How Does Space-Time Curvature Affect Light Near Black Holes?

[rab99]

JesseM wrote

Light doesn't need to have mass to be affected by a black hole, since in general relativity you can explain the motion of light in terms of the the black hole curving spacetime, and light following a geodesic path in this curved spacetime.

I replied

as matter is sucked into a black hole the matter will begin to emanate light(photons). the light emanated may go in any direction even opposite to a vector pointing to the centre of the black hole. 2 problems I see one How does the light that is in no way following a geodeisc path suddenly start follwoing such a path back towards the black hole. 2 What phenomenon turns that photon thru 180 degrees so that it heads back to the black hole if it isn't gravity what is it?

Also curved space time looks, in 2D, like a funny shaped funnel or cone. How do you consturct (pictorially or mathematically) a similar cone in all three dimensions such that light entering from any angle will travell in a geodeisic path? For this to be true the light would have to take a geodesic path regardless of the angle of attack! I find difficulty with this as if you take a single cone shape and rotate it thru all dimensions, cone 1 overlaps cones 2,3,4,5,6,7 etc etc and the cone shape, density, morphology, topology is compeletly destroyed?

Pundits of the curved space theory believe there is no such thing as gravity, do they. They believe the illusion of gravity is created by curved space yes ? If this is so, You say massive objects bend space but any object no matter how tiny, as long as it exerts a garvitational "force" then it has gravity. So even tiny objects ie an electron must bend space, just less, yes ?

 

[JesseM]

Is there any reason you started a new thread for this? It's better not to start new threads for every question you have about a previous comment...

as matter is sucked into a black hole the matter will begin to emanate light(photons). the light emanated may go in any direction even opposite to a vector pointing to the centre of the black hole. 2 problems I see one How does the light that is in no way following a geodeisc path suddenly start follwoing such a path back towards the black hole.

Once an object is inside the event horizon of the black hole, the radial axis becomes the time axis for them--the singularity at the "center" lies in the future rather than in any spatial direction, and the event horizon lies in the past, and they can no more emit light going away from the singularity than we can emit light going backwards in time. If you're familiar with the idea of light cones in spacetime diagrams, you can look at the two images at the very bottom of http://www.etsu.edu/physics/plntrm/relat/blackhl.htm for an illustration of how light cones become "tilted" closer and closer to the horizon, so that once inside the horizon the future light cone only points inward.

If a photon is emitted outside the event horizon, than it can of course be emitted in a direction that takes it away from the BH, in which case it will escape.

Also curved space time looks, in 2D, like a funny shaped funnel or cone.

Strictly speaking the "funnel" picture of a black hole illustrates only the curvature of space, not the curvature of spacetime.

How do you consturct (pictorially or mathematically) a similar cone in all three dimensions such that light entering from any angle will travell in a geodeisic path?

Mathematically you can talk about the curvature of a universe of any number of dimensions using differential geometry (which is the basis for the theory of general relativity), but since we live in 3 dimensional space our perceptual systems don't allow us to imagine anything higher than a 2D surface curved in a 3D "embedding space" (just like we can't imagine colors we've never seen, or how a person blind since birth couldn't imagine images).

Pundits of the curved space theory believe there is no such thing as gravity, do they. They believe the illusion of gravity is created by curved space yes ? If this is so, You say massive objects bend space but any object no matter how tiny, as long as it exerts a garvitational "force" then it has gravity. So even tiny objects ie an electron must bend space, just less, yes ?

Yes, in general relativity this would be true, although if you get down to the planck scale general relativity becomes incompatible with quantum physics, so physicists think we'll need a new theory of "quantum gravity" to describe what happens at very small distances and times and very high energies.

 

[harryjoon]

I have a question (or two) somehow related. An object falling in the direction of increasing curvature within a gravitational field of a massive object is moving from one geodecic line to another. How does it know, or feel this direction and what makes it to go from one curvature to another. Finally what makes it stop since a planetary object does not to spiral to the center and orbits in a fixed radial distance within the gravitational field.

 

[JesseM]

I have a question (or two) somehow related. An object falling in the direction of increasing curvature within a gravitational field of a massive object is moving from one geodecic line to another.

I think you're misunderstanding the term geodesic--the object's entire path through curved spacetime as it falls into the BH is a geodesic, it's following the path that maximizes the proper time (time as measured by a clock carried with the object).

 

[tiny-tim]

How does the light that is in no way following a geodeisc path suddenly start follwoing such a path back towards the black hole. 2 What phenomenon turns that photon thru 180 degrees so that it heads back to the black hole if it isn't gravity what is it?

Hi rab99!

That is really a question about photons.

You're asking, since the photon had the same velocity as the matter that produced it, how does it acquire a different velocity?

The answer is that the photon never existed before … it was created with its own personal speed-of-light velocity!

(and btw, both the free-fall matter and the photon always follow a geodesic)

 

[JesseM]

You're asking, since the photon had the same velocity as the matter that produced it, how does it acquire a different velocity?

