In [[Einstein notation]], contravariant vectors and components of tensors are shown with superscripts, e.g. {{math|''x<sup>i</sup>''}}, and covariant vectors and components of tensors with subscripts, e.g. {{math|''x<sub>i</sub>''}}. Indices are "raised" or "lowered" by multiplication by an appropriate matrix, often the identity matrix.
Coordinate transformation is important because relativity states that there is nonot one correct reference point (or perspective) in the universe that is more favoured than another. On earth, we use dimensions like north, east, and elevation, which are used throughout the entire planet. There is no such system for space. Without a clear reference grid, it becomes more accurate to describe the four dimensions as towards/away, left/right, up/down and past/future. As an example event, assume that Earth is a motionless object, and consider the signing of the [[United States Declaration of Independence|Declaration of Independence]]. To a modern observer on [[Mount Rainier]] looking east, the event is ahead, to the right, below, and in the past. However, to an observer in medieval England looking north, the event is behind, to the left, neither up nor down, and in the future. The event itself has not changed, the ___location of the observer has.
