# Time dilation due to gravity

## Main Question or Discussion Point

If possible can someone tell me what is the formula of time dilation for an object

Which theoretically is Not in orbit, Not moving but close in off to an black hole to be affect by the gravity of it.

It will be greatly appreciated.

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andrewkirk
Homework Helper
Gold Member
Time dilation in the context of gravity is usually used to refer to the ratio T1/T2 where
* T1 is the time between two spacetime events, E1 and E2, as measured by an observer O1 in a strong gravitational field; and
* T2 is the time between E1 and E2, as measured by an observer O2 that is far away and in an inertial reference frame.

A formula for this ratio, assuming the gravitational source is a spherically symmetric, non-rotating mass is

$$\sqrt{1-\frac{r_s}{r}}$$

where ##r## is the Swarzschild radial coordinate of O1 (which is analogous to the distance from the centre of the gravitational source) and ##r_s## is the Swarzschild radius of the source, which is the size to which the source would have to collapse to become a black hole. You can see from this formula that, as observer O1 approaches the event horizon of a black hole from the outside, the ratio heads towards 0.

• bcrowell and Dynamotime
PAllen
2019 Award
Time dilation in the context of gravity is usually used to refer to the ratio T1/T2 where
* T1 is the time between two spacetime events, E1 and E2, as measured by an observer O1 in a strong gravitational field; and
* T2 is the time between E1 and E2, as measured by an observer O2 that is far away and in an inertial reference frame.

A formula for this ratio, assuming the gravitational source is a spherically symmetric, non-rotating mass is

$$\sqrt{1-\frac{r_s}{r}}$$

where ##r## is the Swarzschild radial coordinate of O1 (which is analogous to the distance from the centre of the gravitational source) and ##r_s## is the Swarzschild radius of the source, which is the size to which the source would have to collapse to become a black hole. You can see from this formula that, as observer O1 approaches the event horizon of a black hole from the outside, the ratio heads towards 0.
Actually, for this formula to be valid, both observers must be static - hovering via rockets or resting on a surface. Further, for that form to be valid, O2 must be hovering at 'infinity', stationary with respect to the spherically symmetric source. It is true that at infinity, stationary = inertial, but the characteristic that holds for the generalization where O2 is not at infinity is stationary rather than inertial. Stationary has proper acceleration and is not inertial except at infinity.

• bcrowell and Dynamotime
Thank you so much for info