I don't think that's what he was asking--I think he was asking about a photon that was emitted by matter already inside the black hole, and what would happen to "turn it around" if it was initially emitted in a direction that takes it away from the center. As I said, this is based on the misconception that someone inside the black hole sees the singularity lying in a certain direction in space, rather than in the future direction of time.

 

[Dale]

How does the light that is in no way following a geodeisc path suddenly start follwoing such a path back towards the black hole. ... How do you consturct (pictorially or mathematically) a similar cone in all three dimensions such that light entering from any angle will travell in a geodeisic path? For this to be true the light would have to take a geodesic path regardless of the angle of attack!

An object falling in the direction of increasing curvature within a gravitational field of a massive object is moving from one geodecic line to another. How does it know, or feel this direction and what makes it to go from one curvature to another. Finally what makes it stop since a planetary object does not to spiral to the center and orbits in a fixed radial distance within the gravitational field.

Hi rab and harry, I will try to respond to these together since they both ask about geodesics. First, you can think of a geodesic as "a straight line in a curved space". At each point along the line it is locally straight, but because the space itself is intrinsically curved it is not straight in a global sense. If you were moving about the surface of a sphere you would travel in great arcs which are locally everywhere "straight lines" but curve in a global sense.

Now, if you start at any point on the surface of a sphere you can draw an infinite number of great arcs, one for every direction that you could travel. Thus, there is a geodesic in every direction from any point. But once you start on a geodesic path, if you deviate from that path or "switch geodesics" then your overall path is no longer a geodesic. A free-falling massive object or light each follow a single geodesic path (they don't switch geodesics) that can be completely determined from a knowledge of the starting conditions and the spacetime curvature. For an orbiting body the geodesic is a helix for a circular orbit, or some distorted "almost-helix" for an elliptical orbit.

You might wonder if both light and a satellite follow geodesics then what about a beam of light that is tangent to the satellite's orbit? Isn't it on the same geodesic and therefore shouldn't it orbit as the satellite does? The answer is that in GR it is not space that is curved but spacetime. The light and the satellite are going in the same direction at different speeds in space, but they are going at the same speed in different directions in spacetime. Thus, they are each on a different geodesic through spacetime.

 

[harryjoon]

In general relativity, geodesics generalize the notion of "straight lines" to curved spacetime. This concept is based on the mathematical concept of a geodesic. Importantly, the world line of a particle free from all external force is a particular type of geodesic. In other words, a freely moving particle always moves along a geodesic.

In general relativity, gravity is not a force but is instead a curved spacetime geometry where the source of curvature is the stress-energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space.

You are correct. However, I was referring to orinary planetary systems and not BH. I must appologize for changing the subject a little. My question relates to classical GR in which any objects follows a geodesic line, which is a 4-D straight line, wether it is its orbital path, or its free-falling path of the object. I agree with our friend DaleSpam in his comments
"Thus, there is a geodesic in every direction from any point. But once you start on a geodesic path, if you deviate from that path or "switch geodesics" then your overall path is no longer a geodesic. A free-falling massive object or light each follow a single geodesic path (they don't switch geodesics) that can be completely determined from a knowledge of the starting conditions and the spacetime curvature. For an orbiting body the geodesic is a helix for a circular orbit, or some distorted "almost-helix" for an elliptical orbit."
My question is why the planetary objects in a fixed orbit do not move like a falling object such as a stone which moves in the radial direction of the sphere on which the planets move in their orbit.

 

[belliott4488]

My question is why the planetary objects in a fixed orbit do not move like a falling object such as a stone which moves in the radial direction of the sphere on which the planets move in their orbit.

I might be misunderstanding you, but are you simply asking why objects in orbit around the Earth (for example) follow repeating orbits, whereas stones and other falling objects fall "downward" toward the Earth? The answer is very simple: it's a matter of their velocities. If you could throw a stone with sufficiently high velocity (and there were no atmosphere to hinder its motion), you could throw it into a trajectory that would cause it to orbit the Earth. Conversely, if you were to slow an orbiting satellite to a much lower velocity than it has in orbit, it would drop like a stone.

In my business, which has to do with orbiting satellites, we sometimes hear the statement that "an orbit is just free-fall that keeps missing the Earth." This is meant to emphasize there there is no fundamental difference between a falling rock and an orbiting satellite.

 

[tiny-tim]

… jump … !

"an orbit is just free-fall that keeps missing the Earth."

Exactly!

Or, to put it the other way round:

Anything in free-fall is in orbit … but the orbit gets interrupted by the Earth!​

If a stone drops vertically over the North pole, and if there just happens to be a convenient tunnel through the Earth to the South pole, then the stone will carry on indefinitely in orbit … it'll just be an extremely flat orbit!

So if you jump in the air … right now … try it! … you'll, very briefly, be in orbit!

 

[harryjoon]

Based on Newton's explanation, and to some extent GR's explanation I agree with what you have said. What I am asking is how does an object know it is in a curved space and which direction is increasing or decreasing curvature, since there is no "ATTRACTION FORCE", or poential which is dependent on any property of the falling object. The gravitatioal field or the curved space-time of general relativity is essentially a function of the gravitating mass/energy. By other words how does it know where Earth is and why some miss it and some dont. This might seem nonsensical but needs a mechanism not provided, as far as I know, by general relativity.

