Two observers. One closer to an event horizon.

[longcreepyhug]

I am just an amateur, so go easy on me.

I am simply confused about schwarzschild radii and relativity. It seems to me that if you had two observers, one far away from a black hole, and the other just outside the event horizon, and you allow any small amount of time to pass, they would necessarily be traveling at different velocities due to the different gravitational forces. Because of this, their clocks would run differently. The one closest would run slower relative to the one far away. Yet both observers would still measure the speed of light as 'c' (~300,000,000 meters per second, even though each observer's meters and seconds differ from the other's).

Given the equation for the schwarzschild radius, wouldn't the two observers measure the schwarzschild radius differently? I mean, their numerical answers would both be the same, but since their meters and seconds differ in relative value, wouldn't their results as well?

I am a biologist, and could have just asked some people in the appropriate department, but I figured internet embarrassment would suffice.

 

[pervect]

I am just an amateur, so go easy on me.

I am simply confused about schwarzschild radii and relativity. It seems to me that if you had two observers, one far away from a black hole, and the other just outside the event horizon, and you allow any small amount of time to pass, they would necessarily be traveling at different velocities due to the different gravitational forces. Because of this, their clocks would run differently. The one closest would run slower relative to the one far away. Yet both observers would still measure the speed of light as 'c' (~300,000,000 meters per second, even though each observer's meters and seconds differ from the other's).

Given the equation for the schwarzschild radius, wouldn't the two observers measure the schwarzschild radius differently? I mean, their numerical answers would both be the same, but since their meters and seconds differ in relative value, wouldn't their results as well?

I am a biologist, and could have just asked some people in the appropriate department, but I figured internet embarrassment would suffice.

Welcome to PF.

The only way for the observer near the black hole not to acquire velocity is to thrust with rockets or the equivalent.

But it's not impossible for them to do that. I'm not sure about the "necessary" part - if they accelerate, it's not necessary.

The Schwarzschild radius isn't something you measure physically, and it does not represent a distance. It's just a coordinate.

What you might measure physically is the circumference of the black hole - and if you divide that by 2pi, you'll get the Schwarzschild radius.

By the way - physical meters don't necessarily change size, nor do physical clocks change their time rate. It's really the relation between coordinates and the physical clocks that change. It seems to be a bit ambiguous as to which interpretation one uses, I find that people with the idea that the coordinates reign supreme and the physical clocks don't matter wind up very confused.

 

[longcreepyhug]

Thanks for the quick reply!

Unfortunately, I am even more confused. The schwarzschild radius is not a distance?

Also, in my problem, I am not talking about either observer trying not to acquire velocity. Quite the opposite. I am actually trying to figure out how this scenario would play out if both observers started out (popped into existence, if you will) inert relative to each other and gained velocity in accordance to their different gravitational accelerations.

 

[pervect]

Thanks for the quick reply!

Unfortunately, I am even more confused. The schwarzschild radius is not a distance?

Correct, the Schwarzschild radius is a coordinate. It's like a label on a map that says "you're at E4".

You can convert differences in map coordinates (Schwarzschild coordinates) into distances, but that requires another mathematical operation. A fairly simple one, but enough to make the Schwarzschild coordinates not distances, but coordinates. The conversion from coordinates to distances is done via what's called the "metric", whose main function is just to perform this conversion.

It's rather similar to the way that lattitude and longitude on the surface of the Earth are not distances, but coordinates. Specifying a lattitude and a longitude tells you where you are (specifying the Schwarzschild coordinates does the same thing). And you can figure out the distance between two points if you specify both points lattitude and longitude, but you need to perform some elementary calculations.

So, the Schwarzschild r-coordinate is closely related to the radial distance, but it's not the same thing.

Note that distances (well, at least proper distances!) do not depend on the observer. If you measure the height of the Empire State building with a ruler, locally, and again (by some other means) when you're up 10,000 miles in space, you get the same answer.

