# How strong must gravity be to stretch time?

I'm not sure if this is true or not but I heard that gravity has the ability to stretch time, and I was wandering if this is true.

If so, how intense/powerful does the gravitational force have to be to stretch time? Could the gravity of a planet be enough to alter/stretch time? Or does it need the influence of a black hole like in the movie Interstellar?

• – glS Jan 24 '15 at 11:07
• any object with mass can stretch time... – Hritik Narayan Jan 24 '15 at 11:50
• Gravity is (not the cause of) the 'stretching' of (space)time; mass-energy is the source, i.e., matter tells spacetime how to curve, spacetime tells matter how to move. – Alfred Centauri Jan 24 '15 at 12:40

Any mass will produce a gravitational field which dilates time. The formula for time dilation caused by a non rotating spherical object of mass $M$ is given by:

$$t_\textrm{near object} = t_\textrm{far away}\times \sqrt{1 - \frac{2GM}{rc^2}}$$

where $G$ is the gravitational constant, $c$ is the speed of light, and $r$ is the distance from the centre of the object. This effect only becomes severe in the vicinity of very dense objects (e.g. black holes) - where $M$ can be very large and $r$ can be very small.

The metric describing flat Minkowski space is given by,

$$ds^2=\eta_{\mu\nu}dx^\mu dx^\nu = dt^2-dx^2-dy^2-dz^2$$

For such a system, the stress energy $T_{\mu\nu}=0$. However, now suppose we introduce additional matter content which induces a perturbation in the stress-energy, $T_{\mu\nu} \to T_{\mu\nu}+\delta T_{\mu\nu}$. There is a corresponding change in the metric, $h_{\mu\nu}$ which affects the Einstein tensor describing the geometry of spacetime:

$$G'_{\mu\nu}=\frac{1}{2}(\partial_\sigma \partial_\nu h^\sigma_\mu +\partial_\sigma\partial_\mu h^\sigma_\nu - \partial_\mu\partial_\nu h-\square h_{\mu\nu}-\eta_{\sigma\lambda}\partial_\mu\partial_\nu h^{\sigma\lambda} + \eta_{\mu\nu}\square h)$$

As long as the perturbed system does not have a vanishing stress-energy tensor, you can see anything you add will in some way cause the geometry to deviate from flat Minkowski space thereby 'stretching' spacetime as you would interpret.

Notice we do not have a lower bound on the mass, and in fact it is not the only property that contributes; momentum, stress, pressure and energy are also described by $T_{\mu\nu}.$

Affecting time is one of the primary ways gravity interacts, and can be responsible for 50% of the gravitational effects you see.

For instance, Newton could predict that starlight would be bent by the sun (assume it has a small mass and see how it is bent, then take the limit as the mass goes to zero and you get a result). But Einstein got a result that was twice that by taking the time dilation into account.

Objects get from one place to another by travelling along the path that ages them the most. If the sun is at $(0,0)$ and you wanted to get from $(5,0)$ at $t=0$ to $(4,3)$ at $t=T$ the straight line path gets you closer to the sun, and you age less when you are closer to the sun. But if you took a path that takes you far from the sun but gets you back to $(4,3)$ at $t=T$ then you age little since moving quickly also makes you age slowly. It turns out the path where you age the most is the circular path of radius 5 from $(5,0)$ to $(4,3)$.

And that's why we move on a circle about the sun. We don't notice it because these differences in ageing are 1) a small effect and 2) everything around us is ageing at the same rate, even our clocks. But it happens for every body, not just big ones. And it isn't an additional effect, it is one of the primary effects.