Introduction to Black Holes: Newbie Questions Answered

[Guy From Alberta]

Physics Post 2

Thanks Grizzlycomet for the link to “time dilation.” I am finding this all very interesting. I want to come back to this time dilation theme in a short bit.

I have a number of questions arising; hope it is OK to just spout them out. In this topic, some of us have touched on how gravity does indeed “curve the spatial geometry of the universe.” Can we further define “spatial geometry?”

Now, we have also been talking about “time.” ; and how a definition of time would be dependent upon the frame of reference being used. A “frame of reference;” I presume as meaning a collection of condition, axis, or assumption; which establish how something will be approached or understood. As “Turin” mentioned in a post above “coordinate time;” and “proper time;” and he mentioned how some call “coordinate time, the “fourth dimension;” I came up with more questions, possibly related to some of my earlier questions here about time.

From what I see so far, time is not really the fourth dimension of space; but of “spacetime.” It is within the sphere of general or special relativity where we see time, plus three dimensional space being treated together as a single, four dimensional “manifold,” called “space time.” And spacetime cannot be viewd as a fixed background; but rather, a networking and developing of certain evolving relationships. And, it is a spacetime interval between two events, that results in an entity analogous to distance?

What exactly are a) the dimensions of space?
b) the dimensions of spacetime?

What does it mean to say “a spacetime interval between two events?” How is an “interval” best defined, and what is an example of this kind of “event,” in a case like this?

How are “coordinate time,” and “proper time,” affected by/related to said “Intervals,” and “events?”

Do black holes teach us anything about coordinate time, or proper time?

 

[turin]

For someone who does not have a strong mathematical or physics background, you sure do know who to ask ambitious questions. Let's see what we can say about them.

I have a number of questions arising; hope it is OK to just spout them out.

You(r questions) are one of the reasons I visit the forum. Please ask all the questions you want, or I'll start to get bored.

In this topic, some of us have touched on how gravity does indeed “curve the spatial geometry of the universe.” Can we further define “spatial geometry?”

Gravity curves space-time, as well as space in itself. Geometry is what gets curved. Imagin a volume. This volume is filled with points. Each point has an identity. They form a set. There is also some notion of points being close to each other and points being far away. This notion (called topology, I think, I'm still learning this stuff) is not sufficient to support a notion of curvature. So the points are endowed with a more specific notion of, not only close or far, but how close or how far. This notion is called geometry. (This notion does not have to arise from the same closeness relationships as the topology, as I understand it, but I might not understand it.) It was inspired by Euclid, but later on, people like Gauss and Riemann came along and decided that Euclid put unnecessary restrictions on his geometry.

You can think of geometry as the collection of distances between all the points together with those points between which the distance is defined. Spatial geometry implies that these distances are always positive definite for any two distinct points.

A “frame of reference;” I presume as meaning a collection of condition, axis, or assumption;

Not exactly. There is a more exact definition in the context of relativity. This is one of the things to which I think Einstein (the man himself) gives a decent portrayal. He gives a working definition of a frame of reference is a set of intersecting planes. The (perpendicular) distance that would be measured to these planes gives the (spatial) coordinates of any point in this frame. If you imagine that two sets of such intersecting planes can exist without mutual interference, and then further imagine that they move wrt each other, then these two sets of intersecting planes give two distinct frames of reference.

A frame of reference is something to which you refer to give meaning to your expression. In relativity, this means a set of intersecting planes that allow you to label points with a set of numbers.

From what I see so far, time is not really the fourth dimension of space; but of “spacetime.” It is within the sphere of general or special relativity where we see time, plus three dimensional space being treated together as a single, four dimensional “manifold,” called “space time.”

Exactly

spacetime cannot be viewd as a fixed background;

This is a GR notion. In SR, spacetime is fixed for sure. Since you really want to talk about BHs, then you are correct to say that the geometry is dynamical. Whether spacetime itself, on which the geometry is endowed, is dynamical is more of a philosophical debate (as I understand it). Actually, this is another item for which I rather liked the treatment of Einstein. (See Relativity, 5th ed., pp. 135-57, "Appendix V: Relativity and the Problem of Space", or the first chapter in The Meaning of Relativity, I forget what it's called.)

but rather, a networking and developing of certain evolving relationships.

