The Black Hole War

In the presence of mass, spacetime curves in four dimensions according to the GTR. The 'Falling straight in' looks 'straight' from our vantage point in three dimensions, but in four dimensions the trajectory is conforming to a curved geodesic. Let's abstract this bit into a purely mathematical realm, one which should be pretty familiar to most folks here: logarithms.

Let's say you've a sheet of lin-log paper with the x-axis being the linear one. You've some quadrille (Cartesian) paper as well and you've got a function which you plot on both sheets: y = log(x) [log base 10, but CR4's editor mishandles sub- and superscripts under Google Chrome and so I'm omitting the subscript altogether. Just assume it's there].

In Cartesian Space the locus is curved. In Log Space it's a straight line. In a manner of speaking, we live in Log Space. An object falling 'straight in' looks straight to us but is in fact following a curved trajectory.

When considered from a 4-dimensional POV, the 3-dimensional world in which we live is a surface. A curved one when in the presence of mass.

Is this clear as mud?

CR4's editor mishandles sub- and superscripts under Google Chrome

You are right about this. I often use MS Word to get the sub & superscripts in, then paste in my comment with Chrome. The rest of this comment was clear as mud as you said.

If anybody is interested, there is a part 2 of this discussion here.

"... is as small as the smallest possible size, the Planck length size."

This isn't strictly the case. There is no 'smallest possible size' really, the Planck Length speaks more of the limit at which we can measure the size to any meaningful degree and this is a different proposition altogether! The energies required to measure anything smaller than a Planck Length will result in the creation of black holes and no information will be gained.

There is no 'smallest possible size' really, the Planck Length speaks more of the limit at which we can measure the size to any meaningful degree

He talked about the measurement problem, but in another area he stated it was the smallest size. That would mean that the universe is not a continuum, but is quantized. This is something every physicist should want to know.

The energies required to measure anything smaller than a Planck Length will result in the creation of black holes and no information will be gained.

True. He said the Planck length, Planck time, & Planck mass are the size, half life and mass of the smallest possible black hole, but I don't think the term half life is appropriate. Anyway, this may be why it is thought to be the smallest possible size.

"If the lake analogy is correct, then gravity is the flow of space-time into objects that have mass." [emphasis mine]

This immediately raises the question: "How then is spacetime aware of the location of mass into which it is flowing? There must be some sort of gradient in place in order for this to happen, and that gradient cannot be one in spacetime itself, because that is what is flowing, yes?"

Neither the GTR nor QM make mention of this kind of flow. With respect to what coordinate frame is the flow measured, for instance, and indeed, what does it mean for spacetime to 'flow'? If such is the case, then the 'flow' should be measurable but, before that can happen, 'flow' in this sense must be defined in at least conceptual terms first before it can be measured in quantifiable terms, yes?

How then is spacetime aware of the location of mass...?

I didn't know the universe has consciousness. HTRN will be very interested in this.

If such is the case, then the 'flow' should be measurable but, before that can happen, 'flow' in this sense must be defined in at least conceptual terms first before it can be measured in quantifiable terms, yes?

Gravity is the 'last frontier' of science. My only answer at this point is that it is measured by it's force.

europium: "How then is spacetime aware of the location of mass...?"

StandsardsGuy: "I didn't know the universe has consciousness. HTRN will be very interested in this."

'aware' figuratively speaking.

Regarding point 2, the hypothetical 2000 mile tall falling man experiences a differential force between his head and his feet. He gets stretched, similar to a tidal force. His feet get pulled more than his midsection, and his midsection gets pulled more than his head. So using his midsection as the frame of reference, his head and feet both move further away.

As I understand it, Einstein mostly considered small objects in a localized region of a field. He understood tidal forces and I think he would have agreed with Susskind (assuming you are quoting Susskind correctly, of course). Curvature is a proper term for a geodesic, even one that appears straight over some nominal distance.

I agree. And there is curvature regardless, even if that curvature is zero ('flat' spacetime).

Whilst the man is being stretched radially, he is also being compressed tangentially. This compression is the basis for the heating of an accretion disk around a black hole.

To better picture perhaps what is happening to the man in a curved-spactime, 'geodesic' sense, we 'map' the man onto a patch of flat spacetime and then watch what happens to him as he approaches a black hole. He becomes distorted, true, but only from an external observer's POV, not from his own. From the man's POV, his shape hasn't changed at all - very confusing to the man because he doesn't have a clue as to why he is dying, and in the most wretched way. The radial stretching is relative to the length of his geodesic in flat spacetime; likewise with his tangential geodesic. The man eventually arrives at the singularity where the curvature in all directions is infinite. Keep in mind that throughout all of this I am conveniently neglecting altogether what is happening to his perception of time meanwhile: from his own POV the man never reaches the singularity.

Why is he "compressed" tangentially?

