Why Does Covariant Index Represent Partial Derivative in Special Relativity?

[GR191511]

I‘m reading the chapter 4 《Perfect fluids in special relativity》of《A First Course in General Relativity》.In the process of deriving conservation of energy-momentum,it said:$\frac {\partial T^0{^0}} {\partial t}=-\frac {\partial T^0{^x}}{\partial x}-\frac {\partial T^0{^y}}{\partial y}-\frac {\partial T^0{^z}}{\partial z}$$\;$then it writes:$T^0{^0}{_0}+T^0{^x}{_x}+T^0{^y}{_y}+T^0{^z}{_z}=0$...I wonder why the partial derivative is represented by covariant index？

 

[PAllen]

Are you sure it doesn't have a comma before each lower index? That comma indicates partial derivative, making it different than a tensor index. In this older notation, a semicolon would indicate a covariant derivative.

 

[malawi_glenn]

There is a comma indeed.

This notation is introduced in the book (2nd edition) in equation (3.19)

 

[PeterDonis]

I‘m reading the chapter 4 《Perfect fluids in special relativity》of《A First Course in General Relativity》.In the process of deriving conservation of energy-momentum,it said:$\frac {\partial T^0{^0}} {\partial t}=-\frac {\partial T^0{^x}}{\partial x}-\frac {\partial T^0{^y}}{\partial y}-\frac {\partial T^0{^z}}{\partial z}$$\;$then it writes:$T^0{^0}{_0}+T^0{^x}{_x}+T^0{^y}{_y}+T^0{^z}{_z}=0$...I wonder why the partial derivative is represented by covariant index？

Because the derivative operator acts like a covector, i.e., a thingie with a lower index. For example, you can contract the derivative operator with a vector to get a number (this number is usually called the "divergence" of the vector).

 

[Ibix]

Incidentally, Carroll's lecture notes state this notation and then go on to say how easy it is to make exactly the mistake OP did, especially in handwriting. That's why he largely uses $\partial_0T^{00}$ in preference to $T^{00}{}_{,0}$.

 

[PeterDonis]

Incidentally, Carroll's lecture notes state this notation and then go on to say how easy it is to make exactly the mistake OP did, especially in handwriting. That's why he largely uses $\partial_0T^{00}$ in preference to $T^{00}{}_{,0}$.

Yes, I've always preferred that as well, even though my favorite GR textbook, MTW, insists on using commas and semicolons instead of partials and nablas. I've never really understood why: it's not as though partials and nablas are scarce resources.

 

[malawi_glenn]

I've never really understood why

I never understood the need of the $\dot y$ notation for $\dfrac{\mathrm{d}y}{\mathrm{d}t}$

 

[PeterDonis]

I never understood the need of the $\dot y$ notation for $\dfrac{\mathrm{d}y}{\mathrm{d}t}$

Perhaps it's meant to induce eyestrain. It certainly does a good job of that for me.

 

[PAllen]

Talk about extreme concern about ink, I'm am always annoyed by one of my favorite old references - Synge's GR book - he (with warning early in the book) routinely just uses lower indexes for either partials or covariant derivatives wherever he thinks "context" should make it clear; even in cases where context might be 5 pages earlier! (I've have never seen any other author follow this 'convention').

 

[Orodruin]

Incidentally, Carroll's lecture notes state this notation and then go on to say how easy it is to make exactly the mistake OP did, especially in handwriting. That's why he largely uses $\partial_0T^{00}$ in preference to $T^{00}{}_{,0}$.

I’ll usually mention that the comma/semicolon notation exists and then happily go on using partials and nablas. It is just clearer to me.

I never understood the need of the $\dot y$ notation for $\dfrac{\mathrm{d}y}{\mathrm{d}t}$

This, on the other hand, I have no particular issue with for some reason.

 

[Ibix]

I've never really understood why

Compactness. Also, I think there's a degree of sense to it because "the partial derivative of $T$" is one "thing", so I understand the desire to notate it as one "thing" without having to introduce some arbitrary new letter for it. It's kind of analogous to the $\dot x$ notation, in fact, which I do use.

But I personally find the commas too easy to lose among the other indices. I wonder if this particular notational preference is well correlated with the strength of the physicist's astigmatism.

Edit: cross-posted with several others, I see.

 

[Ibix]

This, on the other hand, I have no particular issue with for some reason.

Because there's nothing else above the letter for the dot to become visually lost in, I suspect. I'd avoid $\dot i$ and $\dot j$, though.

 

[PAllen]

Because there's nothing else above the letter for the dot to become visually lost in, I suspect. I'd avoid $\dot i$ and $\dot j$, though.

But second derivatives might be fine ... but maybe not in German (?)

 

[Ibix]

But second derivatives might be fine ... but maybe not in German (?)

$\ddot{\ddot{e}}$

 

[Orodruin]

I’ll say this though. In PDE literature it is quite common to use subscripts to denote partial derivatives (I also do it). For example, the wave equation for $u(x,t)$ would be
$$
u_{tt} -c^2 u_{xx}=0.
$$
It is quite convenient and fine as long as you only ever deal with scalars. The problems start to arise when you want to combine that with tensors in index notation …

Context is important.

 

[vanhees71]

I never understood the need of the $\dot y$ notation for $\dfrac{\mathrm{d}y}{\mathrm{d}t}$

That's the old quarrel between Newton and Leibniz. The intoduction of Leibniz's notation in England by Maxwell was anrevolution ;-).

 

[Orodruin]

That's the old quarrel between Newton and Leibniz. The intoduction of Leibniz's notation in England by Maxwell was anrevolution ;-).

It is said that they both worked independently, but I find both their works a bit … derivative …

 

[malawi_glenn]

It is said that they both worked independently, but I find both their works a bit … derivative …

And now it has been integrated into our standard math curriculum. If only the people responsible would know their limits...

 

[Orodruin]

And now it has been integrated into our standard math curriculum. If only the people responsible would know their limits...

I don’t know … I may be partial …

 

[malawi_glenn]

I don’t know … I may be partial …

maybe, but I need to get to the kernel with this

(that was lame, even for my standards)​

 

[Orodruin]

maybe, but I need to get to the kernel with this

(that was lame, even for my standards)​

It seems we reach these types of conversations at discrete intervals. I guess that makes a difference.

 

[PAllen]

On a more serious note, I looked through a French calculus textbook from circa 1725 once - I read no French. I could follow it easily, the notation and even order of presentation of topics was already similar to texts of my era. I wonder whether this is a good thing.

 

[PeterDonis]

Where's the :groan: emoji?