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So, I was wondering; I realize that Einstein tried to picture what it would be like from the point of view of the photon, and as he went faster, the photon did not go any slower. I also realize that the speed of light is the same in all inertial reference frames. If the speed of light is always c no matter how fast you are going (compared to other intertial reference frames) then cannot it be concluded that you are always at rest in the intertial reference frame of light? Can't we say that from the point of view of light, everything else is at rest?

Also, people always say that in non-inertial reference frames it's different, how is it different? What changes?

This is exactly why I need to get back to writing the intro to str. Here's a quick summary of the deal. My apologies that this take the time to explain things fully, but I hope you have enough familiarity with these concepts that it will do.

In Euclidean (x,y)-space, the metric is ds² = dx²+dy², which is just the Pythagorean theorem or distance formula in differential form. Rotations about the origin preserve the distance from the origin. Curves of constant distance r from the origin are {r² = x²+y²}, which are circles--this is taken directly from the metric. Thus, rotations in Euclidean space translate points along these curves--they "go around in circles", if you will.

The Minkowski metric is ds² = dt²-dx². The curves of constant r from the origin are {r² = t²-x²}, why are now hyperbolas (familiarity with conic sections helps a lot here). Rotations in Minkowski spacetime translate points along hyperbolas instead of circles. These rotations are called Lorentz boosts, and they inertial reference frames that share an origin (if a translation of is also involved, they are known as Poincaré transforms). Now, hyperbolas have asymptotes--in these case, the asymptotes are t = ±x, which represent the speed of light. This is critical: since they are asymptotes, no amount of rotation (Lorentz boost) will reach these points.

Immediately, it follows that no amount of acceleration (Lorentz boost) will allow one to reach lightspeed. Their complete disconnect from every inertial frame means that they are not proper inertial frames at all--i.e., light has no inetial reference frame. It doesn't make much sense to talk about how the universe "looks" from the "point of view" of a light beam.

By considering constant rotation in Euclidean coordinates (in a circle), one can also derive constant rotation (acceleration!) in Minkowski spacetime, since they are related by a Wick rotation (here, x→ix), which will be again by along a hyperbola. If the particle being accelerated is far enough away, its hyperbolic wordline will be below the line t = x--in other words, there will be an horizon preventing an observer at the origin from sending signals to that particle (but not vice versa).

Yes, there are event horizons in special relativity. The difference in GTR is that some event horizons are no longer observer-dependent--they're absolute.

yeah everything was fine, but I also lost you at the second last paragraph, explain in more depth please. I didn't understand very much in that paragraph at all.

I mean the second last paragraph in your post. here:



So now I don't understand what the rotation in Euclidian coordinates and t constant rotation in Minkowski spacetime is. I understand the whole hyperbola thing and the equations for spacetime.

Bah, at some point this summer I'm going to get a good book and attempt to fill the GTR hole in my education [I think my special rel. could do with a dust down as well, to be honest]. Vorpal, any recommendations for good texts on GTR?

I keep hearing good things about Schutz, but since I don't own it, no comment.

The majority of the books I own on GTR and related topics are actually shifted towards the latter (esp. differential geometry), but I'll assume you're more interested in physics. I'm now sure of your current level of mathematics, so I'll simply say this: check out Carroll first, and if it seems too hard, fall back on d'Inverno. If it is clear and you would like to know more, Wald very good and MTW will serve you for years to come. Additionally, try to see if you can get a glimse of Weinberg--that one definietly feels like it's written by a particle physicists for particle physicists.

The first thing to do is to look at this book:

• Carroll, S.: Spacetime and Geometry: An Introduction to General Relativity. Addison-Wesley, 2004. [www] [gr-qc/9712019].

Not only is it a very good introductory book to GTR, but it is actually available online. It covers a wide variety of topics, including black hole thermodynamics and curved spacetime QFT. Best of all, it's free, which is why I mention it first. Take a look at it to gauge your current level. If Carroll looks like it might give you trouble, I the best remedy for that is:

• d'Inverno, R.: Introducing Einstein's Relativity. Oxford: Clarendon Press, 1992.

This book begins with essentially nothing--it starts with the Bondi k-calculus approach to STR (the k-factor is actually exp(T), where T is the Minkowski angle as above, making them multiplicative instead of additive); it develops the elementary mathematics needed for GTR in a very clear, although inelegant, way, and then goes on to GTR topics like the Schwarzschild solution, black holes, gravitational waves, and the Friedmann-Lemaître-Robertson-Walker family of cosmological solutions. It does not cover thermodynamics or QFT. Further cons are that the exercises are boring and the mathematics are inelegant. Despite this, d'Inverno is simply the best book for a complete beginner.

On the other hand, if you're fairly experienced in the background mathematics already, a better try might be:

• Wald, R.: General Relativity. Chicago: University of Chicago Press, 1984.

Wald is one of the better books to learn modern GTR from. It coveres a wide variety of topics, the only conspicuously missing piece being gravitational waves. I think that for serious study, Wald is one of the best.

The Bible:

• Misner, C.; Thorne, K.; Wheeler, J.: Gravitation. San Francisco: W. H. Freeman and Company, 1973.

If you can afford it (in my experience, it is almost impossible to check it out from libraries--I was once on a waiting list for over half a year before I simply purchased it), this tome will serve you very well. It has almost everything you would need, starting from the basics. My only complaint is that the style doing mathematics seems very inconsistent--perhaps because it was written by three people. MTW does not cover black-hole thermodynamics or curved-spacetime QFT, but covers just about everything else, beginning with STR, in more depth than should be legal. (Except global methods, which is a shame.)

MTW may be a bit overwhelming. I'd recommend shelling out the cash for that one only if you will have frequent contant with GTR in the future. Speaking of global methods...

• Hawking, S.; Ellis, G.: The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press, 1975.

What MTW lacks in global methods, this book has. It assumes a near-mastery of tensors, however, and draws on many other mathematical topics like topology. You probably won't need it, but I include it here because it is one of my favorites. It is not a textbook, but a monograph.

Since you are in particle physics, you may enjoy the following:

• Weinberg, S. Gravitation and Cosmology: Principles and Applicatins of The General Theory of Relativity. John Wiley & Sons, 1972.

Weinberg doesn't like geometry in GTR, and consequently I could never get very deep into this book before finding it tiresome. It may be alright for you, given your background in particle physics. One of the focuses of this book is physical experiments, in which it is fairly unique, although overall it is still very theoretical (I'm not sure how you feel about that). It is also the only introductory GTR book I own that covers gravitons in any significant amount of detail. I'd recommend seeing if this book is available at a library or elsewhere first--it is very mathematically demanding, so don't commit unless you're certain!

Finally, the following is a differential geometry book:

• O'Neill, B.: Semi-Riemannian Geometry. San Diego: Academic Press, 1983

This book is nothing short of geometry of GTR (i.e., semi-Riemannian) done right. It makes all the others' treatments look like mere cookbooks. I don't expect you're interested in the mathematics particularly, but if you are, this book is one of the better ones for GTR.

ok I understand what the Euclidian and Mikowski rotations are now, mathematically... but (and forgive me if this is somehow inherent in what you've posted) what are they used for? and why do you need them?

ah, very cool. smile