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Units, Constants and Useful Formulas
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Last modified: January 26, 2018
Vierbein (Frame) Fields
-----------------------
The action for a point particle is:
S = -mc∫ √(-gμν(∂xμ/∂τ)(∂xμ/∂τ))dτ
We postulate that there is another action that is equivalent to this.
S' = (1/2)∫ (gμνe-1(∂xμ/∂τ)(∂xμ/∂τ) - em2) dτ
This form of the action makes it look as if we have coupled the
worldline theory to 1D gravity, with the field e(τ). The equation
of motion for e is:
e2 = gμν(∂xμ/∂τ)(∂xμ/∂τ)/m2
If we plug this back into S' we get:
S' = (1/2)∫ (gμν√(m2/gμν(∂xμ/∂τ)(∂xμ/∂τ))(∂xμ/∂τ)(∂xμ/∂τ)
- ((∂xμ/∂τ)(∂xμ/∂τ)/m2)m2) dτ
= m∫ (√(gμν(∂xμ/∂τ))(∂xμ/∂τ)) - gμν(∂xμ/∂τ)(∂xμ/∂τ) dτ
= -m∫ (√(gμν(∂xμ/∂τ))(∂xμ/∂τ)) dτ = S
e in this case is referrred to as an EINBEIN FIELD. This idea can
be extended to more dimensions. For 2 indeces we have a ZWEIBEIN
FIELD, for 3 indeces we have a DREIBEIN FIELD and for 4 indeces we
have a VIERBEIN FIELD. The latter case is also referred to as a
TETRAD.
The vierbein field can be interpreted in the following manner.
Consider a point, P, on a curved manifold. P is defined in the
coordinate frame, x, and there is a metric gμν(x) associated with
it. However, it is also possible to construct a Lorentz frame
with coordinates, ξa(x), in tangent space that is locally flat at
P. A vector on the manifold at P, can now be mapped into this new
frame and expanded as a linear combination of orthonormal basis
vectors on a Lorentz manifold.. The vierbein field, therefore,
has two kinds of indices: those that label the spacetime
coordinates and those that label the frame coordinates.
The vierbein field theory is the most natural way to represent
a relativistic quantum field theory in curved space and can be
regarded as a gauge field for gravity. While it plays the role
of a gauge field it not behave in the same way as the vector
potential field.
We can view the vierbein eμa as the transformation matrix
between arbitrary coordinates, x, and inertial coordinates, ξ.
dξm = (∂ξm/∂xμ)dxμ ≡ eμm(x)dxμ
and the inverse,
dxμ = (∂xμ/∂ξm)dξm ≡ emμ(x)dξm
Therefore, since the inner product has to be the same in both
frames:
dξm = eμm(x)dxμ = eμm(x)emμ(x)dξm
ds2 = ηmndξmdξn = ηmneμmeνndxμdxν
The vierbeins satisfy the following relationships:
eμaeaν = δμν = I
eaμeμb = δab = I
gμν = eμaeνbηab and gμν = eμaeνbηab
ηab = eμaeνbgμν and ηab = eaμebνgμν
det[gμν] = det[eμaeνbηab]
det[gμν] = det[eμa]det[eνb]det[ηab]
det[gμν] = (det[eμa])2(-1)
√(-det[gμν]) = det[eμa]