 

[tiny-tim]

… it "blindly" follows a geodesic …

Hi harryjoon!

It doesn't "know where Earth is" … all it "knows" is the bit of space it's in at the moment.

It can't see ahead!

It obeys Newton's first law … no force is acting on it, so it "blindly" follows the geodesic it happens to already be on.

It doesn't "ask why", or need to know why!

 

[belliott4488]

Based on Newton's explanation, and to some extent GR's explanation I agree with what you have said. What I am asking is how does an object know it is in a curved space and which direction is increasing or decreasing curvature, since there is no "ATTRACTION FORCE", or poential which is dependent on any property of the falling object. The gravitatioal field or the curved space-time of general relativity is essentially a function of the gravitating mass/energy. By other words how does it know where Earth is and why some miss it and some dont. This might seem nonsensical but needs a mechanism not provided, as far as I know, by general relativity.

I agree with tiny-tim's response, but I'll add that I believe it is a postulate that Newton's First Law should be translated from motion in a straight line to motion along a geodesic. In other words, it is not a result derived from more fundamental principles, but rather is an observation of nature.

I honestly don't recall if this is presented as a postulate of GR, however - can someone answer that for me?

 

[harryjoon]

Hi belliott4488
I agree in GR Newtons laws hold locally. OK I put it even more nonsensically, Why do I fall down after I jump up! Where I am standing I am happy in my geodesic. I jump, i.e. I put myself in a new geodesic which is in the direction of decreasing curvature, which I would follow eternally had there not been an Earth so close by. What you are saying is that if I jump high enough so that I reach a geodecic which does not collid with the Earth I would be in a permanent orbit around the earth. After I am in orbit if I were to lose my orbital velocity say by radiating a definite amount of energy the extent of which is totally at my disposal, I would reverse the above process and fall down to earth. This change of velocity over time, acceleration, would not be limited to that of gravitational accelration, this would be in contradiction to the observed fact in gravitational systems all bodies fall with the same acceleration, g, independent of their internal contitution.

 

[JesseM]

Hi belliott4488
I agree in GR Newtons laws hold locally. OK I put it even more nonsensically, Why do I fall down after I jump up! Where I am standing I am happy in my geodesic.

If you're standing, you're not on a geodesic path--only a free-falling object whose path is influenced solely by gravity follows a geodesic, on the ground you're being pushed up by the electromagnetic force between the floor and your feet, which keeps you from falling on a geodesic path to the center of the Earth (and up the other side) as you would if gravity were the only force acting on you.

I jump, i.e. I put myself in a new geodesic which is in the direction of decreasing curvature, which I would follow eternally had there not been an Earth so close by.

When you jump, it's electromagnetic forces between atoms again changing your path, although once you're in the air you are following a geodesic until you hit the ground (at least if we ignore air resistance).

What you are saying is that if I jump high enough so that I reach a geodecic which does not collid with the Earth I would be in a permanent orbit around the earth.

It has nothing to do with height--it would be possible to "orbit" the Earth at sea level if your initial velocity in the sideways direction were high enough and air resistance could be ignored. And even if your jumping velocity is small, if there were no forces acting on you other than gravity so that you could fall straight through the ground towards the center of the Earth, you would pass by the center at high speed and pop up to ground level on the other side, then fall down again, over and over, so this would itself be a kind of highly elliptical orbit.

After I am in orbit if I were to lose my orbital velocity say by radiating a definite amount of energy the extent of which is totally at my disposal, I would reverse the above process and fall down to earth. This change of velocity over time, acceleration, would not be limited to that of gravitational accelration, this would be in contradiction to the observed fact in gravitational systems all bodies fall with the same acceleration, g, independent of their internal contitution.

If you cause yourself to deviate from a geodesic path by "radiating energy" in some form, then you are not in freefall, and the "observed fact in gravitational systems all bodies fall with the same acceleration" only applies to objects in freefall (and at about the same position in the gravitational field, so that differences in the strength of the field don't come into play).

 

[harryjoon]

If you cause yourself to deviate from a geodesic path by "radiating energy" in some form, then you are not in freefall, and the "observed fact in gravitational systems all bodies fall with the same acceleration" only applies to objects in freefall (and at about the same position in the gravitational field, so that differences in the strength of the field don't come into play).

Thank you JesseM for you comments. Would you explain in what way I would be different from a falling stone, a falling satalite or airplain etc, and how is the gravitational acceleration differentin this case. Galileo found it to be the same.

 

[yuiop]

If you cause yourself to deviate from a geodesic path by "radiating energy" in some form, then you are not in freefall, and the "observed fact in gravitational systems all bodies fall with the same acceleration" only applies to objects in freefall (and at about the same position in the gravitational field, so that differences in the strength of the field don't come into play).

Gas and particles falling into the accretion disc of a black hole are thought to radiate huge amounts of energy in the form of x-rays. Some sources say possible up to 100% of the mass of the particles is converted into energy by this process. The principle is that particles undergoing extreme acceleration radiate and this is observed in cyclotrons. What puzzles me is that particles falling towards a black hole are presumably in free fall and therefore should not "experience" acceleration in theor own frame. If the particles do not "feel" acceleration, how do they "know" to radiate?

