Undergrad Dirac comment on covariant derivatives

[Kostik]

TL;DR

Dirac says "Even if one is working with flat space ... one must write one's equations in terms of covariant derivatives if one wants them to hold in all systems of coordinates." But in flat space, covariant derivatives = ordinary derivatives. What does Dirac mean?

Dirac in "General Theory of Relativity" (top of p. 20) says "Even if one is working with flat space ... and one is using curvilinear coordinates, one must write one's equations in terms of covariant derivatives if one wants them to hold in all systems of coordinates."

This comment follows his conversion of the d'Alembert equation $$\Box V = \eta^{\mu\nu}V_{\mu\nu}=0$$ to covariant form $$g^{\mu\nu} V_{;\mu;\nu} = g^{\mu\nu} \left( V_{,\mu\nu}-\Gamma^\alpha_{\mu\nu}V_{,\alpha} \right) =0.$$ In flat space, all the $\Gamma^\alpha_{\mu\nu}=0$, so covariant derivatives are the same as ordinary (partial) derivatives. For instance, if we work in the Euclidean plane but with polar (curvilinear) coordinates, the metric will change accordingly, but still the $\Gamma^\alpha_{\mu\nu}=0$.

What exactly is Dirac trying to say here?

 

[Ibix]

The Christoffel symbols are only zero in Cartesian coordinates on flat spacetime. In other coordinate systems (e.g. polars on flat spacetime) they are not zero, so you need the full covariant form of derivatives.

 

[Kostik]

Right! Stupid question. Thank you. (Edit: actually the Christoffel symbols vanish in any system of rectilinear coordinates, not just cartesian...?)

 

[DrGreg]

The Christoffel symbols are only zero in Cartesian coordinates on flat spacetime. In other coordinate systems (e.g. polars on flat spacetime) they are not zero, so you need the full covariant form of derivatives.

And, of course, it's not just the spatial coordinates that you need to consider, it also applies to non-inertial coordinates such as Rindler coordinates (in which an accelerating rocket is at rest) or Born coordinates (in which a rotating disk is at rest). The time-related Christoffel symbols can be interpreted as representing "fictitious forces" such as "g-force" or "centrifugal force" respectively.

 

[Ibix]

actually the Christoffel symbols vanish in any system of rectilinear coordinates, not just cartesian...?

They definitely don't need to be normalised the same in all dimensions, true, since $\partial_ag_{bc}=0$. I'm not certain if non-orthogonal lines give zero Christoffel symbols without actually doing the maths.

 

[Kostik]

In rectilinear coordinates the metric entries $g_{\mu\nu}$ are constants (but with off-diagonal terms), all derivatives are zero, so the Christoffel symbols vanish.

After I read your post I considered what $g$ and $g^{-1}$ look like in simple plane polar coordinates, and realized my original post was wrong.

 

[Ibix]

In rectilinear coordinates the metric entries are constants (but with off-diagonal terms)

Ah yes - then agreed.

 

[Orodruin]

Right! Stupid question. Thank you. (Edit: actually the Christoffel symbols vanish in any system of rectilinear coordinates, not just cartesian...?)

In any affine coordinate system.