Probably dumb question. problem understanding formula

[moe darklight]

ok, i realize this is probably a really simple/dumb question, so I should explain that I don't have any formal learning in any math. my last math class was years ago in grade 11, and it was basic algebra. everything else I know is on a need to know basis (when i see a symbol i don't know, formula i don't understand and is necessary for whatever I'm reading, I find out what it is)... so I probably skipped over some basic things... i know it's not good, but my career path is not science, i like learning about these things just for fun so I didn't bother to learn some things i probably should have.

i only know the concepts of SR and the very basic formulas (time contraction, etc.). figure i'd learn the basics then get into the harder stuff... anyway, i recently decided to try and get into the harder stuff (harder for me at least lol hard=numbers, I've dyscalculia) and got "a first course in general relativity" by schutz.



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I'm having trouble understanding the reasoning on page 11 ("1.6 invariance of intervals"). it comes after a spacetime diagram of a light-ray being reflected and how it is seen from the point of view of 2 observers (the slope of the ray is the same for both obviously, and what changes is t and x).

then it gives the formula: ( http://aycu28.webshots.com/image/13067/2004495092710270588_rs.jpg ) . (the formula also equals 0 because v=1=c). ok, so far so good, i get that.

then it talks about how those coordinates are linear combinations of their counterparts, and how ?s^2 (s with the bar on top ... the first part of pic. 1, before "=") is a quadratic function of the unbarred coordinate increments. ok.

this is where it loses me:

1) "we can therefore conclude that ( http://aycu16.webshots.com/image/13015/2001170268022097477_rs.jpg ) for some numbers {M; a,b (<-greek letters)=0,...,3}, which may be a function of v, the relative velocity of the two frames"

can anyone explain step by step how they arrive at this and what it stands for? ... I'm guessing I'm missing something very basic here. but I'm pretty good at figuring out stuff by one example and extrapolating the concept... just need the words to explain what the hell is going on up there.


problem 2) "and that if ?s^2=0 then ( http://aycu28.webshots.com/image/11867/2003091507785314037_rs.jpg )" ... this is probably the dumber question, but where did the ?r come from? r is usually radius, but i don't see how that would apply here..


again, sorry if this is newb stuff, but this is my first attempt at getting into the math of it all.

 

[cristo]

1. So you're OK with equation (1.1) and its barred counterpart? So, since each barred coordinate is a linear function of the unbarred coordinates, i.e. $$\Delta \bar{x}^a= f\Delta t +g\Delta x+\cdots$$, then [itex](\Delta \bar{x}^a)^2[/tex] will be a quadratic function. If you plug each of these into the barred form of (1.1) you will obtain some quadratic function that can be writted in the form (1.2).

If you can't see it, try expanding (1.2).

2. Here, since ds^2=0, he has taken $(\Delta t)^2$ to the other side and square rooted. Then he has replaced delta t with delta r. He's done this, since now the right hand side of the equation after (1.2) is the usual euclidean distance between two points, which is better denoted by a symbol other that delta t. There is no notion of radius here!

 

[mjsd]

commenting on p.10-11 of Schutz:

This is a rigorous proof of why $$\Delta s_a^2 = \Delta s_b^2$$ for inertial observers a and b. The first assumptions used here is: the fact that speed of light is the same for all inertial observers, in this case a and b. The consequence of this is that if $$\Delta s_a^2 = 0$$ for two events in frame a then $$\Delta s_b^2 =0$$ for the same two events in frame b; the second assumption is that the relationship between the coordinates of frame a and b is linear, which leads to those statements about "quadratic function". Now
$$\displaystyle{\Delta s_b^2 =\sum_{\alpha=0}^{3} \sum_{\beta=0}^{3} M_{\alpha\beta} (\Delta x_a^\alpha)( \Delta x_a^\beta)}$$
where here subscript b is the "barred" system in Schutz, and subscript a is the "unbarred". this is just the most general way of representing $$\Delta s_b^2$$ if it is a quadratic funtion of the "a" coordinate system.

now, the next series of workings (+exercises) is simply using the $$\Delta s_a^2 = 0 = \Delta s_b^2$$ condition to deduce the entries of $$M_{\alpha\beta}$$... that's all

and the "r" is indeed the "distance" in 3D $$r=\sqrt{\Delta x^2+ \Delta y^2+ \Delta z^2}$$

cross posted with cristo...unintentionally (slow computers )

 

[moe darklight]

o ok i get the $$\Delta r$$ part now... still need help tho

I know this is routine basic stuff for people who've been doing this for a while, so please remember that my first encounter with physics of any sort was when i joined this forum 3 months ago, and this is my first attempt at the math behind it... before that the only sciences i payed attention to were biology and philosophy, plus, as i mentioned, i would stay as far away from numbers as possible for obvious reasons lol. so please take into account that I'm pretty new to all of this.

for that first question, my problem is i don't have the intermediate steps (and the book doesn't provide any with its answers.. it assumes the reader is already familiar with this. usually seeing an example worked out is what i need), so i don't know what I'm doing wrong or what I'm missing.

in the back the book recommends reading into linear algebra and matrices, so I've looked into the basics of both (the very basics). what else should i know?

