Exploring the Curvature of Space-Time in Relation to Black Holes

[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?

 

[tiny-tim]

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.

But aren't you contradicting yourself?

You are saying that the particle (or the geodesic) follows a future time direction, which is the same as "there is no space direction to follow"!

Did you read my response in post #38?

Yes, and in particular, not only the above passage, but the following from http://gregegan.customer.netspace.ne...te/Planck.html which you quote, presumably with approval:

We won't hit the singularity face-first, we'll hit it future-first …

erm … if my head goes through the event horizon before my feet, then surely I will hit the singularity head-first!

In what sense do I not hit the singularity head-first? In what sense do I hit it "future-first"?

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?

Sorry, but "enforced falling" is a fact.

Our job is to choose the mathematics or physics with which to describe that fact.

rab99 and I think that talking about movement along a time direction is confusing and against common-sense.

Especially when the time direction (whose projection in three dimensions is radial) is in the same three-dimensional direction as the space direction from my feet to my head!

I would be happier with the following explanation:

Outside an event horizon, an object follows a time-like geodesic (in space-time). The time-like geodesics go in every space direction.

Inside an event horizon … it's the same … an object follows a time-like geodesic! Except that the time-like geodesics now do not go in every space direction, but are confined within a cone. ​

This uses the same physics … unlike saying "we must now move along time directions instead of space directions" … the physics is that movement is always along time-like directions, but those directions are arranged differently inside an event horizon compared with outside!

 

[Dale]

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?

"Common Sense" is not a scientific theory, and this is a gross misapplication of Occam's razor. Occam's razor is only applicable in the case where you have two scientific theories that each explain the data equally well. In that case, since there is no experimental reason to prefer one theory over the other it is best to use the simpler theory. So, for example, Occam's razor allows you to reasonably prefer Special Relativity over Lorentz' Ether Theory since they make the same experimental predictions and Special Relativity is simpler.

Occam's razor does not allow you to discard GR simply because it is not "common sense" or because you are confused about it. If you are going to invoke Occam's razor, then what alternate theory are you choosing instead which explains the data equally well and is simpler than GR?

 

[JesseM]

You are saying that the particle (or the geodesic) follows a future time direction, which is the same as "there is no space direction to follow"!

Huh? All geodesics "follow a future time direction", unless I'm misunderstanding what you mean by that phrase. Put it this way, the geodesic extending from an event inside the horizon lies inside the future light cone of that event, just as would be true for geodesics outside the horizon; the point is that the light cones tip over as they get closer to the horizon, as depicted in the diagrams at the bottom of http://www.etsu.edu/physics/plntrm/relat/blackhl.htm which I have pointed to before.

erm … if my head goes through the event horizon before my feet, then surely I will hit the singularity head-first!

I would guess it's probably true that if you synchronize clocks at your head and feet using the usual SR technique when far from the hole, then fall in head-first, your head would indeed be crushed earlier according to the clock there. But I think this is sort of complicated by gravitational time dilation--if you were to synchronize a different pair of clocks at your feet and head using the same method after having let the first pair run for a while, I think you'd find that according to the newly-synchronized pair, the clock at your head would show less time than the one at your feet thanks to gravitational time dilation (the same would be true if you just stood on your head for a long time in a strong gravitational field--your head would age slower than your feet, in effect). It may be that if you synchronize a new pair of clocks at your head and feet very shortly before being crushed by the singularity, then the difference in time that each was showing at the instant of being crushed would be very small, and that the closer the new synchronization was to the moment of being crushed, the less the time difference.

Also, even if your feet and head are at some sense crushed at different times (one would have to specify a particular coordinate system for the falling observer to make sense of this), if you think in terms of Egan's description of the approaching the singularity being like the collapse of a hypercylindrical universe, then it may just be that the collapse seems asymmetrical in this coordinate system, with the region of space at your head having collapsed to zero radius in the two finite directions before the region of space at your feet has collapsed to zero radius in these directions. You'd still be hitting the singularity "future-first" in this picture.

Sorry, but "enforced falling" is a fact.

Our job is to choose the mathematics or physics with which to describe that fact.

