I Curvature of space and spacetime

[gpran]

General relativity suggests that path of light is curved around sun. This curvature is not dependent upon frequency of the photon.
What is the physical difference between 'curvature of space' and 'curvature of space-time' ? We can make measurements at two points in space at same time. But there will be some time difference between two points in space-time. Will this time difference be equal to R/c ?

 

[phinds]

Actually, the path is NOT curved when properly described. It travels in a straight line in the geometry that described space-time (pseudo-riemann Geometry). It is only when we try to force-fit Euclidean geometry onto space time that we then talk about it being curved.

 

[gpran]

Thanks. Does the pseudo-reimann geometry represent the space-time space and the euclidean geometry represent the ordinary space? Then does it also mean that two points in spacetime are related by R/c?

 

[phinds]

Thanks. Does the pseudo-reimann geometry represent the space-time space and the euclidean geometry represent the ordinary space?

I don't know how to answer that. Euclidean geometry works fine as a representation of small-scale issues like building bridges, just as does Newtonian gravity.

[quote\Then does it also mean that two points in spacetime are related by R/c?[/QUOTE]I don't know what you are asking.

 

[gpran]

Basically trying to understand the physical concept of spacetime. Just trying to visualise the difference between two locations in ordinary space and spacetime geometry.

 

[FactChecker]

What you call "ordinary space" does not exist. The only thing that really exists is the Riemannian geometry, which is curved. The curvature can be detected and measured within the Riemannian Geometry without reference to any Euclidean geometry world. Some simple ways that curvature can be detected and measured are by measuring that the angles of a triangle do not add up to 180 or that the circumference of a circle is not 2πr. So without looking at any other geometry, we can see the effects of General Relativity in the Universe. In particular, the exact orbit of Mercury makes sense with GR.

 

[gpran]

Thanks.

 

[FactChecker]

You can try to visualize GR within Euclidean space. A book I like for casual understanding is Relativity Visualized by Lewis Carroll Epstein. But you can not take that approach very far. Really deep understanding requires another approach.

 

[Ibix]

Spacetime is a four dimensional entity. How you choose to split it into "space" and "time" is up to you. Just like in ordinary Euclidean three dimensional space you just pick an x direction, a y direction and a z direction you can just pick four directions. The unusual thing about pseudo-Riemannian geometry is that one of the directions has a different behaviour from the others, and you'd be smart to call this a time-like direction - but there are still infinitely many ways of choosing your four directions. Incidentally, this means that you cannot "make measurements at two points in space at same time" in any non-arbitrary sense, since "time" in this sense is just a direction you picked.

You can always choose four directions and build a small set of axes (three rods and a clock). You can always do it twice, at slightly different places, and then ensure that they are oriented in the same directions in some sense. And you can always extend your rods and keep your clocks ticking. If the rods are the same perpendicular distance apart always and the clocks always tick at the same rate then you have flat spacetime. If either condition is violated then you have curved spacetime. You always get the latter if you measure carefully enough.

 

[gpran]

Kindly guide me if there is any time difference between two points on this triangle? What is the time difference between two points on time space?

 

[gpran]

I am really trying to understand: "You can always choose four directions and build a small set of axes (three rods and a clock). You can always do it twice, at slightly different places, and then ensure that they are oriented in the same directions in some sense. And you can always extend your rods and keep your clocks ticking. If the rods are the same perpendicular distance apart always and the clocks always tick at the same rate then you have flat spacetime. If either condition is violated then you have curved spacetime. You always get the latter if you measure carefully enough".But wonder if in 4 dimension space, we can define separation between two points in terms of exact values of distance and time.
Also it may be difficult to define two simultaneously occurring events if such events occur at different locations.

 

[Ibix]

Kindly guide me if there is any time difference between two points on this triangle? What is the time difference between two points on time space?

Your questions are like asking "is there a difference in the z coordinates of the corners of a triangle". It depends on the triangle's orientation with respect to your coordinate system.

But wonder if in 4 dimension space, we can define separation between two points in terms of exact values of distance and time.

Of course. But only in an arbitrary way, just as the x,y,z separation of two points in Euclidean space is arbitrary.

Also it may be difficult to define two simultaneously occurring events if such events occur at different locations.

Again, it's easy to define it, but there is no absolute answer to whether two events are simultaneous or not. Google for "relativity of simultaneity".

 

[gpran]

Thanks for your reply. The space- time has properties which are very different from our normal understanding of ordinary space. This discussion was very useful in getting some idea about it.