# Why is it lowering indices?

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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？

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.

dextercioby, GR191511 and malawi_glenn
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There is a comma indeed.

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

PeroK and GR191511
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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).

vanhees71, GR191511 and malawi_glenn
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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}##.

vanhees71, Dale, GR191511 and 1 other person
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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.

vanhees71, Dale, GR191511 and 1 other person
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I've never really understood why
I never understood the need of the ## \dot y## notation for ## \dfrac{\mathrm{d}y}{\mathrm{d}t} ##

vanhees71
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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.

vanhees71 and malawi_glenn
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').

vanhees71 and malawi_glenn
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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.

Dale and malawi_glenn
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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.

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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. 😁

malawi_glenn and 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 (?)

malawi_glenn and Ibix
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But second derivatives might be fine ... but maybe not in German (?)
##\ddot{\ddot{e}}## 😁

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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.

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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 ;-).

dextercioby
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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 …

vanhees71 and PeterDonis
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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...

vanhees71 and Orodruin
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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 …

vanhees71
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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)​

vanhees71 and Orodruin
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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.

vanhees71