Covariant Derivatives in Curved Spacetime
Covariant derivative = Normal derivative
derivative of the metric.
The connection or Christoffel symbol (
)
handles the derivatives of the metric.
";" the semicolon convention for covariant derivatives.
![](Images/d_equals_normal.jpg)
Derivative = normal - change in metric.-
![](Images/co_contra_derivative.jpg)
Covariant derivative of a contravariant vector. -
![](Images/antisymmetric_derivative.jpg)
Independent of the metric because
. -
![](Images/symmetric_derivative.jpg)
Depends on metric.
Omission: The details of the Christoffel symbol are not discussed here.
Slide 25 hopes for a cure.