GR vs SR: Reconciling Contravariant & Covariant Vector Components

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SUMMARY

The discussion focuses on the reconciliation of contravariant and covariant vector components in General Relativity (GR) and Special Relativity (SR). In GR, the distinction is based on how vector components transform under coordinate changes, while in SR, it is primarily a notational convenience that aids in deriving the invariant interval. The use of Minkowski coordinates in SR simplifies the distinction, but this is a specific case of flat spacetime. The connection between the two theories is fundamentally linked through the metric tensor.

PREREQUISITES
  • Understanding of General Relativity (GR) principles
  • Familiarity with Special Relativity (SR) concepts
  • Knowledge of vector components and their transformations
  • Basic grasp of the metric tensor and its role in relativity
NEXT STEPS
  • Study the transformation properties of vector components in General Relativity
  • Explore the implications of the metric tensor in both GR and SR
  • Learn about Minkowski coordinates and their application in Special Relativity
  • Investigate coordinate transformations that introduce non-constant metric components
USEFUL FOR

Students and professionals in physics, particularly those studying relativity, as well as mathematicians interested in the applications of tensor calculus in theoretical frameworks.

nigelscott
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I am trying to reconcile the definition of contravariant and covariant
components of a vector between Special Relativity and General Relativity.

In GR I understand the difference is defined by the way that the vector
components transform under a change in coordinate systems.

In SR it seems that it is more of a notational thing that allows for the
derivation of the invariant interval.

How are the 2 things related?
 
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Do you have a source for the different treatment in SR? It can be made to look like it's just a notational thing in SR by using Minkowski coordinates where ##g_{\mu\rho}=g^{\mu\rho}## so many of the differences between contravariant and covariant components disappear - but that's just taking advantage of the fact that flat spacetime is an especially nice special case (which is why we call it "special" relativity).

It's a good exercise to choose some perverse coordinate transform in which the metric acquires off-diagonal and non-constant components even in flat space-time, work a few otherwise-easy problems in those coordinates, just so that you can see the machinery working. You'll find that the contra/co distinction appears even in SR.
 
OK, soon after I posted I realized that the connection is through the metric tensor.
 

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