# The Naked Scientists Forum

### Author Topic: Can there be a circle in SpaceTime?  (Read 6720 times)

#### yor_on

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##### Can there be a circle in SpaceTime?
« on: 26/10/2011 05:36:15 »
If you add time coordinates to the spatial coordinates, can there be a circular motion in SpaceTime? That one simplifies a lot to me, so I'm curious to if you agree with me in you needing all coordinates to define a position relative the observer.

There are no circles :) instead we find 'spirals'. Also, how would a Lorentz contraction be described in such a 'motion', would it crack up the spinning wheel or not. And think of a planet from this. Wouldn't every point in that planet would then describe a 'spiral'?

How about a sphere, 'free falling' (no acceleration) and being at rest with you. There every point would follow a 'straight line', if we define the 'straightest lines' as those described by geodesics. But if it was rotating in some direction, whilst still being at rest with you, 'free falling'?

How would each point of that ball then look, counting in all coordinates, spatial and time.
==

Alternatively, assume a ring spinning, at rest with you, free falling. Then assume that we rotate the ring, 90 degrees at a time, all directions possible, what would that do to the coordinates, graphically.
« Last Edit: 26/10/2011 05:41:17 by yor_on »

#### MikeS

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##### Can there be a circle in SpaceTime?
« Reply #1 on: 26/10/2011 06:48:54 »
If you add time coordinates to the spatial coordinates, can there be a circular motion in SpaceTime? That one simplifies a lot to me, so I'm curious to if you agree with me in you needing all coordinates to define a position relative the observer.

There are no circles :) instead we find 'spirals'. Also, how would a Lorentz contraction be described in such a 'motion', would it crack up the spinning wheel or not. And think of a planet from this. Wouldn't every point in that planet would then describe a 'spiral'?

How about a sphere, 'free falling' (no acceleration) and being at rest with you. There every point would follow a 'straight line', if we define the 'straightest lines' as those described by geodesics. But if it was rotating in some direction, whilst still being at rest with you, 'free falling'?

How would each point of that ball then look, counting in all coordinates, spatial and time.
==

Alternatively, assume a ring spinning, at rest with you, free falling. Then assume that we rotate the ring, 90 degrees at a time, all directions possible, what would that do to the coordinates, graphically.

I suspect you mean helix?  If so, anything travelling in a circular motion in space is travelling in a helix in space time.

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #2 on: 26/10/2011 07:35:07 »
Yep, a helix Mike. Nice word too.

#### JP

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##### Can there be a circle in SpaceTime?
« Reply #3 on: 26/10/2011 13:18:22 »
This question seems familiar.  I think we've had this asked before on the forum, but I can't find it.

A circle in Cartesian coordinates is just the set of all points equidistant from the center of the circle.  In special relativity, because distance is measured slightly differently, a circle is a hyperboloid.  This is because in Cartesian coordinates, distance is just given by the Pythagorean theorem Distance2=x2+y2+x2.  If you use time in the proper relativistic way, then Distance2=x2+y2+x2-time2.  That minus sign makes a world of difference and turns circles into hyperboloids.

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #4 on: 27/10/2011 05:16:17 »
Want to dissect that one JP?

Especially Distance2=x2+y2+x2-time2.

Why -?

#### MikeS

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##### Can there be a circle in SpaceTime?
« Reply #5 on: 27/10/2011 08:02:35 »
JP
Interesting.  I wondered why power station cooling towers were that shape.

#### imatfaal

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##### Can there be a circle in SpaceTime?
« Reply #6 on: 27/10/2011 12:35:08 »
Want to dissect that one JP?

Especially Distance2=x2+y2+x2-time2.

Why -?

This is the metric tensor of lorentz / special relativity. in flat minkowski space time the spacetime interval is dS and we can relate that to the co-ordinates in 4 d space time with the following metric

dS2 =  - cdT2 +  dx2 + dy2 + dz2

this is equivalent to pythagorus in euclidean space
dS2 = dx2 + dy2 + dz2

There is a nice summary of space time intervals here http://en.wikipedia.org/wiki/Spacetime_interval#Spacetime_intervals
and on metric tensors here http://en.wikipedia.org/wiki/Metric_tensor

In warped space the metric tensors get really weird!

