length of a piece of string

emu-addict

emu-addict wrote:

...the length of a piece of string is the maximum distance that its ends can be parted by....

this seems like a good definition enough but i was wondering, is this a definition that only applies in euclidean/flat space? would it also apply in the vicinity of a black-hole, say?

Son Goku

It wouldn't entirely apply because different sections of the string would have different distance scales.

The vicinity of a black hole has a "metric field", rather than a single metric like euclidean space.

Although unless the gradient was pretty severe, i.e. a black hole, your definition holds.

emu-addict

so, can you come up with a generalized definition? something to do with integrating the metric over the string?

Son Goku

What is called the "affine parameter" parameter measures length in curved space-times.

Which is:

(Affine) = (Constant)(Proper-time) + (constant).

The square of this is, in the most base case, obtained using quanternions. (although there are several other methods)

The problem is this isn't invariant under a boost transformation.
So if the string is moving at high velocities this doesn't hold.

In a sense there is no universal definition of length, because there is no absolute frame of reference.

emu-addict

Son Goku wrote:

...In a sense there is no universal definition of length, because there is no absolute frame of reference....

so basically, the length of a piece of string is whatever you want it to be as long as its moving in the right way and placed in the right place.

though, of course, it gonna be easier just to cut it to the right length, right?

egan007

Er....the distance between one end and the other?
This answer has been around a while

Son Goku

egan007 wrote:

Er....the distance between one end and the other?
This answer has been around a while

Except it's wrong.

so basically, the length of a piece of string is whatever you want it to be as long as its moving in the right way and placed in the right place.

though, of course, it gonna be easier just to cut it to the right length, right?

Basically it will be a length based on your physical conditions at the time.

nesf

Son Goku wrote:

Basically it will be a length based on your physical conditions at the time.

I disagree. The string itself remains constant, only our perspective of it changes. So in essence the lenght of a piece of string does not change. Only the observers perception of lenght.

The assumption that it changes because the observer's perception changes is demonstrative of the human ego. The world does not revolve around us and so forth.

Of course, if you want you can take the view that the "lenght" of "anything" can only be defined by an observer in a certain frame of referance and nothing has an inherent correct lenght, only the lenght that it can be observed with having.

Edit: The whole idea being that no one frame is more correct than any other. So there isn't a "correct" lenght, since there is no way to determine somethings lenght outside of a frame of reference. If you take this perspective as Son Goku has (i think) you come to a different realisation.

All a matter of perspective really. Personally I don't think that the "lenght" changes, I just think that it's not measureable

Son Goku

What I'm kind of saying is that length is in actuality, an old concept.

There is no "length" of the string as such a thing only applies in a space with a single metric.

nesf

Son Goku wrote:

What I'm kind of saying is that length is in actuality, an old concept.

There is no "length" of the string as such a thing only applies in a space with a single metric.

I agree completely, just was having a little fun with the last part. It's not easy for someone not aufait with physics to accept that something like lenght isn't constant and immutable. Was trying to use layman's terms for any non-techies reading the thread.


My take on the "how long is a piece of string" question is this: I would interpret the question as questioning the assumption that everything has "lenght" and that this can be measured accurately, although this is probably not apparent to people unfamiliar with physics.

My thinking is that there is a concept of what lenght is, and this concept is some immutable and unchangeable constant. I don't think that it is a measureable concept though, since the act of measuring brings in change. It's like the Heisenberg Uncertainty Principle, in that it's impossible to know it. It doesn't mean that it doesn't exist, it's just impossible to measure it accurately.

If that makes sense.

Son Goku

I get you now.

To people who don't know what a metric is it is basically a mathematical object which describes a space.
(to be technical it is a symmetric second rank version of a larger class of objects called Tensors)

If a string is a space with the following metric:

and it has a length of 1 meter, when it move into a space with this metric:

to a person in the first metric it will look like it has a length of two meters.
To the person back in the second metric the piece of string will look like it has a length of one meter and if it is moved back into the first metric it will look like it has a length of 0.5 meters.
If the string is between the boundaries it will have a mixed length and look like it is 1.5 meters to the guy in the first metric and 0.75 to the guy in the second metric.

However this is a spinning Black Holes metric:

where:

r = distance from the hole.
As you can see the string will have an infinity of different lengths depending on where it is.

nesf

Purty

Will assume they are correct, my memory is ****e when it comes to formulae

egan007

Son Goku wrote:

Except it's wrong.



Basically it will be a length based on your physical conditions at the time.

How could it be wrong the distance from one end to the other may vary depending on conditions but you would get the measurement for that piticular environment..

Son Goku

I should point out that there is no fundamental length of the string which is expanded or contracted depending on what metric it is in.
The metric defines the length. There is no meaning of the length out side the metric.

(Well one can't exist outside a metric so I guess that last sentence is superfluous)

Son Goku

egan007 wrote:

How could it be wrong the distance from one end to the other may vary depending on conditions but you would get the measurement for that piticular environment..

