Covariance, and its counterpart contravariance, has been one of the things I've found a bit hard to understand. So I borrowed this book: Black Hole Physics Frolow and Zelnikov, OUP. It says the covariance principle is part of a 'method' used to upgrade flat spacetimes to those that include gravitational interactions. But it includes the caveat (footnote p. 53): "Since the covariant derivatives do not commute in a curved spacetime, ... their different ordering might give different answers [:the "which happens first" problem]. The resulting difference [in the answers with different orderings] is proportional to the spacetime curvature. In a general case in order to control such terms, describing direct interaction of the matter with the curvature, additional information based on the observations is required.". It also says (p. 68) that in flat spacetimes "the structure of the local nullcones is 'rigid': Each ... local nullcone can be obtained from another one by parallel transport.". Then explains how this rigidity is absent in curved spacetimes. And: " .. in this mathematical introduction we do not restrict ourselves to four dimensions. We assume the number of dimensions is arbitrary.". Oh great. Thinks "maybe time isn't one-dimensional, but has say, at least two abstract directions, like a complex number does . . .?".
Arfa brane, as I'm not sure what the question is in this post, I may not be able to respond in the way that you want. Covariant and contravarient components,which describe how a vector transforms, aren't related to the principle of covariance. However, I will elaborate on their meaning anyway. A vector is, generally, a linear combination of basis vectors e₁, e₂, e₃, etc. If the coefficients are V, then the vector can be written as \(\mathbf{V} = \sum \ V^{n} \mathbf{e_{n}}\) Such a vector, containing components with an upper index, is referred to as a covariant vector. Why? Imagine performing, on the coordinate system, a particle transformation. For the purpose of example, consider a transformation which doubles the distance in between each increment of distance on the three axes. Then, the result is that the vector V will end up twice as long as it was in the beginning, just like the coordinates. Therefore, the vector transforms with the coordinate system, hence the "co-" in "covariant". Now, instead, consider a dual vector space in which the basis consists of covariant vectors. In such a vector space, we can describe the basis vectors of the previous example. If we declare the basis vectors to be ω¹, ω², ω³, etc. then the vector e can be expressed in terms of the basis as \(\mathbf{e} = \sum \ e_{n} \mathbf{\omega^{n}}\) Due to the fact that the components of this vector possess a lower index, the vector is called a contravariant vector. Once again, consider the coordinate transformation from the previous example. If all of the basis vectors from the original vector space are doubled, then they are halved in this dual space. Therefore, the contravariant vectors transform in the opposite way as compared to the coordinates. Hence the prefix "contra". Now, onto general covariance..... Yes. If you assert general covariance, then a (uniformly) accelerated frame of reference must yield the same physics of an observer in not only a uniform gravitational field, but also a local region of a varying gravitational field. One can then proceed, using special relativity, to derive the coordinates of an accelerating observer, the Rindler metric. Geometrically, the Rindler metric is curved. Therefore, spacetime in a gravitational field must be curved. Alternatively, one could take the approach asserting that, by the consideration of thought experiment, light curves in an acclerated. Hence, it would need to do so in a gravitational field. Since light rays always define, in spacetime, the shortest distance in between two points ( a geodesic), spacetime must b Yes. Specifically, "geodesic deviation" is described by the Christoffel connection. In curved spaces, the effect reveals itself more prominently. If two vectors, located on the equator of a sphere, are initially parallel, they will converge when brought to the pole. That is solely due to the curvature of the sphere. Right. As described above, parallel transportation of vectors, in this case the null trajectories of light rays, on curved surfaces results in geodesic deviation. This generality is useful when developed the mathematical principles. When you make the transition to a specific case, simply plug in the number of dimensions. People have toyed with idea, but the results are quickly determined to be absurd.