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #7 on: 27/10/2011 21:55:32 »
Thanks Imatfaal, so where does the - comes from in JP:s definition?

This seems more or less like a definition that must change with the metric you use. I got curious and tried to look up a graphic representation of a hyperboloid

« Last Edit: 27/10/2011 21:58:55 by yor_on »

#### JP

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##### Can there be a circle in SpaceTime?
« Reply #8 on: 27/10/2011 22:11:06 »
This seems more or less like a definition that must change with the metric you use.

Exactly!

Metrics basically tell you how to calculate lengths.  Let's say we have coordinates (x,t).  In a Euclidean metric, you calculate length by s2=x2+t2.  In a Minkowski metric (which is what you use for special relativity) s2=x2-t2

In a general case, you might have s2=ax2+bt2+cxt, so in general you need 3 numbers, a,b and c to describe distances--these are described by the "metric tensor" which has 4-components, although 2 of them are the same.

(I'm actually cheating a little.  All the distances and times should be infinitesimally small, since the definition of distance might vary from point to point and is only valid at a particular infinitesimally small patch of space-time.  The fact that things are small is why you can't have higher order powers of x and t appearing.)

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #9 on: 28/10/2011 00:18:08 »
If I assume that a distance between two points A and B always must be positive, which seems rather reasonable to me? As to get it otherwise I will have to introduce time going backwards, at the very least? Or how would I do it otherwise? Then the length of something, no matter its metric, can't be negative

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #10 on: 28/10/2011 02:15:16 »
That should mean that a circle must be a spiral/ helix, or hyperbolic if I imagine the motion starting as a thought up point at the lower rim of that hyperbole then finding its way 'up' in a four dimensional way (that is 'space' as we see it, and add time as the fourth coordinate), am I wrong there?

And 'gravity' will always define those coordinates we find, 'buckling' its metric. But as 'times arrow' is a 'positive' one to me, as in having a defined temporal direction the motion if we split that circle into points, should become a wiggly spiral :)

That because, as you move that point forward in time, the 'gravity' it exist in may change, and with that comes a length contraction and time dilation relative its next point, defining that circle to us, also assuming that a Planck length will be what makes the smallest understandable difference, well, maybe somewhat larger considering HUP.

Is that too weird?

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #11 on: 28/10/2011 03:13:13 »
The reason I see that 'spiral/helix' is that I see time as having both a value and a direction. And that is not wrong, although not the usual way to define it. Normally time is defined as a scalar, having only a value. From a entropic point of view in where you expect 'time' to evolve in each point, according to some entropic principle, you might want to argue that this is the best definition.

But my view is that time has a lot to do with 'c', and using that we will find that although we all find others 'time' to vary, we also can agree on that the 'local clock' we then use is, in some means, 'universal', meaning that all 'local frames' have a same defined 'ground state' as defined by 'c' locally (clock).

And that gives SpaceTime a 'direction', not only locally.
« Last Edit: 28/10/2011 03:14:48 by yor_on »

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #12 on: 28/10/2011 03:37:36 »
You might ignore the universality there, I usually do, as I define it all locally to keep it simple. But on the other hand, if that 'ground state' didn't exist we would have a lot of trouble assuming that 'entropic view' for example, as we then might find that there was no clear definitions of 'entropy' as in a radioactive decay relative another expected to be the 'exact same'. Also we have the fact that all atoms behave the same, and that one interaction with one atom is interchangeable with a same action on another 'same' atom, etc.

So it is a important principle to me, and if we let our imagination run free for a moment all points in SpaceTime then will be 'moving', locally at a same temporal speed, but in this case containing a vector at that 'ground state'.

#### JP

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##### Can there be a circle in SpaceTime?
« Reply #13 on: 28/10/2011 15:42:48 »
You brought up a lot of points, and I don't have time to answer them all right now, but I'll try to answer a couple.

