Comments on: An introduction to special relativity for a high school math circle

By: Tõnu Eevere

Tõnu Eevere — Thu, 21 Feb 2013 11:48:18 +0000

Mistahes üksühene seos (vastavus) on avaldatav funktsionaalselt, nii nagu f(x) = y; nii ka g(y) = x; vastavalt niisiis teisendusfunktsioon f(ct) ja teisenduse pöördfunktsioon. Hulgateoreetiliselt nii f kui ka g laienevad koguhulkadele: fE) = F; ja g(F) = E;
Kui meil on seos üldliikmete x ja y vahel, siis need elemendid/hulgad – on olemas (Vt. Valiku e. Zermelo aksioom).
Vaatleme Galilei teisendust sihil v (teljel x), alghetkest, mil AB = r = ct;
x`= x – vt; f(ct) = ct(1 – v/c); ja g(ct) = ct/(1- v/c);
On kerge näha, et funktsioon f määrab relatiivse ruumi F; g – aga määrab nn. “sündmuste-vahelise ruumi” E*, mis ei tarvitse olla samane hulgaga E.
Ometigi, kui küsida vastust Zenoni apooriale: kuskohas saab Achilleus kätte kilpkonna, kui vastavad kiirused on v ja c, kusjuures Achilleus on (kunagi) mõõtnud ära selle esialgse vahemaa kui R =ct.
Vastus: Achilleus saab kilpkonna kätte Achilleuse ruumis,
kohal g(ct) = ct/(1 – v/c).
Üldine Galilei ruumiteisendus on: f(ct) = ct[1 - (v/c)cosa];
selle laotus Cartesiuse ristkoordinaadile avaldub dimensionaalsete seostega: x`= ctcosa – vt; y`= k ctsina; z`= k ctsina;
milles k – on k = 1/L, milles L – on Lorentz-tegur.

By: Luqing Ye

Luqing Ye — Tue, 22 Jan 2013 14:40:31 +0000

Dear Prof.Tao,

In Exercise 4.1,(g),It seems that “Doppler .s.h.i.t.s” should be “Doppler shifts”…

[Gah, that is fixed now. Well, I guess some of the high school students will have been amused by that typo... -T.]

By: Luqing Ye

Luqing Ye — Mon, 21 Jan 2013 09:58:18 +0000

Dear Pro.Tao, In figure 4,I found that those black lines which represent sound are not Parallel with each other,nor are they symmetry regarding the t-axis,this means that the |velocity| of the sound changes,but in fact the |velocity| of the sound does not change.Is it your problem or my problem...... [Yes, some of the image elements were not aligned properly; it should be fixed now. -T.]

By: Víkendové surfovanie « life in progress

Víkendové surfovanie « life in progress — Sun, 13 Jan 2013 20:45:26 +0000

[...] Terry Tao o špeciálnej teórii relativity – pripravené pre stredoškolákov [...]

By: Josh Samani

Josh Samani — Sat, 12 Jan 2013 21:33:52 +0000

Hi Terry,

I am a physics grad (you’re actually on my grad committee at UCLA), and I think this is an important, subtle point. In your notes you write

“Actually, it turns
out that the situation is symmetric: from Bob’s point of view, Alice experiences
time dilation, while from Alice’s point of view, Bob also experiences time dilation.”

This is true when both Alice and Bob are inertial observers, but when they start spatially coincident but one of the twins accelerates away from earth and then returns, the symmetry is broken, and that is why it makes sense for one twin to be older and the other younger once they are again spatially coincident in the end.

My apologies if this was addressed somewhere else, and I didn’t notice.

[Fair enough, I'll add a comment to this effect in the next revision of the ms. -T.]

By: Siddhant Saraf

Siddhant Saraf — Tue, 08 Jan 2013 06:02:09 +0000

Last line on page 10 (footnote 6) , “… consists of a particle together its antiparticle…”
Missing a ‘with’ there.

[Thanks, this will be corrected in the next revision of the ms -T.]

By: John

John — Tue, 01 Jan 2013 15:10:54 +0000

I just found out that wikipedia has a wonderful explanation of how twin paradox is resolved. I used to think general relativity is needed to explain the twin paradox. Glad to be proved wrong. I guess the highly motivated students (and readers) would be grateful to know an external reference to this mind boggling problem?

By: Weekly links for December 30 « God plays dice

Weekly links for December 30 « God plays dice — Mon, 31 Dec 2012 06:17:12 +0000

[...] Terry Tao wrote an introduction to special relativity for a high school math circle. [...]

By: Alan Cooper

Alan Cooper — Sun, 30 Dec 2012 20:57:33 +0000

I like this but do have some suggestions.

One thing that always troubles me about the focus on spacetime diagrams as the essence of special relativity is the fact that they are perfectly compatible with the classical case in which there is a well-defined spacelike direction, and indeed the setup of drawing space and time axes is counter to the “mixing” of space and time that is essential to the theory. The essence of relativity is that the axes only represent one person’s point of view and that a moving observer will define them quite differently, and I would suggest making that point right at the outset.

Also I think Stuart is right about the twin “paradox” (being resolved by the fact that it is impossible to have the twins ever meet to compare ages without violating the condition that both live in inertial frames)

The derivation of E=mc^2 is cute but kind of begs (part of) the question by starting with the assumption that a material mass can in fact be converted to photons. It might be worth mentioning that there are other derivations which do not require either quantum theory or the convertibility of matter into radiation. (One of the great puzzles of history and paedagogy of modern physics is the extent to which ideas from two theories which we actually find very difficult to combine into a consistent whole are often used to motivate one another’s development.)

By: An introduction to special relativity for a high school math circle « Guzman's Mathematics Weblog

Sat, 29 Dec 2012 22:08:38 +0000

[...] An introduction to special relativity for a high school math circle. [...]