Apeiron, Vol. 13, No. 4, October 2006
462
Einstein Equations for Tetrad
Fields
Ali Rıza ŞAHİN, R. T. L. Istanbul (Turkey)
Every metric tensor can be expressed by the inner product of
tetrad fields. We prove that Einstein equations for these fields
have the same form as the stress-energy tensor of
electromagnetism.
Keywords: Einstein equations, tetrad fields, metric tensor,
stress-energy tensor, electromagnetism
It is agreed that gravitation can be best described by general relativity
and that it cannot be explained by using fields as in electromagnetism
or as in the case of any other interaction. Furthermore, it has been
assumed that the metric tensor is the best mathematical argument to
use to study on gravitation. Such opinions lead physicists to
concentrate more on only the metric tensor and, hence, to change it
according to circumstances. As a result, this method provides some
important results about gravitation. However, it is also obvious that
these results are not enough to understand gravitation as well as,
perhaps, other interactions.
In the present paper, instead of concentrating on the metric tensor,
we shall focus on tetrad fields. Our first objective will be to find some
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Apeiron, Vol. 13, No. 4, October 2006
463
reasonable mathematical results with these fields. The complete
interpretation of the results will be out of the scope of this paper.
Gravitation curves the space-time and this effect is related to the
line element or invariant interval as
ds 2 = g μν dx μ dxν
where g μν is the metric tensor and its elements are some functions of
the space-time.
The metric tensor with tetrad fields is given by [1], [2]
g μν = e μ •eν
(1)
where e μ are basis vectors or tetrad fields, and these are some
functions of the space-time also ( μ ,ν = 0,1, 2, 3 ).
Similar to (1), the inverse metric tensor can be written as
g μν = e μ •eν
where e μ are basis vectors of the dual space or cotetrad fields.
However, we will refer to these fields as inverse fields throughout this
work.
There are some useful features of and equations for the tetrad
fields and inverse fields. First
g μα gαν = δ μν
e μ •eα gαν = δ μν
e μ •eν = δ μν
(2)
Other equations and all detailed calculations are given in the appendix
section.
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Apeiron, Vol. 13, No. 4, October 2006
464
If the metric tensor is determined, it is well-known that it is
demanding work to find the Einstein equations. The Christoffel
symbols for the metric tensor (1) are
1
1
Γα μν = f αν •e μ = f α μ •eν
2
2
where f α ν = ∂α eν − ∂ν eα .
The Riemann tensor for the above Christoffel symbols is
1
1
1
Rα μβν = ∂α f βν •e μ + f αν •f βμ + f α β •fμν .
2
4
4
the Ricci tensor is
1
Rμν = jν •e μ + f α ν •fαμ ,
4
and the Ricci scalar is
1
R = jβ •e β + f αβ •fαβ
8
1
where jν = ∂α fαν = ∂α ∂α eν .
2
Finally the Einstein Tensor can be expressed as
Gμν =
1⎡ α
⎛1
⎞⎤
f ν •fαμ − g μν ⎜ f αβ •fαβ + jα •eα ⎟ ⎥ .
⎢
4⎣
⎝4
⎠⎦
(3)
The expression in square brackets is the same as the stress-energy
tensor of electromagnetism except for the inner products. Despite this
difference, the equations of motion of the tetrad fields have the same
form as the Maxwell equations; that is ∂α ∂α eν = jν .
Several results can be obtained from (3). However, the most
significant of these is that the Einstein equations for the tetrad fields
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Apeiron, Vol. 13, No. 4, October 2006
465
certainly give the electromagnetic stress-energy tensor. More
precisely, the general relativity reveals that there are some inherent
constraints for tetrad fields. This means there are also definite limits
for the metric tensor. Since every metric tensor can be written in
terms of tetrad fields, metric tensors cannot be chosen or adjusted
arbitrarily. Instead, metric tensors must be found as inner products of
tetrad fields after these fields are determined to be consistent with
∂α ∂α eν = jν .
Appendix
In this section detailed calculations and some useful equations are
given for convenience, although some of these can be found from
several sources and in different forms.