If I assume that a distance between two points A and B always must be positive, which seems rather reasonable to me? As to get it otherwise I will have to introduce time going backwards, at the very least? Or how would I do it otherwise? Then the length of something, no matter its metric, can't be negative

The square of the distance can be negative in special relativity (and presumably general as well).  What that tells you is that nothing can travel between those points in space-time because to do so would mean travel faster than the speed of light.  Distance of zero means travel between them is possible only at light speed, and positive distance means sub-light speed travel is possible.  There's more here: http://en.wikipedia.org/wiki/Spacetime#Spacetime_intervals

That should mean that a circle must be a spiral/ helix, or hyperbolic if I imagine the motion starting as a thought up point at the lower rim of that hyperbole then finding its way 'up' in a four dimensional way (that is 'space' as we see it, and add time as the fourth coordinate), am I wrong there?

What's your definition of "circle" and what's your definition of space-time.  Under the definition of a circle as all points an equal distance from some center, and working in special relativistic space-time, a circle becomes a hyperbola.  A helix is given by all points equidistant from some center in space, where time is always moving forward, which would be the Newtonian picture, but it's definitely not a circle in space-time, since time is independent of space.

You might ignore the universality there, I usually do, as I define it all locally to keep it simple. But on the other hand, if that 'ground state' didn't exist we would have a lot of trouble assuming that 'entropic view' for example, as we then might find that there was no clear definitions of 'entropy' as in a radioactive decay relative another expected to be the 'exact same'. Also we have the fact that all atoms behave the same, and that one interaction with one atom is interchangeable with a same action on another 'same' atom, etc.
I'm not sure what you're getting at here.  What do particles have to do with circles in space-time?

Regarding the "locality" of general relativity, it is a universal law.  Sure, some of the math works out by considering small patches of space-time, but it also gives you the tools to connect those patches to build up motion over large regions.  It's also the case that the same laws of physics hold everywhere, so it's universal in that sense as well.

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #14 on: 30/10/2011 07:35:57 »
Negative as in finding FTL?

Well, that's one point I haven't discussed at all? I'm expecting that there are no such things as 'negative distances', except possibly mathematically?

As for my definition of a circle :)

I will go with the one where 'a circle as all points an equal distance from some center' for this. (In a uniform motion, being 'at rest' with it). And I'm not sure at all, if I assume four positional definitions for each point, except, possibly when 'at rest'?

As for the rest.

What I started to think about was how to define 'time moving' and that made me think of my own idea of a local 'universality' of time, probably should have kept that one apart from this.

« Last Edit: 30/10/2011 07:39:12 by yor_on »

#### JP

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##### Can there be a circle in SpaceTime?
« Reply #15 on: 30/10/2011 17:44:15 »
1) Negative intervals (interval is the word for "distance" in space-time, since time is now involved) are real things, in that they measure something physical about the universe.  A negative interval between events means they can't be causally linked--i.e. no signal can pass between them because it would have to do so at faster than the speed of light.

2) "All points an equal distance from some center," where distance is measured as a space-time interval, is going to always give you a hyperboloid in special relativity, because intervals are defined in such a way so that it doesn't change as you change reference frame.  In general relativity, it's going to be messier.  You can define a "circle" in the same way, but the exact shape it takes is going to depend on where you are in space-time and how "big" it is.

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #16 on: 30/10/2011 18:13:17 »
Defined that way I have no problem with calling it 'negative' JP. As for me discussing that circle it's a given to me that I include gravity.

#### MikeS

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##### Can there be a circle in SpaceTime?
« Reply #17 on: 02/11/2011 15:33:53 »
"Can there be a circle in SpaceTime?"

Looked at in one unit of panck time, yes, why not?

#### JP

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##### Can there be a circle in SpaceTime?
« Reply #18 on: 02/11/2011 16:15:02 »
"Can there be a circle in SpaceTime?"

Looked at in one unit of panck time, yes, why not?

Why not?

