Gravity Time Dilation interior to a planet Regarding Gravity Time Dilation:  We know that, in a stronger gravitational field, time passes more slowly.   As one descends below the Earth's surface, how does the intensity of gravitational pull change, and if it is neutralized at the Earth's center (because of it being equally attractive from all directions), is the Gravitational Time Dilation also neutralized?
 A: Gravitational time dilation has an important feature (see "Important features of gravitational time dilation" at https://en.wikipedia.org/wiki/Gravitational_time_dilation) that says - The time dilation in a gravitation field is same as the dime dilation due to speed that is equal to escape velocity from that gravitational field.
That way the time dilation on surface of earth is same as that due to speed of 11.2 kilometer per second (escape speed from surface of earth).
This way, the time dilation will be greater inside earth and maximum at center of earth. Because, some speed is needed to reach surface, and then 11.2 km/s to escape from surface of earth. Suppose escape velocity at the center of earth is 13.7 km/s (I am not sure of exact value though). You can put speed of 13.7 in the time dilation formula $$t'=t\times \sqrt{1-v^2/c^2}$$ to find gravitational time dilation at the center of earth.
So, to answer your question, the time dilation does not depend upon the gravitational force (which would be zero at center of earth). It depends upon gravitational potential, and thus on escape velocity. Time dilation will not be neutralized at center, on the contrary, it will be maximum.
A: The gravitational force you experience at any particular depth only depends on the mass below you and the distance to the centre ($F(R)=G(\int_0^R 4\pi \rho(r) r^2 dr)/R^2$, where $\rho(R)$ is the density profile, if we use Newtonian mechanics). But the gravitational time dilation depends on the potential ($\Phi(R)=-\int_R^\infty F(R) dr$), so although there is no force at the centre, it is still there and actually maximal in magnitude.  
For a detailed calculation see The young centre of the Earth by Uggerhøj, Mikkelsen, & Faye. They find that due to the dilation the centre of the Earth is 2.5 years younger than the surface. 