Very good. When you get right down to it, physics can only go so far as to make epistemic judgements.

What exactly are a) the dimensions of space?
b) the dimensions of spacetime?

The dimensions of space are the degrees of freedom that a point particle may utilize. A point particle can go up, right, or out. Any other behavior of a strict point particle is a combination of scaled versions of these behaviors (including negative and vanishing multiples). The dimensions of spacetime are not quite so understandable. They do not indicate degrees of freedom for a point particle. The reason why spacetime is treated like a 4-D space is that, there are results of a certain type of transformation (Lorentz transformation), which are very much like what rotations would be in 4-D space.

Imagine just a 1-D space, like a line. A particle in this space has 1 degree of freedom. It can go, let's say, to the right. (Going to the left is just a negative scaled multiple thereof.) Now, let us consider the behavior wrt time. This behavior can most readily be considered from a 2-D type construct called the 1+1 D spacetime for this particle. If we further posit onto this structure the rules of SR, then there is a characteristice wedge in this 2-D construct. I will hereafter refer to this wedge as the "lightcone." The interior of the lightcone is allowed, however, the exterior is forbidden. This lightcone follows the particle through the 2-D construct. It would look kind of like a tiny car with its headlights shining out front with a spread beam (though this is not why it is called a lightcone). The car is allowed to drive wherever the beam shines. 2 stipulations on the car: 1) Regardless of the direction of the car, the beam always shines in the same direction, 2) the car always moves forward at the same speed. Clearly, this car does not have the freedom to enjoy every point in this 2-D construct, so the time dimension doesn't exactly add a degree of freedom, at least, not like a spatial degree of freedom, not a direction that a particle can choose.

What does it mean to say “a spacetime interval between two events?” How is an “interval” best defined, and what is an example of this kind of “event,” in a case like this?

The only way I really know how to field this one is with some math. An event is a "point" in spacetime. Return again to the 1+1 D spacetime (forget about the car for the moment). Imagine 2 distinct points in this spacetime (any two will do). Imagine the straight line segment connecting them. This line segment represents the interval. In Euclidean geometry, this interval would be quantified by the Pathagorean theorem. In spacetime geometry (which is not Euclidean), the Pathagorean theorem is generalized to a metric.

Imagine that the time axis is verticle and the space axis is horizontal. Then, qualitatively, the more verticle the interval, the greater it is. Intervals that are not verticle are shorter, to the limit that the interval coincides with the edge of the lightcone (as in the car example, the edge of the head light beam). If the interval is even more horizontal than the edge of the lightcone, then it is negative. This is clearly not possible using the Pathagroean theorem. (Note: you may have heard of a treatment that invokes an ict axis, and that claims the validity of the Pathagroean theorem therefrom. This is OK if all you ever want is a cursory understanding. The idea of non-Euclidean geometry is mathematically more sophisticated and will take you further. I have never tried to understand GR in terms of the ict notion, but I don't think that the ict notion can even be carried that far, so, even for our purposes, it is useless.)

How are “coordinate time,” and “proper time,” affected by/related to said “Intervals,” and “events?”

There is a concept called a "worldline." This is the collection of all events that represent the existence of a particle through time. Returning once again to the 1+1 D spacetime, any particle in the 1-D space will actually be a line (or, in general, a curve) in the 1+1 D space time. This line, which can be considered the posisition of the particle at all points in time, is the worldline. According to the rules of relativity, two distinct events on the worldline of a particle are separated by an interval that is always more horizontal than the edge of the wedge. Therefore this interval is positive. The proper time (between these two events) is one way of quantifying the length of this interval. The proper time can be directly related to the coordinate time interval (Δt) by the time dilation (Lorentz transformation).

The proper time usually implies an interval that does not "violate" the light cone. For intervals that do extend outside the light cone (more horizontal than the edge of the lightcone), the value changes sign (becomes negative), and it is usually denoted as the proper distance instead of the proper time. Fundamentally, proper time and proper distance are the exact same thing (with the occasional exception of a sign convention), and it is just a matter of context that determines the particular label.

Do black holes teach us anything about coordinate time, or proper time?