The nature space-time isn't known on sub-Planck scales.  You could make up a definition, but it wouldn't be scientific until you knew how it related to larger scales.  All my comments above have to do with scales that are big enough that you can define space-time as a smooth, classical structure.

#### damocles

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##### Can there be a circle in SpaceTime?
« Reply #19 on: 02/11/2011 23:06:04 »
Can there be a circle in space time co-ordinates?
Well, yes. That is a very easy one -- you simply have to trace a circle on a Minkowski diagram.
Can that circle be related to any likely behaviour that can be observed for any system?
Probably not, although pair creation (electron + positron) followed by annihilation in a strong magnetic field might come close.
The point is that in most ways (if not all -- I am not a particle physicist) a positron may be regarded as an electron moving backward in time. So the electron produced in the pair creation event could be regarded as moving forward through time until it meets its annihilation event, which reverses its direction in time until it can be made to meet its own creation event, travelling a different spatial course as the "positron" to the creation destination.

Can there be a circular motion in space time co-ordinates?
There is a fundamental semantic problem here:
On one interpretation any circular motion occurs in space time co-ordinates, though the shape it traces out on a Minkowski diagram would be helical.
Any attempt at interpreting a "circular motion" as one that is closed on a Minkowski diagram runs into a problem: any normal definition of "motion" involves the change in spatial co-ordinates along an increasing time axis. The whole idea of a closed motion, let alone a circular one, becomes a contradiction.

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #20 on: 03/11/2011 03:34:32 »
Yes, that was my thought too :)

#### yor_on

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##### Can there be a circle in SpaceTime?
« Reply #21 on: 03/11/2011 03:44:22 »
Eh, the last one, but I think it hold in reality for anything you do, including drawing a circle on a paper. Einstein defined it as SpaceTime, not as space and time. And physicists expect it possible to move 't' in their equations, as if our arrow isn't here. But even mathematicians grow old :)

#### Soul Surfer

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##### Can there be a circle in SpaceTime?
« Reply #22 on: 03/11/2011 09:24:47 »
Why are you particularly interested in a "circle"  technically a circle is a two dimensional object  a sphere is a three dimensional object.  a conventional four spatial dimensional object can only be visualised by its projection on to a three dimensional object in that a circle that expands from a point reaches a maximum and contracts back to a point is a preojection of a three dimensional sphere on a two dimensional surface.  so you could think of an object in for dimensional space that expands into a sphere and then contracts as you move through the forth dimension.  this ref goes some way towards this but is not quite right  http://www.foundalis.com/phy/4Dsphere.htm  and this goes the whole hog to n dimensions.  http://en.wikipedia.org/wiki/N-sphere.

An orbit is a circle (or ellipse to be accurate) in space time  so if you are ultimately thinking about cyclic structures that could contain and localise energy it would be better to think of a vortex.

#### JP

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##### Can there be a circle in SpaceTime?
« Reply #23 on: 03/11/2011 14:07:31 »
Can there be a circle in space time co-ordinates?
Well, yes. That is a very easy one -- you simply have to trace a circle on a Minkowski diagram.
Can that circle be related to any likely behaviour that can be observed for any system?
Probably not, although pair creation (electron + positron) followed by annihilation in a strong magnetic field might come close.

Sounds like a 1-loop Feynman diagram:

http://en.wikipedia.org/wiki/One-loop_Feynman_diagram

#### damocles

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##### Can there be a circle in SpaceTime?
« Reply #24 on: 03/11/2011 21:29:20 »
Can there be a circle in space time co-ordinates?
Well, yes. That is a very easy one -- you simply have to trace a circle on a Minkowski diagram.
Can that circle be related to any likely behaviour that can be observed for any system?
Probably not, although pair creation (electron + positron) followed by annihilation in a strong magnetic field might come close.

Sounds like a 1-loop Feynman diagram:

http://en.wikipedia.org/wiki/One-loop_Feynman_diagram

It is almost exactly what it is, except that it is based around a significant delay and spatial shift between the creation and annihilation events.

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##### Can there be a circle in SpaceTime?
« Reply #24 on: 03/11/2011 21:29:20 »