# Time dilation due to two cancelling gravitational potential?

We know that the formula for gravitational time dillation is $$T_{dilated}=\sqrt{1-\frac{2Gm}{Rc^2}} \cdot T_{without gravity}.$$ But if there are two stars (of equal mass) and an atomic clock is placed at the centere of the line which connects both star's centre of mass then the gravitational potential will be zero on the clock then does it experience gravitational time dilation relative to a clock far away from the stars?

If so then what causes it since the above formula doesn't seem to work in this case.

## 2 Answers

You’ve confused potential and force.

Midway between two stars, the gravitational force is zero: the forces are equal and opposite vectors, so sum to zero.

The potential is not a vector: the potentials from the two stars just add.

[if] an atomic clock is placed at the centere of the line which connects both star's centre of mass then the gravitational potential will be zero

The gravitational potential is always negative, so it never cancels. The point you are describing is a saddle point in the potential. So the gradient of the potential is zero, but the potential itself is negative.

Note, in general relativity not all spacetimes have potentials. So the answer above is not a general answer but it is valid for this specific question