GR vs SR: Reconciling Contravariant & Covariant Vector Components

In summary, the difference between contravariant and covariant components of a vector in General Relativity is defined by the way they transform under a change in coordinate systems, while in Special Relativity it is more of a notational difference that allows for the derivation of the invariant interval. However, this notational difference is related to the use of Minkowski coordinates in SR, where the metric tensor has special properties. Even in flat spacetime, the distinction between contravariant and covariant components is still present.
  • #1
nigelscott
135
4
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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  • #2
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.
 
  • #3
OK, soon after I posted I realized that the connection is through the metric tensor.
 

1. What are contravariant and covariant vector components?

Contravariant and covariant vector components are two different ways of representing the same vector. Contravariant components are defined with respect to the basis vectors of the coordinate system, while covariant components are defined with respect to the dual basis vectors. In general, contravariant components transform in the opposite way to the coordinate system basis vectors, while covariant components transform in the same way as the dual basis vectors.

2. How do general relativity and special relativity reconcile contravariant and covariant vector components?

In special relativity, the laws of physics are the same in all inertial reference frames, so the use of contravariant and covariant components is interchangeable. However, in general relativity, the laws of physics are dependent on the gravitational field, and the use of contravariant and covariant components allows for a more precise description of the effects of gravity on objects.

3. What is the significance of using contravariant and covariant vector components in general relativity?

Using contravariant and covariant components allows for the formulation of the equations of general relativity in a covariant form, meaning that they are independent of the choice of coordinates. This is important as it allows for a consistent description of the effects of gravity on objects, regardless of the reference frame or coordinate system used.

4. How do contravariant and covariant vector components affect the metric tensor?

The metric tensor in general relativity is defined in terms of both contravariant and covariant components. The contravariant components determine how the metric tensor transforms under a change of coordinates, while the covariant components determine the curvature of spacetime. Together, they provide a complete description of the geometry of spacetime.

5. Can contravariant and covariant vector components be used in other areas of physics?

Yes, the use of contravariant and covariant components is not specific to general relativity and can be applied in other areas of physics. They are commonly used in differential geometry and other fields of mathematics, and have applications in fields such as electromagnetism and quantum mechanics.

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