Why are there two delta x^u/v in equation (1.16) on page 9 of Carroll's GR book?

[OS Richert]

I am attempting to learn GR (for fun) using Sean Carroll's text. I have made it ALL the way to . . . . page 9

I don't understand the equation on the bottom of page 9; (1.16) (namely, why there are TWO delta x ^u and delta x^y; but to type out my question I will need to know how you guys format in Physics notation. Can someone give me a link to that thread and then I will return here and type my question more appriopatly.

(If someone thinks they can understand my question anyway and wants to answer it; it is this. W have a 4x4 matrix (the n with the uv lower idicie) mulitplied by a 4x1 matrix delta x^u which itself is a single scalar which is exactly the spacetime interval (equation 1.10 on page 7). BUT, there are two delta x ^ u/v. What's up with that?)

 

[pervect]

Do you mean eq 1.9 of http://xxx.lanl.gov/PS_cache/gr-qc/pdf/9712/9712019.pdf
?

Distance is a quadratic form. Try the wikipedia on the Einstetin summation convetion, hopefully that will help

http://en.wikipedia.org/wiki/Einstein_notation

 

[OS Richert]

yes, 1.9 of those online lecture notes. It loooks like your wikipedia link will solve my problem. I will look at it tommorrow, as it is 2am here and I have already had a little too much to drink. Thanks!
Pete

 

[Integral]

https://www.physicsforums.com/showthread.php?t=8997" is the thread that explains the LaTex formating.

 

[Thrice]

$ds^2 = (dy^1)^2 + (dy^2)^2 +...+(dy^N)^2$

$ds^2 = \delta_{kj} dy^k dy^j$

 

[Thrice]

Carroll's text isn't very good for fun learning. Do you have a bit of disposable income for ordering texts, or is there a college near you?

 

[actionintegral]

Thrice, what book do you recommend for learning GR?

 

[Thrice]

Thrice, what book do you recommend for learning GR?

Your mileage will vary. I personally found https://www.amazon.com/dp/012200681X/?tag=pfamazon01-20 one to be very helpful, but I'd say you're best off going to some college library and scanning through their relativity section.

 

[OS Richert]

Thank you, Thrice, for providing the instructions for Latex. I have done my best to ask again as wikipedia and your explanation fell short on my feable brain.

Equation 1.10 is the spacetime interval.

[tex]( \Delta s )^2 = -( c \Delta t )^2 + ( \Delta x )^2 + ( \Delta y )^2 + ( \Delta z )^2[/tex]

Carroll then introduces the following notation.

[tex]x^\mu : x^0 = ct, x^1 = x, x^2 = y, x^3 = z[/tex]

and the 4x4 metrax metric

[tex]\eta_\mu_\nu = \left(\begin{array}{cccc}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right)[/tex]

Finnally, Carroll give us (1.16) and says the content is the same as (1.10) above.

[tex]( \Delta s )^2 = \eta_\mu_\nu \Delta x^\mu \Delta x^v[/tex]

my only guess here, is that $$x^\mu$$ and $$x^\nu$$ are equal, and that is where we are getting the square terms from. And then we sum every role of the matrix with these two (somewhat like matrix multipliation though I don't know what kind of vector $$\Delta x^\mu \Delta x^\nu$$ is supposed to be, so we end up with

-1 (ct)(ct) + 0(x)(x) + 0(y)(y)... + 1(y)(y) + 0(z)(z)...

Is that correct?

Carroll's text isn't very good for fun learning. Do you have a bit of disposable income for ordering texts, or is there a college near you?