P.S. It's amazing that Galileo discovered something about gravity that was considered accurate for a century, simply by rolling cylinders down a plank and timing them with a water clock.

 

[belliott4488]

Thank you JesseM for you comments. Would you explain in what way I would be different from a falling stone, a falling satalite or airplain etc, and how is the gravitational acceleration differentin this case. Galileo found it to be the same.

Sorry, it's belliott4488 (Bruce) this time ... The airplane is different if it is flying, because it depends on the atmosphere for lift, which pulls it off of the geodesic it would otherwise follow. If you get rid of the atmosphere, then you, the falling satellite, and the airplane all follow geodesics. You will probably follow different geodesics, though, even if you start at the same point, due to your different velocities - this is what is different from Galileo's case. If you are inside the airplane, then you will float, apparently weightless, as you and the airplane follow the same geodesic. Perhaps you've seen pictures of the 747 airliner that NASA uses for training? The plane flies in such a way that it follows the geodesic that it would follow with no atmosphere (it must use power to overcome the air resistance), and the passengers inside float weightlessly. They are all on geodesics.

 

[tiny-tim]

What puzzles me is that particles falling towards a black hole are presumably in free fall and therefore should not "experience" acceleration in theor own frame. If the particles do not "feel" acceleration, how do they "know" to radiate?

Puzzles me too.

wikipedia on black holes says that xrays are radiation caused by heating caused by pressure friction etc http://en.wikipedia.org/wiki/Black_hole#Accretion_disk:

Black holes give off radiation because matter falling into them loses gravitational energy which may result in the emission of radiation before the matter falls into the event horizon

But wikipedia on x-ray astronomy says that xrays are caused by loss of gravitational energy http://en.wikipedia.org/wiki/X-ray_source#Astronomical_sources_of_X-rays:

Black holes give off radiation because matter falling into them loses gravitational energy which may result in the emission of radiation before the matter falls into the event horizon.

Which is right?

 

[JesseM]

Thank you JesseM for you comments. Would you explain in what way I would be different from a falling stone, a falling satalite or airplain etc, and how is the gravitational acceleration differentin this case. Galileo found it to be the same.

I don't understand the question. When you say "in what way I would be different", are you imagining yourself being in frefall, or yourself being on the ground? And what do you mean when you say the gravitational acceleration is different?

 

[harryjoon]

I don't understand the question. When you say "in what way I would be different", are you imagining yourself being in frefall, or yourself being on the ground? And what do you mean when you say the gravitational acceleration is different?

The equation of geodesic(http://en.wikipedia.org/wiki/Geodesic_equation) relates the acceleration directly to the curvature of space-time. The greater the curvature the greater the acceleration, a fact which is observed in gravitational field of Earth (or any other). Not only this acceleration can not be detected locally(in the rest frame of the object) by using any local parameter, analogous to the relative velocity of an object, it increases as we approach the center of the gravitational field. The lateral acceleration of an object moving on or around earth, increases the further we go away from the center of gravitational field. The former is a property of the gravitational field only, while the latter is a property of the moving object only. Furthermore, the method of energy loss of the object ( friction etc) should not affect our definition of a free-falling object, since even in the absence of air a fallining object loses energy, of course far more slowly if it is is a stable orbit. Hence, my question remains, how is a falling man any different from a falling stone? and why do I fall back down to earth, not according to Newton, but according to GR.

 

[Dale]

If you are a person jumping, a rock falling, or a satellite orbiting you are traveling on a geodesic in GR. If the geodesic intersects the worldline of the Earth then you fall back down, if not then you orbit or escape. It is as simple as that

 

[harryjoon]

There are number of points which I believe suggest that it is not that simple;
1)-Worldline of objects may or may not intersect. If it does it is given that they will meet.
2)-Worldline of an object may or may not be along the geodesic line of the curved space-time field produced by Earth's mass.
3)-A free-falling object travels along a geodesic of the curved space-time field produced by Earth's mass, which is also its worldline.
4)-The world line of the Earth is NOT along a geodesic of its curved space-time field (produced by Earth's mass).
5)- Earth must carry its curved space-time field produced by its mass, along its worldline, which means that the world line of an object orbitting say one meter above Earth surface, i.e following a geodesic of the field which is a meter above the surface of the earth, will always be one meter above the Earth surface, independent of the position of Earth along its worldline. A contrarry sugestion would mean objects will be left behind as Earth travels along its worldline, which is contrary to our observation.

 

[Dale]

There are number of points which I believe suggest that it is not that simple;

You are over-doing things here. It is that simple (except for the process of actually calculating a geodesic).

1)-Worldline of objects may or may not intersect. If it does it is given that they will meet.
2)-Worldline of an object may or may not be along the geodesic line of the curved space-time field produced by Earth's mass.
3)-A free-falling object travels along a geodesic of the curved space-time field produced by Earth's mass, which is also its worldline.
4)-The world line of the Earth is NOT along a geodesic of its curved space-time field (produced by Earth's mass).

All true, but none of it contradicts what I said above.

