1
$\begingroup$

One site I came across says Einstein's 10 Field Equations use 20 variables, while another said 40.

There are four variables in spacetime - three for space and one for time, right?

But there are two index symbols for each tensor, usually denoted by the Greek letters mu and nu. I don't know if the rank of a tensor is related to the number of variables in relativity.

$\endgroup$
4
  • 1
    $\begingroup$ No, the rank of a tensor is the number of indices. The indices are then summed over all four variables. The metric tensor is a rank two tensor based on a four-variable spacetime, hence it can be represented by a 4x4 matrix. The full Riemann-tensor is a rank 4 tensor (4^4=64 entries), Christoffel symbols are rank 3 etc. The field equations are therefore 16 equations for (naively) 64 variables. One then needs symmetries and other tools to reduce and solve this system of equations. $\endgroup$ Dec 18, 2021 at 1:39
  • 4
    $\begingroup$ @AtmosphericPrisonEscape You seem to be answering in a comment, something strongly discouraged on SE generally. $\endgroup$ Dec 18, 2021 at 6:31
  • $\begingroup$ @StephenG Thanks for morally highgrounding on the internet, I just wasn't sure about the numbers I quoted, as my GR is quite a while ago. So my rationale was to wait for a "Nah, but.." before answering. $\endgroup$ Dec 18, 2021 at 11:43
  • 1
    $\begingroup$ @AtmosphericPrisonEscape It's not "moral highgrounding", it's simply drawing it to your attention. Given that votes up on answers earn more reputation than comments you're really being reminded that you can gain more reputation and give the community an opportunity to vote on your response. Comments on SE can (and are) be deleted (outside of human control as I understand it) and so an answer in comments should not be considered persistent. That's my understanding of how it works. $\endgroup$ Dec 18, 2021 at 14:40

1 Answer 1

5
$\begingroup$

A single vector equation like $\vec{F} = m \frac{d^2 \vec{r}}{dt^2}$ is actually three equations one for each of the $x,y,z$ components of the 3-vector $\vec{r}$.

In the spacetime of general relativity there are four directions to worry about, e.g., $t,x,y,z$. A vector is a rank-1 tensor and has four components. So a vector equation in GR would really be four equations. In common notation the components of the spacetime position 4-vector are $x^\mu$, where $\mu$ is an index designating which of the four components you are talking about.

In general relativity the gravitational field is represented by the metric tensor, which is a rank-2 tensor. In component notation it is written $g_{\mu\nu}$, where each index, $\mu$ and $\nu$, can take one of four values. This means it has 16 components. An equation, like the Einstein field equation, is actually 16 equations, one for each component of the metric.

The metric is a symmetric tensor, meaning $g_{\mu\nu}=g_{\nu\mu}$. This means some of the 16 equations are duplicates. Einstein's field equations are 10 independent equations that fully determine all 16 components of $g_{\mu\nu}$.

In general a rank-$N$ tensor has $d^N$ components, where $d$ is the dimensionality of the space, so $4$ for spacetime.

The Riemann curvature tensor is a rank-4 tensor, $R_{\alpha\beta\mu\nu}$, with $4^4 = 256$ components. Like the metric tensor, it has some internal symmetries (and anti-symmetries) so not all components are independent. It turns out there are only 20 independent components of the Riemann curvature tensor.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .