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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:

$$\begin{align*} \hat F &= - G \frac{m M}{r^2} \hat e_r,\quad G = 6.67\times 10^{-8} cm^2/g^2\\ \frac{\hbar G}{c^3} &= l_P^2(\text{Planck length}) \\ &= (1.61624(12) \times 10^{-33} cm)^2 \end{align*}$$

An easier way to describe this is through potential:

$$\begin{align*} \hat F &= -m \vec{\nabla}\phi \end{align*}$$

where $\phi$ is the gravitational potential, which only depends on the position. For point mass:$$ \phi(\vec x) = - \frac{MG}{|\vec x -\vec x_0|}\,, $$

for mass distribution:$$ \phi(\vec x) = -\int\,d^3\vec x'\,\frac{G\rho(\vec x)}{|\vec x -\vec x'|} $$

It means that we can introduce a gravitational field:$$\begin{align*} \vec G &= -\vec \nabla\phi\\ \vec\nabla\cdot\vec G &= -4\pi G\vec\nabla \rho(\vec x)\quad(\text{Gauss})\\ \Delta\phi &= 4\pi G\rho\quad(\text{Poisson})\\ \end{align*}$$

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. ($E=h\nu$)

- 1912 Equation for the "c-field", which is way off:
$$\Delta c = \kappa \left(c \sigma + \frac{1}{2\alpha}\frac{(\vec\nabla c)^2}{c}\right)$$

In the c-field, the speed of light is modified by the gravity. - 1912 Start of Marcel Grossmann collaboration.
- Metric tensor $g_{\mu\nu}(x)$.
- $g_{00}=-[1+2\phi(x)]$.

## Principal of Equivalence

Statement: If you have a particle in a gravitational field:

$$ m_I \vec a = m_G \vec g $$

where the inertial mass $m_I$ equals gravitational mass to $m_G$. It is tested to be true in the level of $10^{-12}$.Example:

$$ m \frac{d^2x}{dt^2}= m \vec g + (\text{other force}) $$

By applying coordinate change:$$ x = x - \frac 12 g t^2\\ t = t $$

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 $\xi^\mu$, and transform the coordinate system to an arbitrary coordinate system $x^\mu$.

In the freely-falling frame:

$$ \frac{d^2\xi^\alpha}{d\tau^2} = 0 $$

In the lab frame:$$ x^\alpha = \left(x(\xi)\right)^\alpha $$

where $x$ is the function that maps $\xi$ to $x$. From the first equation (in fact, 4 equations parameterized by $\alpha$):$$\begin{align*} 0 &= \frac{d}{d\tau}\left(\frac{\partial \xi^\alpha}{\partial x^\mu}\frac{d x^\mu}{d \tau}\right)\\ &= \frac{\partial \xi^\alpha}{\partial x^\mu}\frac{d^2 x^\mu}{d \tau^2} + \frac{\partial^2 \xi^\alpha}{\partial x^\nu\partial x^\mu}\frac{d x^\mu}{d \tau}\frac{d x^\nu}{d \tau} \\ \end{align*}$$

multiply (contract) by ${\partial x^\lambda}/{\partial \xi^\alpha}$ (now we have 4 equations parameterized by $\lambda$):$$\begin{align*} 0 &= \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial \xi^\alpha}{\partial x^\mu}\frac{d^2 x^\mu}{d \tau^2} + \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\alpha}{\partial x^\nu\partial x^\mu}\frac{d x^\mu}{d \tau}\frac{d x^\nu}{d \tau} \\ &= \frac{d^2 x^\lambda}{d \tau^2} + \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\alpha}{\partial x^\nu\partial x^\mu}\frac{d x^\mu}{d \tau}\frac{d x^\nu}{d \tau}\\ \\ \text{Define(Affine Connection):}\quad\, \Gamma^\lambda{}_{\mu\nu} &\equiv \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\alpha}{\partial x^\nu\partial x^\mu}\\ \\ 0 &= \frac{d^2 x^\lambda}{d \tau^2} + \Gamma^\lambda{}_{\mu\nu} \frac{x^\mu x^\nu}{d^2\tau} \end{align*} $$

This is called the "**Equation of free fall**" or "**Geodesics Equation**". This is apparently not the end of the manipulation because $\Gamma^{\lambda}{}_{\mu\nu}$ still depends on $\xi$. There has to be a way to express the affine connection with properties related to $x$.

