I Why Does Covariant Index Represent Partial Derivative in Special Relativity?

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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?
 
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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.
 
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There is a comma indeed.

1658289810079.png


This notation is introduced in the book (2nd edition) in equation (3.19)
 
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GR191511 said:
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).
 
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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}##.
 
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Ibix said:
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. :wink:
 
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PeterDonis said:
I've never really understood why
I never understood the need of the ## \dot y## notation for ## \dfrac{\mathrm{d}y}{\mathrm{d}t} ## :oldbiggrin::headbang:
 
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drmalawi said:
I never understood the need of the ## \dot y## notation for ## \dfrac{\mathrm{d}y}{\mathrm{d}t} ## :oldbiggrin::headbang:
Perhaps it's meant to induce eyestrain. It certainly does a good job of that for me. :wink:
 
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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').
 
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Ibix said:
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.

drmalawi said:
I never understood the need of the ## \dot y## notation for ## \dfrac{\mathrm{d}y}{\mathrm{d}t} ## :oldbiggrin::headbang:
This, on the other hand, I have no particular issue with for some reason.
 
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PeterDonis said:
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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Orodruin said:
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. 😁
 
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Ibix said:
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 (?)
 
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PAllen said:
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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drmalawi said:
I never understood the need of the ## \dot y## notation for ## \dfrac{\mathrm{d}y}{\mathrm{d}t} ## :oldbiggrin::headbang:
That's the old quarrel between Newton and Leibniz. The intoduction of Leibniz's notation in England by Maxwell was anrevolution ;-).
 
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  • #17
vanhees71 said:
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 …
 
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  • #18
Orodruin said:
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...
 
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  • #19
drmalawi said:
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 …
 
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Orodruin said:
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)​
 
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drmalawi said:
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.
 
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  • #22
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.
 
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Where's the :groan: emoji?
 
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