THus when you measure a distance, everyone agrees. (There is an important exception, this is due to the relativity of simultaneity. THe phenomenon is known as length contraction. It's a velocity / simultaneity dependent effect. Sorry if this is confusing, but I don't know how to avoid it. Studying SR before one tries to tackle GR is one way to help minimze the confusion.)

Also, in my problem, I am not talking about either observer trying not to acquire velocity. Quite the opposite. I am actually trying to figure out how this scenario would play out if both observers started out (popped into existence, if you will) inert relative to each other and gained velocity in accordance to their different gravitational accelerations.

Well, if you don't try to fight gravity, then the observer close to the black hole will accelerate relative to the one who is not close to the black hole. For observers who are fairly close by, this is called "geodesic deviation" and there's a simple formula for it (well, relatively simple).

You might look at https://www.physicsforums.com/showpost.php?p=4175529&postcount=15

There's some online resources at http://www.eftaylor.com/download.html#general_relativity, Taylor has the first two chapters of "exploring black holes" online. You'll have to buy or borrow the book to see the rest of the chapters, though.

 

[longcreepyhug]

Don't even proper distances depend on some hypothetical sufficiently removed observer?

 

[pervect]

Don't even proper distances depend on some hypothetical sufficiently removed observer?

No - it will depend on the notion of simultaneity used - but if you have a rigid object, there's really only one choice for that.

 

[longcreepyhug]

I feel that I am not phrasing my question properly. It seems that you are giving me answers that are far too basic, in a very complex phrasing. I may be just a biologist, and therefore, out of my element, but I know what a coordinate is and I have repeatedly read the special and general theories. My problem is that, based on what I have read, I am having trouble grasping how a black hole could ever be a real object, as the critical threshold seems to be only an apparent one, that would change depending on the placement, motion and/or mass of each observer.

 

[Nugatory]

My problem is that, based on what I have read, I am having trouble grasping how a black hole could ever be a real object, as the critical threshold seems to be only an apparent one, that would change depending on the placement, motion and/or mass of each observer.

Without reference to any observers or any coordinates, I can say that the event horizon is a surface in four-dimensional spacetime. I can calculate whether my path through spacetime intersects that surface, and if it does, I'll feel the same sense of urgency that I'd feel if you told me that my path down the highway is about to intersect the path of a heavy truck coming the other direction at high speed. That's enough to satisfy me that it's real.

I can also calculate (after choosing a coordinate system that's convenient for the problem at hand, and that I can change at will, but the result of the calculation will not change) exactly what will happen to anything or anybody in the general vicinity of the black hole and what they will observe. Again, that seems "real" enough to satisfy me.

What I cannot do is establish a single global standard of time and distance that everyone can agree to use locally.

 

[Nugatory]

Don't even proper distances depend on some hypothetical sufficiently removed observer?

No - they are invariant, the same for all observers and all coordinate systems. The confusion around this point comes from two factors:
1) To compute proper times and distances, you need to choose some coordinate system, just to get numbers that you can crunch. Otherwise you're left with a correct and coordinate-free but unevaluateable path integral of the abstract metric tensor.
2) Schwarzschild coordinates have the very nice property that at a sufficient distance from the central mass, the time coordinate is the proper time and the spatial coordinate distance is the proper distance.

Put these together, and you'll see a lot of people speaking and writing as if the distant observer is somehow uniquely connected to proper time and proper distance. But that's a shortcut, potentially confusing - he's not really.

 

[pervect]

I feel that I am not phrasing my question properly. It seems that you are giving me answers that are far too basic, in a very complex phrasing.

I'm surprised that my phrasing was too complex. But I *was* indeed trying to give very basic answers.

I may be just a biologist, and therefore, out of my element, but I know what a coordinate is and I have repeatedly read the special and general theories.

