# Would Gravitational Time Dilation still apply between two gravitational sources?

As explained on this Wikipedia page, when an object is experiencing strong gravitational forces, time passes slower for it. If, in theory, you had two objects exerting the same amount of gravitational pressure on an object in opposite directions, effectively cancelling each other out, would they cancel out the time dilation, or would they compound, making the time dilation more prevalant?

• The effect of time dilation gets 'compounded'. – Praneet Srivastava Apr 21 '16 at 6:56

The time dilation is not dependent on the gravitational force, but rather on the gravitational potential energy.

As long as we are well away from black holes we can use an approximation called the weak field limit. In this approximation the time dilation relative to an observer at infinity is given by:

$$\frac{dt}{dt_0} = \sqrt{1 + \frac{2\Phi}{c^2}}$$

where $\Phi$ is just the Newtonian gravitational potential (per unit mass) given by the expression we learned in school:

$$\Phi = -\frac{GM}{r}$$

Note that $\Phi$ is negative, so the right hand side is less than one and $dt \lt dt_0$ so we get time dilation.

Suppose we take a system like you suggest where you are positioned exactly between two masses so the net force on you is zero: Your gravitational potential is the sum of the potentials due to the two masses, so it is:

$$\Phi = -\frac{GM}{r} + -\frac{GM}{r} = -\frac{2GM}{r}$$

And your time dilation relative to infinity is therefore:

$$\frac{dt}{dt_0} = \sqrt{1 - \frac{4GM}{c^2r}}$$

So even though there is no net force on you there is still a time dilation.