Appendix 3B: Components with sign inversion of the metric tensor

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 $[S1]$, which is defined as follows:

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

Transformed quantities are marked with an accent. For the metric

${\overset {\smile }{g}}_{\mu \nu }=-{g}_{\mu \nu }$

 

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

$\displaystyle \Gamma _{\ \ ij}^{k}={\frac {1}{2}}g^{kl}(\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij})={\overset {\smile }{\Gamma }}_{\ \ ij}^{k}$

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

$\partial _{\mu }\left(-g_{\mu \nu }\right)=-\partial _{\mu }\left(g_{\mu \nu }\right)$

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 $[S1]$, but can be defined independently with a positive or negative sign $[S2]$:

$R_{\ \ ikp}^{m}=[S2]\cdot \left(\partial _{k}\Gamma _{\ \ ip}^{m}-\partial _{p}\Gamma _{\ \ ik}^{m}+\Gamma _{\ \ ip}^{a}\Gamma _{\ \ ak}^{m}-\Gamma _{\ \ ik}^{a}\Gamma _{\ \ ap}^{m}\right)={\overset {\smile }{R}}_{\ \ ikp}^{m}$

 

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

$R_{\mu \nu }=[S2]\cdot [S3]\cdot R_{\ \ \mu \lambda \nu }^{\lambda }={\overset {\smile }{R}}_{\mu \nu }$

 

Ricci scalar with sign inversion of the metric
However, the Ricci scalar, which is defined directly by the metric, changes its sign depending on $[S1]$:

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

 

Einstein tensor with sign inversion of the metric
The trace-inverted Ricci tensor (Einstein tensor) thus remains the same under change of $[S1]$:

$\displaystyle \varepsilon _{ij}=R_{ij}-{\frac {1}{2}}Rg_{ij}={\overset {\smile }{R}}_{ij}-{\frac {1}{2}}{\overset {\smile }{R}}{\overset {\smile }{g}}_{ij}={\overset {\smile }{\varepsilon }}_{ij}$

But:

Failed to parse (syntax error): {\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}) }

 

References

 

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