There are some environments where that measurement wouldn't even be consistant along the string's length.
Near a black hole even the string wouldn't agree on its length.

nesf

egan007 wrote:

How could it be wrong the distance from one end to the other may vary depending on conditions but you would get the measurement for that piticular environment..

I think you missed the point.

Let me ask you this. What is lenght? Is it the physical dimensions of an object or one's measurement of said dimensions?

nesf

Son Goku wrote:

The metric defines the length. There is no meaning of the length out side the metric.

You are 100% correct. Lenght as defined as a one dimensional measurement of a particular object's physical dimensions (one in particular) is dependent on the metric. It's dependent on the measurement and the measurement is dependent on the metric.

But from a philosophical point of view, could you agree that a conceptual true "lenght" exists that is not necessarily observable in the physical world? (Just a meandering though rather than a serious point)

Son Goku wrote:

(Well one can't exist outside a metric so I guess that last sentence is superfluous)

Hehe... we should gve that statement to the philosophers and see what they can do with it.

"I am in a metric, therefore I am"

Son Goku

nesf wrote:

But from a philosophical point of view, could you agree that a conceptual true "lenght" exists that is not necessarily observable in the physical world? (Just a meandering though rather than a serious point)

Yes I could agree to that.
Even scientifically there would seem to be some underlying "property" which manifests itself as different lengths in different metrics.
The string appears to posses something which the metrics operate on to produce different lengths to different observers.
What it is, I do not know, although if anything it would be the true "length".

nesf wrote:

Hehe... we should gve that statement to the philosophers and see what they can do with it.

"I am in a metric, therefore I am"

They'd have a field day over it.

nesf

Son Goku wrote:

Yes I could agree to that.
Even scientifically there would seem to be some underlying "property" which manifests itself as different lengths in different metrics.
The string appears to posses something which the metrics operate on to produce different lengths to different observers.
What it is, I do not know, although if anything it would be the true "length".

Cheers, that sums up my thinking quite nicely actually. I feel like I've achieved something today on an intellectual level, my understanding of the world around me is just that little bit more accurate now

Capt'n Midnight

The length of a piece of string would be measured from it's frame of reference otherwise we'd be talking about the apparant length.

since a black hole was mentioned do we need to take into account spagettifaction and hookes law or will the string be non elastically deformed ?
if it's a very large black hole and a short string you could probably assume that the string would not be stretched or assume it's moving sideways.

If it is floating in free space then you could assume it's ramdomly orientated and not stretched and at a guess the length is pobably of the order of the distance between the ends squared (random walk) or if you move the ends apart with the minimum of force then the length may well be PI times the distance (meandering)

'course these strings would be a different shade of horses pajymas altogether http://www.boards.ie/vbulletin/showthread.php?t=247223

Son Goku

Capt'n Midnight wrote:

The length of a piece of string would be measured from it's frame of reference otherwise we'd be talking about the apparant length.

In curved space-time an object doesn't really have a consistent frame of reference due to the presence of a metric field.

In other words each infinitesimal piece of the string will occupy a space of a different shape to the preceding and following pieces.

Capt'n Midnight

Would the observer on the string be able to detect the curvature in space time away from a black hole ?

Another way would be to put a mirror at one end of the stirng and see how long it takes to see the reflection eg: with a spinning mirror at the observers end. The length is half the time by c. Assumption here is that it's a vacuum.

If part of the string had crossed the event horizon then an observer on the safe side wouldn't be able to see or measure the string.

Son Goku

Capt'n Midnight wrote:

Would the observer on the string be able to detect the curvature in space time away from a black hole ?

Yeah, if his measurements were sensitive enough.

Capt'n Midnight wrote:

Another way would be to put a mirror at one end of the stirng and see how long it takes to see the reflection eg: with a spinning mirror at the observers end. The length is half the time by c. Assumption here is that it's a vacuum.

Unfortunatly the speed of light varies in curved space.

nesf

Son Goku wrote:

Unfortunatly the speed of light varies in curved space.

Stupid universe. Refusing to fit into an Eucleadian space. *kicks*

Capt'n Midnight

Son Goku wrote:

Yeah, if his measurements were sensitive enough.

Ours aren't yet. What if the difference is too small to measure relative to the length of the string - like the Planck distance .

Unfortunatly the speed of light varies in curved space.

as does the length - so looks like it might be impossible to answer the question about "how long is a piece of string ?"

nesf

Capt'n Midnight wrote:

as does the length - so looks like it might be impossible to answer the question about "how long is a piece of string ?"

To be fair his answer of "It depends where the string is" is fairly accurate.

And it would be a judicious approximation to assume that we won't want to know what the lenght of string is when it's near a black hole.

Could we settle on, it depends on the observes relationship to the string? ie to account for non-inertial frames of reference and such?

Does the question really need to encompass the exotic behaviour found around black holes?

Son Goku

nesf wrote:

Could we settle on, it depends on the observes relationship to the string? ie to account for non-inertial frames of reference and such?

Yeah, that seems fine.
Unless we're near spatial anomolies it's completely correct.

Sev

Can one not take the string to an area of space-time that is 'flat', and measure its length there? or is there something fundamentally flawed in that?