5)- Earth must carry its curved space-time field produced by its mass, along its worldline, which means that the world line of an object orbitting say one meter above Earth surface, i.e following a geodesic of the field which is a meter above the surface of the earth, will always be one meter above the Earth surface, independent of the position of Earth along its worldline. A contrarry sugestion would mean objects will be left behind as Earth travels along its worldline, which is contrary to our observation.

Neglecting air resistance, and assuming a perfectly spherical Earth and circular orbit, yes. It still doesn't contradict the above.

 

[JesseM]

There are number of points which I believe suggest that it is not that simple;
1)-Worldline of objects may or may not intersect. If it does it is given that they will meet.

If they interact through any non-gravitational forces, like the electromagnetic forces between atoms, then this can cause them to deviate from a geodesic when they meet. Idealized non-interacting particles will just pass by (or through) each other when they meet, traveling on different geodesics because their velocities at the point they meet are different.

2)-Worldline of an object may or may not be along the geodesic line of the curved space-time field produced by Earth's mass.

As long as the object is not being acted on by non-gravitational forces, it will always follow a geodesic (at least if its mass is small compared to the Earth--I'm not sure if two objects which are both massive enough to significantly affect each other's motion are still following geodesics, you'd have to ask someone with more expertise in GR).

3)-A free-falling object travels along a geodesic of the curved space-time field produced by Earth's mass, which is also its worldline.
4)-The world line of the Earth is NOT along a geodesic of its curved space-time field (produced by Earth's mass).

The worldline of the center of the Earth might indeed be a geodesic in the curved space-time produced both by the Earth and the Sun, though I'm not totally sure about this for the reason mentioned above.

5)- Earth must carry its curved space-time field produced by its mass, along its worldline, which means that the world line of an object orbitting say one meter above Earth surface, i.e following a geodesic of the field which is a meter above the surface of the earth, will always be one meter above the Earth surface, independent of the position of Earth along its worldline. A contrarry sugestion would mean objects will be left behind as Earth travels along its worldline, which is contrary to our observation.

Sure, spacetime is curved by the presence of mass and energy in GR, so it makes sense that the curvature would move along with a moving mass.

 

[harryjoon]

Therefore, could we not conclude that the worldline of ONLY those objects, which are in a stable orbital path such as planets in orbit around the sun, orbiting satalites etc., corresponds to a geodesic of the curved space-time produced in the presence of the a mass, i.e only planetary objects free-fall along a geodesic in their worldline. All other objects which are not in a stable orbit, as they travell along their worldline, they move orthogonal to the the geodesic lines, from reigons with less curvature to those with greater curvature, in the radial direction of the curved space-time around Earth. The qustion I asked (posted) earlier was what is the mechanism, according to GR, for this natural (inertial, unprovoked, whatever is the right word) change of geodesic of falling objects.

 

[JesseM]

Therefore, could we not conclude that the worldline of ONLY those objects, which are in a stable orbital path such as planets in orbit around the sun, orbiting satalites etc., corresponds to a geodesic of the curved space-time produced in the presence of the a mass, i.e only planetary objects free-fall along a geodesic in their worldline.

No, any small object in free-fall (again, I'm not totally sure about large objects which contribute significantly to the spacetime curvature themselves) stays on a geodesic as long as it is not being acted on by non-gravitational forces. For example, if you jump into the air and we neglect air resistance (perhaps you are jumping on the moon), then for the period when you're not touching the ground, you are following a geodesic, but as soon as you hit the ground electromagnetic forces between the atoms of the ground and the atoms of your feet cause you to stop following this geodesic path (if you were made out of neutrinos you wouldn't have any electromagnetic interaction with the ground so you'd continue falling right through on something close to a geodesic, although it wouldn't be perfect because there is still some interaction of neutrinos with other particles via the weak force).

 

[Dale]

The easy way to determine if you are on a geodesic experimentally is to use an accelerometer. As long as your accelerometer reads 0 you are on a geodesic. Of course, if you are on a geodesic that intersects with the Earth you will abruptly leave said geodesic and the accelerometer will register a very large acceleration. But while the accelerometer reads 0 you are on a geodesic.

 

[MeJennifer]

The easy way to determine if you are on a geodesic experimentally is to use an accelerometer. As long as your accelerometer reads 0 you are on a geodesic. Of course, if you are on a geodesic that intersects with the Earth you will abruptly leave said geodesic and the accelerometer will register a very large acceleration. But while the accelerometer reads 0 you are on a geodesic.

That is incorrect in two ways.

If we consider a system with test object T orbiting a test mass M then their geodesics always intersect (except for cases with a positive cosmological constant). Second under no circumstance will the accelerometer register anything, except for the moment when T crashes onto M.

Note that geodesics are spacetime paths not paths in space parameterized by time.

 

[Dale]

under no circumstance will the accelerometer register anything, except for the moment when T crashes onto M.

Which is the moment that T leaves the geodesic.

 

[MeJennifer]

Which is the moment that T leaves the geodesic.

Yes. But that is not an intersection of geodesics.

An intersection of geodesics is for instance the case I described with objects T and M. Eventually both test objects T and M meet without any accelerations.

