# Geometry of minkowski spacetime

since the first moment I've started studying the theory of relativity i thought that the minkowski metric represents a flat spacetime (a 4D euclidean space) but while I was surfing the WWW , I arrived to an interactive applet the helps you visualise the idea of spacetime curvature is GR , here it is : http://www.adamtoons.de/physics/gravitation.swf
in this applet when i putted the "spherical mass" bar to zero , it should have given me a flat spacetime ....... i can see the flatness is space but i see a curved path in time ..... shouldn't it be flat in both space and time ??!!

seeing this also made me remember an old question i had when i first started relativity , why do we use hyperbolic functions "sinh .. cosh ... etc" in the rotation equations of lorentz transformation instead of ordinary trigonometric functions ??

Hurkyl
Staff Emeritus
Gold Member
One rationalization is that the level curve of the Euclidean distance function is a circle
x² + y² = r²​
and the level curve of the Minkowski distance function is a hyperbola
x² - y² = r² or y² - x² = r²​

Of course, in the end, the right answer is "because it works".

They use a trick as they impose periodic boundary conditions in the time direction. If you start out with no mass an no initial velocity you see that you come back to the same point in space and time.

Minkowski space isn't really curved in the geometrical sense. The "weirdness" comes from the fact that it has a different signature when you compare it with a Euclidean space.

Ich
You're lucky, the author of this website (A.T.) is a member of PF. He should be the one who explains it best.

A.T.
since the first moment I've started studying the theory of relativity i thought that the minkowski metric represents a flat spacetime (a 4D euclidean space) but while I was surfing the WWW , I arrived to an interactive applet the helps you visualise the idea of spacetime curvature is GR , here it is : http://www.adamtoons.de/physics/gravitation.swf
This is not Minkowski space-time (which is pseudo-Euclidean), but space-propertime (which is Euclidean). It is a slightly different type of diagram, or geometrical interpretation. Here is a simple version of it:

in this applet when i putted the "spherical mass" bar to zero , it should have given me a flat spacetime ....... i can see the flatness is space but i see a curved path in time ..... shouldn't it be flat in both space and time ??!!
A cylinder is flat in terms of intrinsic curvature. The extrinsic curvature you see doesn't affect someone living within the cylindric surface, that represents space-propertime .

However, it is important to understand about that visualization, that it is not a closed cylinder, but rather a multi layered roll of space-propertime. After each cycle you arrive at a new layer of the diagram.

Press the "Help" button for explanations.

seeing this also made me remember an old question i had when i first started relativity , why do we use hyperbolic functions "sinh .. cosh ... etc" in the rotation equations of lorentz transformation instead of ordinary trigonometric functions ??
Well, that is how Minkowski space-time is defined, it is pseudo-Euclidean. That is why some things are better visualized with the Euclidean space-propertime.