q^{μν (S)} = q^μq^ν     (= q^μ_a q^ν_b η^ab).                                                                 (2.73)

D_ρq_μν^(s) = D_ρq^μν(S) = 0.                                                                 (2.170)

D_νq^μ = ∂_νq^μ + Γ^μ_νλ q^λ = 0                                                         (2.182)

The Evans wave equation and lemma are derived [1-3] from the tetrad postulate of differential geometry [9,10]:

D_μ q^a_ν = ∂_μq^a_ν + ω^a_μb q^b_ν − Γ^λ_μνq^a_λ = 0                                                 (8.13)

Equations (2.179) and (2.180) are the tetrad postulate of differential geometry [11]:

D_ρq_μ = 0.                                                                 (2.179)

D_ρq^μ = 0.                                                                 (2.180)

where D_ρ is given by

D_ρq^μ_a = ∂_ρq^μ_a + Γ^μ_ρσ q^σ_a                                                 (1.59')

... consider the equation of metric compatibility in geometry [11]:

D_ρq_μν^(s) = D_ρq^μν(S) = 0.                                                                 (2.170)

D_ρq^μ_a = ∂_ρq^μ_a + Γ^μ_ρσq^σ_a                                                                 (1.59')

Using the definition of the symmetric metric

q_μν^(S) = q_μν,                                                                 (2.171)

q_μν^(S) = q_μq_ν,                                                                 (2.172)

the equation of metric compatibility becomes

D_ρ(q_μq_ν) = (D_ρq_μ)q_ν + q_μ(D_ρq_ν) = 0,                                                 (2.173)

where the Leibnitz rule . . .

Multiply both sides of Eq. (2.173) by q^ν, to obtain from the wrong Equ.(2.149) (see [3; Sec.6]

− 2 D_ρq_μ + q_μq^νD_ρq_ν = 0.                                                                 (2.175)

Now multiply both sides of this equation by q^μq_ν:

q_μq_νD_ρq_μ = 2 D_ρq_ν.                                                                 (2.176)

Similarly, it can be shown that

q^νq_μD_ρq_ν = 2 D_ρq_μ.                                                                 (2.177)

When μ = ν, these equations become . . .

D_ρq_μ = − D_ρq_μ,                                                                 (2.178)

whose only solution is

D_ρq_μ = 0.                                                                 (2.179)

Recall that q_μ is the tetrad q^a_μ with unwritten index a, and q^μ is the inverse tetrad q^μ_a Equations (2.179) and (2.180) are the tetrad postulate of differential geometry [11]. . . .

D_ρq^μ = 0.                                                                 (2.180)

The equation of metric compatibility for q^μ, written out in terms of the Christoffel symbol, is

D_νq^μ = ∂_νq^μ + Γ^μ_νλ q^λ = 0.                                                         (2.182)

Multiplication of both sides of this equation by q_λ gives

q_λ D_νq^μ = q_λ ∂_νq^μ + q_λq^λ Γ^μ_νλ = 0.                                                 (2.183)

(4.1) The tetrad postulate: D_νe^a_μ = 0,                                                                 (4.6)

where D_ν denotes the covariant derivative [2], is true for any connection, whether or not it is metric compatible or torsion free.

Equation (4.5), the wave equation for the vielbein as eigenfunction, follows from the tetrad postulate [2]:

D_ρ e^a_μ = 0,                                                                 (4.15)

which holds whether or not the connection is metric-compatible or torsionfree. Differentiating Eq. (4.15) covariantly gives Eq. (4.5):

D^ρ(D_ρe^a_μ) := D^ρD_ρ e^a_μ = ( + D^μ Γ^ρ_μρ + 2 Γ^λ_μμ D_λ) e^a_μ = ( + D^μ Γ^ρ_μρ) e^a_μ = 0.                 (4.16)

D_ρq_μν^(S) = 0                                                                                 (4.27)

A third type of wave equation can be obtained using the definition [1]

q^(S)_μν = q_μq_ν = η_abq^a_μq^b_ν in extended notation.                                                 (4.29)

Covariant differentiation of products is defined by the Leibniz theorem [2]; therefore the metric compatibility of the symmetric metric tensor, Eq. (4.27), implies that

D_ρ(q_μq_ν) = q_μ(D_ρq_ν) + (D_ρq_μ)q_ν = 0 .                                                                 (4.30)

The equation of metric compatibility (4.32) can be derived independently as a solution of the equation of parallel transport [2] written for the inverse metric four-vector q^μ:

^Dqμ/_ds := ^dqμ/_ds + Γ^μ_νλ ^dxν/_ds q^λ = 0,                                                 (4.35)

(ds)² = q^μq^νdx_μdx_ν                                                                 (4.36)

∂q^μ/ ∂x^ν = − Γ^μ_νλ q^λ.                                                                 (4.56)

. . . and the equation of metric compatibility for q^μ is

D_νq^μ = ∂_νq^μ + Γ^μ_νλq^λ = 0.                                                                 (5.85)

A novel and fundamental wave equation of general relativity can be deduced [1] from the metric compatibility condition

D_μq^ν = ∂_μq^ν + Γ^ν_μλ q^λ = 0                                                                 (6.1)

on the metric four-vector q_μ used in the standard definition [2,3] of the metric tensor of the Einstein field equation

q_μν = q_μq_ν = q^a_μÄ q^b_ν in extended notation                                                 (6.2)

as the outer product of two metric vectors.