I realize I'm doing things kind of backwards here so sorry 'bout that. but I'm doing this as a pastime not as a career or anything like that... so i can afford to be ignorant of some of the intricacies of it all, as long as i get the gist of it.

 

[jtbell]

Frankly, I think you're trying to learn how to run while you're still at the crawling stage. I suggest that you back up and either learn more about SR and its mathematical framework before moving on to GR, or find a lower-level introduction to GR.

For SR, I sugggest Taylor and Wheeler's "Spacetime Physics." Maybe someone else can suggest a similar book on GR if one exists.

 

[mjsd]

yeah, Schutz is a graduate level textbook, if you just want to learn SR, any first year uni text will have it covered. Taylor and Wheeler is nice too. To do GR you must at least learn "tensors"

 

[moe darklight]

http://www.math.odu.edu/~jhh/counter2.html -- would this be a good place to start? (the free pdf book at the end), I found it in the links section here at PF.

I didn't realize the schutz book was meant for more advanced readers when i bought it... i kept on reading and some things I get, others I don't.. probably the blotchy nature of my knowledge.

On SR I read einstein's book (minus the GR section), and a couple of online texts/tutorials. enough to understand the main ideas (time dilation/length contraction), and be able to do some of the simpler exercises --

which is really all I'm interested in now... just enough so that when i read a science article i have a deeper understanding of what it's talking about and its implications.. so that when i read something where a physicist talks about wormholes, I don't just go "wow! that sounds fascinating! if only i knew what the hell they're talking about!" lol and actually have some idea of what's going on.

thanks for the patience :)

 

[jtbell]

As part of getting ready to study GR seriously, I think you should at least understand the Lorentz transformation equations of SR (which are not the same thing as simply length contraction and time dilation), and the formulation of physical quantities and laws in terms of "four-vectors" (e.g. $x_\mu = (ct, x, y, z)$. For understanding four-vectors, it may in turn be helpful if you study ordinary "three-vectors" (e.g. $\vec x = (x, y, z)$) if you haven't learned about them before.

 

[moe darklight]

thanks, workin' on it. good to know what i need to know... pretty much up to now my main problem is I don't know where to start, my only way of knowing is when I stumble on something i don't understand or looks alien, so I know I'm missing something.

by chance I found this website: http://ocw.mit.edu/index.html this should definitely help me a lot, I didn't even know it existed! it's pretty awesome... they have lots of video lectures-- which is good because my eyes are going to drop out of their sockets if I keep reading out of a screen like this

 

[moe darklight]

Hey, back just to make sure I'm on the right track here.

So here's my "lesson plan:"
(I'm using these as guides, but obviously not my only source of info -- I'll also be looking at more SR)

1) I'm on these right now on linear algebra: http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome/index.htm . -- It's pretty straight forward so far so no problem here.. a good warm up seeing as the last time I was anywhere near a number was 3 years ago.

2) then I'll go on to the differential equations course: http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/CourseHome/index.htm

3) then introduction to tensors: http://www.math.odu.edu/~jhh/counter2.html

4) THEN? I'll be able to understand more of GR? <-- notice uncertain question marks of doom

would this get me where I want? am I missing anything? I hope I'm at least on the right path...

thanks for the help, this is about as out of my element as I can get: I start film school in July ...

 

[masudr]

Yes. Perhaps the most important part of SR to take away with you to GR is the notion of tensors on Minkowski spacetime. I would say, don't move on to GR until you can see how that neatly summarises almost all of SR (of course, by doing that, ideally you will have learned lots of other things too).

 

[moe darklight]

Perhaps the most important part of SR to take away with you to GR is the notion of tensors on Minkowski spacetime.

First of all I have to get the notion of tensors :rofl: -- I haven't looked at those yet

thanks! good to know I'm on the right track. I'm really glad I joined PF, I've learned more in the past 4months than in 5 years of high school (although... I would have probably learned more back then had I actually shown up for a class :uhh: ) .