3 ships and a rather long piece of string - SRT

Hi Guys,
Just had an amusing thought about SRT and how this may make an interesting gendanken for those who are pssionate about A.E.s SRT

We have three ships in space. all traveling at relative V
they are all connected together by a a piece of string long enough to accomodate the various distances involved.

Think about it for a bit and show that time is relative? [ the rate of time may be relative due to dillations but the present moment for all ships must be the same moment [ absolute] or the string must time travel as some stage along it's length.

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please excuse the scribble...

maybe this is crazy but hey...what the hell...

 

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Well, the idea behind special relativity is that reference frames only make sense if they're point-like. (This is something that a poster called zanket never realized.) This means that at each point along the string, you can define a good coordinate frame, but you cannot define a good coordinate frame for ``the string'' as a whole.

So the resolution is that you are not considering the fact that the string is an extended object, and therefore you need multiple reference frames to describe it.

 

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Either you mean something other with point-like than I think, or I'm missing something: A reference frames gives coordinates to all events in spacetime.

It makes perfect sense for me to give coordinates to something that isn't centered on me...

 

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This is confusing. How can a reference frame give coordinates for all point-events in spacetime? This runs counter to all that is known about both special and general relativity. Specifically, the default would be that, to each point-event in spacetime, the maximum necessary reference frame is the point itself; it may be "larger", but it need not.

Also, if you care, and more significantly, it is also counter to the general construction of manifolds. (Unless, of course, you think that spacetime is not a manifold as classically defined)

Which I have tried to explain elsewhere......

 

QuarkHead is correct---there is no one coordinate frame that can cover an entire manifold.

And, reading back on my response, I should have said that OBSERVERS have to be point-like, and an observer defines a reference frame. What each observer sees depends on his reference frame---so Observer A looking at some arbitrary point B on the rope sees something different than an observer living at point B does when he looks to point A.

 

So you're saying that for a given reference frame there are events that cannot be given coordinates in that reference frame?

God, I hated differential geometry...

Sure, that's not contested.

[Edit: Actually, rereading that last response, I was being overly strict in my reading of your first post. Please ignore my tired, confused babble...]

 

say the rope is made of copper strands and is used as an intercom system between ships. would it work?
sending messages into the future sounds like a great idea... or even more intiguing: what if the rope had a fillament of optical fibre...would it work?
I tend to to think that SRT would force our optic fibre transmissions to have variable light speed as the info travellled back and forth along the rope or cable.
Arguing that the optical fibre at any given point become an specific reference frame thus maintaining invariance of light.
But given that we are talking about infinitely small divisions of points along the cable this seems like an absurd proposition and another example of just how far we as rational beings will go to accomodate photon theory.

The bottom line of course is that regardless of theory the ships are connected by a cable or rope whilst traveling at relative V.
According to SRT this connection bridges the gap between relative time lines. A bit like a father writing to his son before the father is born.

 

Well, I'm sorry to hear that, I think it's rather fun, at least on a superficial level. As it's my day off, lemme try and explain to all what's going on here.

We will think of spacetime as a manifold. We only need to know that this means it is always possible to choose coordinate "neighbourhoods" that are indistinguishable from "flat" spacetime in the Euclidean sense.

Recall that a point in spacetime, in our manifold, is an event. We now need to know exactly what we mean by saying 2 such points are the same or different. For this we will need what's called a separation axiom. The only such axiom that is not in some sense pathological is the so-called Hausdorff property; for any pair of points \(x \ne y\), I can always choose coordinate neighbourhoods \(X \ni x,\; Y \ni y\) such that \(X \cap Y = \emptyset\). This is by definition.

Then, for \(x\) and \(y\) to talk meaningfully about their spacetime vectors is impossible, since, simply by virtue of their neighbourhoods being disjoint, there can be no coordinate transition function \(X \to Y\).

Let us now introduce a third point \(z\), and choose an arbitrary neighbourhood \(Z \ni z\), such that \( x,\; z \in X \cap Z\). Now these 2 guys can compare spacetime vectors by means of the transition function \(\sigma: X \to Z\) which maps each coordinate set onto to the other.

But, by Hausdorff, I can still find a neighbourhood \(Z' \ni z\) such that \(X \cap Z' = \emptyset\), and we're back to square one!

Yeah, OK, that was so rough-and-ready as to be nearly wrong (but not quite!)

 

Horses for courses, no doubt. I simply found algebra (plus logic and discrete math) more fun, which probably has quite a lot to do with why I became a computer scientist instead of a mathematician.

Did I mention topology wasn't my favourite, either? I did enjoy large parts of Munkres' "General Topology", but sometimes it just got crazy (and crazy difficult, usually) with no warning.