Equation (6.1) states that the covariant derivative of the metric vector q^ν vanishes. It has been shown that the metric compatibility condition (6.1) leads to the usual metric compatibility condition of the metric tensor

D_νq_μν = 0                                                                 (6.3)

used in Riemann geometry [3] to relate the Christoffel symbol Γ^ν _μλ and metric tensor q_μ^ν.

The tetrad can be defined by the equation

V^a = q^a_μ V^μ,                                                                 (9.5)

where V^a is any contravariant vector in the orthonormal basis, indexed a, and V^μ is any contravariant vector in the base manifold indexed µ. For example, V can represent position vectors x, so the tetrad becomes

x^a = q^a_μ x^μ,                                                                 (9.6)

and has sixteen independent components, the sixteen irreducible representations of the Einstein group [8-12]. The vector V can also represent metric vectors q [3-7], so the tetrad can be defined by

q^a = q^a_μ q^μ.                                                                 (9.7)

This means that the symmetric metric [2-7] can be defined in general by the dot product of tetrads:

q_μν^(S) = q^a_μ q^b_ν η_ab,                                                                 (9.8)

where η_ab is the metric diag(−1, 1, 1, 1) of the orthogonal space [2]. The antisymmetric metric is defined in general by the wedge product of tetrads:

q^c_μν^(A) = q^a_μ Ù q^b_ν,                                                                 (9.9)

and the general metric tensor with sixteen independent components is defined by the outer product of tetrads:

q^ab_μν = q^a_μq^b_ν = q^a_μ Ä q^b_ν .                                                                 (9.10)

The dot product is the gauge invariant gravitational field, the cross product the gauge invariant electromagnetic field, . . .

In the Minkowski space M⁴ there exists no "cross product" × : M⁴ × M⁴ → M⁴ .

. . . and the outer product combines the two fields and defines the way in which one influences the other.

The tetrad can also be defined [2] by basis vectors; for example,

ê_(μ) = q^a_μ ê_(a),                                                                 (9.11)

for covariant basis vectors, or

θ^(a) = q^a_μ θ^(μ)                                                                 (9.12)

x^a'_μ = q^a'_μ x^μ.                                                                 (9.17)

The well-known tetrad postulate [2-7] is

D_μq^a_ν = ∂_μq^a_ν + ω^a_μb q^b_ν − Γ^λ_μνq^a_λ = 0                                                 (9.24)

and is the basis of the Evans lemma and Evans wave equation [3-7] of differential geometry, the most powerful and general wave equation known in general relativity, and thus in physics.

Therefore gravitation is described by the symmetric metric q^μq^ν and electromagnetism by the anti-symmetric metric defined by the wedge product q^μÙq^ν.

The symmetric metric tensor is defined by the symmetric tensor product of two metric four-vectors:

q^{μν (S)} = q^μq^ν (= q^μ_a q^ν_b η^ab = g^μν)                                                                 (3.4)

and the anti-symmetric metric tensor by the wedge product:

q^{μν (A)} = q^μ Ù q^ν (= q^μ_a Ù q^ν_b η^ab = O)                                                                 (3.5)

where the metric four-vector is:

q^ν = (h⁰, h¹, h², h³)                                                                 (3.6)

If the term η^ab would be omitted in the extended notation of (3.5) then the "antisymmetric metric" would not be invariant under changes of the tetrad by Lorentz transforms in the tangent space.

The symmetric metric tensor is then defined through the line element, a one form of differential geometry:

ω₁ = ds² = q^{ij (S)}du_i du_j ,                                                                 (3.12)

and the anti-symmetric metric tensor through the area element, a two form of differential geometry:

ω₂ = dA = − ½ q_ij ^(A)duⁱ Ù du^j .                                                                 (3.13)

These results generalize as follows to the four dimensions of any non-Euclidean space-time:

ω₁ = ds² = q_μν ^(S) du^μ du^ν,                                                                 (3.14)

ω₂ = *ω₁ = − ½ q_μν ^(A)du^μ Ù du^ν .                                                                 (3.15)

In differential geometry the element du^σ is dual to the wedge product du^μ Ù du^ν.

The symmetric metric tensor is:

                                                h₀²   h₀h₁   h₀h₂   h₀h₃
                                                h₁h₀   h₁²   h₁h₂   h₁h₃
                q_μν ^(S) =                                                                                                                 (3.16)
                                                h₂h₀   h₂h₁   h₂²   h₂h₃
                                                h₃h₀   h₃h₁   h₃h₂   h₃²