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Author Topic: What becomes of a straight line when Conceptually challenged?  (Read 12179 times)

Offline yor_on

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So, assuming that light 'bends' to gravity, also assuming that they take shortest path in a curved SpaceTime. Think of a perfect sphere of a perfectly homogeneous mass, equal gravity at all points of its surface. Then think of a object orbiting it (just as inhumanly perfect), being in a free fall around it as we might say. Are you with me so far? For simplicity's sake also assume that it's at some place in our universe where all other 'gravity's' equal out, aka some sort of Lagrangian 'space'.

What about that object orbiting?

It's actually performing a circle, at the same time it take the shortest path through SpaceTime, isn't it? Now, normally we would associate a straight line with a shortest path, right?

1. But, how does that circle become a straight path?

And then turn it around, if Einstein is right, then what is a circle?

2. An infinite amount of 'straight lines'?

Heh.
« Last Edit: 11/09/2010 22:38:25 by yor_on »


 

Offline tommya300

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What becomes of a straight line when Conceptually challenged?
« Reply #1 on: 12/09/2010 01:50:48 »
Here is a dynamic and crude explanation.
Ideally, an infinite number of iterations of points plotted of an infinite number of tangent lines that intersect the circumference of a circle that has a fixed diameter rotating about a single fixed point called "true center". Removing the fixed center the tangent line will not rotate about will continue ideally straight.
.

 

Offline JP

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What becomes of a straight line when Conceptually challenged?
« Reply #2 on: 12/09/2010 02:36:46 »
It's not a circle in space-time.  It would have to go through the same point in space and time to do that. It's moving forward in time, even though it hits the same coordinate in space repeatedly.
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #3 on: 12/09/2010 03:27:21 »
Nice explanation Tommya :) And the graphic was really cool, but the thing was that the thingie orbiting in fact was thingiyng away in a 'straight' line already. Well, according to the definition that a 'bent path' due to gravity in fact is a straight path, just following SpacetTimes geodesics, while orbiting :)

That's the thing I liked most about it.

And JP, that's a very clever point? You sure you're a honest man? Not that I mind scientists, but really, I mean? Would you like one to marry your sister? The chalk powder alone ..

Anyway, you got a real nice point there.
Just a quicky, where exactly can I draw that circle?

I mean, for it to be a real circle??
Ouch..
 

Offline tommya300

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What becomes of a straight line when Conceptually challenged?
« Reply #4 on: 12/09/2010 04:07:27 »
Thanks Yor_on!
 I emphasize, I expressed only in an ideal model of points plotted relative to the circumference of a circle.
I did not address orbits and the spacetime issue!
I think the issue with "Almost Circular Orbit", that it has a sooner decay rate, and is shortly lived.
Sort out some of your subquestions. Confusion can set in went addressing a question if there is to many questions at once.

Shortest distance between 2 points is a stright line.
My animation refers to a fixed point on a tangent line in reference to an offset fixed distant center point. This is no way explaining an orbit or spacetime....

JP is addressing spacetime, he knows what he is talking about!

We can read it as, you can walk in the same footprint, but never can repeat it in the same moment in time, because time has passed, and as we speak of the present, we are continuing to fall into the future.
Natural time travel.
« Last Edit: 12/09/2010 05:33:31 by tommya300 »
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #5 on: 12/09/2010 04:32:54 »
All to true Tommya, I'm just making it a 'idealized' situation, isolating those two 'perfect spheres' from all other gravity to make my Q understandable. So it's a thought experiment only, although I expect it to be possible to describe mathematically.

But if no circle can be made without SpaceTime becoming involved as JP suggests here, and that one was a unexpected one JP, my jocularity notwithstanding :)

Also supposing the statement I made  about geodesic paths are correct, which I to the best of my knowledge believe it to be. Then the question still seems unsolved, with JP adding to the complexity here :) And yet, it really is such a simple question?

Heh.

 ...ahem
 

Offline JP

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What becomes of a straight line when Conceptually challenged?
« Reply #6 on: 12/09/2010 05:33:50 »
Fortunately, I don't have a sister, so I don't have to worry about keeping all those scientists at bay!

Let's first agree on how to define a "straight line".  If we're dealing with flat 3D space (not space-time), and I give you two points, the shortest path between them is a straight line.  This is the definition we want to keep in SR and GR. 

