Brief note before running off to a long meeting…
Can’t work out whether today’s second long class on the structure of Special Relativity was scary or not. Last lecture we did the classic thought (and real) experiments that lead to the deduction of the Lorentz transformations, and the realization that the words “space” and “time” really should be combined into “spacetime”, and that they need to be thinking in those terms. At the end of the lecture and for most of today, I spent a lot of time emphasizing why Special Relativity is really not weird at all, and developed everything from there on as a very simple analogy to rotations in Euclidean space. The only thing that’s different is this minus sign in front of the “t” part, which is either annoying or beautiful. I urged them to pick the latter since without it there’d be just boring Euclidean four-space, with no yesterdays or tomorrows…
Of course, a bit of time must be spent (forgive the unintended pun) developing some slight wrinkles on the usual rotations formalism to keep track of the minus sign. This has its ups and downs (pun intended), and I worry at times that it might scare the student who is not willing to suspend disbelief and practice it a bit to discover that it is really quite easy… But maybe that is true about everything non-trivial one teaches.
So here’s a question. I like to keep the minus sign explicit by using [tex]\eta_{\mu\nu}={\rm diag}(-1,1,1,1)[/tex] for my basic dot product to get [tex]ds^2=-(ct)^2+dx^2+dy^2+dz^2[/tex], while others like to make things complex and write four vectors and so forth with an [tex]i[/tex] in the time part so as to stick to using the Kronecker [tex]\delta_{\mu\nu}={\rm diag}(1,1,1,1)[/tex].
Which one are you? And why? I think bringing in complex numbers makes Special Relativity seem even more mysterious than it really is.
Is this a dog-people vs cat-people thing?
-cvj
Use Jim’s book. It’s a breeze. See my earlier post.
-cvj
Defintely, don’t use i. The transition to general relativity is much easier once one gets use to different metrics. I am also teaching GR at Queen Mary now, its a great course to give but a worry. I have just done the covariant derivative today, I always am unsure how many of them get it.
reagrds
DSB
Yes… that’s exactly what I like to emphasize.
-cvj
Avoid the “i” whenever possible. It obscures the physics, even though it “aids” in computations. I learned it first with the i’s, but it always looked artificial. Also, formulas for rotations using sines and cosines just get replaced by hyperbolic sines and cosines of a rapidity variable. That is a much more useful notion close to the speed of light than the velocity of a particle itself.
Also, the excessive use of i’s also makes us forget that the Lorentz group is non-compact and this leads very easily to nonsense.
Gosh, yes, [tex]x^0=ct[/tex] of course. I can’t imagine not doing that, since without it the Lorentz transformations don’t look nearly as neat and symmetric.
Best,
-cvj
BTW, I don’t like ict, but by all means use x_0=ct: it is necessary for the picture of spacetime that all coordinates carry the same units. Also, the speed of light then becomes just a unit of measurement, a social convention (as emphasized by Einstein), good lesson to be equipped with when reading about varying speed of light theories in the popular press.
Clifford,
No i’s, please! It just obscures the fundamental geometric difference between space and spacetime. Plus, it doesn’t really make things more intuitive when you try to generalize to curved spacetime, since you’re not going to end up with something that looks like a familiar rotation. And it’s not clear whether bringing in imaginary numbers is conceptually easier than just switching signs in the Pythagorean theorem, anyway. MTW’s box 2.1, “Farewell to ict”, gives other justifications which I remember agreeing with.
Tommy,
Gravitational physicists like -+++ because it makes spacelike hypersurfaces have an ordinary +++ metric. Particle physicists like +—- because it makes timelike vectors have positive squared norm (thus, you don’t need to stick in extra minus signs when computing invariants like mass).
Mitch,
I think imaginary time makes the transition to GR harder; see my response to Clifford.
Anonymous Hero,
Interestingly, +— and -+++ are not actually equivalent (also here), mathematically speaking … but the difference only shows up when studying spinors in non-orientable spacetimes, so this is kind of esoteric. 🙂
I am currently an undegrad student in my last year.. and i do prefer the lorentzian metric as compared to introducing the i. It also causes a problem while reading relativistic quantum mechanics, especially an old text like sakurai. Where, the i from relativity, and the i from quantum mechanics need to be treated differently… it causes a lot of confusion!
Einstein’s ancient presentations were using the imaginary unit and it might emphasize the similarity with the Euclidean spacetime which is very useful psychologically to properly understand what’s going on and how much symmetry there is, I think. A dog?
By the middle of a first course on special relativity, students should understand why these things are equivalent, much like the observation that +— -+++ are equivalent and two conventions for the same physics.
In the long run, I am a cat. It is more convenient to use real numbers as coordinates. Nevertheless, complex time might be important for some quantum cosmology questions.
That’s right Clifford, all the cool kids like their metrics Lorentzian, it is well known.
(and cats are better, also well known).
Yes….. in other words, cats are just better. As I always thought.
Discuss.
-cvj
I don’t like the complex notation too much, it bring in too soon the mathematically convenient but physically meaningless concept of Euclidean time. Better delay introducing students to artificial mathematical constructs until they are absolutely necessary. Occasionally you have to remind even experienced people that Euclidean time doesn’t really exist…
I think that the complex notation, while a little challenging, makes the transition to GR and its metric tensors less of a leap.
The better question is to ask why anyone ever uses the (+,-,-,-) signature anymore? Things are so much uglier and more complicated using it. Its like people who insist on not using the metric system and tell you something is 3.2 yards long.
Of course you should stick with the COMMON notion of a non definite metric. Using the delta with a mysterious ‘i’ somewhere will only confuse the students even more.