It sounds like you may think my statement that coordinates are map references is "unfair" , wrong, or misleading somehow. I'm afraid I'm not sure why you think that's unfair, or what you think coordinates are if you don't agree with my summary.

Or perhaps you want some references to the literature - the particular literature which influenced my statements. In that case I'll offer Misner's "Precis of General Relativity" http://arxiv.org/abs/gr-qc/9508043One of the non-very-technical points I was trying to make is that the problems of assigning coordinates to a curved space-time is rather similar to the problem of assigning coordinates to the curved surface of the Earth. Certain properties that you'd like a coordinate system to have are not globally possible. Among these desirable but globally impossible properties are having a change in coordinate directly have physical significance as a distance.

Einstein says some similar things, though I"m not sure if he'd phrase things the way Misner did (probably not). For instance read Einstein's musings about rods, heated disks, coordinate systems, and generalized coordinates:

http://www.bartleby.com/173/24.html

After describing in detail how one might create a square grid out of little rods, he muses

It is a veritable wonder that we can carry our this business without getting into the greatest difficulties.

He didn't draw the obvious analogy of mentioning that you couldn't carry out the above procedure on a curved surface such as the surface of the Earth. It's an interesting omission, but perhaps he thought it was too obvious to mention. He did go so far as to specify that the slab was flat, before describing his network of rods used to set up his coordinate system.

In the next section Einstein goes on to talk about Gaussian coordinates. And he goes into detail about generalized coordinates, which he calls "u" coordinates and the "v" coordinates, and how you get distances out of these generalized coordinates via the metric.

My problem is that, based on what I have read, I am having trouble grasping how a black hole could ever be a real object, as the critical threshold seems to be only an apparent one, that would change depending on the placement, motion and/or mass of each observer.

There are a variety of flavors of "event horizons", most of which are observer dependent as you note. This doesn't make the underlying geometry any less "real" (where "real" is read as observer independent).

A lot of people have problems with black holes, and 99% of them are the result of taking coordinates too seriously. Which is why I started mentioning these basic issues, when I saw the general direction in which you were heading.

 

[SteveDave]

Einstein's theory of relativity strictly adheres to the observer. You will always get a different answer from clocks or potential Astronauts as to where the Event Horizon may be, how long it took to reach that point, if it is possible to orbit a Black Hole without being "sucked in" and so on. I believe (and could easily be wrong myself) that you are asking a question many have asked before and all ended up with the same answer, "Who is taking measurements and with what from where"? The truth is no one can actually answer these questions until we as mankind "meet" a real Black Hole and come back with consistent data. That being said, this same hole would have to be visited over and over again by many different observers and all the data analyzed by many countries and so on until a general consensus was reached. The truth is, we have many theories and equtions to explain this phenomenon, but there really is no known truth as of yet. I know it sounds assinine but there are several good Youtube videos related to this topic that explain in good Lamen's terms.

 

[PeterDonis]

based on what I have read, I am having trouble grasping how a black hole could ever be a real object, as the critical threshold seems to be only an apparent one, that would change depending on the placement, motion and/or mass of each observer.

Perhaps this will help: the critical threshold is not determined by how far away you are from some particular point. The fact that we ordinarily speak of the threshold in terms of a distance is, unfortunately, misleading in that respect. As pervect noted, the "radius" of the horizon is really the circumference of the horizon divided by 2 pi; it is not a distance that is measured radially. It is certainly not a measured distance from the singularity at the center of the hole.

Strictly speaking, the singularity is not even "at the center" of the hole, because it is in the future; asking how far away you are from it is like asking how far away you are from tomorrow. Your "distance" from the singularity when you are at the horizon is properly measured as how much time it will take you, by your own clock, to fall to the singularity. And that *does* depend on your state of motion; different observers who fall through the horizon with different velocities, or who accelerate differently once they are inside, will take different amounts of time to reach the singularity according to their own clocks. So the dependence of the "distance" on the observer's state of motion is there; it's just not quite what you might have been expecting.