 

[rab99]

I think you're misunderstanding the term geodesic--the object's entire path through curved spacetime as it falls into the BH is a geodesic, it's following the path that maximizes the proper time (time as measured by a clock carried with the object).

Hi JesseM

as I understand it a geodiesic is just a geomertrical shape that looks like a spiral down a funnel yeh ? or is there more mystery to it ?

I am singulary disatissfied with your response about future time direction the like it sounds like magic to me. What you are saying is that common sense ends at the event horizon I am a little too hard nosed for that proposal. I think there would be direction in a blach hole after all how does matter know which direction is towards the singularity. How does gravity know in which direction to act if there is no direction etc ect

You are proposing that all the laws of physics break down in a black hole they may be exotic but I don't think they are that exotic.

Lets say the diameter of a black hole is 2 light years so it a 2 light year sphere and at the center of the sphere is the magical singularity which is very small so there is no distance and direction from the edge of the balch hole to the singularity at its center ... sounds like magic to me

 

[Dale]

Yes. But that is not an intersection of geodesics.

I never said it was.

I really don't understand your point here. If an accelerometer reads 0 it is traveling along a geodesic. If the accelerometer does not read 0 it is not traveling along a geodesic.

I don't understand the relevance of your comments about two intersecting geodesics. The worldline of the ground is not a geodesic (an accelerometer on the ground reads g, not 0), so when I was talking about a geodesic intersecting the worldline of the Earth I was not talking about two geodesics intersecting. I was talking about a geodesic intersecting a non-geodesic worldline.

 

[rab99]

I prefer occams razor rather than a surreal explanation

 

[rab99]

the problem I have with this theory is if you have a 2 dimensial geodiesic and you rotate it thru all three dimensions then the overlapping of the geodeisics cancell each other out and you end up with a nett effect of a big fat zero. Ill post a pic soon to show what I mean

 

[rab99]

Look at the picture this is the bending of space by a massive object located at point A. When an object falls onto this bent surface the path it follows towards point A is described by this surface. Now rotate the bent space thru 360 degrees about point A, you end up with an homogenous circle no distinguishing shape, no surface where is the surface that is to be followed. How does the falling object know which surface is to be followed ?

 

[rab99]

if an object approached from the x direction how does it know what surface to follow ?

 

[JesseM]

as I understand it a geodiesic is just a geomertrical shape that looks like a spiral down a funnel yeh ? or is there more mystery to it ?

It isn't very helpful to picture a geodesic as a path through curved space, because geodesics in GR aren't minimizing the spatial distances, they're maximizing the proper time through curved spacetime. But sure, a geodesic path might look like a spiral down the black hole, or it might look like a straight line into it along the radial direction.

I am singulary disatissfied with your response about future time direction the like it sounds like magic to me. What you are saying is that common sense ends at the event horizon I am a little too hard nosed for that proposal. I think there would be direction in a blach hole after all how does matter know which direction is towards the singularity. How does gravity know in which direction to act if there is no direction etc ect

Of course there is direction! There are three spatial dimensions and one time dimension, just like always. It's just that the singularity now lies in the future time direction rather than in any spatial direction, much like the Big Crunch that would be the end of time for a collapsing universe (the mirror image of the Big Bang, which lies in our own past time direction but not in any particular spatial direction).

You are proposing that all the laws of physics break down in a black hole they may be exotic but I don't think they are that exotic.

Er, no I'm not. I'm just telling you what the theory of general relativity predicts about the inside of black holes, using exactly the same laws that apply outside the black hole.

Lets say the diameter of a black hole is 2 light years so it a 2 light year sphere and at the center of the sphere is the magical singularity which is very small so there is no distance and direction from the edge of the balch hole to the singularity at its center ... sounds like magic to me

Before making dismissive comments like this, could you try to make sure that you actually understand what I'm telling you? Once inside the black hole, the observer finds himself in what is in effect a collapsing universe, where there are still 3 spatial dimensions but one of the three (or two of the three, I've forgotten) is finite (it wraps around, like the circumference of an infinite tube), and constantly shrinking (the tube gets thinner and thinner, crushing objects on its surface together), until at some time the size of this space goes to zero and all matter is crushed to infinite density. General relativity could be said to "break down" at the exact moment of the singularity, but until that point it gives a perfectly sensible description of what's going on, and in any sufficiently small region the laws of physics look exactly the same for an observer inside the horizon as they do for an observer outside.

There's a neat sci-fi story about a dive into a black hole by Greg Egan, an author who is well-versed in general relativity, here:

http://gregegan.customer.netspace.net.au/PLANCK/Complete/Planck.html

In one scene, the characters discuss some of the points I've been talking about while running a simulation of a fall into a black hole:

Gisela highlighted a vertical section of their world line, where they'd hovered on the three-M shell. “Outside the event horizon — given a powerful enough engine — you can always stay fixed on a shell of constant tidal force. So it makes sense to choose that as a definition of being ‘motionless’ — making time on this map strictly vertical. But inside the hole, that becomes completely incompatible with experience; your light cone tilts so far that your world line must cut through the shells. And the simplest new definition of being ‘motionless’ is to burrow straight through the shells — the complete opposite of trying to cling to them — and to make ‘map time’ strictly horizontal, pointing towards the centre of the hole.” She highlighted a section of their now-horizontal world line.

