Understanding the Ricci and Riemann Curvature Tensors in Tensor Calculus

[Jack3145]

The Ricci Tensor comes from the Riemann Curvature Tensor:

$$R^{\beta}_{\nu\rho\sigma} = \Gamma^{\beta}_{\nu\sigma,\rho} - \Gamma^{\beta}_{\nu\rho,\sigma} + \Gamma^{\alpha}_{\nu\sigma}\Gamma^{\beta}_{\alpha\ rho} - \Gamma^{\alpha}_{\nu\rho}\Gamma^{\beta}_{\alpha\sigma}$$

The Ricci Tensor just contracts one of the indices:

$$R_{\nu\rho} = R^{\beta}_{\nu\rho\beta}$$

What is the function of the Ricci Tensor and the Riemann Curvature Tensor? How does the contraction of indices change the effect?

 

[tiny-tim]

Welcome to PF!

What is the function of the Ricci Tensor and the Riemann Curvature Tensor? How does the contraction of indices change the effect?

Hi Jack3145! Welcome to PF!

The Riemann Curvature Tensor is the curvature … it tells us everything!

Contraction of indices loses information … but it's information we don't need for for Einstein's field equations, for example.

The Ricci tensor is the "trace" part of the Riemann Tensor … it has 10 independent components, out of the Riemann Tensor's 20 (the other 10 are in the Weyl tensor, the "tracefree" part of the Riemann Tensor).

The Ricci tensor is all we need to know for Einstein's field equations …

$$R_{\mu\nu}\ -\ \frac{1}{2}\,R\,g_{\mu\nu}\ =\ 8\pi\,T_{\mu\nu}$$ or $$T_{\mu\nu}-\ \frac{1}{2}\,T\,g_{\mu\nu}\ =\ \frac{1}{8\pi}\,R_{\mu\nu}$$​

The Ricci tensor, loosely speaking, describes the relative volumes of elements of space, while the Weyl tensor describes their relative shear and twisting.

 

[Naty1]

I previously saved the following since it gave a nice insight:

"The Ricci tensor-- or equivalently, the Einstein tensor-- represents that part of the gravitational field which is due to the immediate presence of nongravitational energy and momentum.

The Weyl tensor represents the part of the gravitational field which can propagate as a gravitational wave through a region containing no matter or nongravitational fields.

Regions of spacetime in which the Weyl tensor vanishes contain no gravitational radiation and are also conformally flat, which implies for example that light rays passing through such a region exhibit no light bending."

What I haven't found yet is how these were originally formulated... how did they figure out the components in the first place?

 

[Mentz114]

What I haven't found yet is how these were originally formulated... how did they figure out the components in the first place?

If I remember correctly, the Riemann tensor emerges when you do a double covariant derivative of the separation between two worldlines. The relative acceleration of the two points depends on the Riemann tensor only.

 

[Naty1]

Mentz, while I don't understand the full implications of your post it makes a lot of sense since world lines provide a measure of space curvature...some start out parallel and stay that way, others converge,others diverge...so derivatives would seem to measure that rate of change/ degree of curvature...Thank you!

 

[tiny-tim]

geodesic deviation equation

If I remember correctly, the Riemann tensor emerges when you do a double covariant derivative of the separation between two worldlines. The relative acceleration of the two points depends on the Riemann tensor only.

Yes, the LHS of the geodesic deviation equation is the double covariant derivative of the separation between two initially parallel (ie with the same 4-velocity) worldlines, and the RHS is the Riemann tensor acting on the 4-velocity and the initial separation:

$$\frac{D^2\,\delta x^{\alpha}}{D\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}$$

for 4-velocity $V$ and separation $\delta x$

where $\tau$ is proper time and $R$ is the Riemann curvature tensor