# Bei's Study Notes

### General Relativity 05Last updated: 2016-08-14 19:21:06 PDT.

Reference: UCI OpenCourseWare - GR 06 (Playlist)

### Equation of free fall from Lagrangian

Last time we wrote down the equation of free fall:

where is the Christoffel symbol of the first kind, which can be written as a function of metric tensor:

Today we are going to derive the equation of free fall from the variational principal.

Action:

the principal is that the equation of motion can be derived from:

By this, it means a particle goes from to from a trajectory where is an parameter. If we perturb the trajectory with an arbitrary finite variation multiply by an infinitesimal factor of , then the action remains the same. We can write it down:

The boundary term is and integrate by partial:

Plug it in,

Since is arbitrary, we get again the equation of free fall:

By definition, this is trajectory of the stationary path (in terms of proper time) in 4 dimentional spacetime. This is the Geodesics. Therefore we replaced gravitational force with the geometry of spacetime.

There has a clear consequence with QM because of path integral:

### Newtonian approximation

For slow moving particles:

For weak, stationary fields:

If both conditions are met:

This looks a lot like the the Newtonian gravitation:

where is the gravitational potential (which is always negative),

At the boundary, the gravitation field is , the should also be , therefore the constant should be as well.

Remark: 1. goes to at the event horizon. 2. The boundary gravitational field may not be .

In the case of stronger gravitational field (which is dimensionless):

1. at the surface of a white dwarf;
2. at the surface of the sun;
3. at the surface of the earth;
4. at the surface of an proton;

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### Time Dilation

We can caluculate which is the factor of time dilation:

If the :

comparing two clocks:

The frequency of the two stationary clock:

In the previous conditions: