Curved space

[Widdekind]

In special relativity, momenergy is calculated in a single reference frame. The 4-displacement is measured in that frame, as you indicate. But the proper time is invariant -- it is the same for all (uniformly moving) frames of reference.

"mom-energy" =

[

"space-time 4-displacement" between events A,B

]

divided by

[

"proper time"

]

I understand you to be telling me, that "proper time" -- which, in my mind, is specific to that reference frame "threading through" both events A,B -- is better thought of, as the "invariant interval [math]dS = \sqrt{dS^2} \approx \sqrt{t^2 - x^2}[/math].

 

In my mind,

 

"universally invariant interval"

dS

between events A,B

is distinct from

 

"proper time"

which is limited, to that special reference frame, in which the whole entire invariant interval dS is "perceived" to be "elapsed time" dS = dt_p, i.e. in every other reference frame, the invariant interval, although constant, is composed of a time part and a space part, whose difference remains invariant.

 

In my wordage, I understand you to be telling me, that

 

"momenergy" = "4-displacement" / "invariant interval

dS

"

Intuitively, that seems a better definition, on the 'grounds' that the invariant interval dS is "directly accessible" to every reference frame, without any naive notions, of the perceived-to-be-moving particle somehow "reporting its proper time". To my mind, distinguishing "invariant interval" (same for all observers) from "proper time" (specific to a special reference frame) helps clarify this concept.

[IM Egdall]

"mom-energy" =

[

"space-time 4-displacement" between events A,B

]

divided by

[

"proper time"

]

Intuitively, that seems a better definition, on the 'grounds' that the invariant interval dS is "directly accessible" to every reference frame, without any naive notions, of the perceived-to-be-moving particle somehow "reporting its proper time". To my mind, distinguishing "invariant interval" (same for all observers) from "proper time" (specific to a special reference frame) helps clarify this concept.

 

I agree.

[DrRocket]

I don't think such a configuration can be found easily in outer space: the Universe has no poles and random points in space cannot be found at the antipodes of the universe.

 

 

What you think or believe is irrelevant. It is what you can support with evidence or prove from established theory that counts.

 

You might consider the fact that one of Einstein's preferred models for the universe modeled space as a 3-sphnere.

 

A geodesic is an arc according to this: http://en.wikipedia.org/wiki/Geodesic

 

And the shortest distance between two points in 4D spacetime could be a wormhole........equivalent to boring through the earth from one point on the surface to another.

 

Correct?

 

Wiki strikes again. The second definition is the correct one. On a Riemannian manifold there is always an affine connection, the Levi-Civita connection that produces the metric.

 

A geodesic is only length minimizing locally -- in a sufficiently neighborhood of a point. This is not a good definition, but rather is a consequence of the proper definition noted above.

 

On manifolds that are not geodesically complete there are points that are not connected by geodesics. On other manifolds there may be points connected by more than one, and perhaps infinitely many geodesics.

 

I would sugges that some people in this thread (not particularly you) might want to actually learn some differential geoometry prior to opening their mouths and making fools of themselves.

Edited February 10, 2012 by DrRocket

[Santalum]

What you think or believe is irrelevant. It is what you can support with evidence or prove from established theory that counts.

 

You might consider the fact that one of Einstein's preferred models for the universe modeled space as a 3-sphnere.

 

 

 

Wiki strikes again. The second definition is the correct one. On a Riemannian manifold there is always an affine connection, the Levi-Civita connection that produces the metric.

 

A geodesic is only length minimizing locally -- in a sufficiently neighborhood of a point. This is not a good definition, but rather is a consequence of the proper definition noted above.

 

On manifolds that are not geodesically complete there are points that are not connected by geodesics. On other manifolds there may be points connected by more than one, and perhaps infinitely many geodesics.

 

I would sugges that some people in this thread (not particularly you) might want to actually learn some differential geoometry prior to opening their mouths and making fools of themselves.

 

Can I take it that, on a 2D curved surface a geodesic = an arc but that this does not hold for higher dimensional curves?

