1. NPTEL β Physics β Mathematical Physics - 1
Lecture 26
Covariant and Contravariant Vectors
Let us remind ourselves of the scalar function. A scalar function is
physical quantity, such as height of a hill, temperature of a system etc.
which are function of coordinates in a particular coordinate system. Thus let us
define a scalar function π·(π₯1, π₯2, π₯3) in a π3 . In another (barred) coordinate
system, this is defined as π·(π₯Μ 1, π₯Μ 2, π₯Μ 3). But the value of the scalar function
should be independent of the coordinate system. Hence
π·(π₯Μ 1, π₯Μ 2, π₯Μ 3) = π·(π₯1, π₯2, π₯3)
If we want to find the gradient,
ππ·Μ ππ· ππ₯1 ππ· ππ₯2 ππ· ππ₯3 ππ· ππ₯π
ππ₯Μ π =
ππ₯1 ππ₯Μ π +
ππ₯2 ππ₯Μ π +
ππ₯3 ππ₯Μ π =
ππ₯π ππ₯Μ π
=
ππ₯π π
π·
ππ₯Μ π ππ₯π
(1)
Compare this with the following case, consider a curve in space parameterized
in a Cartesian Coordinate system, π₯π = ππ (π‘), i = 1, 2, 3
π1(π‘), π2(π‘) and π3(π‘) are smooth functions of the parameter t. The tangent to
π
this curve which is a vector has components π₯π = ππ₯
= π β²
(π‘).
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ππ‘ π
Again consider a new coordinate system,
π₯Μ π = ππ(π₯1, π₯2, π₯3)
The curve can be represented in terms of new coordinates,
π₯Μ π = ππ(π1(π‘), π2(π‘), π3(π‘))
= βπ(π‘)
The components to the tangent to the curve in the primed coordinate system is
given by,
2. NPTEL β Physics β Mathematical Physics - 1
π₯Μ π = hiο’ (π‘) =
πππ ππ1 πππ ππ2 πππ ππ3
ππ₯1 ππ‘ ππ₯2 ππ‘ ππ₯3 ππ‘
+ +
= ππ₯Μ
π
ππ₯
π
π₯
π
(2)
Transformation according to Eq. (1) is definitely distinct than that of Eq. (2).
Based on such transformations, we can define two kinds of vectors, such as,
(a)One whose components transform according to Eq. (1)
(b)Other whose components transform according to Eq. (2)
To emphasize on the ongoing discussion, let us look at the dot product of two
vectors. Let π΄β and π΅ββ be vectors that transform according to Eq. (2),
βπ΄
βββπ
=
ππ₯Μ π
ππ₯π π΄π
;
βπ΅
βββπ
=
ππ₯Μ π
ππ₯π π΅π
Then the dot product is π΄ββββπ
π΅ββπ
(with sum over repeated indices
understood). In terms of the unprimed coordinates,
βπ΄βββπ β
π΅βββπ =
ππ₯Μ π ππ₯Μ π
ππ₯π ππ₯
π
π΄π π΅π
=
ππ₯Μ π ππ₯Μ π
ππ₯π ππ₯
π
π΄π π΅
π
(3)
The RHS of Eq. (3) does not reduce to a dot product.
Next consider two vectors (πββ and πββ) where πββ transforms according to
Eq. (1) and πββ transforms according to Eq. (2). So,
πΜ π = ππ ;
ππ₯
π
ππ₯Μ π
βπ
βββπ
=
ππ₯Μ π
ππ₯π π
π
Now take the dot product,
πΜ ππΜ π = ππ ππ =
ππ₯Μ π
ππ₯π ππ₯Μ π
ππ₯
π
ππ₯π ππ₯Μ π
ππ₯Μ π ππ₯π
ππ π
π
Thus πΜ ππΜ π = ππ ππ = πΏπ
π
ππ π
π
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ππ₯
π
ππ₯
π
= ππ π
π
3. NPTEL β Physics β Mathematical Physics - 1
Now the RHS is indeed a dot product. Thus we shall get a dot product only if
one of the vectors transform according to Eq. (1) and the other according to Eq.
(2).
It may be noted that the analysis can trivially be extended to an n- dimensional
space, namely, ππ. To be specific general theory of relatively demands a four
dimensional space time.
With the above notation in mind, we define two kinds of vectors namely,
π΄β = {π΄1, π΄2 β¦ β¦ β¦ . π΄π} and π΅ββ = {π΅1, π΅2 β¦ β¦ β¦ π΅π}
Which are defined as the contrvariant and covariant vectors respectively. They
transform according to,
π΄Μ π = ππ₯Μ
π΄π and
π
ππ₯π ππ₯Μ π
π΅ = ππ₯
π΅
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Μ π
π
π
They are thus distinguished from placement of indices. Only when
a contravariant vector (with upper index) appears with a covariant vector (with
a lower index) in a sum, the result is independent of the coordinate system.