QFT, QED, Allgemeine Relativität, elastisch, Materialgesetz, GUT, Gravitation, Wechselwirkungen

The transformation properties of the components of the constitutive equation are examined under inversion of the sign [math]\displaystyle{ [S1] }[/math], which is defined as follows^[1]:

[math]\displaystyle{ g_{\mu\nu}=[S1] \cdot \mathrm{diag}(-1,+1,+1,+1) }[/math]

Transformed quantities are marked with an accent. For the metric

[math]\displaystyle{ \overset{\smile}{g}_{\mu\nu}= - {g}_{\mu\nu} }[/math]

Christoffel symbols under sign inversion of the metric
The Christoffel symbols of the second kind

[math]\displaystyle{ \displaystyle \Gamma^k_{\ \ ij} = \frac{1}{2} g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}) =\overset{\smile}{\Gamma}^k_{\ \ ij} }[/math]

are independent of the choice of the sign [math]\displaystyle{ [S1] }[/math], since the sign in both [math]\displaystyle{ g_{\mu \nu} }[/math] and [math]\displaystyle{ g^{\mu \nu} }[/math] switches and transforms the partial derivative with the metric sign change as follows:

[math]\displaystyle{ \partial_\mu \left( -g_{\mu\nu}\right) = -\partial_\mu \left( g_{\mu\nu}\right) }[/math]

As a result, the signs cancel each other out.

Riemann tensor under sign inversion of the metric
As a consequence, the Riemann tensor is also independent of [math]\displaystyle{ [S1] }[/math], but can be defined independently with a positive or negative sign [math]\displaystyle{ [S2] }[/math]:

[math]\displaystyle{ R^m_{\ \ ikp} = [S2] \cdot \left( \partial_k \Gamma^m_{\ \ ip} - \partial_p \Gamma^m_{\ \ ik} + \Gamma^a_{\ \ ip} \Gamma^m_{\ \ ak} - \Gamma^a_{\ \ ik} \Gamma^m_{\ \ ap} \right) = \overset{\smile}{R}^m_{\ \ ikp} }[/math]

Ricci tensor with sign inversion of the metric
Also, the Ricci tensor does not change its sign when changing [math]\displaystyle{ [S1] }[/math]. Its sign depends on the definition of the Riemann tensor and the constitutive equation:

[math]\displaystyle{ R_{\mu\nu} = [S2] \cdot [S3] \cdot R^\lambda_{\ \ \mu\lambda\nu} = \overset{\smile}{R}_{\mu\nu} }[/math]

Ricci scalar with sign inversion of the metric
However, the Ricci scalar, which is defined directly by the metric, changes its sign depending on [math]\displaystyle{ [S1] }[/math]:

[math]\displaystyle{ R=g^{\mu\nu}R_{\mu\nu} = [S1] \cdot \mathrm{diag}(-1,+1,+1,+1) \cdot [S2] \cdot [S3] \cdot R^\lambda_{\ \ \mu\lambda\nu} = - \overset{\smile}{g}^{\mu\nu}\overset{\smile}{R}_{\mu\nu} = -\overset{\smile}{R} }[/math]

Einstein tensor with sign inversion of the metric
The trace-inverted Ricci tensor (Einstein tensor) thus remains the same under change of [math]\displaystyle{ [S1] }[/math]:

[math]\displaystyle{ \displaystyle \varepsilon_{ij} = R_{ij} - \frac{1}{2}R g_{ij} = \overset{\smile}{R}_{ij} - \frac{1}{2}\overset{\smile}{R} \overset{\smile}{g}_{ij} = \overset{\smile}{\varepsilon}_{ij} }[/math]

But:

[math]\displaystyle{ \mathrm{tr}(\mathbf{\overset{\smile}{ε}}) = \overset{\smile}{g}^{ab}\overset{\smile}{\varepsilon}_{ab} = - g^{ab}\overset{\smile}{\varepsilon}_{ab} = - g^{ab}\varepsilon_{ab} = - \mathrm{tr}(\mathbf{ε}) }[/math]