What does the Riemann tensor represent?
The Riemann curvature tensor is a tool used to describe the curvature of n-dimensional spaces such as Riemannian manifolds in the field of differential geometry. The Riemann tensor plays an important role in the theories of general relativity and gravity as well as the curvature of spacetime.
Why is curvature a tensor?
For each pair of tangent vectors u, v, R(u, v) is a linear transformation of the tangent space of the manifold. It is linear in u and v, and so defines a tensor. Thus in the general case of non-coordinate vectors u and v, the curvature tensor measures the noncommutativity of the second covariant derivative.
How did Einstein find field equations?
Einstein made two heuristic and physically insightful steps. The first was to obtain the field equations in vacuum in a rather geometric fashion. The second step was obtaining the field equations in the presence of matter from the field equations in vacuum.
How many independent components does the Riemann tensor have?
20 independent components
In four dimensions, therefore, the Riemann tensor has 20 independent components.
How do you find scalar curvature?
The sectional curvature of an n-sphere of radius r is K = 1/r2. Hence the scalar curvature is S = n(n − 1)/r2. The parameter r is a geometrical invariant of the hyperbolic space, and the sectional curvature is K = −1/r2. The scalar curvature is thus S = −n(n − 1)/r2.
Is the Riemann tensor symmetric?
The symmetries of the Riemann tensor mean that only some of its 256 components are actually independant.
Is Riemann tensor symmetric?
Why Einstein equation is nonlinear?
The nonlinearity of the Einstein field equations stems from the fact that masses affect the very geometry of the space in which they dwell. And this is the fundamental insight of (1): mass curves the geometry of spacetime, and the geometry of spacetime in turn tells masses how to move.
Why Christoffel symbol is not a tensor?
It is important to note, however, the Christoffel symbol is not a tensor. Its elements do not transform like the elements of a tensor.
How many independent components does the Riemann curvature tensor have in 2 dimensions?
1 independent component
In dimension n = 2, the Riemann tensor has 1 independent component. It is therefore entirely determined by the Ricci scalar, or scalar curvature: Rαβγδ = R gα[γgδ]β. In dimension n = 3, the Riemann tensor has 6 independent components, just as many as the symmetric Ricci tensor.