Cordelia's expression of puzzlement began to give way to astonishment. “So when the light cones tip over far enough … the definitions of ‘space’ and ‘time’ have to tip with them?”

“Yes! The centre of the hole lies in our future, now. We won't hit the singularity face-first, we'll hit it future-first — just like hitting the Big Crunch. And the direction on this platform that used to point towards the singularity is now facing ‘down’ on the map — into what seems from the outside to be the hole's past, but is really a vast stretch of space. There are billions of light years laid out in front of us — the entire history of the hole's interior, converted into space — and it's expanding as we approach the singularity. The only catch is, elbow room and head room are in short supply. Not to mention time.”

Cordelia stared at the map, entranced. “So the inside of the hole isn't a sphere at all? It's a spherical shell in two directions, with the shell's history converted into space as the third … making the whole thing the surface of a hypercylinder? A hypercylinder that's increasing in length, while its radius shrinks.” Suddenly her face lit up. “And the blue shift is like the blue shift when the universe starts contracting?” She turned to the frozen sky. “Except this space is only shrinking in two directions — so the more the angle of the starlight favours those directions, the more it's blue-shifted?”

“That's right.”

...

Cordelia raised the binoculars and looked sideways, around the hole. “Why can't I see us?”

“Good question.” Gisela drew a light ray on the map, aimed sideways, leaving the platform just after they'd crossed the horizon. “At the three-M shell, a ray like this would have followed a helix in spacetime, coming back to our world line after one revolution. But here, the helix has been flipped over and squeezed into a spiral — and at best, it only has time to travel half-way around the hole before it hits the singularity. None of the light we've emitted since crossing the horizon can make it back to us.

 

[rab99]

JesseM

so the 2d picture I drew of curved space doesn't look anyhitng like reality? What would the 2d pic look like assuming it can be drawn?

I mean I have seen those pics of a bowling ball on a trampoline as an anology, and then you roll a marble on the trampoline and it makes a geodiesic path to the bowling ball, is that a sufficiently true analogy in which case that is what I have drawn?

 

[JesseM]

Look at the picture this is the bending of space by a massive object located at point A. When an object falls onto this bent surface the path it follows towards point A is described by this surface. Now rotate the bent space thru 360 degrees about point A, you end up with an homogenous circle no distinguishing shape, no surface where is the surface that is to be followed. How does the falling object know which surface is to be followed ?

I don't understand what you mean by "bent surface". The surface in an embedding diagram is supposed to be space itself with one dimension taken away, not a surface that lies in 3D space. If you can picture a universe with only two dimensions like in the story flatland, and then picture a 2D surface being curved by gravity so that planets lie in depressions on this surface, then if you take a cross section of this surface in a plane that lies at right angles to the 2D universe, you'll get a curved 1D line--this would be an embedding diagram for the curvature of 2D space that could be visualized by 1D beings. Similarly, when a gravity well in 3D space is pictured as an actual depression in a 2D plane, the idea is the same--see http://www.bun.kyoto-u.ac.jp/~suchii/embed.diag.html on the meaning of embedding diagrams.

 

[MeJennifer]

JesseM

so the 2d picture I drew of curved space doesn't look anyhitng like reality? What would the 2d pic look like assuming it can be drawn?

I mean I have seen those pics of a bowling ball on a trampoline as an anology, and then you roll a marble on the trampoline and it makes a geodiesic path to the bowling ball, is that a sufficiently true analogy in which case that is what I have drawn?

Note that a line does not have intrinsic curvature, you need at least a plane for that.

 

[rab99]

I may be wrong but I think you just described my trampoline analogy and the 1d line is a cross section thru the trampoline yeh? But this is for the purposes of visualisation only as in reality the curve is in three dimension not 2 or 1 as nothing can exist in 1 or 2 dimensions yeh?

 

[JesseM]

I may be wrong but I think you just described my trampoline analogy and the 1d line is a cross section thru the trampoline yeh? But this is for the purposes of visualisation only as in reality the curve is in three dimension not 2 or 1 as nothing can exist in 1 or 2 dimensions yeh?

The curved 1D line (and I agree with MeJennifer that the curvature here is not intrinsic, it's just how the line appears in 2D space) is a cross section through the curved 2D trampoline surface, and in the same way, the curved 2D trampoline diagram is supposed to represent the curvature in a cross section of the 3D space around a massive object.

 

[tiny-tim]

… speed-paths …

JesseM

so the 2d picture I drew of curved space doesn't look anyhitng like reality? What would the 2d pic look like assuming it can be drawn?

I mean I have seen those pics of a bowling ball on a trampoline as an anology, and then you roll a marble on the trampoline and it makes a geodiesic path to the bowling ball, is that a sufficiently true analogy in which case that is what I have drawn?

Hi rab99!

A geodesic is easiest to define as a null curve in four-dimensional space-time.