So what are the issues that come up in applying this definition to SR and GR?  One problem is that space-time is curved.  Another is that space-time has time in it, and this makes it behave very differently than space alone.

Thinking about curved space for a moment--assume that the earth is a sphere and you're stuck to its surface.  If I give you two points on the earth and ask you to draw the shortest path between them (remembering that you have to stay on its surface), then you draw a curved arc between the two points.  This isn't what we'd think of as a straight line, but it's the form that concept takes if you're required to travel on this particular curved surface. 

Just like the definition of a line changes on a curved surface in space, definitions of geometric objects also change if you're dealing with flat space-time, simply because time behaves differently from space.  For example, a circle in space is made up of all points equidistant from a common center.  If you try to construct that in space-time, you get a hyperbola.  This is because the time coordinate doesn't behave like the space coordinates.  (In computing distances, it gets subtracted so that any points connected by a path at light speed have zero distance between them.)  Still, concepts based on distance can be defined and computed.  It's just that what we consider to be the definition of a circle in space isn't a circle in space-time. 

So in summary, a straight line in flat space is some sort of curved line in curved space, and a circle in flat space is a hyperbola in space-time.  GR combines both of these factors so that you have curved space-time.  You can still define a generalization of "straight lines" to be the shortest path between two points, but you would expect it to have a very odd shape in space-time. 

I'm not sure that answers your question since its all very mathematically abstract, but I hope it helps.
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #7 on: 12/09/2010 09:02:54 »
No, not really. Well, a little then :)

But my point, ahem, is that the the straight line in this 3D idealization (+ time) actually is a closed circle and that even thought they 'come back' to the same space coordinates in a different time, the same can be said for any type of circle made, as long as times arrow is involved. And doing it without involving time(s arrow) seems impossible to me?

So? a closed circle representing a straight line in SpaceTime, but it all hanging upon if my understanding is correct, that 'all geodesics represent the shortest path in a curved SpaceTime'. As for 'flat SpaceTime' I understand that to be a generalization, for cases where one don't need to take into consideration curved SpaceTime, but i may be wrong there?

You wrote "For example, a circle in space is made up of all points equidistant from a common center.  If you try to construct that in space-time, you get a hyperbola." Can you give me an example of why that is so? And are you talking about lifting a two-dimensional figure up in a 3D space? No you're not are you, you're talking about using time or not using time. But that becomes even weirder, I'm sure that you can do it mathematically, but considering doing anything inside our SpaceTime without involving time becomes impossible, unless we are talking about 'changing' time in adapting it to different frames of reference? And having no-time at all is then only given to photons/light? And that frame isn't allowed to use, is it? But even so, the point is still that whenever you create a 'closed circle' it should stay closed, no matter how it will transform itself geometrically in a different space/time. I doubt that you can 'open' it by simply placing it 'elsewhere'. On the other hand, considering that it should be the equivalence to a straight line it have to be open when transformed into ??

Ah well, I like it but I will need to think about it. And if you see some way to clear my confusion JP, you're most welcome to it :)
==

In some way it actually fits in how I think about space sometimes, like some 'surface' presenting a 3D image for us in times arrow. I've thought about that before, wondering if space just is 'surface(s)' as all those paths do exist simultaneously. No matter if light 'travels' or not, space will still express itself the same if we would to measure a photons path. For example, sending one photon at a time splitting it in equal distances from source to sink, constantly backing the sink, will present us one 'surface'. Move the laser a little and do it again, and you will have another description of how space bends, with another 'surface' depicted. Like invisible layers but not arranged upon each other at all, more like very weirdly folded in all directions, and then wondering how you could fold that pattern out of a one-dimensional surface? And now I'm perfectly certified, am I not :)
==

One might argue that for to call it a 'real circle' all paths in it have to be taken simultaneously, Possibly? Thinking of drawing a circle on paper and when then looking on it its path will be covered simultaneously in all its parts? If you see how I think. A circle can only be a circle when there are no openings in it under times arrow, and to cover that situation one would have to build a ring circling the perfect sphere instead? But how would one argue then, considering that this perfect rings path also should represent a straight line in a curved SpaceTime..

