The Principle of Relativity: Invariance and Covariance (Barut 1964, I.6)
Peter
I have found the following sentences in A. O. Barut, Electrodynamics and
Classical Theory of Fields and Particles, New York: MacMillan 1964, which may
be helpful for clarifying the relationship between the form of an equation
and its relativistic (non-)invariance.
"As a generalization of ordinary vector calculus (which expresses the fact
that the vector equations are valid in any three-dimensional coordinate
frame), we can write the fundamental equations of physical theories as four-
vector, tensor or spinor equations such that the *form* of these equations is
the same for all observers... We will then say that the equations are
*covariant*.
However, it is important to realize that the *relativistic invariance*
does not necessarily imply covariance." (p.35)
"Or, an equation may not be manifestly covariant because it has been
reduced to a particular frame. If a covariant equation reduces to a familiar
equation in a particular frame, then the former is a unique covariant form of
the latter, for if an equation is valid in one frame, it must be valid in all
frames." (p.36)
What do you think?
Does this mean that the Helmholtz decomposition of the microscopic Maxwell
equations does not contradict/prevent their relativistic invariance?
Thank you very much in advance,
Peter
Oh No
>
>I have found the following sentences in A. O. Barut, Electrodynamics and
>Classical Theory of Fields and Particles, New York: MacMillan 1964, which may
>be helpful for clarifying the relationship between the form of an equation
>and its relativistic (non-)invariance.
>
> "As a generalization of ordinary vector calculus (which expresses the fact
>that the vector equations are valid in any three-dimensional coordinate
>frame), we can write the fundamental equations of physical theories as four-
>vector, tensor or spinor equations such that the *form* of these equations is
>the same for all observers... We will then say that the equations are
>*covariant*.
> However, it is important to realize that the *relativistic invariance*
>does not necessarily imply covariance." (p.35)
> "Or, an equation may not be manifestly covariant because it has been
>reduced to a particular frame. If a covariant equation reduces to a familiar
>equation in a particular frame, then the former is a unique covariant form of
>the latter, for if an equation is valid in one frame, it must be valid in all
>frames." (p.36)
>
>What do you think?
I think it is very funny to say that invariance does not imply
covariance, since invariance is just a trivial special case of
covariance. Perhaps he means that covariance does not imply invariance.
>
>Does this mean that the Helmholtz decomposition of the microscopic Maxwell
>equations does not contradict/prevent their relativistic invariance?
>
This is true. The Helmholtz decomposition is frame dependent.
Regards
--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)