Understanding the Minkowski Metric: Explained Step-by-Step for Beginners

[VantagePoint72]

To me, it looked like you were saying that the GR texts teach all the math and that you don't need to learn any diff geo afterwards. But I misinterpreted your statement then.

Fair enough. By 'necessary differential geometry' I meant 'necessary to understand the rest of the textbook', not 'necessary for your future life as a physicist'.

 

[VantagePoint72]

I know what you said and I'm saying that if you go into a physics book without knowing the mathematics beforehand from an actual math text, then you will not have an adequate understanding of the math as far as the mathematics is concerned. You will only be able to apply it to the level of physics problems presented in that specific text. Schutz was my first GR text, then I used Carroll and Wald in conjunction and I didn't learn differential geometry beforehand so I'm speaking from experience. I had to go back and learn the math properly from a math text in order to clear up the horribly hand-wavy presentations of the mathematics presented in books like Carroll and Schutz (Wald is quite an exception in the realm of undergrad / first year grad physics texts since he is actually rigorous with the mathematics).

Well, this is a happy coincidence: I followed literally the exact same sequence when I first began studying GR. Clearly you and I hold this approach in different regard. For me, I appreciated the broad (but increasingly more detailed with each of those books) understanding of GR that came from viewing geometry through the lens of physics. When I went on to study differential geometry on its own, I found it much more satisfying to be able to see immediately how most of the things I was studying enhanced my understanding of relativity. My physical knowledge made it clear exactly which bits were most relevant and merited the deepest study. In the end, studying differential geometry not only helped me better understand the GR I already knew; already knowing a fair amount of GR helped me better understand the differential geometry I was studying. Which has, of course, been useful well beyond just GR.

I am happy that that is the path through GR and geometry that my undergraduate and graduate education lead me on. Apparently you are not. That's fine: as I've said from the very beginning, there is a significant element of personal taste here. However, surely you can't sustain the position that the sequence of GR and then more differential geometry that I'm advocating invariably leads to inadequate understanding. I would presume you think you turned out fine.

 

[WannabeNewton]

Well, this is a happy coincidence: I followed literally the exact same sequence when I first began studying GR. Clearly you and I hold this approach in different regard. For me, I appreciated the broad (but increasingly more detailed with each of those books) understanding of GR that came from viewing geometry through the lens of physics. When I went on to study differential geometry on its own, I found it much more satisfying to be able to see immediately how most of the things I was studying enhanced my understanding of relativity. My physical knowledge made it clear exactly which bits were most relevant and merited the deepest study. In the end, studying differential geometry not only helped me better understand the GR I already knew; already knowing a fair amount of GR helped me better understand the differential geometry I was studying. Which has, of course, been useful well beyond just GR.

I certainly agree that viewing the geometry through the lens of physics is quite important and very satisfying (this is possibly one of the bigger reasons I got into physics). I would never want a fleshed out exposition of differential geometry in Schutz for example because he provides such an awesome intuition for the physics that I certainly think would be ruined by adding too much math. I don't think it would be good to add too much math to the physics book itself, I'm saying it might help to learn it before using the book. I can't say that learning GR has helped me personally understand differential geometry better but I can say that it has helped me appreciate it more, that's for sure.

I am happy that that is the path through GR and geometry that my undergraduate and graduate education lead me on. Apparently you are not. That's fine: as I've said from the very beginning, there is a significant element of personal taste here. However, surely you can't sustain the position that the sequence of GR and then more differential geometry that I'm advocating invariably leads to inadequate understanding. I would presume you think you turned out fine.

I am not unhappy as much as I am confused when I read math in physics texts where the mathematical facts are not proven / exposed rigorously. This, I agree, is a matter of purely personal taste. I certainly don't disagree with your advice that learning GR then learning differential geometry is one advantageous route to take; I'm just saying that there can be people out there who are not satisfied with the exposition of the math in the GR texts to begin with (as I was). Looking back, I do agree that there is quite a bit of personal taste involved in this. However you can agree that just learning the relevant geometry through physics texts cannot be the end all be all of the education.

 

[VantagePoint72]

I think perhaps an analogy is in order:

Calculus is very much the language of mechanics. However, students (in the educational systems with which I'm familiar) don't typically begin working with it until the final year before they begin university, and really get to delve into until first year uni. Nonetheless, we do not wait until then to begin educating people in physics. We present an algebraic version of it throughout high school to begin fostering physical intuition and motivating further study. Indeed, my first year physics class as undergraduate even included precisely the sort of 'on-the-fly' instruction in calculus that I am describing with differential geometry. Obviously that introduction was not the end of calculus education at my undergrad institution. However, it made it possible to start working with the fundamental machinery of classical mechanics. The same thing happened with my first dedicated course in electromagnetism: a vector calculus preview was necessary since the multivariable course was taught concurrently. For those students interested in mathematics primarily for its use as the language of physics, the physical knowledge provides excellent context for the deeper mathematical study. More importantly, it makes it possible to start understanding the physics as quickly as possible.

I think a dogmatic insistence on a rigorous study of differential geometry prior to embarking on a study of GR [note: this is not directed at anyone in particular, I'm just emphasizing my original point] is precisely as wrong-headed as insisting high school students not be taught a lick of physics, nor undergraduates until they've finished with real analysis and partial differential equations. I think it fundamentally confuses logical dependence with pedagogical dependence. There will, of course, be students interested in the mathematics for its own sake. These are students who will pursue a dedicated education in mathematics alongside their physics education. However, for the students primarily interested in the application of mathematics to physics, I do not believe any favours are done by placing as much abstract mathematics as possible between them and their exploration of physical concepts.

 

[VantagePoint72]

However you can agree that just learning the relevant geometry through physics texts cannot be the end all be all of the education.

I do (and have already said) that I readily agree with that.

 

[WannabeNewton]

More importantly, it makes it possible to start understanding the physics as quickly as possible.

To conclude, I think this is the whole crux of it all. If the goal is to learn the physics then yeah go right into the relevant physics text but there will be people, me included, who are just uncomfortable in learning physical theories at the level of GR or QM etc. without some exposition to the math as presented in a math text, beforehand.

However I think GR is funny in the sense that even if you do learn differential geometry from a math text beforehand, you will still not be able to do any of the problems involving tensor calculus (like the one I posted a solution to in post #7 of this thread) until you actually pick up a GR book and start reading it because as far as I've seen, no modern differential geometry text even touches the index based tensor calculations that you see so pervasively in e.g. Wald.

 

[shounakbhatta]

Hello,

Thank you for referring the source. I would start shortly with Hartle's Gravity. Honestly speaking, I am a self-learner. There are some concepts which are not clear in my mind. Hence, when I ask questions, naive, it might seemed idiotic. So, I refrain. Definitely I would go ahead with the references. Differential geometry would take some time, I believe. Anyway...

Thanks for all the help.