Yes. For instance, "inside" a BH, the distance from the center becomes another time like axis. That is, once a particle enters a BH, it loses another degree of freedom, because it now not only must move forward in time, but its distance from the center of the BH must consistently decrease. At the center, the worldline of the particle terminates.

 

[Guy From Alberta]

For someone who does not have a strong mathematical or physics background, you sure do know who to ask ambitious questions...
You(r questions) are one of the reasons I visit the forum. Please ask all the questions you want, or I'll start to get bored.

Thanks Turin...I appreciate the vote of confidence. My mother always used to say something similar - that I asked a lot of questions which kept her hopping. But, believe it or not; I am just a simple trucker type guy who is really interested in certain scientific fields, such as physics. When a person is really interested in something; it seems to help. I have about 14 years experience in nursing, before what I do now, so the sciences seem to ring a bell with me.

One of the several reasons I am trying to explore the black hole angle, in relation to time, and/or space time; is that I believe human beings can understand themselves better; when they understand the "physics" of who they are; and where they are in space/time. This is why I mentioned in the beginning of this thread that some of the many ideas ruminating in my cerebrum, may at some point delve into the philysophical end of things here.

But, before I can really explain all that; I feel I need to study more into some of the basic, physical concepts. There are some things that I would like to understand better.

For now; I just have one, quick, clarifying question about black holes. My recent studies indicate that there are atleast 4 different kinds of black holes...the Kerr Black Hole; the Kerr-Newman, the Reisner-Nordstrom; and the Swarzchild Black Hole.

Now, in your last post to me; you stated: ""inside" a BH, the distance from the center becomes another time like axis." I apologize; but I don't quite get what is meant by a "time axis;" or a "time like axis." What is meant by "axis" in this context? And, if we can figure me out that far; I find myself wondering about some pertinent differences:

a) between black holes that are rotating (Kerr-Newman), and black holes which have an "angular momentum." (Reisner-Nordstrom).
b) between BHs that are "charged" and BHs which have zero charge; (as in Kerr-Newman vs the Swarzchild BH).
c) If BHs teach us something about "time," or "space-time;" how would the things we can learn about time differ, with each of these 4 types of BHs?

I have some other questions; but I need to take this all just a bit at a time. If I shift gears too fast; I'll stall my "rig." And I am thinking my current questions may change a bit, if we can answer these ones.

Thanks everyone for taking the time to reply here. I am enjoying it.

 

[turin]

in your last post to me; you stated: ""inside" a BH, the distance from the center becomes another time like axis." I apologize; but I don't quite get what is meant by a "time axis;" or a "time like axis." What is meant by "axis" in this context?

There are a few ways to look at it.

You don't have access to the future (classically speaking). You only gain information in the future by "going there;" processes do not flow from the future to the past, and neither do you. In fact, the future doesn't even really exist. You move along and constantly "lay down" your worldline as you go, like building a road right in front of you (or maybe like leaving footprints). The only stipulation is that you do not violate the light cone. This is not a philosophical issue, this is more of a definition. There is a fundamental principle in (classical) physics called "causality." When we speak of a path that is time-like, we mean that the points at larger values of the parameter along this path are a direct result of the points at the lesser values of the parameter, thus the parametrization. Another consequence is that we may orient this axis with the time axis in a meaningful way. Essentially, any path/line/curve that doesn't violate the lightcone is (potentially) somebody's timeline.

Let's call the r-axis (in the Schwarzschild geometry) the distance from the center (of the coordinate system/BH). There is a characteristic radius, outside of which, there are three spatial degrees of freedom and one temporal dimension, the coordinate of which is always increasing for physical processes. This is the "flow of time" if you will. This is the feature of time that I think will make things most clear. You must increase your time coordinate value. You must pass from one hyperplane of 3-space onto another with a greater value of t. You cannot pass to a hyperplane with a lesser value of t. This is a timelike quality of t, or of the temporal axis/direction. (Of course, you can design a clock that moves its hands counterclockwise, but that is not the point. The real point is not that time always increases, but that it changes monotonically. You can say that the parameter you have chosen is actually decreasing throughout a physical process; that is arbitrary. The convention is to denote the variation as an increase; whether the parameter be increasing or decreasing, the parameter is consistently doing so throughout the entire physical process. This is what makes it a time-like parameter, the fact that the variation cannot take place in the other sense.)

