Posted: Mar 02, 2011 8:13 pm
Xaihe wrote:I know what you're trying to explain, but I don't think you understand my point exactly. What I'm questioning is whether there is an assumption here and whether that assumption is valid.
First, my own assumption is that mass curves spacetime and that this accounts for gravity and time dilation. Then, the assumption necessary for the picture you laid out is that spacetime at a particular point can be curved in multiple (possibly opposing) directions at once. And this opposite curvature would then account for the apparent decreased acceleration and increased time dilation. I'm just having some trouble wrapping my head around this idea (which is no argument).
If my interpretation is wrong, please let me know where (and if you would, why).

Is it something like this that's troubling you:

"There has to be a gravitational field to produce the time dilation effect, but at the centre of the planet, there is clearly no gravitational field, because an object there feels no force at all, so how the fuck can we see time dilation ?"

One issue is that thinking in terms of a "gravitational field" gets confusing in relativity unless you're very specific about what you mean by it. The object in relativity that is most analogous to the Newtonian "gravitational field" is the connection, (also called the Christoffel symbols). In Newtonian theory the thing you differentiate to get the gravitational field is the potential, and the relatvitistic analog of the potential is the metric. The first derivatives of the metric give the connection coefficients.

Newton
.............................................Einstein

Potential...........................................Metric
....|.....................................................|......
.differentiate....................................differentiate
....|.....................................................|......
....V.....................................................V.....
..Field...........................................Connection

Now, the reason why the field (connection) can be a bit misleading is that at any point, you can find a coordinate system which makes it vanish at that point. Effectively, you choose a local freely falling frame (geodesic normal coordinates), and hey presto, what you thought was a gravitational force vanishes there (if you have a tidal component to your field you can detect it by noting how the paths of freely falling particles diverge or converge there).

However, note that there's more to physics than just the first derivative of the metric at a point. There's also the second derivatives, from which you can construct the curvature. I haven't tried computing it for the Schwarzschild interior metric, but I'm sure the curvature doesn't vanish at that point.

Anyway, for the purposes of analysing the time dilation in the OP, we want to compare two identical clocks, one at the centre and one at the surface. The thing is, to get from the centre to the surface, we have to travel a fair old distance, climbing out of the gravity well in the process. So the gravitational field (connection) value at a single point (in this case the earth's centre) is NOT determining the dilation effect. It's the comparison of the metric (potential) at the two points that determines this.