But it is easiest to understand as its weighted projection in three-dimensional space, which is a curve coupled with a speed at each point (+ in one direction, - in the other).

(For speed-of-light particles, the speed has to be replaced by energy or frequency … so the curve could simply be coloured according to the perceived colour of the light at each point!)

Let's call that a speed-path.

So, if a geodesic is thought of as a metal wire snaking through space-time, then the speed-path is what you get by melting it and letting the metal "fall through time" onto three-dimensional space (and then solidifying again)!

The faster the geodesic, the more metal falls on that part of the path.

A particle in free-fall at a particular point in space, and with a particular velocity, will follow the speed-path it happens to be on … which is the unique speed-path at that point and in that direction at that speed.

Two particles at the same point and with the same velocity but with different speeds will be on different speed-paths, and will continue to follow them.

"Half" the geodesics produce faster-than-light speed-paths … these are forbidden for ordinary particles in ordinary space.

As for black holes: the equations for the geometry of space-time are the same inside the event horizon as outside it … but some of the coefficients have changed sign, and so the solutions look different.

In particular, some solutions are forbidden (faster-than-light) inside which would not be outside, and vice versa.

So there are geodesics (or speed-paths) for every point and velocity inside the event horizon. But the ones which are forbidden are not the ones we would normally expect.

Nevertheless a particle inside an event horizon certainly "knows" which speed-path to follow.

 

[pmb_phy]

I think you're misunderstanding the term geodesic--the object's entire path through curved spacetime as it falls into the BH is a geodesic, it's following the path that maximizes the proper time (time as measured by a clock carried with the object).

Actually a (timelike) geodesic is a worldline for which the propertime has a stationary value. In general there are more than one geodesics between events.

Pete

 

[Dale]

I prefer occams razor rather than a surreal explanation

As far as Occam's Razor goes, relativity is hard to beat. It has no "tuneable" parameters whatsoever, so it is pretty simple. Also, SR and GR are developed from a very small number of postulates, so even the foundation is simple.

What theory are you referring to that explains the data as well as GR and has fewer free parameters or postulates?

 

[Dale]

as I understand it a geodiesic is just a geomertrical shape that looks like a spiral down a funnel yeh ?

No, this is not what a geodesic is at all. A geodesic is the closest thing you can get to a straight line in a curved space. E.g. on a sphere a geodesic is a great circle. On a cylinder a geodesic is a helix. Other surfaces have other geodesics. IMO, the geodesics on a torus are particularly interesting. For geodesics in spacetime you can no longer picture it as embedded in a higher-dimensional flat space, so you have to define the curvature of the surface intrinsically, which is what the math behind GR is all about.

For the special case of an object in an unstable orbit around a black hole the geodesic is as you describe, but that is one very specific case and not a general description of what a geodesic is.

 

[tiny-tim]

… Occam's razor … time directions …

Hi JesseM and DaleSpam!

I think rab99's Occam's razor was referring to the the following, from his own post #32 four minutes earlier (but interrupted by DaleSpam's intervening post):

I am singulary disatissfied with your response about future time direction the like it sounds like magic to me. What you are saying is that common sense ends at the event horizon

Im a little too hard nosed for that proposal. I think there would be direction in a blach hole, after all how does matter know which direction is towards the singularity. How does gravity know in which direction to act if there is no direction etc ect

You are proposing that all the laws of physics break down in a black hole they may be exotic but I don't think they are that exotic.

which in turn was referring to your post #2:

Once an object is inside the event horizon of the black hole, the radial axis becomes the time axis for them--the singularity at the "center" lies in the future rather than in any spatial direction, and the event horizon lies in the past, and they can no more emit light going away from the singularity than we can emit light going backwards in time. If you're familiar with the idea of light cones in spacetime diagrams, you can look at the two images at the very bottom of http://www.etsu.edu/physics/plntrm/relat/blackhl.htm for an illustration of how light cones become "tilted" closer and closer to the horizon, so that once inside the horizon the future light cone only points inward.

Applying Occam's razor, should there not be a better explanation, consistent with "common-sense", of "enforced falling" inside an event horizon than the "magic" of saying that there is no space direction to follow, only a time direction?

 

[MeJennifer]

A geodesic is the closest thing you can get to a straight line in a curved space. E.g. on a sphere a geodesic is a great circle. On a cylinder a geodesic is a helix.

A cilinder is not a curved but a flat space.

 

[JesseM]

Applying Occam's razor, should there not be a better explanation, consistent with "common-sense", of "enforced falling" inside an event horizon than the "magic" of saying that there is no space direction to follow, only a time direction?

Did you read my response in post #38? It's not that "there is no space direction to follow", there are still 3 space dimensions and 1 time dimension, but for an observer in the horizon the singularity lies in the future time direction, much like the Big Crunch singularity of a collapsing universe. This is just a consequence of apply GR to the region inside the event horizon (Did you look at the diagram of light cones tilting near the horizon near the bottom of http://www.etsu.edu/physics/plntrm/relat/blackhl.htm? Similar diagrams can be seen in many GR textbooks)--surely Occam's razor says the laws of physics should be the same inside as outside, rather than inventing new physics like "enforced falling" inside the horizon?