Write Torsion Tensor: Definition, Metric Tensor & Equation

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The discussion centers on the relationship between the torsion tensor and the metric tensor in differential geometry. It explores whether non-symmetric Christoffel symbols can be expressed in terms of the metric, similar to symmetric ones. The torsion-free condition is highlighted as essential for defining a metric-compatible connection uniquely. The conversation suggests that without this condition, multiple metrics could exist, including those that are torsion-free. Ultimately, the torsion tensor's dependence on the metric tensor remains a complex issue in the context of differential geometry.
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Would it be possible to write the torsion tensor in terms of the metric? I know that a symmetric Christoffel Symbol can be written in terms of the partial derivatives of the metric. This definition of the christoffel symbols does not apply if they are not symmetric. Is it possible to write a definition of the non-symmetric christoffel symbols in terms of a metric, and from that find an equation for the torsion tensor, or is it independent of the metric tensor?
 
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The torsion free condition is required to uniquely define a metric compatible connection. If you let it go, there are several possible metrics (one of which is the torsion free one).
 

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In an inertial frame of reference (IFR), there are two fixed points, A and B, which share an entangled state $$ \frac{1}{\sqrt{2}}(|0>_A|1>_B+|1>_A|0>_B) $$ At point A, a measurement is made. The state then collapses to $$ |a>_A|b>_B, \{a,b\}=\{0,1\} $$ We assume that A has the state ##|a>_A## and B has ##|b>_B## simultaneously, i.e., when their synchronized clocks both read time T However, in other inertial frames, due to the relativity of simultaneity, the moment when B has ##|b>_B##...
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