Bei's Study Notes

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:

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

History of development of Einstein's GR

  1. ~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.
  2. 1911
    • Bending of light by the sun is observable. His calculation was off by a factor of 2. ($E=h\nu$ )
  3. 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.
  4. 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?