Derivation notation with capital D?

[Emil_M]

Hi, I came across a derivation notation I didn't recognize:
Let ##s## be some four-vector and ##\tau## the proper time. What is the significance of
$$\frac{Ds}{\mathrm{d}\tau}?$$

I know ##Ds## can be used to mean the Jacobian, but I've never come across the notation above. Does someone recognize it?

 

[Orodruin]

Where did you come across this notation? It should be explained there what is meant by the notation. If we cannot see the text we are just stabbing in the dark.

 

[Emil_M]

I came across this notation in one of my General Relativity scripts, but I checked the entire text before posting and this notation is not introduced in the script. I guess the author believes the notation is commonplace enough not to need an introduction. The only time ##D_X## was used in the context of derivations was in a prove for the linearity of covariant derivatives as an alternative symbol for ##\nabla_x##.

However, ##D## really doesn't make sense in the context of a covariant derivative here, as I wouldn't know what the operator ##\frac{D}{\mathrm{d}\tau}## means?

Specifically, the above notations appears in a chapter about Gyroscopic Precession:

 

[Emil_M]

Ah ok, the notation is introduced three pages further down... I guess this is a just a formatting error of the author

Thanks for the help, though!

 

[martinbn]

It is not a formatting error. It is a somewhat standard notation for the induced connection.

 

[Orodruin]

It is not a formatting error. It is a somewhat standard notation for the induced connection.

Just to be a bit more specific; the derivative along a curve ##\gamma## with respect to the curve parameter, i.e.,
$$
\frac{Ds}{d\tau} = \nabla_{\dot\gamma} s = \dot x^\mu \nabla_\mu s,
$$
for the induced Levi-Civita connection.