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The Fisher Files » 2008 » September

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Doppler Shift


The Twin Paradox and the Invariant Interval


Extra dimensions


In Problem #4 of PS2, there was a remark that a photon follows a trajectory given by

x^2+y^2+z^2-c^2t^2=0.

I was thinking about this and thought of something interesting.  Suppose the parametric equations for a photon’s trajectory are

x(t)=v_x*t

y(t)=v_y*t

z(t)=v_z*t.

Putting these in to the equation above gives

(v_x^2+v_y^2+v_z^2)t^2-c^2t^2=0

or

v_x^2+v_y^2+v_z^2=c^2.

So, the first equation above is for a photon starting at the origin and traveling in the direction

(v_x/c,v_y/c,v_z/c).

Now, suppose there is a fourth spatial dimension called v.  At every point in space, there would be a fourth direction we could go, the v direction.  Why we are unaware of it I don’t know.  What I do know is that we would know about it (if relativity still held) because we would have

v_v^2+v_x^2+v_y^2+v_z^2=c^2 so that v_x^2+v_y^2+v_z^2<c^2

for the analysis above, implying there would be photons moving at less than the speed of light.  Well, that does not seem to happen, but there is not reason to think that the v direction would be an infinite line, but rather a little loop of some radius a.  So, a each point in space, you could remain at the same (x,y,z) point and travel around a little loop, going at distance 2 pi a.  Really, nothing wrong with that, if a is small enough.  How small is small enough?  Start back at the first equation

v^2+x^2_y^2+z^2-c^2t^2=0

The three parametric equations would be the same, but since the v direction is a little loop, it parametric equation would have to be

v(t)=mod(v_v*t,2*pi*a)

where mod(a,b) is the modulus and is equal to the fractional part of a/b.  Thus, the largest v could be is 2*pi*a.  Then, putting in the parametric equations gives

mod(v_v*t,2*pi*a)^2+(v_x^2+v_y^2+v_z^2)t^2-c^2t^2=0 or

c^2=v_x^2+v_y^2+v_z^2+mod(v_v*t,2*pi*a)/t^2

The last term still makes the velocities along the three spatial dimensions less than c, but, since the largest the last term can be is (2*pi*a/t)^2, its effect gets small with time, so after some time t_o, it doesn’t make any difference.  suppose a=r_e=2*10^-15 m, the Compton radius of the electron.  Then the last term will not matter when 2*pi*a/t_o<<c or t_o>>2*pi*a/c=4*10^-23 s.  This time is comparable to the very fastest nuclear decays.

Extra dimensions sounds like a weird idea, but it is a very hot topic in theoretical physics these days.

Lorentz Transformations as Rotations


In Thursday’s lecture, Tali started by describing how a rotation in Euclidean space left the distance between two points unchanged. In spacetime, the Lorentz transformation leaves the interval s between two events

s^2=(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-c^2 (t_1-t_2)^2

unchanged.  The Lorentz transformation is a hyperbolic rotation in space time.

To see how this works, let’s define an angle alpha such that tanh(alpha)=beta=v/c, where v is the relative velocity between the two frames.  Recalling the relations between the hyperbolic functions *

cosh^2-sinh^2=1  and 1-tanh^2=1

you can show that gamma=1/sqrt(1-beta^2)=cosh(alpha) and beta*gamma=sinh(alpha).  The transformation matrix lambda:

| gamma            beta*gamma     0      0 |

| beta*gamma       gamma          0      0 |

|         0                    0                1      0 |

|         0                    0                0      1 |

then looks like

| cosh(alpha)    sinh(alpha)   0     0 |

| sinh(alpha)     cosh(alpha)  0     0 |

|       0                      0          1     0 |

|       0                      0          0     1 |

which looks similar to the matrix for a rotation in Euclidean space (with some important sign differences).

So, Lorentz transformations just represent geometric operations in spacetime: rotation and translation in space, translation in space and time and hyperbolic rotation in spacetime.  The represention above has no more practical value than using beta and gamma, but it does serve to illustrate the unified nature of the trnasformations.  What Einstein really recognized was that his predecessors were just talking about the wrong space and, once you figure out the right space to use, it is all very simple.  From this perspective, special relativity is just geometry.

What is also important to see is that time is now on a nearly equal footing with the the three spatial coordinates.  Nearly equal because there is a minus sign in the metric tensor in the time component.

