# Gravitational Time Dilation

1. Apr 3, 2013

### Brute Force

Could somebody explain me the following:
According to GR time dilation due to gravitational field is expressed as:
T$_{g}$=T$_{f}$*$\sqrt{1-\frac{2GM}{rc^{2}}}$
where Tg is time with gravitation,
Tf is time somewhere without gravitation
G - gravitational constant
M - mass
r - radial coordinate of observer
and c - light speed.

Lets assume two explorers are working on two different planets with the same gravitation.
My understanding is that gravitation could be expressed as g=G$\frac{M}{r^{2}}$.
Mass M for spherical planet is equals to: M=$\frac{4}{3}$$\pi$r$^{3}$$\rho$, where $\rho$ is density.
If two gravitations are the same then:
g$_{1}$=g$_{2}$ and $\rho$$_{1}$r$_{1}$=$\rho$$_{2}$r$_{2}$ or $\rho$r=const
Getting back to the time dilation formula and replacing M with V*$\rho$:
T$_{g}$=T$_{f}$$\sqrt{1-\frac{2G}{c^{2}}\frac{4}{3}\pi{r^{2}\rho}}$
After replacing all constants with k (remember that $\rho$r is also a constant:
T$_{g}$=T$_{f}$$\sqrt{1-k*r}$

What a surprise: same gravitation, but different time dilation! (assuming planets radii are different)
Did I missed something?
Thanks.

2. Apr 3, 2013

### Jonathan Scott

Gravitational time dilation depends on gravitational potential rather than gravitational field (which is the gradient of the potential), so it is certainly possible to have equal time dilation in different fields, or different time dilations in the equal fields.