Imagine 'lines of gravitational force' radiating outward from a central mass (the radials corresponding to geodesics and the spacing an analogous measure of the metric of that patch of spacetime). Imagine further two small objects (particles, if you will) lying on adjacent radials and equidistant from the mass. As the objects move toward the mass they grow closer together. This is a highly simplified view of things of course and not a formal description in the least (for which I apologise), but hopefully it gives a general idea.

In the environment near a black hole the metric follows the same pattern but here the radials become unimaginably dense (measured tangentially). Anything 'riding' them into the black hole is going to get squished, and hard.

Cygnus X1 is a good example of a relatively nearby binary system comprised of a black hole and an ordinary star. The two stars orbit in such close proximity that the black hole (which is physically much smaller but has the greater mass) is able to siphon matter from its companion. The infalling matter is heated to incandescence by tangential compression as it spirals in, eventually becoming so hot that it emits X-rays.

In an abstract way you can say compressing, but it is space that is getting smaller.

Consider that the universe is expanding - right now. You would not consider yourself as stretching, yet the space between every particle is expanding.

The example you cited where the person is stretched in length comes from the difference in gravitational force between one end of the body to the opposite end. It is a gradient in the vertical direction, but here is no gravitational gradient in the horizontal direction, so there is no actual force causing a compression.

AH wrote: "It is a gradient in the vertical direction, but here is no gravitational gradient in the horizontal direction, so there is no actual force causing a compression."

There are 'gradients' in both directions, as E explained. A cube of free falling particles with no forces between them (other than the gravitational 'force' of a single massive body), just following their own geodesics, are moving farther apart in the vertical direction and closer together in the horizontal directions.

-J

E wrote: "Cygnus X1 is a good example of a relatively nearby binary system comprised of a black hole and an ordinary star. The two stars orbit in such close proximity that the black hole (which is physically much smaller but has the greater mass) is able to siphon matter from its companion. The infalling matter is heated to incandescence by tangential compression as it spirals in, eventually becoming so hot that it emits X-rays"

AFAIK, there is a measure of good old friction involved as well, especially when dust spirals in. Accretion layers at different radial distances move at different orbital speeds, pressure and friction heat them, losing mechanical orbital energy. This causes a slow inspiral to r = 6GM/c², where the orbits become unstable and a rapid inspiral at very high temperature follows.

-J

very confusing to the man because he doesn't have a clue as to why he is dying, and in the most wretched way...I am conveniently neglecting altogether what is happening to his perception of time meanwhile: from his own POV the man never reaches the singularity.

Actually this man is falling to earth, but you have the last part wrong for a black hole. He sees himself going right in, but someone far away sees him never reaching it. Susskind has a very different version, but I didn't want to get into it, except maybe a "part 2" of this discussion.

europium wrote: "To better picture perhaps what is happening to the man in a curved-spactime, 'geodesic' sense, we 'map' the man onto a patch of flat spacetime and then watch what happens to him as he approaches a black hole. He becomes distorted, true, but only from an external observer's POV, not from his own. From the man's POV, his shape hasn't changed at all - very confusing to the man because he doesn't have a clue as to why he is dying, and in the most wretched way."

If I understand correctly what you were saying here, I can't quite agree. The mapping onto flat (Schwarzschild) spacetime will reveal the man to be (apparently) compressed in all directions. Schwarzschild space and time get denser closer to a BH; a coordinate-choice effect and not 'real'.

In his own local coordinates, the man gets radially stretched by tidal gravity, which is a real force operating on him - his proper length will increase, a classical effect, not necessarily relativistic.** This stretch, when mapped back to the flat coordinates, will cancel some of the apparent coordinate compression, but I don't think a real stretch will ever be visible in Schwarzschild coordinates.

Or did I misread you?

-J

** There is a relativistic correction in strong gravity, of course.

I got relativistic corrections in elementary school. I swear that ruler was moving at the speed of light!

As I understand it, Einstein mostly considered small objects in a localized region of a field. He understood tidal forces...

So then you think that modern physics text books have misquoted Einstein. That is a possibility.

As I understand it, Einstein mostly considered small objects in a localized region of a field. He understood tidal forces and I think he would have agreed with Susskind

I have Einstein's book Relativity. The word force is not in the index. The only place I found it was in a footnote about a refutation of Newton. If you have any evidence of your statement, please share it with us.

S wrote: "So gravity and acceleration are not equivalent, but they are absolutely equivalent. (mathematical absolute gets rid of the minus sign)."

Einstein did point out that the equivalence between gravity and acceleration only holds precisely for hypothetical uniform gravitational fields (GM/r, not GM/r²). Or as an approximation in a very small piece of a normal gravitational field. Loosely speaking, his elevator had to be small compared to the distance to the planet.

Long elevators being uniformly accelerated length-wise, either by pushing or pulling,** suffer an effect similar to tidal stretching, but it is quantitatively different in magnitude than for gravity (see e.g. http://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox).

-J

** Ignoring mechanical compression/decompression - here referring to a purely special-relativity effect (related to Lorentz contraction).

