gravity's effect on time

[elihushaw]

so I don't know much about spacetime or physics in general but I'm watching a Nova video on the illusion of time.

 

On Earth, if you are closer to the ground time will move slower for you. Time will move faster for you if you are higher up relatively from the ground. Why does this happen?

[swansont]

It's called gravitational time dilation, and it's not a simple explanation. But, as it turns out, the way light behaves (the speed of light being constant in inertial frames of reference) has implications for space and time — they are not absolutes. It's also true of gravity, which warps space and time.

[Endercreeper01]

It is a consequence of general relativity.

In special relativity, acceleration bends space-time. Einstein reasoned the same must be true for gravity, based on the equivalence principle.

One way to define this curvature is by having an equation for the square of an infinitesimal length in space-time. This is called the metric, and is written as [latex]ds^2[/latex].

When there is no gravitation, the metric is given by (in spherical coordinates)

[latex]ds^2=-dt^2+dr^2+r^2(d\theta ^2+ sin^2 (\theta ) d\phi ^2)[/latex]

However, this is changed when space-time is curved. The metric of curved space-time is given by

[latex]ds^2=\sum g_{ab} dx^a dx^b[/latex]

This represents the sum of coordinates multiplied by the metric tensor component for that certain coordinate.
Time is one coordinate, and the curvature of space-time makes time dilated. Time dilation is given by

[latex]\frac{d\tau }{dt} = \sqrt{g_{00}}[/latex]

which means time is dilated by a factor of the square root of the time component of the metric tensor.

This is why time is dilated.