Now you tell me! I dropped fifty dollars on this book used, bar far the most I have ever spent on a textbook I was buying just for fun. I took my reccomendation from http://math.ucr.edu/home/baez/RelWWW/reading.html#str" site. I don't think I can spend another 60 dollars unless I sell this book off. I have also purchased, but not yet received, "Spacetime Physics" by Edwin F. Taylor and John Archibald Wheeler, so maybe they will introduce this notation with a more verbose explanation (indeed, I haven't gotten out of the sr section yet )

 

[yenchin]

$$( \Delta s )^2 = \eta_\mu_\nu \Delta x^\mu \Delta x^v$$

my only guess here, is that $$x^\mu$$ and $$x^\nu$$ are equal, and that is where we are getting the square terms from. And then we sum every role of the matrix with these two (somewhat like matrix multipliation though I don't know what kind of vector $$\Delta x^\mu \Delta x^\nu$$ is supposed to be, so we end up with

-1 (ct)(ct) + 0(x)(x) + 0(y)(y)... + 1(y)(y) + 0(z)(z)...

Is that correct?

$$( \Delta s )^2 = \eta_\mu_\nu \Delta x^\mu \Delta x^v$$ means you sum over $$\mu$$ and $$\nu$$. That means it is equal
to

$$\eta_{00} {\Delta x^0}{\Delta x^0} + \eta_{01} \Delta x^0 \Delta x^1 +\eta_{02} \Delta x^0 \Delta x^2 + \eta_{03} \Delta x^0 \Delta x^3 + \eta_{10} \Delta x^1 \Delta x^0 + ... + \eta_{33} \Delta x^3 \Delta x^3$$

There are 16 terms in the sum. It is just that $$\eta_{\mu \nu}$$ is zero for $$\mu \neq \nu$$. By $$\eta_{\mu \nu}$$ here I am referring to the $$\mu\nu$$ entry of the matrix and not the matrix itself. The meaning should be clear from the context when you get the hang of it. My apology. So you only have the square terms instead of cross term. This need not hold in an arbitrary metric.

 

[OS Richert]

That makes perfect sense yenchin! Thank you very much!

 

[robphy]

Carroll's text isn't very good for fun learning. Do you have a bit of disposable income for ordering texts, or is there a college near you?

Now you tell me! I dropped fifty dollars on this book used, bar far the most I have ever spent on a textbook I was buying just for fun. I took my reccomendation from http://math.ucr.edu/home/baez/RelWWW/reading.html#str" site. I don't think I can spend another 60 dollars unless I sell this book off. I have also purchased, but not yet received, "Spacetime Physics" by Edwin F. Taylor and John Archibald Wheeler, so maybe they will introduce this notation with a more verbose explanation (indeed, I haven't gotten out of the sr section yet )

Generally, one would start with something at the level of "Spacetime Physics" then proceed on to something at the level of Carroll's text (which, by the way, has a draft version at http://pancake.uchicago.edu/~carroll/notes/... a related video is available at General Relativity Primer ).

Some other new GR textbook suggestions are in my "[URL contribution to the blog
[/URL]

Second, from Jul 20-21, there was the http://www.aapt-doorway.org/TGRU/" . I’m sure that there are some future textbook authors among the rest of the participants. A lot of good ideas were discussed: “Physics first?”, teach GPS?, teach Gravitational Waves? teach Black Holes? teach Cosmology? teach tensors? [how? … and when?], using visualizations and simulations, confronting misconceptions, etc… Stay tuned.

 

[Thrice]

Now you tell me! I dropped fifty dollars on this book used, bar far the most I have ever spent on a textbook I was buying just for fun.

Ouch. I thought you were working with his draft version.

 

[Daverz]

Now you tell me! I dropped fifty dollars on this book used, bar far the most I have ever spent on a textbook I was buying just for fun. I took my reccomendation from http://math.ucr.edu/home/baez/RelWWW/reading.html#str" site. I don't think I can spend another 60 dollars unless I sell this book off. I have also purchased, but not yet received, "Spacetime Physics" by Edwin F. Taylor and John Archibald Wheeler, so maybe they will introduce this notation with a more verbose explanation (indeed, I haven't gotten out of the sr section yet )

Gee, I think the Carroll book is fun. Granted, a very geeky kind of fun. You got a very good deal on the book. I paid $70 for it used, and the binding was broken. Grrrr, got to be careful with those Amazon Marketplace sellers.

Don't forget the online lectures Carroll did (should be linked off his website).

Starting with Taylor and Wheeler to build up some geometrical intuition is a very good idea.