It's also a fact that according to GR, the future light cone of any event that happens inside the horizon contains only events which lie at a smaller Schwarzschild radius, none which lie at a greater one. I guess you can call this "enforced falling" if you so choose, but it seems like a counterintuitive description to me, since from the perspective of a locally inertial coordinate system surrounding that event, you can still go in any direction of any of the three spacelike dimensions.

rab99 and I think that talking about movement along a time direction is confusing and against common-sense.

Who said anything about "movement along a time direction"? The observer inside the horizon has three space dimensions and one time dimension, and the worldline must be timelike as always (meaning that every point on the worldline lies in the past or future light cone of every other event on it). It's just that the light cones are tilted so that every point in the future light cone of some event inside the horizon will be at a smaller Schwarzschild radius than that event. And in Schwarzschild coordianates, the t-coordinate is a space direction inside the event horizon, while the r-coordinate is now the new time dimension; this is just a property of Schwarzschild coordinates though, it is possible to find other coordinate systems where the same coordinate is used for time both inside and outside the horizon.

I would be happier with the following explanation:

Outside an event horizon, an object follows a time-like geodesic (in space-time). The time-like geodesics go in every space direction.

Inside an event horizon … it's the same … an object follows a time-like geodesic! Except that the time-like geodesics now do not go in every space direction, but are confined within a cone. ​

Outside or inside the event horizon, objects follow timelike geodesics. Outside or inside the horizon, if you pick some event, and then pick some 3D spacelike surface which lies within the 4D future light cone of that event (like the base of an ordinary 3D cone), then there are geodesics going from the original event to every point in that spacelike surface, which is what we mean by "geodesics go in every space direction". And exactly the same thing is true inside the horizon--if you pick any 3D spacelike surface which lies within the future light cone of an event inside the horizon, then there are geodesics from that event to any point within the surface (it is probably easier to visualize if you imagine a universe with only 2 space dimensions and one time, so light cones look like ordinary 3D cones and a spacelike surface would just be a flat 2D 'bottom' to a cone whose pointy end represents the event that this is the light cone for). So, it doesn't make any sense to argue that there are spatial directions which the falling observer can't go in. Such a notion would violate the equivalence principle, which says that in any local region of spacetime picked from a larger curved spacetime, it must be possible to find a locally inertial coordinate system in that region where a freefalling observer is at rest and the laws of physics work exactly the same as in SR. Since there are no restrictions on which spatial direction you can move in SR, it can't be true that you'd see such a restriction in GR.

This uses the same physics … unlike saying "we must now move along time directions instead of space directions" … the physics is that movement is always along time-like directions, but those directions are arranged differently inside an event horizon compared with outside!

No, the Egan quote does use the same physics. I suggest that you look at some actual GR textbooks and see what they have to say about the subject before making these confident but uniformed statements about what the theory predicts. For example, a tilting light-cone diagram almost identical to the one I keep linking to at the bottom of http://www.etsu.edu/physics/plntrm/relat/blackhl.htm can be found on p. 829 of the Misner-Thorne-Wheeler textbook Gravitation, and on p. 823 they write of the problem with using Schwarzschild coordinates for events inside the horizon:

The most obvious pathology at r=2M is the reversal there of the roles of t and r as timelike and spacelike coordinates. In the region r > 2M, the t direction, $$\frac{\partial}{\partial t}$$, is timelike ($$g_{tt}$$ < 0) and the r direction, $$\frac{\partial}{\partial r}$$, is spacelike ($$g_{rr}$$ > 0); but in the region r < 2M, $$\frac{\partial}{\partial t}$$ is spacelike ($$g_{tt}$$ > 0) and $$\frac{\partial}{\partial r}$$ is timelike ($$g_{rr}$$ < 0).

What does it mean for r to "change in character from a spacelike coordinate to a timelike one"? The explorer in his jet-powered spaceship prior to arrival at r=2M always has the option to turn on his jets and change his motion from decreasing r (infall) to increasing r (escape). Quite the contrary is the situation when he has once allowed himself to fall inside r=2M. Then the further decrease of r represents the passage of time. No command that the traveler can give to his jet engine will turn back time. That unseen power of the world which drags everyone forward willy-nilly from age twenty to forty and from forty to eighty also drags the rocket in from time coordinate r=2M to the later value of the time coordinate r=0. No human act of will, no engine, no rocket, no force (see exercise 31.3) can make time stand still. As surely as cells die, as surely as the traveler's watch ticks away "the unforgiving minutes," with equal certainty, and with never one halt along the way, r drops from 2M to 0.

