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# Shortest distance between two points

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We all know that the shortest distance between two points is a straight line, well according to Einstein, space-time is curve, so technically, the shortest distance between two points should be a curve line. Am I right? Please comment

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You are (more or less ) exactly right.

all geometric properties (do parallel lines intersect? what is the shortest distence between 2 points? what is the sum of all the angles of a triangle?) Depend on the properties of space. (this is always hard to visualize, so you need a picture... hmm good explanation you can find here: http://library.thinkquest.org/2647/geometry/geometry.htm?tqskip1=1 but no pictures... http://math.youngzones.org/Non-Egeometry/non-E_geometries.html are 2 other geometries then the normal flat ('euclidean'))

Now the part where i think you are not right

- On a curved geometry there still are straight lines; well they are not straight compared to a staight line in flat geometry; but still straight. e.g. take the surface of theearth. A straight line is for example the equator. You can say of course that it is a circle; but that's only from the point of view of an observer in flat 3d space. If you only can live on the surface of the sphere, (and so for you there is no interior of the earth; so no interior for your circle). Another example: take a piece of paper anddraw a straight line on it. Then fold the paper. You see the line now as curved; but from the view of the piece of paper (which is curved itself) the line is still straight.

And guess what? These 'straight lines in a curved geometry' are ecaxtly the paths einstein predicts particles to take. (We can't see the changes in geometry; because we can't see space itself. But we do see particles attracted by gravity, which einstein says is a change of your geometry)

(sorry it's quite early here in holland, so my explanaition is crap.. if you want more detail, please say so).

(Don't worry; it took mankind some 2500 years to realize that there is more then euclidean geometry)

Bo

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wouldnt it be that the shortest between 2 points is the straight line and WITHOUT GRAVITY...???

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Isn't that the shortest distance between 2 points is a geodesic?

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Isn't that the shortest distance between 2 points is a geodesic?

Doesn't that only ring true for the surface of a sphere? I think that we can tunnel through the surface of a sphere and get back to the straight line. Of course we are still left with curved space.

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TeleMad: Isn't that the shortest distance between 2 points is a geodesic?

UncleMartin: Doesn't that only ring true for the surface of a sphere? I think that we can tunnel through the surface of a sphere and get back to the straight line. Of course we are still left with curved space.

But by breaking into and moving through the sphere you've changed to using flat space again, and if I am not mistaken, a straight line in flat space is a geodesic.

PS: Looked it up in a dictionary.

"geodesic: the shortest line beween two points that lies in a given surface."

That would apply to a sphere or a plane.

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Is flat space a real thing? Isn't it just a human contrivance to simplify the whole concept? So we come to a straight line through a sphere being a geodesic in curved space?

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Found this too:

"A straight-line orbit is known as a geodesic; it is the shortest distance in space between two points; it is a straight line in the local geometry, but to an observer elsewhere, whose local geometry is different, it appears curved. A geodesic in flat space (as in special relativity) is the familiar Euclidean straight line..." (Cosmology: The Science of the Universe: Second Edition, Edward Harrison, Cambridge University Press, 2000, p229)

So I was right: the shortest distance between two points is a geodesic...that applies regardless whether we are looking at flat geometry/space or curved geometry/space.

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So Unc's comment is correct - what appears to be a straight line is a geodesic curve in space.

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so if a person draw a straight in the outer space and bring the piece of paper back here, it would appear differently?

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so if a person draw a straight in the outer space and bring the piece of paper back here, it would appear differently?

No, because the piece of paper also 'adopts' so to speak to our geometry.

For the rest i completely agree with Telemad

Bo

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so, if a person is somehow able to look into a piece of paper (a square on earth) in the space from earth, it would look different...???

that means that an image from telescope is not really how it looks? (discounting the gravity lens)

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Don't discount the gravity lens - it is one reason why the sky looks the way it does. Light travels towards us via the shortest possible path. That path is NOT a straight line so what we are seeing is not really what's out there (plus, of course, there is a time issue involved).

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if i understand you correctly: yes.

But this is not as strange as it seems. The most obvious example is e.g. this:

form here we look to say a planet circuling alpha centauri. Then, in our point of view this planet is describing nice ellipses. and if we took the planet here (well let's just say: away from alpha centauri and any other star) it would suddenly move in a straight line. Because the lack of gravity makes the geometry flat.

Bo

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The shortest distance between two points is a worm hole

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The shortest distance between two points is a worm hole

so the fastest way to get my cup of coffee is to travel zillions of miles , in the hope to find a wormhole, have no clue where i pop out; spend millinia trying to find earth back; stay for ages on a highway hitchhiking back to my room and grab my coffee?

Bo

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Originally posted by: Bo

so the fastest way to get my cup of coffee is to travel zillions of miles , in the hope to find a wormhole, have no clue where i pop out; spend millinia trying to find earth back; stay for ages on a highway hitchhiking back to my room and grab my coffee?

Yes.

Anyway...

if we can theorize the difference between following the curvature of the earth compared to tunnelling through it, adding utilization of defined locational worm holes is valid. And shorter.