Does Metric Signature Affect Torsion Definition?

Thread starter binbagsss
Start date Apr 17, 2015
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Definition Metric Torsion

In summary, the metric signature can affect the definition of torsion in various ways, including altering equations, impacting geometric interpretation, and influencing physical implications. There is a difference between Riemannian and pseudo-Riemannian torsion, with the latter being defined on a manifold with a non-positive definite metric. Torsion can be defined in non-Euclidean spaces and has important applications in fields such as general relativity and the study of gravity. Understanding the effects of metric signature on torsion can provide insights into the fundamental properties of space and time.

Apr 17, 2015

binbagsss

I'm looking at 2 sources.

One has it defined as ##T^{c}_{ab}=-\Gamma^{c}_{ab}+\Gamma^{c}_{ba}##
And the other has ##T^{c}_{ab}=\Gamma^{c}_{ab}-\Gamma^{c}_{ba}##

##T## the torsion tensor
and
##\Gamma^{c}_{ab}## the connection.

Or is it more that different texts use different conventions?

Thanks.

Last edited: Apr 17, 2015

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Apr 18, 2015

robphy

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What are those sources?
What are their definitions of Riemann and Ricci?

1. How does the metric signature affect the definition of torsion?

The metric signature can affect the definition of torsion in a few different ways. First, it can alter the mathematical equations used to calculate torsion, as different metrics will have different properties. Additionally, the metric signature can impact the geometric interpretation of torsion, as it affects the curvature of space. Lastly, the metric signature can influence the physical implications of torsion, such as its role in spacetime curvature and gravitational effects.

2. What is the difference between Riemannian and pseudo-Riemannian torsion?

Riemannian torsion is defined on a Riemannian manifold, which has a metric that is positive definite. Pseudo-Riemannian torsion, on the other hand, is defined on a pseudo-Riemannian manifold, which has a metric that is not necessarily positive definite. This difference in metrics can result in different equations and interpretations of torsion.

3. Can torsion be defined in non-Euclidean spaces?

Yes, torsion can be defined in non-Euclidean spaces, including Riemannian and pseudo-Riemannian manifolds. In fact, torsion is an important concept in non-Euclidean geometries, as it describes the curvature and deformation of space in these spaces.

4. How does the metric signature affect the physical interpretation of torsion in general relativity?

In general relativity, torsion is related to the curvature of spacetime and is often interpreted as the non-gravitational component of curvature. The metric signature can affect this interpretation by altering the equations used to calculate torsion and its relationship to curvature. It can also impact the physical consequences of torsion, such as its role in gravitational effects.

5. Are there any real-world applications of studying the effects of metric signature on torsion?

Yes, there are several real-world applications of studying the effects of metric signature on torsion. For example, understanding torsion in non-Euclidean spaces is crucial for accurately describing the behavior of light in curved space, as well as in developing models of gravity that go beyond general relativity. Additionally, studying torsion in different metrics can help us better understand the fundamental properties of space and time.