# Relatively Right in Front of Your Nose

Special relativity holds a special (*ahem*) place in most physicist’s physicists’ hearts. I myself fondly remember learning special relativity from the first edition of Taylor and Wheeler’s Spacetime Physics obtained from my local county library (this edition seemed a lot less annoying than the later edition I used at Caltech.) One of the fun things I remember calculating when I learned this stuff was what “right in front of your nose” meant in different frames of reference.

Suppose you’ve got your own handy dandy inertial frame all set up. You’ve got your rules and you’re your clocks and are all ready to do some measurements. So you perform a measurement and find your nose (yes, your nose) at the origin when your clock strikes midnight on January 7, 2009. At the same time, you also measure that a bug, is just as centimeter and does a backflip. The bug is right in front of your nose, at midnight on January 7, 2009, and it does a flip. Really. In the picture above, your nose at that time is O and the bug doing the backflip is at A.

But now consider the view of you from a rocket passing by (rockets are essential tools for relativity, you know.) Of course this rocket put the origin of its spacetime digram so that the space origin is your nose and its time origin is midnight on January 7, 2009. Cool enough. But because this is special relativity the rocket will see something else. In fact the rock records that the bug doing the backflip has occurred at an earlier time, and at a father location away. This is point B on the diagram above.

What I find cool about this is as follows. On the above plot of drawn the curve x2-t2=1 (time and space can both be measured using centimeters: to convert time to seconds divide by the speed of light.) This represents the set of points that the bug event A could have been transformed into via a Lorentz transform, i.e. if you chose a point on this line, then there is some rocket moving at some relative velocity which will record the bug backflip at this coordinate in its inertial frame. Great. But this also means the opposite. There is a reference frame in which a point in your current inertial frame has a very large value of position and a very large (say negative) value of time.

In other words, there is a reference frame in which what is “right under your nose” is far far away, and just seconds after the big bang (let’s ignore cosmology for now.) There is an event which is a centimeter away from you, which someone who is traveling very very fast will see as ocuring at the same time in that person’s reference frame.

What is this speed (again ignore cosmology) such that, the age of the universe is just under your nose (which we take to be a centimeter away)? Well a simple calculation tells you that this speed is R/sqrt(1+R2)c where R is the time back you want to examine times the speed of light divided by the distance away of under your nose (1 centimeter.) For an time of 13.7 billion years (which is about 430 times 1017 seconds), this gives an R of about 1.3 times 1028 (R is dimensionless.) Now try to punch in this value of R into the above formula. Whoops, not so good for your calculator. But we can re-express the fraction of the speed of light that you need to travel by rewriting the speed as c/sqrt(1/R2+1), which can be Taylor expanded for small values of 1/R as c (1-1/(2R2). In other words, you need to move just under the speed of light, where by just, I mean, about 10-56 fraction slower. In other words 99.99…99 percent of the speed of light with a total of about 56 9s!

So, right under your nose, someone could see an event that happened the age of the universe ago. Where, by someone, we mean someone much much much faster than even Usain Bolt.

1. #1 Ian Durham
January 7, 2009

Ain’t no one faster than Usain.

Seriously though, if you want a terrific pedagogical introduction to SR for teaching purposes (or just for enjoyment), read Tom Moore’s text Six Ideas That Shaped Physics: Unit R. This unit is very short (but the text in it’s entirety is worth reading – it is on par with the Feynman Lectures in that it introduces new insights and really forms the foundation of a whole “philosophy” of physics, I suppose).

At Johns Hopkins we used Bernard Schutz’ A First Course in General Relativity which was awesome. I personally like the geometric approach much better anyway.

2. #2 Thomas
January 8, 2009

Thanks for sharing! I knew about special relativity but had never seen it in quite this way. Very cool.

T.

PS: But still ain’t no one faster than Usain.

3. #3 Killinchy
January 8, 2009

I’m being picky, but “most physicist’s hearts” ??????

4. #4 Dave Bacon
January 8, 2009

In the immortal words of Ralph: “Me fail english? That’s unpossible!”

5. #5 snailclimbingmtfuji
January 8, 2009

Dave, I’m familiar with the way the Lorentz transformations produce outcomes such as you describe with your nose and the bug, where within the frame 2 events are simultaneous but from a frame moving at high velocity they might appear a million (or 13.7 billion) years apart. Physicists like to insist that both perspectives are equally ‘real’, but are they? A person hears a bell next to him ring a second before a distant bell, while a person situated between the two bells hears them as simultaneous. Are both perspectives equally ‘real’? If one perspective gives MEANINGFUL information, information that can be acted upon fruitfully in the real world, and another doesn’t, then how can one argue that they have equal ‘reality’? In your example, in a reference frame at rest with your nose and the back-flipping bug, someone might usefully conclude that your nose was in jeopardy of having an imminent unpleasant encounter and take action, whereas in the rocket’s reference frame the thought would obviously not occur since the nose and bug were separated by unimaginable time and space. Which perspective supplies meaningful information? Physicists are overly infatuated with Einstein’s declaration that there are no privileged positions in the universe�I think he meant it in only a very qualified sense. A reference frame at rest with respect to two events IS privileged with regard to those two events.

6. #6 Pieter Kok
January 9, 2009

Another one: “you’re clocks” (second line below the figure).