In Section 2.2 we proved quite generally that corresponding differentials
in two inertial frames and satisfy the relation

(79) |

(80) |

where and are the coordinates of any two events in and the primed symbols denote the corresponding coordinates in . It is convenient to write

(81) |

(82) |

Let us consider entities defined in terms of four variables

(83) |

Tensor analysis cannot proceed very far without the introduction of
a non-singular tensor , the so-called *fundamental tensor*,
which is used to define the operations of raising and lowering
suffixes (see Eqs. (2.42)-(2.44)). The fundamental tensor is
usually introduced using a metric
, where
is a differential invariant. We have already come
across such an invariant, namely

(84) |

where run from 1 to 4. Note that the use of Greek suffixes is conventional in 4-tensor theory. Roman suffixes are reserved for tensors in three dimensional Euclidian space, so-called 3-tensors. The 4-tensor has the components and when , in all permissible coordinate frames. From now on , as defined above, is adopted as the fundamental tensor for 4-tensors. can be thought of as the

A tensor of rank one is called a *4-vector*. We shall also have occasion
to use ordinary vectors in three dimensional Euclidian space. Such
vectors are called *3-vectors* and are conventionally represented by
boldface symbols. We shall use the Latin suffixes *etc.*
to denote the components of a 3-vector; these suffixes are understood to
range from 1 to 3. Thus,
denotes a velocity
vector. For 3-vectors we shall use the notation
interchangeably; *i.e.*, the level of the suffix has
no physical significance.

When tensor transformations from one frame to another actually
have to be computed, we shall usually find it possible to choose
coordinates in the standard configuration, so that the standard
Lorentz transform applies. Under it, any contravariant 4-vector
transforms according to the same scheme as the difference
in coordinates
between two points in space-time.
It follows that

(85) | |||

(86) | |||

(87) | |||

(88) |

where . Higher rank 4-tensors transform according to the rules (2.30)-(2.32). The transformation coefficients take the form

(89) | |||

(90) |

Often the first three components of a 4-vector coincide with the
components of a 3-vector. For example, the , , in
are the components of , the position
3-vector of the point at which the event occurs. In such cases we
adopt the notation exemplified by
. The covariant
form of such a vector is simply
. The squared
magnitude of the vector is
.
The inner product
of
with a similar vector
is given by
. The vectors
and are said to be *orthogonal* if
.

Since a general Lorentz transformation is a *linear* transformation,
the partial derivative of a 4-tensor is also a 4-tensor;

(91) |

(92) |

(93) |

(94) |

(95) |

(96) |

(97) |

Recall that we still need to prove (from Section 2.2) that the invariance
of the differential metric,

(98) |

(99) |

(100) |

(101) |

(102) |

(103) |

(104) |

(105) |

(106) |