Special Relativity

• Michelson-Morley experiment
• Minkowski space
• Lorentz boost
• Lorentz group
• four-velocity
• time dilation and length contraction
• addition of velocities
• four-current
• four-momentum
• uniform acceleration
• accelerated frames

Seen as a particular case of general relativity

In general relativity, we have a relativistic spacetime $M$, and in a neighborhood of a point we take coordinates $x^{\mu }$. This coordinates give rise to the coordinate frame ${\partial }_{x^{\mu }}$, which is not necessarily a vielbein. We could choose another reference frame $e_{\mu }$ which could even be inertial. If we take a general coordinate transformation $y=F(x)$, in this new coordinates $y^{\mu }$ we have a different coordinate frame ${\partial }_{y^{\mu }}$ which is obtained from ${\partial }_{x^{\mu }}$ by the linear transformation (in every point) $dF$. But the vielbein $e_{\mu }$ is the same (what changes is its expression with respect to the corresponding coordinate frame).
Given two different vielbeins, for every point in spacetime we have a Lorentz transformation relating them.

Now, in special relativity, the spacetime $M$ is the Minkowski space, given with a preferred choice of coordinates $x^{\mu }$, in such a way that the metric has the specific form ${\eta }_{\mu \nu }$, and therefore the coordinate frame ${\partial }_{x^{\mu }}$ is a vielbein. Moreover, it is an inertial frame, as can be checked. Now, if we consider, instead of a general coordinate transformation, a transformation $y=F(x)$ belonging to the Poincare group, the new coordinates $y^{\mu }$ satisfy the same: the coordinate frame ${\partial }_{y^{\mu }}$ is an inertial frame. Both coordinate frames are related by an element of the Lorentz group. We can call to the coordinates obtained from this way inertial coordinates.