Likewise, on p. 3-20 of Taylor and Wheeler's Exploring Black Holes: Introduction to General Relativity they write:

Inside there is an interchange of the character of the t-coordinate and the r-coordinate. For an r-coordinate less than the Schwarzschild radius 2M, the curvature factor (1 - 2M/r) in the Schwarzschild metric becomes negative. In consequence, the signs reverse between the radial part and the time part of the metric, making the dt^2 term negative and the dr^2 term positive. Space and time themselves do not interchange roles. Coordinates do. The t-coordinate changes in character from a timelike coordinate to a spacelike coordinate. Similarly, the r-coordinate changes in character from a spacelike coordinate to a timelike one.

 

[yuiop]

A cilinder is not a curved but a flat space.

That is an interesting observation. Is that because you can glue a flat piece of paper (for example) to a cylinder without having getting any wrinkles, while there is no way you can glue the piece of paper to a sphere without getting wrinkles?

I always thought of a cylinder as having single curvature while a spheres, torus or saddle shape has multiple curvature. Presumably that view is not inline with the formal view of curved space?

 

[tiny-tim]

Occam's razor does not allow you to discard GR simply because it is not "common sense" or because you are confused about it. If you are going to invoke Occam's razor, then what alternate theory are you choosing instead which explains the data equally well and is simpler than GR?

I don't think either rab99 or I are disputing GR, not even inside an event horizon.

We just don't like your explanation of GR inside an event horizon.

Once an object is inside the event horizon of the black hole, the radial axis becomes the time axis for them …

It (talking about "movement along a time direction") isn't common-sense, it isn't clear, and it's one of those explanations which seem to make things more puzzling rather than less.

When rab99 mentioned Occam's razor, I assume he meant that he prefers, for example, an explanation which involves the same concepts inside an event horizon as outside, and not saying, for example, that the radial direction has suddenly become a time direction.

"Common Sense" is not a scientific theory

hmm … Euclid had a number of common-sense rules (I think the standard English translation is "common notions"), which were prior to his five axioms … for example, if A < B < C, then A < C.

Common-sense isn't a scientific theory on its own … but Euclid and I really don't see your objection to it being part of one!

(I've just seen JesseM's post #53 … it seems to spend a long time agreeing with me … but I'll try to find a bit I disagree with, and then come back on it! )

 

[JesseM]

We just don't like your explanation of GR inside an event horizon.

Once an object is inside the event horizon of the black hole, the radial axis becomes the time axis for them …

What if I amended it to say "the radial axis of Schwarzschild coordinates becomes the time axis for them"?

It (talking about "movement along a time direction")

Who has used this phrasing? I'm pretty sure I haven't.

(I've just seen JesseM's post #53 … it seems to spend a long time agreeing with me … but I'll try to find a bit I disagree with, and then come back on it! )

You seemed to be suggesting earlier that the observer inside the event horizon would be in some way restricted in his spatial movements, only being able to move within a cone of space, while I'm saying that he can move in any spacelike direction, but he can't avoid the singularity because it lies in a timelike direction (the radial coordinate in Schwarzschild coordinates is timelike for an observer inside the horizon), and every geodesic in his future light cone (which is quite different from a spatial cone) will end up at the singularity. Is there any of this you disagree with?

 

[MeJennifer]

That is an interesting observation. Is that because you can glue a flat piece of paper (for example) to a cylinder without having getting any wrinkles, while there is no way you can glue the piece of paper to a sphere without getting wrinkles?

Pretty much.

I always thought of a cylinder as having single curvature while a spheres, torus or saddle shape has multiple curvature. Presumably that view is not inline with the formal view of curved space?

Correct,that view is not inline.

However while a cylinder has no intrinsic curvature it does have external curvature. But for GR only intrinsic curvature matters.

 

[Dale]

We just don't like your explanation of GR inside an event horizon.

I certainly haven't made any statements or explanations about GR inside an event horizon.

I personally find the whole discussion of what happens inside the event horizon rather unscientific since, by definition, there is no data from within the event horizon. I am only concerned about how well GR works in regions where we can perform tests and acquire data.

