Is there a connection between covariance, invariance, and dark matter?

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    Covariance Invariance
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The discussion centers on the relationship between covariance, invariance, and their implications in the context of physical laws and dark matter. Covariance is defined as the transformation behavior of quantities under coordinate changes, while invariance indicates that a quantity remains unchanged under such transformations. The Christoffel symbols are highlighted as covariant but not invariant, particularly in non-inertial frames. The assertion is made that the covariance of physical laws is linked to the invariance of tensor equations, despite the differential equations not being invariant under coordinate transformations, which may complicate the understanding of phenomena such as dark matter.

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Covariance and Invariance

We consider the equation:

{\frac {{d}^{2} {x^{\alpha}}}{{d }{{\tau}^{2}}}}{=}{-}{{\Gamma}^{\alpha}}_{\beta\gamma}{\frac{{d}{x^{\beta}}}{{d}{\tau}}}{\frac{{d}{x^{\gamma}}}{{d}{\tau}}}

The covariant form is preserved in all coordinate systems. But the Christoffel symbols may work out to produce different functions in different coordinate systems.So covariance does not guarantee invariance,except in the inertial frames.[The Christoffel symbols work out to zero value in the inertial frames]

Query: Does covariance imply invariance?
 
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Covariance implies invariance of all scalar quantities. Covariance does not imply invariance of the components of a tensor of higher rank in an arbitrary basis.
 
Covariant in this context means "transforms in a particular way (under a coordinate transformation)". If something is invariant it means "does not transform (under a coordinate transformation)". Invariance is a special type of covariance.

In your example you are considering the components of a vector equation. These would be covariant. The components of a tensor are all covariant -- they transform in a "particular way" when you a perform a coordinate transformation. The components of the Christoffel symbols are not covariant.

But the literature can be sloppy when it comes to this. So don't be suprised when something is called invariant when it really should be covariant.
 
Christoffel symbols are perfectly covariant, though its covariance is of second order and not of first order as it is with vectors and tensors. Building invariants from first order objects is easy. Building invariants from second order objects is less trivial. But it is all just algebra, nothing more.
 
 
Would it be appropriate to make the following assertion?

Covariance of the physical laws relates to the invariance of the corresponding tensor equations, notwithstanding the fact that the differential equations coming out of such formulation are not invariant wrt to coordinate transformation.

I am making this assertion in view of the preceding threads as well as the following links:

1) https://www.physicsforums.com/showpost.php?p=3003113&postcount=50
2)https://www.physicsforums.com/showpost.php?p=2999911&postcount=48
 
Maxwell's Equations,I mean the differential equations, may change their form on coordinate transformation[in view of the previous thread].In such a case it would be very difficult,if not impossible to churn out the wave equation as we know it.This may apply to some erratic matter distribution in the galaxies where the metric coefficients have complicated functional forms. Can this be traced to the cause behind dark matter?

[This is a speculation and not a claim]
 

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