 

I suppose it should be obvious that an arc is specific to a 2D curved surace but that a geodesic is more generalised and covers higher dimensions.

 

Perhaps Wikipedia should explicitly state this if it is the case because it currently appears that geodesic = arc may be assumed by some readers.

Edited February 10, 2012 by Santalum

[DrRocket]

Can I take it that, on a 2D curved surface a geodesic = an arc but that this does not hold for higher dimensional curves?

 

no

 

I suppose it should be obvious that an arc is specific to a 2D curved surace but that a geodesic is more generalised and covers higher dimensions.

 

no

 

Perhaps Wikipedia should explicitly state this if it is the case because it currently appears that geodesic = arc may be assumed by some readers.

 

A geodesic is curve. The dimension of the manifold is irrelevant.

 

A geodesic is a smooth curve with the additional costraint that the family of tangent vectors along the curve is parallel along the curve.

 

There is no simple explanation of this. You need to read a book on differential geometry.

[Santalum]

A geodesic is a smooth curve with the additional costraint that the family of tangent vectors along the curve is parallel along the curve. There is no simple explanation of this. You need to read a book on differential geometry.

If you take a given latitude on the earth's surface and draw tangents all the way along that latitude line then they would all converge some where above the north or south pole and form a cone. If you take a given longnitude on the earth's surface and draw tangents along it then none of them will be parellel. So I don't understand how you can have parallel tangents........or have I some how misunderstood what you meant by this? Can you direct me to some sort of diagram that will enable me to visualise what you mean by this? Here is another site that describes a geodesic as a great arc: http://gregegan.cust...warzschild.html

On the surface of the Earth, the closest thing we have to a straight line is known as a great circle, or geodesic. The geodesic between two cities is the route that planes would fly in a perfect world (ignoring various logistical complications), because it always comprises the shortest total distance. In a three-dimensional view of the Earth, the geodesic that joins city A with city B is an arc of the circle with a radius as large as that of the Earth itself, that passes through both cities, and whose centre lies at the centre of the Earth. But we don't need to take that Earth-from-space view; we could equally well define the geodesic from A to B by requiring that everywhere along its length, it acts locally like the straightest possible line: as we walk along it, we won't detect any swerving to the left or right.

As I understand from this the distinction between a great arc and an arc is that an arc is a curved line segment with tangential line segments along its length but a great arc is a curved line segment within a curved surface which has multiple tangential planes along its length. <BR><BR>

 

If you draw lines from two points on the earth's surface and the center of the earth then the line joining those two points on the earth's surface is an arc of a ccircular slice of the earth. So surely in 2D curved surfaces the distinction between an arc and a geodesic is some what blurred.

 

<BR><BR>I obviously can't picture this sort of scenario in 4D spacetime so, without learning the mathematics, I will just have to take your word for it that there is a clear distinction between an arc and a geodesic in that case.

Edited February 10, 2012 by Santalum

[michel123456]

(...)

You might consider the fact that one of Einstein's preferred models for the universe modeled space as a 3-sphnere.

(...)

What is a 3-sphnere?

[imatfaal]

What is a 3-sphnere?

 

The surface of a football or similar (known in maths as a ball) is a sphere - also known as a 2-sphere. It is the locus of all points in 3d space equidistant from a single point (the single point being the centre of the sphere).

 

A 3-sphere is the equivalent shape with an extra dimension. A n-sphere is the locus of all points that are the same distant but in every direction from a chosen point in n+1 space - ie a 2 sphere is all the points in space that are exactly the same distance in 3 dimensions from your given point. it is called a 2 sphere because is it just the surface - in laymans terms if you are on the 2 sphere there are only two directions in which you can move (ie north/south and clockwise/counter-clockwise) even though the construct is in 3d space.

 

I am still curious to see anyone give more than one shortest route between london and new york - I still cannot see how a sphere/2-sphere can have more than one shortest route (except between antipodes)

[DrRocket]

If you take a given latitude on the earth's surface and draw tangents all the way along that latitude line then they would all converge some where above the north or south pole and form a cone.

 

The only latitude that is a geodesic is the equator.

