Is the Earth Flat? An Analysis of Spacetime Curvature

[markintheworld]

This is a bit of a lark, but I'm just wondering if I've understood correctly to draw my conclusion...

It is my understanding that the curvature of spacetime postulated by GR is what causes orbits which appear elliptical. An orbiting body is following its inertial path which might otherwise be a straight line, but due to the curvature of spacetime caused by a nearby massive object, that straight line appears curved, as if the orbiting body were traveling around the edge of a depression. So the moon is traveling in a straight line (its inertial tendency), but because it is near the earth, a massive gravitational object, it APPEARS to be traveling a curved path. Am i correct so far?

Assuming the above is true, that a straight path can appear curved due to the distorting effects of gravity, couldn't it also figure that a flat plane could appear similarly curved? Couldn't it be true that the surface of the Earth is a flat plane, but the curvature of space-time caused by its own gravity is the reason that it appears spherical? Is this why large gravitational objects generally appear to be spherical? Or is there some flaw in my logic?

Insight would be appreciated...

 

[Thrice]

Inertial path doesn't automatically mean straight line. The straight line is a special case. When something follows an inertial path/geodesic/maximizes proper time & all that it may be going straight or not depending on the geometry.

 

[MeJennifer]

An orbiting body is following its inertial path which might otherwise be a straight line, but due to the curvature of spacetime caused by a nearby massive object, that straight line appears curved, as if the orbiting body were traveling around the edge of a depression. So the moon is traveling in a straight line (its inertial tendency), but because it is near the earth, a massive gravitational object, it APPEARS to be traveling a curved path. Am i correct so far?

No in GR theory this is not the case!
An orbiting body is traveling on a geodesic but a geodesic is not necessarily a straight line!
See for a detailed discussion on this subject see the last few pages of: https://www.physicsforums.com/showthread.php?t=126871"

 

[pervect]

Using standard rulers, the surface of the Earth cannot be flat, because it has a non-zero "curvature tensor".

Similarly, space-tme cannot be flat around the Earth, either, because space-time has a non-zero "curvature tensor".

This is assuming that one defines the rather fuzzy English word "flat" with the mathematical definition "zero Riemann curvature tensor".

The longish quote from Einstein in the thread

https://www.physicsforums.com/showthread.php?t=123922

may give some insight. This is the quote about rulers on a heated slab, as per the title of the thread above.

Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab.

For a more mathematical approach that defines what curvature means in 3dimensions, see for instance "Gaussian curvature" in the wikipedia

http://en.wikipedia.org/wiki/Curvature

Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature.

...

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. He runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, he would find C(r) = 2πr. On curved surfaces, the formula for C(r) will be different, and the Gaussian curvature K at the point P can be computed as

$$K = \lim_{r \rarr 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}$$

to be successful at measuring curvature, note that it is assumed that the ants have rulers.