And if you instead consider space as 'visible points', each one deforming under gravity, each point should look differently, depending on from where you looked upon it too, with all points uniquely deformed. But I also think that you will find a 'likeness' to it if you scaled the points up or down, that is, the points deformations would still 'hang together' having a relation. Well, that's rather obvious of course, but still :) Could one see that as a fractal behavior? Naah. It wouldn't be, would it?

oh yes, headache warning indeed :)
« Last Edit: 12/09/2010 10:49:56 by yor_on »
 

Offline LeeE

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What becomes of a straight line when Conceptually challenged?
« Reply #8 on: 12/09/2010 13:26:11 »
When discussing a 'straight line' you need to qualify the dimensional space within which it exists.  As already pointed out, a line on a curved two dimensional surface may be straight within that two dimensional space but will be curved in three dimensional space.

This is the fundamental idea behind 'wormholes', where the wormhole links two points in space via an additional dimension, the distance via the additional dimension being less than the distance constrained by the original lower order dimensional set.

Using the analogy of a sphere, a straight line along its two dimensional surface, from one side of the sphere to the other, will be half the circumference of the sphere in length but a straight line in three dimensions, going right through the sphere, will only be the diameter of the sphere in length.

I think we need to be clear too, about whether we're talking about the existence of circles in space-time, or whether we're talking about travelling in space-time.  There's no reason why circles cannot exist within any dimensional order of space-time, but due to the 'time' factor you can't travel in a circle in space-time.

JP: I'm not sure what you mean by...
Quote
For example, a circle in space is made up of all points equidistant from a common center.  If you try to construct that in space-time, you get a hyperbola.  This is because the time coordinate doesn't behave like the space coordinates.  (In computing distances, it gets subtracted so that any points connected by a path at light speed have zero distance between them.)

Time may be treated 'differently' when calculating speeds, because we measure space against time, but when we measure distances in time we do so just as we measure distances in space; distance is just the displacement between any two points within any dimensional set.  I don't quite see where you get a hyperbola, for this needs the distance to change; shouldn't the path be a helix?
 

Offline JP

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What becomes of a straight line when Conceptually challenged?
« Reply #9 on: 12/09/2010 13:50:05 »
JP: I'm not sure what you mean by...
Quote
For example, a circle in space is made up of all points equidistant from a common center.  If you try to construct that in space-time, you get a hyperbola.  This is because the time coordinate doesn't behave like the space coordinates.  (In computing distances, it gets subtracted so that any points connected by a path at light speed have zero distance between them.)

Time may be treated 'differently' when calculating speeds, because we measure space against time, but when we measure distances in time we do so just as we measure distances in space; distance is just the displacement between any two points within any dimensional set.  I don't quite see where you get a hyperbola, for this needs the distance to change; shouldn't the path be a helix?

The definition of a circle in 2D is the set of all points equidistant from some other point.  The generalization of this to 1 spatial dimension + 1 time dimension is a hyperbola--since time is treated differently from a spatial dimension, a hyperbola is the set of all points equidistant from some center. 

Mathematically speaking, a circle of radius R centered at (0,0) is given by (x2+y2)1/2=R2
The generalization of that to space-time is
(x2-t2)1/2=R2,
which is a hyperbola.

The definition you're thinking of is a circle in space that is traced out at a constant rate.  That's would be helical, like you say.

But one of the big conceptual problems with SR and GR is getting your head around "distances."  Since when you hear the term "shortest path," you naturally think about paths through space, when in fact you have to think about paths in this rather oddly behaved space-time and find the shortest paths there.

So I guess my question would be:

Yor_on, are you talking about circles in space where some object moves around them in time, but comes back to the same spatial point, or are you asking about what a circle would look like in space-time?  If it's the space-time version, what is your definition of a circle?  You could draw a path that comes back to the start, or you could define a hyperbola (the set of all points equidistant from a center). 
 

Offline LeeE

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What becomes of a straight line when Conceptually challenged?
« Reply #10 on: 12/09/2010 14:43:21 »
Yes, good old Pythagoras, but I don't follow that generalisation.   This seems like an odd and unwarranted way to treat time.

You don't need to move at a constant rate to get a helix; you'll just get an irregular helix if you move at varying rates of time (although for the traveller, who won't be aware of any variation of time will see it as a perfect helix)
 

Offline JP

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What becomes of a straight line when Conceptually challenged?
« Reply #11 on: 12/09/2010 15:32:18 »
Yes, good old Pythagoras, but I don't follow that generalisation.   This seems like an odd and unwarranted way to treat time.