 

. So surely in 2D curved surfaces the distinction between an arc and a geodesic is some what blurred.

 

 

Wrong.

 

All geodesics are curves (arcs). Very few arcs are geodesics.

 

What is a 3-sphnere?

 

It is anything topologically equivalent to the set of all points that are some fixed distance from the origin in 4-space.

 

A 2-sphere is anything that is topologically equivalent to the set of all points that are a fixed distance from the origin in 3-space -- like the surface of a globe.\

 

Dimension refers to the object itself, not the dimension of some space in which you might find it embedded. So a 2-sphere is two dimensional because in small local patches it "looks like" a plane.

 

As I understand from this the distinction between a great arc and an arc is that an arc is a curved line segment with tangential line segments along its length but a great arc is a curved line segment within a curved surface which has multiple tangential planes along its length.

 

Apparently you don't understand it at all. This makes no sense.

 

Go read a book on topology and geometry. The book by Singer and Thorpe would be a good place to start.

[Santalum]

 

Apparently you don't understand it at all. This makes no sense.

 

Go read a book on topology and geometry. The book by Singer and Thorpe would be a good place to start.

 

Yes apparently I don't. I am struggling to comprehend what you mean.

 

But now you have me absolutely intrigued even though I have never had any particular interest in maths.

 

I don't suppose there is a website as good as your book that I could read? Anyone?

[DrRocket]

Yes apparently I don't. I am struggling to comprehend what you mean.

 

But now you have me absolutely intrigued even though I have never had any particular interest in maths.

 

I don't suppose there is a website as good as your book that I could read? Anyone?

 

No. But Singer and Thorpe's book is excellent.

 

However, if you have never taken any mathematics beyond, say, high school algebra, you may not find it readable. But then again, you might. While geometry and topology are usually taught to advanced undergraduates or beginniing graduate students (Singer and Thoroe is aimed at undergraduates) there is really no pre-requuisite beyond "mathematical maturity".

 

Unfortunately there is not much that can be done if find the book too imposing at this stage except go back farther and start learning more mathematics. Differential geometry simply requires some background.

 

There is all sorts of stuff on the internet. Some is OK and some is just trash. But you can't go wrong with a book by someone who actually knows what he is talking about. Mathematicians don't come much better than I.M. Singer.

Edited February 11, 2012 by DrRocket

[Santalum]

No. But Singer and Thorpe's book is excellent.

 

However, if you have never taken any mathematics beyond, say, high school algebra, you may not find it readable. But then again, you might. While geometry and topology are usually taught to advanced undergraduates or beginniing graduate students (Singer and Thoroe is aimed at undergraduates) there is really no pre-requuisite beyond "mathematical maturity".

 

Unfortunately there is not much that can be done if find the book too imposing at this stage except go back farther and start learning more mathematics. Differential geometry simply requires some background.

 

There is all sorts of stuff on the internet. Some is OK and some is just trash. But you can't go wrong with a book by someone who actually knows what he is talking about. Mathematicians don't come much better than I.M. Singer.

 

 

http://www.gmat.unsw...fs/navpaths.pdf

 

Found this website explaining rhumb lines, geodesics and gnomic projections etc.

 

What I have been describing is a rhumb line or an small circle arc formed by the intersection of plane and the earth's surface, that does not include the centre of the earth.

 

I can see from their diagram that it is clearly not going to be the shortest distance between two points on the earth's surface.

 

Where as if you tilt that plane until it includes the earth's centre then you have shortened the distance, albeit slightly, between the two points.

 

The penny is starting to drop DrRocket.

 

In fact what I have been assuming is an arc of the circumference of the earth is in fact an arc of the circumference of what would be a smaller earth..........the distance on the earth's surface between two points at a given latitude. This amounts to a tighter curve and when you project that onto the surface of the earth it ends up being a slightly longer distance.

 

Am I correct DrRocket?

 

Also has a perfect decription of the difference between geodesics (ellipsoids and ellipses) and arc (circles and spheres)

 

The 19th century term 'the antipodes' suddenly becomes meaningful to me.

Edited February 11, 2012 by Santalum