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General Relativity 04

Last updated: 2016-08-14 03:35:13 PDT.

Reference: UCI OpenCourseWare - GR 05 (Playlist)

Starting with Newtonian gravity:

An easier way to describe this is through potential:

where is the gravitational potential, which only depends on the position. For point mass: for mass distribution: It means that we can introduce a gravitational field:Gravitational force is special:

- It is the oldest force.
- It couples to everything.
- Gravity cannot be screened.
- It is long range. (Mediated by massless particle if there is any.)
- It is the weakest force.

## History of development of Einstein's GR

- ~1907
- Gravitational fields have relative existence. (Gravitational field can be removed by coordinate transformation).
- There's a need to extend SR to accelerating frames.
- Newtonian gravity is not Lorentz invariant.
- Leads to
**Equivalence Principal**. - Gravitational red-shift (testable!).
- Bending of light.
- Geometry will be curved. By spinning disk.

- 1911
- Bending of light by the sun is observable. His calculation was off by a factor of 2. ()

- 1912 Equation for the "c-field", which is way off: In the c-field, the speed of light is modified by the gravity.
- 1912 Start of Marcel Grossmann collaboration.
- Metric tensor .
- .

## Principal of Equivalence

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 .Example:

By applying coordinate change: The gravitational field is canceled.General statement:

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 : then### My further notes:

The affine connection is also called **Christoffel symbol of the second kind**.

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?