(1626) | |||

(1627) |

Let us define spherical polar coordinates whose axis points along the direction of instantaneous acceleration of the charge. It is easily demonstrated that

These fields are identical to those of a radiating dipole whose axis is aligned along the direction of instantaneous acceleration (see Sect. 9.2). The Poynting flux is given by

(1630) |

This is known as

In order to proceed further, it is necessary to prove two useful theorems.
The first theorem states that if a 4-vector field satisfies

(1633) |

where is an element of the 3-dimensional surface bounding the 4-dimensional volume . The particular volume over which the integration is performed is indicated in Fig. 58. The surfaces and are chosen so that the spatial components of vanish on and . This is always possible because it is assumed that the region over which the components of are non-zero is of finite extent. The surface is chosen normal to the -axis, whereas the surface is chosen normal to the -axis. Here, the and the are coordinates in two arbitrarily chosen inertial frames. It follows from Eq. (1634) that

Here, we have made use of the fact that is a scalar and, therefore, has the same value in all inertial frames. Since and it follows that is an invariant under a Lorentz transformation. Incidentally, the above argument also demonstrates that is constant in time (just take the limit in which the two inertial frames are identical).

The second theorem is an extension of the first. Suppose that a 4-tensor
field satisfies

(1636) |

is an invariant. However, we can write

(1638) |

(1639) |

These two theorems enable us to convert differential conservation laws
into integral conservation laws. For instance, in differential form,
the conservation of electrical charge is written

(1640) |

(1641) |

Suppose that is the instantaneous rest frame of the charge. Let us
consider the electromagnetic energy tensor associated with
all of the radiation emitted by the charge between times and .
According to Eq. (1583), this tensor field satisfies

(1642) |

(1643) |

(1644) |

(1645) |

We can make use of the fact that the power radiated by an accelerating charge
is Lorentz invariant to find a relativistic generalization of the
Larmor formula, (1631), which is valid in all inertial frames. We expect the
power emitted by the charge to depend only on its 4-velocity and
4-acceleration.
It follows that the Larmor formula can be written in Lorentz invariant form as

(1646) |

(1647) |

It follows, after a little algebra, that the relativistic generalization of Larmor's formula takes the form