Statement: If you have a particle in a gravitational field:
where the inertial mass equals gravitational mass to . It is tested to be true in the level of .
By applying coordinate change:
The gravitational field is canceled.
At everything spacetime point, it is possible to choose a "locally inertial frame" where gravity is canceled. In that frame of reference (freely falling
frame), the law is the law of SR.
A more general description of gravitational force
Here we generalize by considering a free-falling particle in its own frame of reference , and transform the coordinate system to an arbitrary coordinate system .
In the freely-falling frame:
In the lab frame:
where is the function that maps to .
From the first equation (in fact, 4 equations parameterized by ):
multiply (contract) by (now we have 4 equations parameterized by ):
This is called the "Equation of free fall" or "Geodesics Equation". This is apparently not the end of the manipulation because still depends on . There has to be a way to express the affine connection with properties related to .
Continue with :
To express in terms of .
Take the derivative respect to , (64 equations parameterized by ):
At this stage, we realize we can solve from just . This is indeed possible:
Now we need to define the inverse of as :
The definitions given below are valid for both Riemannian manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction being made between upper and lower indices (contra-variant and co-variant indices). The formulas hold for either sign convention, unless otherwise noted. Einstein summation convention is used in this article. The connection coefficients of the Levi-Civita connection (or pseudo-Riemannian connection) expressed in a coordinate basis are called the Christoffel symbols.
The first kind is defined as:
Which is amazing since we don't need the inverse metric tensor to express it, as it is simply:
This is so simple that we can write it down as: .
My question will be, can we find such an consistent mapping in a curved spacetime?