How exactly does mass curve space?

[alemsalem]

when we say that the Earth has curved the space around it. is that just an analogy based on the similarity of the mathematics between gravitation and curved spaces?
I'm asking because, nothing stops me (in my imagination:) from stacking perfect parallel planes all around the solar system and that will define a three dimensional Euclidean space. or is there? or is it a property of curved 4 dimensions that i can do that?

thanks a lot..

 

[Zac Einstein]

Mass curve space 'cause of it's energy and gravity...but I'm not sure

 

[pervect]

GR actually tells us that space-time is curved. However, if you take the usual time-slice out of space-time (one that's radially symmetrical around the central body), GR also predicts that space is curved.

If you look at a precise definition of distance - i.e. distance is the shortest path between two points, or an even more precise definition where we replace "shortest" with "extermal", we rarely actually measure distance directly - it would take a very large number of observers to show that some particular path was actually the shortest.

What we actually study is how light and radar signals propagate, and infer the underlying structure of geometry from these observations of light signals.

The results of such experiments involving light and radar signals are consistent with a model where space is curved. The specific amount of curvature can be quantified within a rather general framework called the PPN formalism which exist for the sole purpose of describing in a general fashion how space-time "could be" curved (a more general description than GR which makes more specific predictions about how it IS curved). The value from experiments (such as light deflection by gravity) are consistent with space being curved, and are consistent with the expected mangiutde of the curvature predicted by GR as measured by the appropriate PPN parameter, called gamma. The results are inconsistent with space (using the specified radially symmetrical time-slice) not being curved.

As a rather general way of understanding curvature, you can think of the volume of a sphere as starting to slowly depart from 4/3 pi r^3, where the surface of the sphere is defined by a set of points some distance 'r' from a central point. You can visualizize curvature by thinking about an analogous situation, where you define a set of points that are some distance 'r' away from a central point on , for example, a sphere, where said distances are being measured along curves which are constrained to lie on the surface of the sphere. Then you can see where the circumference of the circle is no longer 2*pi*r.

 

[Passionflower]

As a rather general way of understanding curvature, you can think of the volume of a sphere as starting to slowly depart from 4/3 pi r^3, where the surface of the sphere is defined by a set of points some distance 'r' from a central point.

We can eliminate the notion of r altogether:

In a Euclidean space we have (V is volume and A is area):

$$V =1/6\,{\frac {{A}^{3/2}}{\sqrt {\pi }}}$$

But for instance in a Schwarzschild solution the volume is larger. For instance we can calculate the volume between two shells of a given area if we have the mass of the solution.