Shortest distance between two points

Aki · 09-05-2004

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

Bo · 09-06-2004

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/g....htm?tqskip1=1 but no pictures... http://math.youngzones.org/Non-Egeom...eometries.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

Tim_Lou · 09-06-2004

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

TeleMad · 09-06-2004

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

Uncle Martin · 09-06-2004

Quote:

Originally posted by: TeleMad
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.

TeleMad · 09-06-2004

Quote:

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

Quote:

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.

Quote:

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

That would apply to a sphere or a plane.

Uncle Martin · 09-06-2004

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?

TeleMad · 09-06-2004

Found this too:

Quote:

"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.