Since eα •e λ = δα λ , the partial derivatives of e μ can be written as
∂ν e ρ = ω ρνλ e λ , where ωνρλ are some coefficients. Then
∂ν e ρ •eα = ωρνλ eλ •eα = ω ρνα .
So
∂ν e ρ •eα = ωρνλ eλ •eα = ω ρνα ,
(∂ e
ν ρ
•eα ) eα = ∂ν e ρ .
(A.1)
Similarly it can be shown that
(∂ e
ν ρ
•eα ) eα = ∂ν e ρ .
(A.2)
Another important equation can be derived by starting from
∂ν e ρ •eν = ∂ λ e ρ •eλ . Using (A.1)
∂ν e ρ •eν = ( ∂ λ e ρ •eν ) eν •e λ
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Apeiron, Vol. 13, No. 4, October 2006
466
∂ν e ρ •e = ⎡⎣( ∂ λ e ρ •eν ) e ⎤⎦ •e
ν
λ
ν
∂ν e ρ = ( ∂ λ e ρ •eν ) eλ
The dot product of the last equation with eα is
∂ν e ρ •eα = ( ∂ λ e ρ •eν ) eλ •eα .
Since eα •e λ = δα λ
∂ν e ρ •eα = ∂α e ρ •eν .
(A.3)
Similarly it can be found that
∂ν e ρ •eα = ∂α e ρ •eν .
(A.4)
Another equation can be derived if the derivative of (2) is rewritten
as
∂α e μ •eν = −e μ •∂α eν .
(A.5)
Now we can start to calculate the Einstein equations. The
Christoffel symbols are
1
Γα μν = g αβ ⎡⎣ ∂ μ eν •e β + ∂ μ e β •eν + ∂ν e μ •e β ⎤⎦
2
1
+ g αβ ⎡⎣ ∂ν e β •e μ − ∂ β e μ •eν − ∂ β eν •e μ ⎤⎦ .
2
Using (A.3), we get
1
Γα μν = g αβ ( ∂ μ e β •eν + ∂ν e β •e μ ) .
(A.6)
2
Symmetries and characteristics of the tetrad fields enable to derive
some helpful identities. First using (A.1)
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Apeiron, Vol. 13, No. 4, October 2006
(
)
1 αβ
g ( ∂ μ e β •e ρ ) e ρ •eν + ( ∂ν e β •e ρ ) e ρ •e μ .
2
Similarly, (A.5) enables us to write
1
Γα μν = g αβ − ( ∂ μ e ρ •e β ) e ρ •eν − ( ∂ν e ρ •e β ) e ρ •e μ ,
2
Γα μν =
(
)
(
)
(
)
1
− ( ∂ μ e ρ •eα ) e ρ •eν − ( ∂ν e ρ •eα ) e ρ •e μ .
2
Using (A.3) and (A.4), we can obtain
1
Γα μν = − ( ∂α e ρ •e μ ) e ρ •eν − ( ∂α e ρ •eν ) e ρ •e μ ,
2
Γα μν =
(
)
1 α
∂ e μ •e ρ ) e ρ •eν + ( ∂α eν •e ρ ) e ρ •e μ .
(
2
Also, by using (A.3) and (A.4)
1
Γα μν = ( ∂α e μ •eν + ∂ μ eν •eα )
(A.7)
2
and using (A.5) again
1
1
Γα μν = ( ∂α e μ •eν − ∂ μ eα •eν ) = ( ∂α e μ − ∂ μ eα ) •eν
2
2
can be found.
Although it can be proved easily that ∂α eν = −∂ν eα , for
convenience let
Γα μν =
f α ν = ∂α eν − ∂ν eα .
Then the Christoffel symbols are
1
Γα μν = f α μ •eν .
2
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(A.8)
468
Apeiron, Vol. 13, No. 4, October 2006
α
With similar calculations, Γ
μν
can be found in the following form
1
Γα μν = f αν •e μ .
2
Or using (A.4), (A.7) can be rewritten as
1
Γα μν = ( ∂ν e μ + ∂ μ eν ) •eα .
2
When the Einstein Equations are calculated, one of these Γα μν can be
used.