In any case, your exposition above tells me that spacetime is Hausdorff (i.e. separable), but I'm having difficulty seeing how that matters.

Maybe I'm confused, but I usually think of reference frames in special relativity as each being \(\mathbb{R}^{1,3}\) and related by Lorentz transforms. Not as being subspaces of some "overall" Minkowski spacetime (which seems rather too much like an "absolute reference frame" sort of thing to me). That you shouldn't be able to transform some points in one frame to points in other frames doesn't really match up with that view.

Oh, and I might add the caveat that I haven't actually studied relativity formally (a few weeks during high school physics hardly counts), so I could very well be fundamentally mistaken about this.

 

This is based on a misunderstanding of SR. Because the signal along the wire has a finite speed (namely, speed of light) in ANY reference frame, one cannot ``send a message into the future''.

 

Oh well, I must not have expressed myself quite as well as I might have wished. Lemme try again:

The domain of the Special Theory is Euclidean 4-space , i.e. "flat" in some sense. But, thinking of spacetime as a manifold, we have no reason to assume it is globally flat, in the same sense (this was Ben's earlier point). This seems to imply that the Special Theory is a local theory, as indeed it is

But we do know this nice thing about our spacetime manifold, that we may consider it to be locally Euclidean for any arbitrarily chosen coordinate "patch". And provided only that this patch is non-empty (it never is, of course), we will call it a neighbourhood of some point therein.

Then we may think of the Lorentz transformation as being a coordinate transformation between such neighbourhoods, which I earlier called "transition functions". As I tried to explain, obviously rather poorly, this can only be possible if, for some pair of neighbourhoods, the intersection is not empty. This is how transition functions work, as I am sure you know.

But the Hausdorff property always allows me to choose a pair of non-identical points such that the intersection 0f their neighbourhoods is empty. Thus, no transition functions can possibly exist that sends the coordinates of one to the coordinates of the other - no Lorentz.

It simply remains to assert that, even though any 2 points chosen at random can be "Hausdorffed", the Hausdorff property in no way insists that the neighbourhood of x contains only x. In fact, in general we know this is not the case.

So, the Special Theory holds locally, but not globally.

Umm, I have drink taken, I hope this is clear?

 

but at least at some point along the cable the message must alter it's time line yes? Otherwise light speed would become variant instead of invariant?

 

Yes, it does. Different parts of the string will experiance different tick rates according to some external observer. To demonstrate this: coil a string and lay it on the ground, then grab an end and walk with it. Notice how parts of the coil lay stationary, while other parts move with you? (you are slowly pulling slack off the coil.)
For simple calculations, you can treat the part of the coil that is motionless as one referance frame, and the part that you are pulling on as another. The place where the slack is being consumed is where the referance frames switch.

-Andrew

 

what I meant by time line was that parts of the string would be in the future or past of other parts of the string.
So a message must be traveling through time and not just with time.

 

No. Each section of the string has it's own world line.

 

ahh of course a global reference frame is not allowed under SRT...

so even though the message may be traveling into the future it aint because srt dis-allows it.
hmmmm......but still the ships are connected are they not?
Or are you saying that the ships are not connected somehow even though the rope or cable or string connects them.

Sort of like having your cake and eating it simultaneously...IMO

 

It's no more travelling into the future than a man is on his way to work.

-Andrew

 

if for a moment I declare I have no knowledge of SRT and I tether three ships together with the cable as described why would I accept that each ship observer has his own unique world view when obviously they are connected to my world view by way of the tethered cable.
How would you prove that they would have a different world view?

In other words why should I believe that SRT is correct in this case?

I know ultimately that it is the view that light speed must be held invariant that eventually leads to the notion that this is proof.

For light speed to be invariant SRT must be true is the final assessment IMO.

Provocative role playing:
But given that I do not believe in the existance of photons I still await evidence to support SRT main premise or postulation.
Given that the postulate of lights invariance and therefore photons existance leads to a situation where by the universal constant of inertia and gravity is invalidated.
This means that gravity and inertia are constants only to a unique world view but relative to all other world views.
I fail to see why I should accept this notion especially as it is unable to be proved due to the nature of SRT's unique perspectives for all observers.
So I guess I shall stick to that which makes more sense and look for a way of describing the data and effects we experience that are currently poorly explained using SRT and photon theory.

 

Let me ask you:
What is the difference between 2 ships connected by a string (or fiber optic cable), and 2 ships communicating via radio transmission?

The radio is faster, easier, and won't break as easily.
Otherwise, one can think of the string as a large number of radio transmitters, each staying in contact via exchange of virtual photons (holding the molecules together,) just as the radio transmissions exchange photons.

-Andrew

 

This is a very good way to see things.