Space-Time

In Section 12.3, we proved quite generally that corresponding differentials in two inertial frames and satisfy the relation

(1686) |

Thus, we expect this relation to remain invariant under a general Lorentz transformation. Because such a transformation is linear, it follows that

(1687) |

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

and

The differential , or the finite length , defined by these equations is called the

Let us consider entities defined in terms of four variables,

(1690) |

and which transform as tensors under a general Lorentz transformation. From now on, such entities will be referred to as

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. The fundamental tensor is
usually introduced using a metric
, where
is a differential invariant. We have already come
across such an invariant, namely

(1691) |

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

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
, et cetera,
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: that is, 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 such a transformation, any contravariant 4-vector, , transforms according to the same scheme as the difference in coordinates between two points in space-time. It follows that

where . Higher rank 4-tensors transform according to the rules (1669)-(1671). The transformation coefficients take the form

(1696) | ||

(1697) |

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 .

Because a general Lorentz transformation is a linear transformation, the partial derivative of a 4-tensor is also a 4-tensor:

(1698) |

Clearly, a general 4-tensor acquires an extra covariant index after partial differentiation with respect to the contravariant coordinate . It is helpful to define a covariant derivative operator

(1699) |

where

(1700) |

There is a corresponding contravariant derivative operator

(1701) |

where

(1702) |

The 4-divergence of a 4-vector, , is the invariant

(1703) |

The four-dimensional Laplacian operator, or

Recall that we still need to prove (from Section 12.3) that the invariance of the differential metric,

(1705) |

between two general inertial frames implies that the coordinate transformation between such frames is necessarily linear. To put it another way, we need to demonstrate that a transformation that transforms a metric with constant coefficients into a metric with constant coefficients must be linear. Now,

(1706) |

Differentiating with respect to , we get

where

(1708) |

et cetera. Interchanging the indices and yields

Interchanging the indices and gives

where the indices and have been interchanged in the first term. It follows from Equations (1709), (1711), and (1712) that

(1711) |

Multiplication by yields

(1712) |

Finally, multiplication by gives

(1713) |

This proves that the coefficients are constants, and, hence, that the transformation is linear.