Son Goku

Sev wrote:

Can one not take the string to an area of space-time that is 'flat', and measure its length there? or is there something fundamentally flawed in that?

Basically the flaw is, why would flat-space be any different?

nesf

Sev wrote:

Can one not take the string to an area of space-time that is 'flat', and measure its length there? or is there something fundamentally flawed in that?

Yeah, the problem is that why is it's lenght in "flat" space any more fundamentally correct to it's lenght in curved space? From a "strict" point of view, it isn't any more correct in any one frame of reference than another.

Personally, I don't think that the question need be that complex, as I outlined above.

The question doesn't say where the string is, and that is why curved space-time etc comes up. If you assume the piece of string is on earth then it's a much simpler issue.

Then it's just down to the perspective of the observer relative to the string, which is a much simpler problem than curved space time (special relativity will suffice, and S. R. is much nicer and easier to grasp than general relativity.


Personally I feel that it's a mountain made from a molehill and that the question was never intended to be subjected to a rigourous physical treatment.

stevenmu

Can't believe this thread went 30 posts without someone making a string theory pun

Anyway, I hope I'm not resurrecting a dead issue here but it seems to me that if a piece of string has it's own constant length or not is kind of irrelevant. Trying to determine the length of a piece of string implies measurement and measurement of length is entirly a human construct. Therefore the length of a piece of string is determined by the person measuring it, and if one measurement is taken which takes space-time curvature (and other relativistic effects) into account and another disregards it then are both measurements not equally correct within there own terms of reference ? Kind of like the way an experiment can have an acceptable margin of error, within which the experiment can be a success even if the outcome isn't exactly what was expected ?

Son Goku

stevenmu wrote:

Therefore the length of a piece of string is determined by the person measuring it, and if one measurement is taken which takes space-time curvature (and other relativistic effects) into account and another disregards it then are both measurements not equally correct within there own terms of reference?

A reference frame isn't a point of view though. It is simply a machine for labelling events.

Also in curved space there are no true frames of reference.

stevenmu

I should really be more carefull with my use of terminology on the physics forum By "terms of reference" I'm referring to the terms of the measurement. I think a measurement should really be treated almost like an experiment, as a basic example there could be a case where you need to run a piece of string from the centre of one galaxy to the centre of another another, in that case you would need to take relativistic effects into account or you may end up several million kilometres out. Another example may be that you need a piece of string to go the distance between your sofa and your coffee table, any spatial distortions in this region are almost certainly going to be too trivial to be worth measuring, so disregarding them would still give you a measurement within an acceptable level of error.

To summarise, how long a piece of string is depends on why on you want to measure it in the first place

nesf

stevenmu wrote:

I should really be more carefull with my use of terminology on the physics forum By "terms of reference" I'm referring to the terms of the measurement. I think a measurement should really be treated almost like an experiment, as a basic example there could be a case where you need to run a piece of string from the centre of one galaxy to the centre of another another, in that case you would need to take relativistic effects into account or you may end up several million kilometres out. Another example may be that you need a piece of string to go the distance between your sofa and your coffee table, any spatial distortions in this region are almost certainly going to be too trivial to be worth measuring, so disregarding them would still give you a measurement within an acceptable level of error.

To summarise, how long a piece of string is depends on why on you want to measure it in the first place

See. A physicist will measure it because it's there, an engineer because he has to and a philosopher won't because he'd prefer to talk about it.

Capt'n Midnight

nesf wrote:

If you assume the piece of string is on earth then it's a much simpler issue.

Measuring string at the bottom of a gravity well :eek:
You can't use the speed of light as a reference because it's no longer a vacuum, the speed will vary according to temperature, isobars and humidity ! Also you have to take into account the time dialation effects of the increased gravity (which varies depending on where you are) which makes it very difficult to get an accurate time/speed to turn meters per second back into meters.

Not to mention the humidity would probably affect the length of the string too.

planck2

look if I have a piece of sting, be it an imaginary string, string theory string or an actual string and I am moving with respect to all of you then according to me the length of the string is according to SR is the proper length and rest of you can go do what you like if you are in flat space-time. If its a piece of string in a black hole the piece of string will be torn apart by the a few seconds after it has crossed the event horizon and in any case there'd be no light so you'd have no light to carry out any measurements.

Capt'n Midnight

Is the black hole rotating ?

If the back hole is big enough then the tidal forces won't be that strong over the length of the string and it may pass the event horizon unharmed , taking days to fall into whatever... Hang on - won't it be travelling close to the speed of light by the time it gets there. So that string will have ZERO length for all practical purposes.

nesf

planck2 wrote:

look if I have a piece of sting, be it an imaginary string, string theory string or an actual string and I am moving with respect to all of you then according to me the length of the string is according to SR is the proper length and rest of you can go do what you like if you are in flat space-time. If its a piece of string in a black hole the piece of string will be torn apart by the a few seconds after it has crossed the event horizon and in any case there'd be no light so you'd have no light to carry out any measurements.

Actually. The scale of the string makes a huge difference. It's a different ballgame at quantum and cosmic scales. Best just to keep it in normal scales.