Gravitation page 325 section 13.5, From (1) the Riemann curvature tensor $$R^\alpha{}_{\beta\gamma\delta},$$ one could construct (2) the double dual of Riemann $$G^{\alpha\beta}{}_{\gamma\delta}\equiv \frac{1}{2} \epsilon^{\alpha\beta\mu\nu} R_{\mu\nu}{}^{\rho\sigma} \frac{1}{2} \epsilon_{\rho\sigma\gamma\delta},$$ (3) the Einstein curvature tensor $$G^\beta {}_\delta\equiv G^{\mu\beta}{}_{\mu\delta},$$ (4) the Ricci curvature tensor $$R^\beta{}_{\delta} \equiv R^{\mu\beta} {}_{\mu\delta},$$ (5) the Weyl conformal tensor, (6) the "Bianchi identity", and (7) the "contracted Bianchi identity".

However, it's kind of confusing about their usage. Could you write a little wiki-style introduction to those concepts, where did they appear and what they meant?

  • 2
    $\begingroup$ Momentum, charge, superconductivity, gluons, please explain. $\endgroup$ – my2cts May 26 at 19:03
  • 1
    $\begingroup$ I suggest looking at Frankel's The Geometry of Physics, sections 11.4 and particularly 11.5. Some of these tensors are related to sectional curvatures. $\endgroup$ – Javier May 26 at 19:17
  • 1
    $\begingroup$ This is too broad. Can you please make the question more focused? $\endgroup$ – G. Smith May 26 at 19:26
  • $\begingroup$ @G.Smith perhaps just a little bit of what they meant conceptually. I think I can read Bianchi identity later in section 15, but all those different curvature tensors came out at once and it's been a bit messy. $\endgroup$ – ShoutOutAndCalculate May 26 at 19:35

Most of this stuff comes from Einstein's book: 'The Meaning of Relativity'.

Riemann Curvature Tensor:

Imagine you take a curve, and you place a vector on one end of it. It doesn't matter which direction the vector is pointing. Now, say you displace that vector, to another point very close to it. Let's give them some names. The initial position of the vector is A and the final position is B. The vector itself is, say, $V^\mu$. lets say the difference between the coordinates of A and B is $\zeta$.

Now, this process of displacing a vector is known as parallel displacement. When $V^\mu$ reaches B, it has a change in it's direction ($\Delta V^\mu$). Measuring this change of direction gives us the curvature of the curve. It turns out to be $$\Delta V^\mu = -\frac{1}{2} R^{\mu}_{\sigma \alpha \beta} V^\sigma f^{\alpha \beta}$$ where $f^{\alpha \beta}$ depends on $\zeta$. Now, the bigger tensor above, is the Riemann Curvature Tensor. This tensor completely describes the curvature of the curve.

General Relativity is a theory which describes the curvature of spacetime, so the Riemann tensor has to be necessary in the mathematics of the theory. The problem is: $$R^{\mu}_{\sigma \alpha \beta} = -\frac{\partial \Gamma ^{\mu}_{\sigma \alpha}}{\partial x_{\beta}} + \frac{\partial \Gamma ^{\mu}_{\sigma \beta}}{\partial x_{\alpha}} + \Gamma^{\mu}_{\rho \alpha}\Gamma^{\rho}_{\sigma \beta} - \Gamma^{\mu}_{\rho\beta} \Gamma^{\beta}_{\sigma\alpha}$$

Yes. It's just too tough to calculate. Each of the $\Gamma$s have yet more calculating to do. So it's a lot of work, with calculating each term takes you deeper down the rabbit hole; and so you could rather use:

Ricci Tensor:

This tensor describes the curvature of the curve, but is easier to handle than the Riemann tensor. So, how do you get it? First, let's talk about the Einstein Summation Convention. Basically, it is a shorthand for writing 'sum over indices'. If you have say the expression $$ds^2 = \sum_{\mu, \nu} g_{\mu \nu} dx^{\mu} dx^{\nu}$$ you can simplify and write $$ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}$$ which means that if a particular index occurs on both the top and the bottom of an expression, you 'sum over it'. So, the Ricci tensor is $$R_{\mu \nu} = R^{\rho}_{\mu \rho \beta}$$ You sum over $\rho$ to get a more compact tensor. This is a process known as contraction Now, the Ricci tensor is the one actually used in the:

Einstein Field Equations:

Which are $$R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 8\pi G T_{\mu \nu}$$ Here $R = g_{\mu \nu} R^{\mu \nu}$ (where the Ricci tensor is in the contra-variant form), and I have taken $c = 1$. Now, this is the central set of equations in General Relativity. The left hand side completely describes the curvature of spacetime, caused by the energy described in the right hand side. Or, as said by the oft-quoted line

Spacetime tells matter how to move; matter tells spacetime how to curve - John Wheeler

But the left hand side can be simplified. If we introduce the:

Einstein Tensor:

Defined as: $$G_{\mu \nu} := R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu}$$ Then the field equations become: $$G_{\mu \nu} = 8\pi G T_{\mu \nu}$$ that is another way of obtaining the Einstein Tensor $G_{\mu \nu}$.

In a nutshell:

The three tensors that you specified describe the curvature of space. The Riemann Tensor describes intrinsic curvature completely, while the Ricci Tensor is just a contracted form of the Riemann tensor. The Einstein tensor, is dependent on the Ricci Tensor (and it's trace) and captures the left hand side of the Einstein Field Equations.

Of course, I haven't described the other three things in the question, but I have left a few links in the further info.

Further info:








And a rather good series on tensor calculus for beginners: https://www.youtube.com/playlist?list=PLSuQRd4LfSUTmb_7IK7kAzxJtU2tpmEd3

| cite | improve this answer | |

Not the answer you're looking for? Browse other questions tagged or ask your own question.