That minus sign is what makes SR and GR work and gives them a geometrical interpretation.  If you choose not to follow it, I don't think you'll get the physics of relativity to work out...
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #12 on: 13/09/2010 03:39:55 »
That was a sweet return JP:) Kind'a loved it.
 
"Yor_on, are you talking about circles in space where some object moves around them in time, but comes back to the same spatial point, or are you asking about what a circle would look like in space-time?  If it's the space-time version, what is your definition of a circle?  You could draw a path that comes back to the start, or you could define a hyperbola (the set of all points equidistant from a center)."

Well, let's use my 'new and improved' factual arrangement for me answering your Q LeeE.

TNS proudly present - version 2 of the new major blockbuster ahh, question?

- The Mystical System -

... A perfect ring, orbiting around a perfect sphere, equal gravity at all points of its surface in a Lagrangian 'space'. As it is already there inside SpaceTime I would say that the question still is how to transform its orbit into straight lines on a two-dimensional surface, as I asume that it is there we would see it as a straight line?

Or what am I missing here?

==

As for those questions you take up JP, you know me to well man..
I want to know them too ::))

==

As for my definition of a circle :)
In what manifold? ::))

Seriously though, I would assume anything that is closed, having no end if you were to travel it, also having space around all 'sides'  and being so in all dimensions to be a circle. Like my 'perfect ring', as I expect it to be inside SpaceTime. And in that motto a Moebius strip also represent a ring, even if its surfaces is rotated 180 degrees, as it 'end parts' are nowhere to be seen. But it's hard to define as we speak about rings also when they do not cover all points simultaneously like a 'orbit' of a satellite.
« Last Edit: 13/09/2010 04:15:04 by yor_on »
 

Offline CPT ArkAngel

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What becomes of a straight line when Conceptually challenged?
« Reply #13 on: 13/09/2010 03:45:06 »
yor_on read the article i put in the post about photon sphere radius... out of subject though...
 

Offline JP

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What becomes of a straight line when Conceptually challenged?
« Reply #14 on: 13/09/2010 03:46:02 »
Why do you want to transform it into straight lines?  Do you want it to be real straight lines or do you want it to be the "geodesics" that relativity talks about?  If you want a real straight line, just imagine the orbit as a string.  You can cut it and then unwrap it so it's a straight line.  Without knowing why you want to turn it into a straight line, I can't tell you if that's a useful way to think about it, though.
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #15 on: 13/09/2010 04:30:28 »
Now that one is easy to answer, or maybe not :)

In a curved SpaceTime all paths taken is the straight paths according to Einstein, am I right there? If it is so then my ring will also prescribe a straight path, but, how does it do it?

And the question about if it is geodesics or real straight lines are after all the same as I understands it, as we have a whole mathematical geometric foundation discussing how you define and transform shapes in different dimensions. And what I understand it to hold holy, is that you will have properties that will be the same in every dimensionality, no matter how they get 'folded'? And one of those have to be a closed circle, on that one I'm pretty sure. And therein lies the conundrum for me.

Either I'm mistaken in my premises and jumps wrong in some way, assuming that each part of my free-falling ring takes that straight path through SpaceTime or I'm not. If they do take a 'straight path' the 'gravity surfaces', if we could see them, have to be very very weird, imagining them.

It's so much simpler imagining gravity as arrows, having clear vectors, pointing to a gravitational center, but as those vectors have to be everywhere in SpaceTime, as proved by inertia, in reality I like to think of it more like those 'deforming points' making SpaceTime up, as it gives me a better intuitive picture of how it must be.

Could be wrong though, happened before, and will happen again :)
« Last Edit: 13/09/2010 06:59:46 by yor_on »
 

Offline CPT ArkAngel

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What becomes of a straight line when Conceptually challenged?
« Reply #16 on: 13/09/2010 04:31:30 »
yor_on, you should see it as a heavy ball on a stretching membrane and see the membrane as the spacetime in 2d instead of 3d. I don't see your point about the straight lines. In finite elements, you can represent any object with straight lines...
 