I just get irritated by the consistent and ridiculous misapplications of Occam's razor that I see all the time. Whenever someone cannot be bothered to actually learn a theory or has some random prejudice that it clashes with they shout "Occam's Razor" as though it were some sacred principle that makes all ignorance a scientific virtue. If you want to invoke Occam's razor do it right: as a means of choosing between two scientific theories that fit the data equally well. If you ever try invoking it with only one theory on the table you automatically know that you are using it wrong.

 

[yuiop]

In this thread https://www.physicsforums.com/showthread.php?t=223730&page=2

We discussed the interior Scharzchild metric and the possibility that a true singularity does not actually form by analysing that metric. This possibility is partly supported by papers linked in that thread. Worth a look ;)

 

[tiny-tim]

Huh? All geodesics "follow a future time direction" …

erm … no … geodesics can be time-like or light-like or space-like.

All three types exist at every point in space-time.

An object moves only through space (otherwise, "movement" is meaningless); and its free-fall path in space-time (in which there is no movement) is a time-like geodesic.

I guess you can call this "enforced falling" if you so choose, but it seems like a counterintuitive description to me, since from the perspective of a locally inertial coordinate system surrounding that event, you can still go in any direction of any of the three spacelike dimensions.

I used the phrase "enforced falling" only to describe the subject-matter: my explanation of that subject is:

the time-like geodesics now do not go in every space direction, but are confined within a cone.

You refer to "the three spacelike dimensions" … this is counter-intuitive, since it is not obvious which space directions they correspond to.

I prefer to refer to "every space direction", with its usual meaning. I can then relate those (three-dimensional) space directions to the (four-dimensional) time-like geodesics.

Who said anything about "movement along a time direction"?

I was referring to your:

Once an object is inside the event horizon of the black hole, the radial axis becomes the time axis for them …

in which you seemed to be saying that the radially in-falling object is not moving along a space direction.

What if I amended it to say "the radial axis of Schwarzschild coordinates becomes the time axis for them"?

It's still true, but I still don't like it … it's now even further away from reality. I want explanations which use concrete concepts such as directions, not abstract ones like coordinate axes.

So, it doesn't make any sense to argue that there are spatial directions which the falling observer can't go in.

But there are! I entirely accept that there are, and I also understand why GR requires it … but I don't accept that GR denies the existence of tangential (or, more generally, out-of-cone) space directions!

Such a notion would violate the equivalence principle, which says {snip} the laws of physics work exactly the same as in SR.

I don't think the equivalence principle does require space inside an event horizon to be locally indistinguishable from space outside. The laws of physics must be indistinguishable, but their application need not be.

For example, do you accept that material objects inside an event horizon must travel faster than light, and that that alone distinguishes inside from outside, even for an inertial observer?

For example, a tilting light-cone diagram almost identical to the one I keep linking to …

Yes … but enough with the light-cone diagrams, already!

I have seen them … I've even seen the one in Finkelstein's original 1956 paper!

You seemed to be suggesting … while I'm saying …{you wrote "spacelike" - I assume you meant "timelike"?} … Is there any of this you disagree with?

We're both correct! I'm using three-dimensional space directions to explain why geodesics end in the singularity, and you're using four-dimensional time-like directions for the same purpose.

My only issue is with "the singularity … lies in a timelike direction" … that makes it look as if the singularity is a point in space-time … but it's a line, isn't it, with different bits of it in different timelike directions?

To summarise my approach:

Geodesics are four-dimensional curves (which involve no movement).
They can be projected onto three-dimensional space.
Every free-fall object has a time-like geodesic.
It moves along the three-dimensional projection of that geodesic, but inside an event horizon not all directions are projections of time-like geodesics. ​

 

[JesseM]

erm … no … geodesics can be time-like or light-like or space-like.

I was talking about the geodesics of physical objects--there are no objects which follow spacelike geodesics. And I would interpret the phrase "follow a future time direction" in a way that would cover light-like geodesics (since for any two events which lie on a light-like geodesic, all observers will agree on which came earlier and which came later), though perhaps you would define it differently.

An object moves only through space (otherwise, "movement" is meaningless); and its free-fall path in space-time (in which there is no movement) is a time-like geodesic.

