Simplest question to articulate concepts in General Relativity

[my_wan]

I needed to ask a range of questions about gtr without resorting to the complexity of trying to learn about the formalism on a forum. Many good responses were given to other questions on this forum but the inherit complexity of the situation still left room for misconceptions.

Consider a homogenous hollow inertial mass sphere with no rotation in an otherwise flat region of space. As you approach this sphere the curvature increases along with the time dilation. As you pass inside this sphere acceleration drops to 0 but the time dilation remains the same as on the surface.
Questions;
(1) Is the time dilation uniform throughout the inside of the sphere?
(2) Does gtr define the space inside this sphere as flat.

The answer to these deceptively simple questions would provide myself and possibly others with a framework to learn much more about gtr and the formalism. Much thanks to the many intelligent contributors on this forum.

 

[pervect]

I needed to ask a range of questions about gtr without resorting to the complexity of trying to learn about the formalism on a forum. Many good responses were given to other questions on this forum but the inherit complexity of the situation still left room for misconceptions.
Consider a homogenous hollow inertial mass sphere with no rotation in an otherwise flat region of space. As you approach this sphere the curvature increases along with the time dilation. As you pass inside this sphere acceleration drops to 0 but the time dilation remains the same as on the surface.
Questions;
(1) Is the time dilation uniform throughout the inside of the sphere?
(2) Does gtr define the space inside this sphere as flat.
The answer to these deceptively simple questions would provide myself and possibly others with a framework to learn much more about gtr and the formalism. Much thanks to the many intelligent contributors on this forum.

1)
Yes - if time dilation did not remain uniform throught the inside of the sphere, an object would experience gravitational forces. (This comes from the linearized theory).

2)
The exact meaning of "flat" can be debated, but I can't think of any meaningful sense in which the space inside the sphere is not flat. The Riemann curvature is zero, and the metric coefficients are constant inside the sphere, which means that all the Christoffel symbols are zero (another test for flatness).

The only thing that might be a bit confusing is that time does pass at a different rate inside the sphere than outside - this is due to the effect of the metric coefficients being unity at infinity by convention (a Minkowskian metric), and not being unity inside the sphere.

Note that this relates to the oft-heard statement that time dilation is caused by the gravitational potential energy, not by the gravitational field (both of these terms as used here are used in the Newtoniain sense).

 

[my_wan]

The only thing that might be a bit confusing is that time does pass at a different rate inside the sphere than outside - this is due to the effect of the metric coefficients being unity at infinity by convention (a Minkowskian metric), and not being unity inside the sphere.

Thanks pervect. The above quote explains why certain statements have often been confusing in the past. My personal instincts was that the metric coefficients be set at unity for the observers local. Thanks again.