Upper- and lower-index Levi-Civita tensor/symbol

[bcrowell]

MTW p. 87 defines what they refer to as a Levi-Civita tensor with $\epsilon^{\kappa\lambda\mu\nu}=-\epsilon_{\kappa\lambda\mu\nu}$. They define its components to have values of -1, 0, and +1 in some arbitrarily chosen Cartesian frame, in which case it won't have those values under a general change of coordinates, although it will keep them under a Lorentz transformation. The difference in sign between the upper- and lower-indices version is consistent with what you'd expect from ordinary raising and lowering of coordinates.

Wikipedia has an article "Levi-Civita symbol," which defines it as a tensor density with $\epsilon^{\kappa\lambda\mu\nu}=\epsilon_{\kappa\lambda\mu\nu}$. Their definition implies that it has values of -1, 0, and +1 in any coordinate system. Under this definition it doesn't transform like a tensor, which would presumably be why they call it the "symbol."

MTW don't define a Levi-Civita symbol, and WP doesn't have an article on a Levi-Civita tensor.

So all the terminology seems totally self-consistent in both cases, but the same equation would have different transformation properties depending on whose definition of $\epsilon$ you were using.

Is one way of defining $\epsilon$ more standard than the other? Are there big advantages to one over the other?

 

[Ben Niehoff]

I prefer to use the "Levi-Civita symbol" whose values are {-1, 0, 1} in every coordinate system. If I need to make it a tensor, I put in an explicit metric determinant:

$$\sqrt{|g|} \; \epsilon_{\mu\nu\rho\sigma}$$

However, I prefer to write things in terms of differential forms and index-free notation, so the above is simply the volume form

$$\omega = \frac1{4!} \; \epsilon_{abcd} \; \theta^a \wedge \theta^b \wedge \theta^c \wedge \theta^d$$

where the thetas are the orthonormal frame.

Other people prefer the epsilon symbol to be a tensor. I think this is more typical of people who prefer abstract-index notation; this way, every object with indices is a tensor (except the connection).

I don't think there is a standard either way. Most papers will state which convention they are using. Most papers I've seen use the first convention (with {-1, 0, 1} in all coordinates). I think there is an advantage to this, as then one knows exactly how to do contractions with the epsilon symbol independently of any metric.

Some books even attempt to use both conventions, defining

$$\tilde{\epsilon}_{\mu\nu\rho\sigma} = \sqrt{|g|} \; \epsilon_{\mu\nu\rho\sigma}$$

I think this only gets confusing, though.

 

[bcrowell]

Thanks, Ben, that's very helpful!

Another issue that occurs to me is that it's not obvious whether a tensor version could be defined globally. In some small neighborhood, it can be defined by parallel transporting it from the point at which it was originally defined. But if you go to a larger region, the path-dependence of parallel transport makes bigger and bigger ambiguities, and I can also imagine that some spacetimes would lack global orientability.

-Ben

 

[Ben Niehoff]

Another issue that occurs to me is that it's not obvious whether a tensor version could be defined globally. In some small neighborhood, it can be defined by parallel transporting it from the point at which it was originally defined. But if you go to a larger region, the path-dependence of parallel transport makes bigger and bigger ambiguities, and I can also imagine that some spacetimes would lack global orientability.

Remember that the metric is not necessarily defined globally, either. But the volume form is a tensor and is defined throughout the coordinate patch on which the metric is defined. Orientability comes into play when you try to stitch multiple patches together via transition functions.

I don't really see what this has to do with parallel transport, though.