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Tensors and Relativity: Chapter 2

# Four- vectors

A four- vector is a quantity with four components which changes like spacetime coordinates under a coordinate transformation. We will write the displacement four- vector as:  is a frame independent vector joining near by points in spacetime. means: has components in frame . are the coordinates themselves [ which are coordinate dependent ].

In frame , the coordinates are so: The Lorentz transformations  can be written as where is the Lorentz transformation matrix  and , can be regarded as column vectors.

The positioning of the indices is explained later but indicates that we can use the  summation convention: sum over repeated indices, if one index  is up and one index down. Thus we can write:  is a dummy- index  which can be replaced by any other index; is a free- index , so the above equation is equivalent to four equations. For a general four- vector we can write If and are two four- vectors, clearly and are also four- vectors with obvious components and In any frame we can define a set of four- basis vectors :   and In general we can write where labels the basis vector and labels the coordinate.

Any four- vector can be expressed as a sum of four- vectors parallel to the basis vectors i.e. The last equality reflects the fact that four- vectors are frame independent.

Writing: where we have exchanged the dummy indices and and and , we see that this equals for all if and only if This gives the transformation law for basis vectors : Note that the basis transformation law is different from the transformation law for the components since takes one from frame to .

So in summary, for vector basis  and vector components  we have: Since has a velocity relative to , we have: So is the inverse  of . Likewise It follows that the Lorentz transformations with gives the components of a four- vector in from those in .

The magnitude  of a four- vector is defined as : in analogy with the line element The sign on the will be explained later. This is a frame invariant scalar. is spacelike  if , timelike  if and null  if . The scalar product  of two four- vectors and is: Since  is frame independent. and are orthogonal if ; they are not necessarily perpendicular in the spacetime diagram [ for example a null vector is orthogonal to itself ], but must make equal angles with the line.

Basis vectors form an orthonormal tetrad  since they are orthogonal: if , normalized to unit magnitude: : We will see later what the geometrical significance of is.