Thanks for the link to the Bell paradox. Since it didn't mention gravity AFAIN, I don't see any proof to your point. But since acceleration and gravity have opposite vectors (for a = F/m), it would seem logical that there would be Lorentz expansion for gravity. In any case the contraction is not real locally, it's only apparent for someone in another reference frame.

Yep, I guess one needs a bit more than Bell's paradox to proof that point.

"But since acceleration and gravity have opposite vectors (for a = F/m), it would seem logical that there would be Lorentz expansion for gravity."

I don't know about a term like 'Lorentz expansion for gravity'. As you said, Lorentz contraction is only noticeable in the original reference frame, relative to which the velocity is measured. Likewise, for the usual Schwarzschild coordinates (reference frame), there is a gravitational contraction in the radial direction as you get nearer to a gravitating mass.

Nevertheless, it is possible to view (and observe) a local expansion in an accelerating frame, e.g. while the reference frame distance between the Bell-spaceships are being kept constant, the string between them will stretch and perhaps break. Likewise, if two bodies are radially separated, tied together with a string and then dropped in a gravitational field, the string will stretch and perhaps break under the tidal forces.

Although equivalent at face value, the two stretching forces follow different laws. Hence, gravity and acceleration are not quite equivalent. This was my point, more or less...

S wrote: "This much better than the rubber sheet gravity analogy that uses gravity to explain gravity. If the lake analogy is correct, then gravity is the flow of space-time into objects that have mass."

Just like the 'balloon analogy' cannot really represent 'expanding space', so the 'lake analogy' cannot really represent 'in-falling space' - it is just another nice analogy that helps comprehension up to a point, but no further. Part of the problem is where to draw that line...

Nevertheless, it is clear that black holes suck in normal energy (matter and radiation) from surrounding space, growing 'fatter and heavier', but this is just redistributing existing energy, I think. AFAIK, it neither sucks in dark energy, nor produces it.

-J

PS: Thanks for the short review - I will surely get the book and read it.

Nevertheless, it is clear that black holes suck in normal energy (matter and radiation) from surrounding space, growing 'fatter and heavier', but this is just redistributing existing energy, I think. AFAIK, it neither sucks in dark energy, nor produces it.

I'm sure you don't know it, it was just food for thought (not even a hypothesis). I wasn't just talking about black holes, but all items with mass, such as the earth. Also, I never suggested they suck in dark energy. The idea way maybe they change gravity into dark energy (it's opposite). I don't think anybody has a good explanation for where dark energy comes from, unless you know of someone.

"The idea way maybe they change gravity into dark energy (it's opposite). I don't think anybody has a good explanation for where dark energy comes from, unless you know of someone."

Matter has negative gravitational energy and positive pressure. Dark energy has negative gravitational energy and negative pressure. Apart from the pressure, not quite opposites.

Where it comes from: vacuum energy. Where that comes from, nobody knows, I think...

How about the axial jets out of black hole accretion disks? They are emanating from the center of the disk, in a direction opposite to the gravitational pull. If even light can't escape a black hole, how do these axial jets persist?

If Flatlanders were to contemplate a drain, the 3-D analysis (Flatland has two space dimensions plus time) would conclude that matter was disappearing from Flatland, when it was being extracted in a new dimension.

AFAIK, the axial jets are emitted from outside of the event horizon of the hole, driven by intense magnetic fields set up by the swirling accretion disk. Some of the in-falling materials are apparently gathered around the 'accretion poles' and accelerated away from the hole.

-J

It's the universe's super collider.

One of the best!

Because they're electrically conductive, plasmas tend to entrain magnetic fields. Should the magnetic flux try to move through the plasma, the motion induces a current which opposes the motion, much like a short-circuited generator's armature fights back (figuratively speaking). The more conductive the plasma, the more the flux is 'bolted onto' the plasma's reference frame, so to speak.

If the plasma is moving, the flux moves with it because of this action. A black hole's accretion disk is a rapidly spinning plasma (differentially spinning, as if it weren't complicated enough) and the motion 'twists' the flux much like the fibres in a rope. The dynamics are very complicated, involving even frame-dragging, and poorly understood, but basically the flux lines tend to grow very dense as they're twisted round. Consequently they exert a powerful torque on the plasma, displacing some of it. Bits of the stuff spiral up the twisted flux lines, moving toward the rotational axis and are accelerated, especially near the poles.

In the sense that they're particle accelerators, black holes have no equal. They're compact and efficient.

The LHC is 27 km in circumference and has a design energy of 14 TeV. Taken together with the LHC's brightness (particles per second, roughly) and we're talking about as much power as dissipated by one mossie's beating wings.

A 3.8 solar mass black hole's event horizon, OTOH, is around 24 km in diameter, yet most of the acceleration takes place in one quarter its circumference, or over a distance about 18 km and in one pass. The power dissipated by a stellar black hole's jets depends on how well we feed the accretion disk of course but generally it is many magnitudes greater than the energy demands of our entire planet.

Nice writeup.