What does this have to do with the r-axis being timelike? Simply this: inside a BH, the r coordinate of a particle cannot increase. It is doomed to decrease monotonically. Thus, any physical process (I use the term freely) inside a BH can be parametrized according to its distance from the center, or r coordinate. Thus, this parameter, the r coordinate, can be
isomorphically transformed into time, and is therefore timelike.

If it's philosophy that you're after, then here's a thought (not an answer, but a thought): Physical processes must have a monotonic variation in their timelike parameter. Is consciousness so intimitely connected to physical process that it too must follow such a parameter, or is it possible to survive a conscious eternity within a BH? I suppose this could give some indirect insight into the nature of consciousness.

a) between black holes that are rotating (Kerr-Newman), and black holes which have an "angular momentum." (Reisner-Nordstrom).

I don't understand the difference. I will have to do some research. I am only moderately familiar with the Kerr metric, and I have been under the impression that the only two types of metric are the Schwarzschild and the Kerr. You see, you have stimulated an investigation. This is an example of why I want you to come here and ask your questions.

b) between BHs that are "charged" and BHs which have zero charge; (as in Kerr-Newman vs the Swarzchild BH).

Again, I will had to do some research. I know absolutely nothing about the influence of charge on the metric.

 

[Guy From Alberta]

Quote by Turin

You don't have access to the future (classically speaking). You only gain information in the future by "going there;" processes do not flow from the future to the past, and neither do you. In fact, the future doesn't even really exist. You move along and constantly "lay down" your worldline as you go, like building a road right in front of you (or maybe like leaving footprints). The only stipulation is that you do not violate the light cone. This is not a philosophical issue, this is more of a definition. There is a fundamental principle in (classical) physics called "causality." When we speak of a path that is time-like, we mean that the points at larger values of the parameter along this path are a direct result of the points at the lesser values of the parameter, thus the parametrization. Another consequence is that we may orient this axis with the time axis in a meaningful way. Essentially, any path/line/curve that doesn't violate the lightcone is (potentially) somebody's timeline.

Thankyou for this interesting reply.

I just need to ask re the causality principle for now; and will get back to some of the other items when time permits.

Are you mentioning "causality" here in the context of "time axis;" and if so, how would we reconcile the phenomena or occurrence of "causality problems," (or exceptions)? I know that S. Hawking's "chronology protection conjecture," states that the laws of physics do not allow time machines, for eg., and that "time travel," is only possible "microscopically," whatever that might mean. But in a "causality problem" a real paradox can occurr when some given particle moves faster than the speed of light...the "causality problem" happening from the fact that when 2 events, A and B, happen faster than the speed of light; (for a stationary frame of reference), then, accordingly, event B can happen before event A?

A fellow named K. Thorne, has proposed that something called "wormholes" could be used as "time machines," but I don't understand how a human being could access a "wormhole," since, admittedly they are hypothetical. According to special relativity, from what I understand, traveling faster than light is equivalent to traveling backwards in time? I guess all this is suggesting to me that perhaps, there may be a way we do have access to the future; as we consider "causality loops," where events of the future cause events of the past...it is getting really interesting as I study this; but I do not feel as though I yet understand it all in it's correct context...O well; with more study; I am bound to figure this out better.

Till next time.

 

[turin]

Guy From Alberta,
I looked into the 4 types of BH's. This is what I've got so far:
Schwarzschild:
Is as I described. This is my default BH, so, if I am talking about a BH and I do not specify (or the context does not make the distinction obvious), then I am talking about the friendly Schwarzschild BH, or, more exactly, the associated metric/geometry. The only property that distinguishes one such BH from another is the mass. It is modeled as a singular point at the coordinate r = 0. There is one critical horizon, from r < such a particle cannot theoreticall escape.
Kerr:
This kind of BH has the additional identifying quality of angular momentum, L. Thus, whereas two BH's that both have mass = M are identical in the Schwarzschild case, they can be distinct in the Kerr case in their L's. This one is complicated (to me), so I will leave it to someone else to explain the finer points to you. From what I gathered, it seems that the Kerr BH has two horizons, but that will warrant further investigation.
Reissner-Nordstrom:
This kind of BH is the other extension of the Schwarzschild BH, but, instead of including L, it includes a charge, Q. I have read the justification that these are not expected to exist stably. I don't know if I agree, but, at any rate, their principle existence is not denied by anyone (from whom I have heard). These kinds of BH clearly have two horizons. I am still looking into the meaning.
Kerr-Newmann:
This kind of BH is believed to be the most general form possible in principle. I don't know anything about it.