Offline Geezer

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What becomes of a straight line when Conceptually challenged?
« Reply #17 on: 13/09/2010 04:33:33 »
Could be wrong though, happened before, and will happen again :)

Don't worry about it Yor_on. I used to make mistakes too.
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #18 on: 13/09/2010 04:44:50 »
Okay, think of Space as one continuous one-dimensional surface, or see it as a membrane if you like. Now imagine that it becomes 'knotted', imagine that the 'knotting' made represents what SpaceTimes gravity is, expressed in geodesics. Unravel that membrane and you will find your ultimate straight line, well sort off. And it makes sense to me, But not to my perfect ring :)

Not saying that this is the way Einstein saw it though, it's my own ah, 'idea' that one :) He describes SpaceTime as having geodesics though, and those geodesics are all straight lines in a curved space. And every object of mass or light will follow them so, ignoring the first description, my ring still need to be explained in form of straight lines.

==

Thanks Geezer, that makes us two, right?
Oh, used was it, one then..
                               :)

 

Offline JP

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What becomes of a straight line when Conceptually challenged?
« Reply #19 on: 13/09/2010 05:00:37 »
Don't take the description of geodesics as "straight lines" so literally.  They're really just the shortest paths between points on curved surfaces. 

As an example, just think about the earth (which we'll approximate as a sphere).  If you start off in some direction with a constant velocity, you'll go around the earth in a great circle--this is a geodesic in this case, but it closes on itself and can't be turned into a line without cutting it.  This is an example where geodesics aren't identical to straight lines. 

In GR geodesics are over space-time, so in your example the spatial path is closed, but the object never passes through the same point in space-time twice.  When it arrives at its starting point, its a different time than when it started. 
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #20 on: 13/09/2010 05:36:45 »
Yeah, I thought so to. Like making it into spiraling pattern when time is involved as one of the parameters, which, as it will be never-ending could be seen as a line without end, or, as you see it stop, make an ending, turning in it into a line for sure. Still, will have to think about it some more :) as it, to me, seems perfectly correct. I mean, those geodesics are really straight paths, but it may be the nomenclature that dizzies me there, you could call it paths of least energy expenditure too, couldn't you?
« Last Edit: 13/09/2010 07:00:56 by yor_on »
 

Offline LeeE

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What becomes of a straight line when Conceptually challenged?
« Reply #21 on: 13/09/2010 14:24:26 »
Yes, good old Pythagoras, but I don't follow that generalisation.   This seems like an odd and unwarranted way to treat time.

That minus sign is what makes SR and GR work and gives them a geometrical interpretation.  If you choose not to follow it, I don't think you'll get the physics of relativity to work out...

That response was of no help whatsoever, as well as being offensive.  I was asking a question and at no point did I reject anything.  Your response was just an unwarranted and unjustified accusation, and one that also failed to give the origin and explanation of the generalisation I was querying.

The Lorentz equations at the basis of SR & GR that I'm aware of do not follow the form you gave and instead use the conventional Pythagorean form.

If you can give a link to an article that explains the derivation of that generalisation, then fine, I'll have a read of it.
 

Offline JP

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What becomes of a straight line when Conceptually challenged?
« Reply #22 on: 13/09/2010 15:47:54 »
That response was of no help whatsoever, as well as being offensive.

I'm sorry you thought that.  Sometimes trying to be succinct comes off as being aggressive, I suppose.

You can think in Pythagorean terms, but the geometric interpretation of general relativity (and the geodesics Yor_on has asked about) rely on that extra minus sign on the time coordinate or they don't work.  I don't know of a website that covers it, but textbooks on general relativity will do so.  Many on special relativity will as well.
 

Offline kenhikage

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What becomes of a straight line when Conceptually challenged?
« Reply #23 on: 14/09/2010 14:34:06 »
I may be way off base here, but I thought that plotting circular movement over time resulted in a sine wave~ That is, of course, plotting two-dimensional space against time. I would think then that plotting three-dimensional against time would result in a sort of wave sheet.

Is that why particles also behave like waves? ???

I hope I'm not derailing this topic btw.
 

Offline yor_on

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What becomes of a straight line when Conceptually challenged?
« Reply #24 on: 14/09/2010 19:58:10 »
Not as I can see? There seems to be different possibilities for how to describe it mathematically? But you really need to give some examples to how you think kenhikage so that we uninitiated see what you mean, why it have to be a sine curve in 2D and possibly a wave sheet in 3D? Wave-sheet btw? What would that look like?
 

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What becomes of a straight line when Conceptually challenged?
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