Continue with $d\tau^2$:

$$\begin{align*} d\tau^2 &= -\eta_{\alpha\beta} d\xi^\alpha d\xi^\beta\\ &= -\eta_{\alpha\beta} \frac {\partial \xi^\alpha}{\partial x^\mu} \frac {\partial \xi^\beta} {\partial x^\nu} dx^\mu dx^\nu\\ \text{Define(Metric):}\quad\,g_{\mu\nu} &\equiv -\eta_{\alpha\beta} \frac {\partial \xi^\alpha}{\partial x^\mu}\frac {\partial \xi^\beta} {\partial x^\nu}\\ d\tau^2 &= - g_{\mu\nu} dx^\mu dx^\nu \\ \end{align*}$$

To express $\Gamma^\gamma{}{}_{\mu\nu}$ in terms of $g_{\mu\nu}$. Take the derivative respect to $x^\lambda$, (64 equations parameterized by $\lambda, \mu, \nu$):

$$\begin{align*} \frac{\partial g_{\mu\nu}}{\partial x^\lambda} &= -\eta_{\alpha\beta} \left( \frac{\partial}{\partial x^\lambda} \frac {\partial \xi^\alpha}{\partial x^\mu} \frac {\partial \xi^\beta} {\partial x^\nu} + (\mu\leftrightarrow\nu) \right)\\ &= -\eta_{\alpha\beta} \left( \frac{\partial^2\xi^\alpha}{\partial x^\lambda \partial x^\mu} \frac {\partial \xi^\beta} {\partial x^\nu} + (\mu\leftrightarrow\nu) \right)\\ \end{align*}$$

$$\begin{align*} \frac{\partial^2\xi^\alpha}{\partial x^\lambda \partial x^\mu} &=\delta^\alpha{}_\rho\frac{\partial^2 \xi^\rho}{\partial x^\lambda\partial x^\mu} \\ &=\frac{\partial \xi^\alpha}{\partial x^\rho}\frac{\partial x^\rho}{\partial \xi^\beta} \frac{\partial^2 \xi^\beta}{\partial x^\lambda\partial x^\mu}\\ &=\Gamma^{\rho}{}_{\lambda\mu}\frac{\partial \xi^\alpha}{\partial x^\rho} \end{align*}$$

$$\begin{align*} \frac{\partial g_{\mu\nu}}{\partial x^\lambda} &= -\eta_{\alpha\beta} \left( \Gamma^{\rho}{}_{\lambda\mu}\frac{\partial \xi^\alpha}{\partial x^\rho} \frac {\partial \xi^\beta} {\partial x^\nu} + (\mu\leftrightarrow\nu) \right)\\ &= \Gamma^{\rho}{}_{\lambda\mu}g_{\rho\nu}+ \Gamma^{\rho}{}_{\lambda\nu}g_{\rho\mu} \\ \end{align*}$$

At this stage, we realize we can solve $\Gamma$ from just $x$. This is indeed possible:$$\begin{align*} \frac{\partial g_{\mu\nu}}{\partial x^\lambda} + \frac{\partial g_{\lambda\nu}}{\partial x^\mu} - \frac{\partial g_{\mu\lambda}}{\partial x^\nu} &= \Gamma^{\rho}{}_{\lambda\mu}g_{\rho\nu}+\cancel{\Gamma^{\rho}{}_{\lambda\nu}g_{\rho\mu}} + \Gamma^{\rho}{}_{\mu\lambda}g_{\rho\nu}+\cancel{\Gamma^{\rho}{}_{\mu\nu}g_{\rho\lambda}} - \cancel{\Gamma^{\rho}{}_{\nu\mu}g_{\rho\lambda}}- \cancel{\Gamma^{\rho}{}_{\nu\lambda}g_{\rho\mu}}\\ &= 2\Gamma^{\rho}{}_{\lambda\mu}g_{\rho\nu} \end{align*}$$

Now we need to define the inverse of $g_{\rho\nu}$ as $g^{\rho\nu}$:$$ g^{\nu\sigma}g_{\rho\nu} = \delta^\sigma{}_\rho\,, $$

then$$ \Gamma^{\sigma}{}_{\lambda\mu}=\frac 12 g^{\nu\sigma} \{ \partial_{\{\lambda,} g_{\mu\}\nu} - \partial_\nu g_{\mu\lambda} \} $$

### My further notes:

The affine connection $\Gamma^d{}_{ab}$ 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:

$$ \Gamma_{cab} = g_{cd}\Gamma^d{}_{ab} $$

Which is amazing since we don't need the inverse metric tensor to express it, as it is simply:$$ \Gamma_{cab} = \frac 12(\partial_bg_{ca}+\partial_ag_{cb}-\partial_cg_{ab}) $$

This is so simple that we can write it down as: $[ab,c]$.My question will be, can we find such an consistent mapping $x$ in a curved spacetime?