Are you mentioning "causality" here in the context of "time axis;"

I don't know what you mean. I will define causality (as I understand the physicists to use the term) as the notion that all physical processes follow a common parameter in a monotonic sense, and that this sense is preserved for any transformation to a valid physical frame of reference. This parameter may or may not be the value of the projection on the time axis. One thing I feel I should point out about this notion/principle (and indeed about science in general) is that it is axiomatic and should not be endowed with any deeper meaning. There are good reasons for the postulate, but the reasons have never been (up to this point) absolutely conclusive.

how would we reconcile the phenomena or occurrence of "causality problems," (or exceptions)?

In general, there are two possibilities: 1) show that the problem/exception is merely ostensible, 2) eliminate the axiom that leads to the problem/exception. If you are referring to a specific case, let's hear it.

I know that S. Hawking's "chronology protection conjecture," states that the laws of physics do not allow time machines, for eg., and that "time travel," is only possible "microscopically," whatever that might mean.

If I detect a hint of cynicism here, then I would say that I am right there with you. I have heard/read/seen all kinds of crap (and by "crap" I mean reasoning for the inability to change the sense of one's time'parametrization wrt the rest of the physical system with which one is causally connected). This has been "disproven" on the micro-scopic level (to what I consider an acceptable degree of confidence), and even a mere description of decoherence into a causal mode of parametrization has yet to be clearly demonstrated (so far as I've seen). The basis of assertions of the "chronology protection conjecture" range from qutie convincing (albeit empirically devoid) billiard ball type situations to innane rhetoric such as "where are all the tourists." The resulting assertions have a stinkingly conjectural basis.

... a real paradox can occurr when some given particle moves faster than the speed of light...the "causality problem" happening from the fact that when 2 events, A and B, happen faster than the speed of light; (for a stationary frame of reference), then, accordingly, event B can happen before event A?

I think you've just about got it. Although, I would restate it:

"A real paradox could occur if some given particle moves faster than the speed of light...the 'causality problem' happening from the fact that when 2 events, A and B, happen in such a way that A is to cause B at a point in time much earlier than a light signal could arrive at B from A, ... then, according to the causal structure indicated by SR, event B could happen before event A, which is inconsistent with said causal structure."

A fellow named K. Thorne, has proposed that something called "wormholes" could be used as "time machines," but I don't understand how a human being could access a "wormhole," since, admittedly they are hypothetical.

It is not so much their hypothetical nature as it is the requirement of superluminal velocity, which is hypothetically impossible (is that an oxymoron?), in order to make use of them.

According to special relativity, from what I understand, traveling faster than light is equivalent to traveling backwards in time?

Not quite. In the frame of reference wrt which the velocity is faster than light, the object is still "moving forward through time." The issue is that, upon a Lorentz transformation, traveling faster than light in any inertial frame leads to inconsistent "direction of travel through time" in distinct inertial reference frames (which can result in "travelling backwards in time"). That, together with the principle of causality, is what prohibits superluminal velocity.

I guess all this is suggesting to me that perhaps, there may be a way we do have access to the future; as we consider "causality loops," where events of the future cause events of the past...

What is a "causality loop?" Do you mean a loop-hole in the theory re causality? SR provides no such loop-hole, and I am pretty certain that neither does GR. QM? Maybe. Reality? That's a totally different story.

 

[Guy From Alberta]

Hi Turin

I agree with, and see your points re "causality." I actually do not intend to insinuate any exceptions to "causality" here; but bear with me a few days till I figure out what I do mean by that. I uncovered "causality loop" somewhere in my recent studies; and need to figure out where. I agree, that everything needs to be governed by definite laws of physics. And perhaps, other "laws." Your philosophical question re conscience is one I will defintely get back to as well. Very interesting.

I wonder though...is it possible for anything to travel faster than light? If so; what are some examples?

 

[turin]

I agree with, and see your points re "causality." I actually do not intend to insinuate any exceptions to "causality" here; but bear with me a few days till I figure out what I do mean by that.

