I model survival outcomes using a Cox proportional hazards model and want to evaluate model fit.

Harrel's concordance index C is defined as the proportion of observations that the model can order correctly in terms of survival times. Sometimes C is called the AUC.

The area under the receiver operating characteristic curve AUC is defined as the area under the curve of sensitivity and 1-specificity. At time t the Cox model may be used to obtain a ROC and its AUC. Are C and the ROC-AUC related? How?

I model survival outcomes using a Cox proportional hazards model and want to evaluate model fit.

Harrel's concordance index C is defined as the proportion of observations that the model can order correctly in terms of survival times. When censoring is observed the statistic only includes those patient pairs for which valid comparisons can be made. Sometimes C is called the AUC.

The area under the receiver operating characteristic curve, also called AUC (or AU-ROC), is defined as the area under the curve of sensitivity and 1-specificity; this statistic is equal to the concordance of predicted and observed classes, see here for example. 

At a time point t the Cox model may be used to obtain a ROC and its AUC if there is no censoring. I suppose if there is censoring, the area under the ROC can still be obtained with some additional caclulations (time-dependent AUC).

However, what is the relationship between Cox-Survival (Harrel's) C (AUC) and the AU-ROC? When are they equivalent, when different?