The Riemann tensor defined by
Rα μβν = ∂ β Γανμ − ∂ν Γα βμ + Γα βλ Γ λνμ − Γανλ Γ λ βμ .
using the above Christoffel symbols
1
1
1
1
Rα μβν = ∂ β f α ν •e μ + f αν •∂ β e μ − ∂ν f α β •e μ − f α β •∂ν e μ
2
2
2
2
1 α
1 α
+ f β • ( ∂ν e μ + ∂ μ eν ) − f ν • ( ∂ β e μ + ∂ μ eν ) ,
4
4
1
1
Rα μβν = ∂ β ( f αν •e μ ) − ∂ν ( f α β •e μ )
2
2
1
1
+ ( f α β •eλ ) ( ∂ν e μ + ∂ μ eν ) •eλ − ( f αν •eλ ) ( ∂ β e μ + ∂ μ eν ) •eλ ,
4
4
(
Rα μβν =
)
(
)
1
1
1
⎛
⎞
∂ β f α ν − ∂ν f α β ) •e μ + f αν • ⎜ ∂ β e μ − ( ∂ β eμ + ∂ μ eβ ) ⎟
(
2
2
2
⎝
⎠
1
⎛1
⎞
+ f α β • ⎜ ( ∂ν e μ + ∂ μ eν ) − ∂ν e μ ⎟ ,
2
⎝2
⎠
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Apeiron, Vol. 13, No. 4, October 2006
Rα μβν =
469
1
1
1
∂ β f α ν + ∂ν f β α ) •e μ + f α ν •fβμ + f α β •f μν .
(
2
4
4
As a result of ∂ β f αν + ∂ν fβ α + ∂α fνβ = 0 , the last Riemann tensor can
be simplified. For this, write ∂ β f α ν + ∂ν f β α = −∂α fνβ = ∂α f βν . Thus,
1
1
1
Rα μβν = ∂α f βν •e μ + f αν •f βμ + f α β •fμν .
2
4
4
The Ricci tensor is
1
1
1
Rα μβν = ∂α f βν •e μ + f αν •f βμ + f α β •fμν .
2
4
4
1
Let jν = ∂α fαν = ∂α ∂α eν . Thus the Ricci tensor becomes
2
1
Rμν = jν •e μ + f α ν •fαμ .
4
and the Ricci scalar is
1
R = Rαβ g αβ = jβ •e β + f αβ •fαβ .
8
1
Here f αβ •fαβ is multiplied by because of twofold summation.
2
1
The Einstein tensor is defined as Gμν = Rμν − g μν R . Then,
2
1
1
1
⎡
⎤
Gμν = jν •e μ + f αν •fαμ − g μν ⎢ jα •eα + f αβ •fαβ ⎥ .
4
2
8
⎣
⎦
The last expression can be simplified if we start with
jμ •e μ = jβ •e β .
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Apeiron, Vol. 13, No. 4, October 2006
As a result of (2), e μ •e μ = 4 . So
e μ •e
4
jμ •e = ( jβ •e
μ
jμ •e μ =
1
4
β
)
(e
= 1 and we can write
μ
•e μ )
4
(( j •e ) e ) •e
β
β
470
μ
μ
,
μ
,
1
jβ •e β ) e μ .
(
4
The dot product of the last equation with eν yields
jμ =
1
1
jβ •e β ) e μ •eν = ( jβ •e β ) g μν .
(
4
4
Finally, the Einstein tensor becomes
jμ •eν =
Gμν =
1⎡ α
⎛1
⎞⎤
f ν •fαμ − g μν ⎜ f αβ •fαβ + jα •eα ⎟ ⎥ .
⎢
4⎣
⎝4
⎠⎦
References
[1] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Ch. 13, Freeman (1973).
310 pp.
[2] A. Waldyr, Jr. Rodrigues, Quintino A. G. de Souza, “An Ambiguous Statement
Called ‘Tetrad Postulate’ and the Correct Field Equations Satisfied by the
Tetrad Fields”, arXiv:math-ph/0411085 v11 16 Aug 2006
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