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.
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:
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):
at the surface of a white dwarf;
at the surface of the sun;
at the surface of the earth;
at the surface of an proton;
We can caluculate which is the factor of time dilation: