Relativity

We measure the time by the distance light travel in that amount of time. In other word, we take \( c = 1 \).

We also choose the mostly-positive convention, where the Minkowski metric is taken to be: \[ \eta_{\mu\nu} := \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix}. \]

The hypobolic angle \( w \) for the Lorentz boost.

\[ w = \ln [ \gamma(1+\beta) ] = - \ln [\gamma (1-\beta) ] \]

\[ w = \operatorname{arctanh} \frac{|\mathbf{p}|c}{E} = \frac{1}{2} \ln \frac{E + |\mathbf{p}|c}{E - |\mathbf{p}|c} \]

The coordinate of events changes depending on the observers at the same point in spacetime, according to the Lorentz transformation.

Scalar and Minkowski metric are Lorentz invariant. While tensors are generally covariant.

• 광행차

The phenomenon where celestial objects appear to be displaced forward in the observer's direction of motion. When the velocity of the observer change the amount of displacement changes showing an apparent motion.

The explanation is particularly simple in the framework of relativity. Consider a frame in which the celestial object is moving and the observer stays still. The observer would measure the past trace of the object, therefore the position will be displaced in the direction opposite to the motion of the object.

Some energy increases by a factor of \( \gamma \). For example:

• the two photons moving opposite to each other.
  • which leads to the conclusion that the kinetic energy should be somehow be different.
  • Einstein thought that the mass was subject to change. But apparently it's out of fashion now.

It is the coordinate system of the frame that is accelerating with proper acceleration \( \alpha \).

\begin{align*} t &= \tilde{x}\sinh(\frac{\alpha}{c^2}c\tilde{t}) \\ x &= \tilde{x}\cosh(\frac{\alpha}{c^2}c\tilde{t}) \\ \end{align*}

where \( \tilde{x} \) and \( \tilde{t} \) are the proper length and proper time of the accelerating frame.

Notice the proper time at a point along the constant \( \tilde{x} \) differs from the coordinate time since it accelerates slower:

\begin{equation*} \tau(\tilde{x}, \tilde{t}) = \frac{\tilde{x} \alpha}{c^2}\tilde{t}. \end{equation*}

This is due to the relation between the position and proper acceleration along the gridline:

\begin{equation*} \frac{c\tilde{t}}{D} = \frac{c\tau}{\tilde{x}}. \end{equation*}

4-momentum flux.

\( T^{\mu\nu} \) is the flux of \( \mu \) component of the 4-momentum along the \( \nu \) direction.

The divergence free condition is imposed on the ground of physics, which encodes the local conservation of energy and momentum.

For any metric tensor field \( g_{\rho\sigma} \), we can always find linear transformation \( L_{\mu}{}^{\rho} \) that turns the coordinates into locally Lorentz.

We use Taylor expansion on both metric and transformation: \[ \left(L_{\mu}{}^{\rho}(p) + \Delta\tilde{x}^{\tau}\tilde{\partial}_{\tau} L_{\mu}{}^{\rho}(p) + \Delta\tilde{x}^{\tau}\Delta\tilde{x}^{\upsilon}\tilde{\partial}_{\tau} \tilde{\partial}_{\upsilon}L_{\mu}{}^{\rho}(p)+ \cdots \right) \left( g_{\rho\sigma}(p) + \Delta\tilde{x}^{\tau}\tilde{\partial}_{\tau} g_{\rho\sigma} + \Delta\tilde{x}^{\tau}\Delta\tilde{x}^{\upsilon} \tilde{\partial}_{\tau}\tilde{\partial}_{\upsilon}g_{\rho\sigma} + \cdots \right)\left(L_{\mu}{}^{\rho}(p) + \Delta\tilde{x}^{\tau}\tilde{\partial}_{\tau} L_{\mu}{}^{\rho}(p) + \Delta\tilde{x}^{\tau}\Delta\tilde{x}^{\upsilon}\tilde{\partial}_{\tau} \tilde{\partial}_{\upsilon}L_{\mu}{}^{\rho}(p)+ \cdots \right) . \] The expansion is done in local Lorentz frame, and \( \tilde{x}^{\mu} \) is the coordinates of that frame.

The zeroth order term can always be transformed to \( \eta_{\mu\nu} \):

The zeroth order term can always be set to \( \eta_{\mu\nu} \) \[ \eta_{\mu\nu} = L_{\mu}{}^{\rho}(p)\,g_{\rho\sigma}(p)\,L_{\nu}{}^{\sigma}, \] because there is 10 constraints, reduced from 16 to 10 by the symmetry of metric, and 16 degrees of freedom in \( L_{\mu}{}^{\rho} \). The remaining 6 degrees of freedom is precisely the 3 rotations and 3 boosts included in the Lorentz transformations.

The first order term can also always be set to 0, \[ 0 = \Delta\tilde{x}^{\tau} \left( \tilde{\partial}_{\tau}L_{\mu}{}^{\rho}(p)\, g_{\rho\sigma}(p)\,L_{\mu}{}^{\rho}(p) + L_{\mu}{}^{\rho}(p)\, \tilde{\partial}_{\tau}g_{\rho\sigma}(p)\,L_{\mu}{}^{\rho}(p) + L_{\mu}{}^{\rho}(p)\, g_{\rho\sigma}(p)\,\tilde{\partial}_{\tau}L_{\mu}{}^{\rho}(p)\right) \] because we have additional 40 (4×10) constraints from \( \tilde{\partial}_{\tau}g_{\rho\sigma} \), and 40 (10×4) additional degrees of freedom from \( \tilde{\partial}_{\tau}L_{\mu}{}^{\rho} \). Notice \[ \tilde{\partial}_{\tau}L_{\mu}{}^{\rho} = \left( \pdv[2]{x^{\rho}}{\tilde{x}^{\tau}}{\tilde{x}^{\mu} \right) \] and \( \tau \) and \( \mu \) are symmetric indices.

On the second order term, there is 100 (10×10) constraints from \( \tilde{\partial}_{\tau}\tilde{\partial}_{\upsilon}g_{\rho\sigma} \), and 80 (20×4) degrees of freedom from \( \tilde{\partial}_{\tau} \tilde{\partial}_{\upsilon}L_{\mu}{}^{\rho}\). We have 20 degrees of freedom that determines the curvature of spacetime.

\begin{equation*} G_{\mu\nu} - \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} \end{equation*}

where \( G_{\mu\nu} \) is the Einstein tensor defined as:

\begin{equation*} R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}. \end{equation*}

Einstein tensor is required, to make the divergence zero.

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