Following Misner, Wheeler and Thorne.
Foundations of General Relativity
Basic Definitions
- Events: Things that happen at a particular point in spacetime
- Coordinates: How you label events in spacetime. There is no unique way of assigning coordinates
- Written as $x_{\mu}$, where $\mu$ ranges from 0 to 3
- A coordinate transformation is a set of 4 equations. Each one determines how the new coordinate depends on the old 4 coordinates
- Coordinate singularities can occur (think of the north pole of the earth on latitude and longitude)
- These can be dealt with via multiple patches of coordinates
- Vectors: The seperation between two events in spacetime. In flat spacetime, this is the difference between the two coordinates assigned to each event. This falls apart in curved spacetime, but is an good approximation for arbitrarily close events
- These transform under coordinate transformations as: $\epsilon^{\beta} = \frac{\partial x^{\alpha}}{\partial x^{\beta}} \epsilon^{\beta}$
- Follows from Taylor expansion
- These transform under coordinate transformations as: $\epsilon^{\beta} = \frac{\partial x^{\alpha}}{\partial x^{\beta}} \epsilon^{\beta}$
- Summation Notation: $\epsilon_{\alpha}\gamma^{\alpha} = \Sigma_{\alpha=0}^{3} \epsilon_{\alpha}\gamma^{\alpha}$