I didn't say anything about an object "moving" through time, did I? And yes, of course I agree that a non-massless particle in free-fall will follow a timelike geodesic. Why are you telling me this? Do you think I was saying something different in what I wrote?

I used the phrase "enforced falling" only to describe the subject-matter: my explanation of that subject is:

the time-like geodesics now do not go in every space direction, but are confined within a cone.

I can't think of any way to interpret this statement in a way that doesn't make it nonsense. Do you agree that different objects passing through a particular point in spacetime can end up at any point in the future light cone of that point in spacetime depending on their velocity, regardless of whether the point is inside or outside the event horizon? Do you also agree that objects going through a given point can never end up at a point outside the light cone of that point, outside the horizon as well as inside? If so, in what sense do you think objects can "go in every space direction" outside the horizon but not inside the horizon, and in what sense are they "confined to a cone" inside but not outside?

You refer to "the three spacelike dimensions" … this is counter-intuitive, since it is not obvious which space directions they correspond to.

An observer is obviously free to orient his three spatial axes in any spacelike direction he wants, the point is that he can come up with some locally inertial coordinate system in his local region that has three spatial coordinates and one time dimension, such that the usual laws of SR apply in this region.

I prefer to refer to "every space direction", with its usual meaning.

What is the "usual meaning"? And do you agree that an observer inside the horizon is indeed free to move in any spatial direction, but there is no spatial direction that takes him further from the singularity in Schwarzschild coordinates?

I was referring to your:

Once an object is inside the event horizon of the black hole, the radial axis becomes the time axis for them …

in which you seemed to be saying that the radially in-falling object is not moving along a space direction.

As I mentioned, I meant "radial axis" to refer to the usual Schwarzschild coordinates.

What if I amended it to say "the radial axis of Schwarzschild coordinates becomes the time axis for them"?

It's still true, but I still don't like it … it's now even further away from reality. I want explanations which use concrete concepts such as directions, not abstract ones like coordinate axes.

It's difficult to make meaningful statements about space and time that don't refer to coordinate systems. And at least when talking about locally inertial coordinate systems, the coordinates do have a very simple physical meaning--they represent measurements on a grid of rulers and clocks moving inertially.

But there are! I entirely accept that there are, and I also understand why GR requires it … but I don't accept that GR denies the existence of tangential (or, more generally, out-of-cone) space directions!

Can you define "tangential" without referring to a coordinate system like Schwarzschild coordinates? I suppose "out-of-cone" is a start, but I haven't claimed that there are no events on the event horizon which lie out of the light cones of an event inside the horizon--of course there are! But that doesn't mean the horizon lies in any particular spatial direction for an observer inside the horizon--for this observer I think it would be a spacelike surface that lies in their past (as defined in whatever coordinate system they're using inside the horizon, not all parts of the surface would lie in their past light cone), much like the spacelike surface consisting of the set of all events that happened precisely 10 billion years after the Big Bang in comoving cosmological coordinates. Do you agree this surface lies in our past, not in any particular spatial direction for us? Do you also agree that there are plenty of events on this surface which don't like in our past light cone?

I don't think the equivalence principle does require space inside an event horizon to be locally indistinguishable from space outside. The laws of physics must be indistinguishable, but their application need not be.

I don't understand your distinction between the "laws of physics" and their "application". Do you agree that any experiment done in a small windowless room over a small period of time will have the same result regardless of whether the room is inside our outside the horizon, provided the region of spacetime is small enough that there are no significant tidal forces?

For example, do you accept that material objects inside an event horizon must travel faster than light, and that that alone distinguishes inside from outside, even for an inertial observer?

No. They may have a coordinate speed greater than c in Schwarzschild coordinates (which is different from 'faster than light', since a light beam in the same region will have a greater coordinate speed), and even in SR if you use non-inertial coordinate systems objects can move faster than c, but in any local region it's possible to use freefalling rulers and clocks in that region to create a locally inertial coordinate system in that region, and nothing will move faster than c in this coordinate system.

You seemed to be suggesting … while I'm saying …{you wrote "spacelike" - I assume you meant "timelike"?} … Is there any of this you disagree with?

