?

It appears to slow down, ie its speed goes to zero (as seen by an outside observer).
It approaches c in the sense that, if dropped at rest from infinity, and we have a series of stationary observers all the way to the event horizon, its speed as measured by the observer it is passing goes to c as it gets closer to the event horizon.

[Jeff Root]

Actually that was not me, that was Noclevername.

Oooops!!!

I conflated two different people!

Sorry, noncryptic! Sorry, Noclevername!

-- Jeff, in Minneapolis

[caveman1917]

Explanation appreciated, caveman. Is this a general result of GR or specifically a Penrose* interpretation?

It's a general plain GR result of doing it completely in schwarzschild coordinates. Schwarzschild coordinates become singular at the event horizon (and are thus seen as only "valid" outside the event horizon), and that's why the question is usually answered by switching to different coordinates (such as GP, as Phil does in the video linked to earlier) that aren't singular at the event horizon. However this somewhat begs the original question, since the question implicitly specifies that schwarzschild coordinates must be used, it refers to "to the outside observer the infaller never crosses the event horizon".

As i'm sure you know at the event horizon in schwarzschild coordinates the time and radial dimension switch places, so that the result of doing the analysis in those coordinates is going to get weird shouldn't be so surprising. I think the point to take home isn't so much the weirdness per se, but how everything still stays consistent even doing it that way, ie that the twin coming back from the infinite future only appears the moment it becomes impossible to stop it from travelling there in the first place.