b {\displaystyle X} (a number called the rank of the tensor). If I found it here, and if an alien measured it, and we compared our answers, they would be scalar multiples of each other (choice of Parisian metre stick for me, choice of Imperial foot for the alien, or, vice versa..). given a metric, the connection is determined by the metric. Notions of parallel transport can then be defined similarly as for the case of vector fields. general relativity but relies on the somewhat arbitrary choice of a time coordinate. I would rather say. A However, in general relativity, it is found that derivatives which are also tensors must be used. So, it isn't a condition, it is a consequence of covariance derivative and metric tensor definition. Consider the analogy with Newtonian gravity. = ) G &= \partial_\rho \left( \frac{\partial \xi^i}{\partial x^\mu}\frac{\partial \xi^i}{\partial x^\nu} \right) - g_{\mu \sigma} \frac{\partial x^\sigma}{\partial \xi^i} \frac{\partial^2 \xi^i}{\partial x^\nu \partial x^\rho} - g_{\sigma \nu} \frac{\partial x^\sigma}{\partial \xi^i} \frac{\partial^2 \xi^i}{\partial x^\mu \partial x^\rho} \\ turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative. B If the tangent space is n-dimensional, it can be shown that In this world, there is only one metric tensor (up to scalar) and it can pretty much be measured. ... A. Einstein, The Foundation of the General Theory of Relativity, in H. A. Lorentzet al., eds.,The Principle of Relativity (Dover, New York, 1952). The EFE relate the total matter (energy) distribution to the curvature of spacetime. a https://physics.stackexchange.com/questions/47919/why-is-the-covariant-derivative-of-the-metric-tensor-zero/62394#62394, https://physics.stackexchange.com/questions/47919/why-is-the-covariant-derivative-of-the-metric-tensor-zero/411664#411664. The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equations). a Mathematically, tensors are generalised linear operators - multilinear maps. General Relativity, at its core, is a mathematical model that describes the relationship between events in space-time; the basic finding of the theory is that the relationship between events in the same as the relationship between points on a manifold with curvature, and the geometry of that manifold is determined by sources of energy-momentum and their distribution in space-time. CITE THIS AS: Weisstein, Eric W. "Covariant Derivative." The exact nonzero value of the covariant divergence of the Ricci tensor (in spacetimes where it … d General Theory of Relativity is a great theory, conﬁrmed by all existing data (see, ... Covariant derivatives allow to formulate invariant under general transforma-tions of coordinates basic equations of the General Theory of Relativity. Actually the above calculation is also valid if you consider a higher dimensional flat space $i,j=1,...,N$ where the variety is embedded $\mu,\nu,\rho,\sigma=1,...,M$ with $M

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