We're both correct! I'm using three-dimensional space directions to explain why geodesics end in the singularity, and you're using four-dimensional time-like directions for the same purpose.

When I wrote "while I'm saying that he can move in any spacelike direction, but he can't avoid the singularity because it lies in a timelike direction", I did mean "any spacelike direction"; in other words, if he constructs a locally inertial coordinate system, he can move along any of the three orthogonal rulers, his movements are not restricted to a cone in space as I was thinking your quote was suggesting.

My only issue is with "the singularity … lies in a timelike direction" … that makes it look as if the singularity is a point in space-time … but it's a line, isn't it, with different bits of it in different timelike directions?

Yes, but that's why I emphasized the part in Egan's quote about the "approach to the singularity" looking like a collapsing hypercylinder from the perspective of an observer inside. If you picture a 2D universe on the surface of a regular cylinder, and the radius of the cylinder is shrinking until it hits zero at some moment, then this is a line singularity rather than a point, but it still lies in a timelike direction for a flatlander living on the cylinder...before the cylinder has collapsed, there's no spatial direction the flatlander on the surface can point to and say "singularity that-a-way".

To summarise my approach:

Geodesics are four-dimensional curves (which involve no movement).
They can be projected onto three-dimensional space.
Every free-fall object has a time-like geodesic.
It moves along the three-dimensional projection of that geodesic, but inside an event horizon not all directions are projections of time-like geodesics. ​

"not all directions are projections of time-like geodesics" is wrong if "directions" is meant to refer to spatial directions--if you foliate a black hole spacetime into a stack of spacelike hypersurfaces, then all directions in a given hypersurface will be a projection of a time-like geodesic. I think the issue with Schwarzschild coordinates is that the set of all events at a particular coordinate time t does not represent a spacelike hypersurface, only the portion outside the event horizon would be spacelike.

When you say "projected onto three-dimensional space" this is just too vague without a particular coordinate system and a particular definition of simultaneity (since you are obviously talking about position in space changing over time, which requires us to have a meaningful notion of what space looks like at a particular time). But if you do pick a coordinate system which assigns every event in the spacetime a time-coordinate, then there are two possibilities:

1. the set of all events at a single time-coordinate is always "spacelike" in the physical sense (no event in the set lies within the light cone of any other in the set), in which case every event will have valid timelike geodesics going in every direction in space.

2. The set of all events at a single time-coordinate is not a spacelike surface, so it doesn't make sense to say that projections of geodesics onto this surface qualifies as projecting the geodesics "onto three-dimensional space".

So, either way, I think I disagree with your summary above.

 

[yuiop]

A cilinder is not a curved but a flat space.

Hi again Jennifer,

Does that mean an almost infinitely long body with most of its mass in a central cylindrical core parallel to the main axis (similar cross section to that of the Earth) can be handled by Special relativity (eg using Minkowski spacetime) ?

I do not want to hijack this thread so could you reply to thread I started here https://www.physicsforums.com/showthread.php?t=225573&page=3 which is where my question relates to? (basically it asks if a horizontally moving object will fall at the same rate as a purely vertically falling object as measured by an observer at rest with such a gravitational body)

Thanks :)

 

[MeJennifer]

Does that mean an almost infinitely long body with most of its mass in a central cylindrical core parallel to the main axis (similar cross section to that of the Earth) can be handled by Special relativity (eg using Minkowski spacetime) ?

I would say no, it seems to me that such a configuration would still give a curved spacetime.

 

[yuiop]

I would say no, it seems to me that such a configuration would still give a curved spacetime.

Is that because we are talking about concentric cylinders and moving from one to the other rather than staying on the surface of one cylinder?

I am, by the way only talking about motion parallel to the main axis of the cylinder and not motion around it. I am also talking about a fall distance dr that is infinitessimal compared to radius (R) of the massive body, so we can consider R to be aproximately constant. For example on Earth, considering the acceleration to be a constant 1g is a reasonable aproximation when we are talking about a fall of a few meters.

 

[JesseM]

Hi again Jennifer,

Does that mean an almost infinitely long body with most of its mass in a central cylindrical core parallel to the main axis (similar cross section to that of the Earth) can be handled by Special relativity (eg using Minkowski spacetime) ?

