Relativity: Tensor Algebra
1. Expression of the basis vectors
gμ and gμ
1.1. In the frame x:
The vector position r has the components x1 and x2 in the frame x
with the basis vectors e1 and e2:
r = x1 e1 + x2 e2
The differential element dr is:
dr = dx1 e1 + dx2 e2
The total differential is:
dr = (∂r/∂x1) dx1 + (∂r/∂x2) dx2
Therefore:
e1 = ∂r/∂x1, and
e2 = ∂r/∂x2
or
ei = ∂r/∂xi
1.2 In the frame y:
The vector position r has also the components y1 and y2 in the frame y
with the basis vectors a1 and a2:
r = y1 a1 + y2 a2
dr = dy1 a1 + dy2 a2
dr = (∂r/∂y1) dy1 + (∂r/∂y2) dy2
a1 = ∂r/∂y1
a2 = ∂r/∂y2
or
ai = ∂r/∂yi
Therefore, we use the following convention:
ei = ∂r/∂xi = ∂yi/∂xi = gi
for covariant basis vectors.
ai = ∂r/∂yi = ∂xi/∂yi = gi
for contravariant basis vectors.
Covariant basis vectors:
gi = ∂yi/∂xi
Contravariant basis vectors:
gi = ∂xi/∂yi
2. Covariant and contravariant components
For the covariant basis vectors:
r = x1 e1 + x2 e2 =
= x1 g1 + x2 g2
=
xi gi
For the contravariant basis vectors:
r = y1 a1 + y2 a2
= x1 g1 + x2 g2
= xi gi
Contravariant basis vectors:
r = xi gi
Covariant basis vectors:
r = xi gi
3. Metric Tensor
gi = ∂yi/∂xi
gi = ∂xi/∂yi
ym are the components of the vector r in
the gi basis vectors.
We have the following products:
3.1. Contravariant-covariant product:
gi gj = ∂xi/∂yi . ∂yi/∂xj =
δij
gi gj = gij = δij
3.2. Contravariant-contravariant product:
gi. gj = ∂xi/∂ym .
∂xj/∂ym = ∂xi∂xj/∂ym∂ym
gi. gj = gij = ∂xi∂xj/∂ym∂ym
3.3. Covariant-covariant product:
gi . gj = ∂ym/∂xi .
∂ym/∂xj
gi . gj = gij = ∂ym∂ym/∂xi∂xj
4. Changing indices
We have:
r = xi gi
and
r . gj = xi gi . gj =
Therefore:
xi δij = xj
r . gj = xj
We have:
r = xi gi
and
r . gj = xi gi . gj =
Therefore:
xi δij = xj
r . gj = xj
5. Lowering and raising indices
5.1. Lowering::
We have:
xj = r . gj
and
r = xi gi
Therefore:
xj = xi gi . gj
= xi gij
xj = gij xi
5.2. raising::
We have:
xj = r . gj
and
r = xi gi
Therefore:
xj = xi gi . gj
= xi gij
xj = xi gij
6. Dot product of vectors
6.1. Scalar product of two vectors::
We have:
r = xi gi
and
s = xj gj
Therefore:
r. s = xi gi . xj gj =
xi xj gi. gj =
xi xj gj. gi =
xi xj δji =
xi xi
r = xi gi
s = xj gj
r.s = xi xi
6.2. Tensorial product of two vectors::
We have:
r = xi gi
and
s = xj gj
Therefore:
T = r s = xi gi xj gj =
T = xi xj gigj =
xi xj gij = Tij gij
Similarly,
r = xi gi
and
s = xj gj
Therefore:
U = r s = xi gi xj gj =
U = xi xj gi gj =
xi xj gij = Uij gij
T = Tijgij
U = Uklgkl
T U = Tij gij Ukl gkl =
Tij Ukl gij gkl =
Tij Ukl gi gj gk gl =
Tij Ukl gi δjk gl
=
Tij Ujl gi gl
=
Tij Ujl δil
=
Tij Uji
=
Tij Uij
T = Tijgij
U = Uklgkl
TU = TijUij
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