I appologize if my responses seemed like an affront. My style may be a little obstreperous sometimes. You may have a better definition than I; I merely wanted to rigorously indicate the definition that I have in my mind when the term causality is used in the context of relativity.

I agree, that everything needs to be governed by definite laws of physics. And perhaps, other "laws."

Well, I'm not so sure that I strictly agree with that. Perhaps there are physical laws; but, perhaps they are mental constructs that enable the human beast to cope with its existence in an orgainzed way that promotes evolution. Introspectively, I cannot rule out the latter. Furthermore, in my own mind, I cannot comprehend of a set of laws, no matter how fundamental, without a "legislature," nor can I comprehend of every object in the universe adhering to these laws without an immense "bureaucracy."

Your philosophical question re conscience is one I will defintely get back to as well.

Just keep in mind that I in no way claim to know the answer. One reason why I posed the question was to seed the distinction between the rigorous relativistic definition of time, and your personal phenomenology of time.

...is it possible for anything to travel faster than light? If so; what are some examples?

Of course, if you allow for a loose definition of "anything" and "travel." The most immediate example that comes to mind is phase velocity.

Imagine a wave (in the ocean) that is coming into shore at a slight angle. Associate the speed of the wave with the velocity of light. Then, from some perdendicular distance from the shore, d, the wave will take some time to arrive at the shore, t. d/t is less than the velocity of the wave, so it can be associated with less than the velocity of light. Now consider the incidence. The wave hits some point at one end of the shore, and, as the wave continues to move in, that point slides down to the other end. This sliding of the point of incidence happens extremely quickly, in fact, much more quickly than the speed of the wave. You can associate this speed with the phase velocity along the shore.

The phase velocity does not causally connect events. The event of the wave hitting the shore at one end has no influence on the wave hitting the shore at the other end. Both events had some exterior cause out in the distant ocean. This is one model that people have used to justify the phenomenon of entanglement (of course, with matter waves rather than ocean waves).

 

[Guy From Alberta]

Quote by Turin

What does this have to do with the r-axis being timelike? Simply this: inside a BH, the r coordinate of a particle cannot increase. It is doomed to decrease monotonically. Thus, any physical process (I use the term freely) inside a BH can be parametrized according to its distance from the center, or r coordinate. Thus, this parameter, the r coordinate, can be
isomorphically transformed into time, and is therefore timelike.

If it's philosophy that you're after, then here's a thought (not an answer, but a thought): Physical processes must have a monotonic variation in their timelike parameter. Is consciousness so intimitely connected to physical process that it too must follow such a parameter, or is it possible to survive a conscious eternity within a BH? I suppose this could give some indirect insight into the nature of consciousness.

This is just another quick question - I find myself wondering how/if this "nature of consciousness" would be affected when we consider whether or not the person is kind and loving; or whether they are mean and wicked? How might personality affect this monotonic variation; when we consider the nature of consciousness?

Also, perhaps more in the philosophy realm; is another question sort of related to "time;" and the "monotonic variation" it may have is the concept of "forever;" and related terms. Some of our studies make me wonder what "forever" might really be. That is, does "time" ever end?

I will be back likely on the week end.

 

[turin]

Be careful not to get too philosophical in here. I request that we please try to stick to the physics. I know that I'm the one who brought it up, and I appologize. I just know that these philosophical discussions tend to irritate a lot of posters here (including myself, usually). If you would like to keep talking about this issue, then PM me, or maybe you could start another thread about it (I think there's a philosophy board somewhere around here) and then give me the link.

 

[Guy From Alberta]

Be careful not to get too philosophical in here. I request that we please try to stick to the physics. I know that I'm the one who brought it up, and I appologize. I just know that these philosophical discussions tend to irritate a lot of posters here (including myself, usually). If you would like to keep talking about this issue, then PM me, or maybe you could start another thread about it (I think there's a philosophy board somewhere around here) and then give me the link.

Hi Turin

I understand your request in the last post above; no problems here on my end. I was more or less just responding to your one comment above. I am sorry for taking so long to get back here; but in the nicer weather, plus with my studies; I will have to be away from time to time. I will be back later tonight, or tomorrow with some more physics questions/comments.