You have to distinguish between the notion of an ordinary physical cylinder in 3D space, and the notion of 3D space itself being represented as a hypercylinder in a 4D embedding diagram. MeJennifer was talking about 3D space having a hypercylindrical shape (which as she said involves no intrinsic curvature, so it's really just flat space with an unusual topology that makes it finite in one or two directions), I would think a physical cylinder would cause some intrinsic curvature in GR.

 

[MeJennifer]

MeJennifer was talking about 3D space having a hypercylindrical shape (which as she said involves no intrinsic curvature, so it's really just flat space with an unusual topology that makes it finite in one or two directions), I would think a physical cylinder would cause some intrinsic curvature in GR.

Basically correct, but in you explanation you are one dimension short.

GR curves a 4D not a 3D space.

 

[JesseM]

Basically correct, but in you explanation you are one dimension short.

GR curves a 4D not a 3D space.

GR does talk about curvature of 4D spacetime, but the cylinder represents a topology for flat 3D space, similar to the different possible finite topologies for space discussed in this article (though the topologies discussed there are finite in all directions, while the a space with the topology of a cylinder would be finite in some directions and infinite in others).

 

[yuiop]

You have to distinguish between the notion of an ordinary physical cylinder in 3D space, and the notion of 3D space itself being represented as a hypercylinder in a 4D embedding diagram. MeJennifer was talking about 3D space having a hypercylindrical shape (which as she said involves no intrinsic curvature, so it's really just flat space with an unusual topology that makes it finite in one or two directions), I would think a physical cylinder would cause some intrinsic curvature in GR.

Surely, if Minkowski spacetime can handle the case of an accelerating rocket, then it can handle the motion of a falling particle in the idealised cylindrical planet?

Does not the Equivalence principle require that they are the equivalent?

An accelerating rocket can not duplicate the the gravity of an spherical massive body, but surely it can duplicate an idealised gravitational flat gravitational body that has planar symmetry horizontally? Otherwise, what is the point of the EP?

[EDIT] Maybe I should phrase it another way. What hypothetical gravitational body is equivalent to an acccelerating rocket? If the answer is none, it makes the EP invalid.

 

[JesseM]

Surely, if Minkowski spacetime can handle the case of an accelerating rocket, then it can handle the motion of a falling particle in the idealised cylindrical planet?

An accelerating rocket doesn't curve spacetime to any significant degree, a cylindrical planet would.

Does not the Equivalence principle require that they are the equivalent?

Only if you zoom in on a very small region of the curved spacetime where the curvature was negligible.

An accelerating rocket can not duplicate the the gravity of an spherical massive body, but surely it can duplicate an idealised gravitational flat gravitational body that has planar symmetry horizontally? Otherwise, what is the point of the EP?

[EDIT] Maybe I should phrase it another way. What hypothetical gravitational body is equivalent to an acccelerating rocket? If the answer is none, it makes the EP invalid.

The point of the equivalence principle is that the laws of physics in the local spacetime neighborhood of a freefalling observer in curved spacetime must reduce to the laws of physics in inertial frames in flat SR spacetime. There's no way the laws of physics in a large region of curved spacetime where tidal forces are significant can be treated as equivalent to the laws of SR.

 

[yuiop]

An accelerating rocket doesn't curve spacetime to any significant degree, a cylindrical planet would.

Sure it would curve it around the cylinder, but parallel to the cylinder it would be horizontally flat. What if we replaced the cylinder with flat body with "almost" infinite horizontal dimensions?

Only if you zoom in on a very small region of the curved spacetime where the curvature was negligible.

The point of the equivalence principle is that the laws of physics in the local spacetime neighborhood of a freefalling observer in curved spacetime must reduce to the laws of flat SR spacetime.

A vertically free falling observer in a falling elevator would observe that a horizontal light beam, a horizontally moving particle and a released stationary particle do not fall relative to the elevator. They behave as if the elevator was far away from any gravitational body. An observer that was not free falling, would observe that the particles and the light beam and the elevator all appear to be falling at the same rate. So why does GR predict that a particle moving horizontally falls faster than a particle without horizontal motion, even when we consider a flat gravitational body?


P.S Does that mean Rindler spacetime is only valid for accelerating rockets and cannot be applied to even hypothetical gravitational bodies?