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© The scientific sentence. 2010

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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