User:Ric.Peregrino

wiki notes

Initial page

Hi,

I'm new to this, and just started this page, after having posted a talk to a page on the Levi-Civita connection titled "The Asymmetric Metric":

Cheers, Ric

Further Review

After further research on the subject, if I understand it, the approach of contorsion maintains a symmetric metric, and adds a symmetric component in addition to the Levi-Civita connection, to compensate to keep metricity in the presence of Torsion. This symmetric compensation then would affect geodesics, giving Torsion an indirect affect on matter macroscopically, instead of just coupling to fermion intrinsic spins. I need to read up, it seems a question of definitions.

Back on the original subject, note that an antisymmetric component of a metric does not affect distance or geodesics.

From contorsion:

$$D_{\mu }V_{\nu }^{\ a}=\partial _{\mu }V_{\nu }^{\ a}+{{\omega _{\mu }}^{a}}_{b}V_{\nu }^{\ b}-\Gamma _{\ \nu \mu }^{\sigma }V_{\sigma }^{\ a}$$

Hmm, is it the case that the spin connection affects tensors of rank 2 or higher and doesn't affect vectors or scalars, like the connection affects vectors and higher ranked tensor but doesn't affect scalars? Or is this second non-greek index something I'm not getting?

Index Ordering

The ordering of indices of symbols and contractions seem to differ amongst the sources. Einstein and Spain both have the Christoffel symbol of the 1st kind as:

$$[ij,k]={\frac {1}{2}}\left(\partial _{j}g_{ik}+\partial _{i}g_{jk}-\partial _{k}g_{ij}\right)$$

and contract with the 2nd index of the contravariant metric to get the Christoffel symbol of the 2nd kind:

$$\Gamma _{ij}^{l}=g^{lk}[ij,k]$$

Contorsion defines:

${\bar {\Gamma }}^{l}{}{}_{ij}={\tfrac {1}{2}}g^{lk}(\partial _{i}g_{kj}+\partial _{j}g_{ik}-\partial _{k}g_{ji})$

And I have the definition I propose, and contraction with the 1st index.

$\Gamma ^{l}{}{}_{ij}={\tfrac {1}{2}}g^{lk}(\partial _{i}g_{kj}+\partial _{j}g_{ik}-\partial _{k}g_{ij})$

A bit of a mess really. I've seen someone quote index order from MTW. Let's check that out...

Oh, and I need to study differential geometry.

Updates

Update: 5/19/24

Consider complex vector fields, where a Hermitian metric may make sense. The symmetry requirement of a metric for real vector spaces apparently becomes a conjugate symmetric requirement.

Update: 5/8/24

ended up getting useful feedback on physics.SE. It seems a contorsion tensor is added back to the symmetric component of the connection, changing it from the Levi-Civita connection to compensate and keep metricity.

Using this space to capture some formatted text:

$${d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0\$$

summation convention Levi-Civita connection Christoffel symbols

${\vec {\Psi }}:\mathbb {R} ^{d}\supset U\to \mathbb {R} ^{n}$

$$g_{ij}=\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{i}}},{\frac {\partial {\vec {\Psi }}}{\partial x^{j}}}\right\rangle .$$