# How is time affected by the bending of space-time?

By Newton's Laws, we can calculate the effect of gravity, but he didn't give the actual reason for gravity. I searched the internet for it, but all the answers have this "bending of space-time" stuff. If the space is bent by the mass of the body, how does it affect the time around it, If the body bends both space and time

• Are you asking specifically about gravitational time dilation, or are you asking how spacetime curvature causes gravity? – John Rennie Jun 11 '18 at 16:01
• Gravitational time dilation – Raj Shukla Jun 11 '18 at 16:17
• Not only how space-time curvature causes gravity, but what is curvature and how does mass bend it? – Bill Alsept Jun 11 '18 at 18:27

Before we get to gravitational time dilation, let's introduce some general relativistic notation:

1. $\tau$ is proper time between two events $A$ and $A'$ for a slow-ticking clock within a gravitational field.
2. $t$ is the time between two events $A$ and $A'$ for a fast-ticking clock infinitely far away from the mass creating the gravitational well.

These are related thusly for a spherically-symmetric non-rotating mass (see Schwarzschild solution): $$\tau = t \sqrt{1 - \frac{2GM}{rc^{2}}},$$ where $M$ is the mass, $G$ is Newton's gravitational constant, $c$ is the speed of light, $r$ is the radial coordinate of the observer, essentially the distance from the observer to the center of $M$. You can play around with this equation by plugging in the mass of the Sun and our distance from its center.

Moreover, if you actually want to know where this equation comes from, look up the Schwarzschild metric: $$ds^2 = c^{2} d\tau^{2} = \left ( 1 - \frac{2GM}{rc^{2}} \right )c^2 dt^2 - \left (1 - \frac{2GM}{rc^{2}} \right )^{-1} dr^2 - r^2 (d\theta^2 + \sin^2 \theta d\varphi^2).$$ In simple terms, the Schwarzschild metric allows you to calculate distances in the curved spacetime around $M$ analogous to how the pythagorean theorem $ds^2 = dx^2 + dy^2$ allows you to calculate distances in Euclidean space. Finally, the fundamental equations of General Relativity are the Einstein field equations (EFEs): $$R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = \frac{8 \pi G}{c^4}T_{\mu \nu},$$ where the left-hand side describes the curvature of spacetime, given that $R$, $R_{\mu \nu}$, $g_{\mu \nu}$ are parameters in differential geometry, and the right-hand side describes the density of energy and momentum in spacetime. The solutions of the EFEs are metrics $g_{\mu \nu}$. In sum, the solutions of the EFEs, which allow us to calculate distances in various spacetimes, can be used to derive a relationship between $\tau$ and $t$.

Bending/curving should not be taken literally. It is sometimes used for demonstration purposes. It also fits very well with mathematics of general relativity. It also has been demonstrated practically via gravitational lensing and solar eclipse experiments. Still no one knows what it actually is. It is "something" that changes property of spacetime in that region which causes all the observations that we have made. These observations include time dilation in the vicinity of mass/energy.

If we apply the gravitational acceleration on light (which Newton did not think of), the accumulated effect over path of the light would probably give similar results as gravitational lensing caused by curving.

There are things like "Precession of the Perihelion of Mercury" can only be explained via GR, i.e. via mathematics of curving of spacetime.

We do not know what exactly that curving is and why it causes time dilation or other effects.

Physics does not answer "why" or even "how", it does answer "how much".