* Hyperbolic trigonometric functions: sinh x=(exp(x)-exp(-x))/2, cosh x=(exp(x)-exp(-x))/2, tanh x=sinh x/cosh x.

Stellar Aberration


The phenomenon of stellar aberration was early evidence agains the ether.  What was observed was that, relative to fixed coordinates of the Earth, distant stars seems to move in a small ellipse, completing on revolution over the course of a year.  The first observations were carried out by Bradley in 1729 (*).

Stellar aberration results from the motion of the Earth perpendicular to the direction of light coming from a distant star.  To see how, it works, put yourself in the rest from of the star, so the telescope appears to be moving with a velocity v perpendicular the the vector from the telescope to the star.  A photon that makes it down the long tube of a telescope must have a component of velocity v parallel to the direction in which the telescope moves and a total velocity of c.  So, the photon makes an angle tan alpha~alpha=v/c with respect to the vector connecting the telescope and the star.  The telescope must be inclined at this angle for the photon to make it through.

Another way to think about stellar aberration is think about running through the rain.  If the rain falls with a velocity w and you run with a velocity v, the rain appears to you (the runner) as falling with an angle beta, tan beta=v/w.

Numerically, the velocity of the Earth around the sun is v=150×10^9×2xPi m/(365×24x60×60 s)=30000 m/s, so alpha=v/c=30000 m/s/3×10^8 m/s=10^-4 rad. 1rad=57.3 degrees=206,000 arc-src, so alpha=20 arc-sec, so over six months, the star shifts by an angle of 40 arc-sec, which is just what Bradley observed.

That’s an analysis from the light moves like little particles.  What about the ether?  Suppose there is an ether and the Earth is at rest in it (i.e. the Earth is “dragging” it along with it).  Light hitting the Earth’s ether form outside would be dragged along with the ether and stellar aberration would not be observed.  To be honest, I’m not sure how to analyze this if the stat is at rest in the ether frame.  To be really honest, I’m really not interested in figuring it out; the ether theory is just wrong!

This is another interesting aspect of the ether theory: I think it is the only thing we spend  lot of time teaching you that is just plain wrong.  You learn classical mechanics that later gets modified by quantum mechanics and relativity, but classical mechanics is still very useful and an active area of research in its own right.  We have to teach way the ether is wrong so you understand why relativity is right.

Stellar aberration is very nicely explained in French, p. 41-46.

*Astronomers frequently express distances as angles, since that is what they measure with a telescope.  They use degrees, minutes and seconds, with one minute being 1/60th of a degree and one second being 1/60th of a minute.  Thus, an arc-second is 1/3600th of a degree.  The best telescopes, like the Magellan Telescope in Chile (where several MIT astronomers work) have angular resolutions of 0.15 arc-sec.

The Fizeau Experiment


I was reading about the Fizeau experiment in which the Fresnel drag coefficient was measured.  The idea was that if light propagated in water (index of refraction n=1.33), the light would propagte with a velocity of c/n+w(1-1/n^2), where w=water velocity.  This was in fact observed as predicted by Fresnel.

The index of refraction comes from the polarization of atoms and molecules in the medium.  An applied electric field polarizes the dipoles, lining them up in the same direction.  This alignment cancels some of the field, slowing it down.  It is a very simple picture, but a very powerful way of looking at the effect of an electric field on matter.

The Fizeau experiment made me think of how this might be applied to looking for dark matter.  Dark matter, which comprises 23% of the matter of the universe, could have a very small dipole moment and this would lead to an index of refraction type effect in vacuum.  It owuld be very tiny, but maybe measurable over galactic distances.

The Ether


I was reading Resnick last night about some of the either experiments (MM, of course, but also the Fitzeau experiement).  I was struck by how primative the thinking was at the time.  At the time of MM (1881), physicists could not image any other way for a wave to propagate than through some medium that gets disturbed, like the way sound compresses air or ripples in water.  It took Einstein’s explanation of the photo-electric effect as light quanta (photons) moving through the vacuum to really solidify the idea that electromagnetic radiation moved through the vacuum like particles.

These days, physicists and astronomers are grapping with a similar sort of problem: dark energy.  When we observe a distant superpnova explode, it seems to move faster then cosmologists expect from the amount of matter in the universe.  The best guess right now is that the universe is filled by dark energy which is a very strange substance.  If you take a normal gas on Earth, the pressure is positive (pushing out) and increases with density, while for dark energy, the pressure is negative (pulls in) and decreases (gets more negative) with density.  There is about 20 times as much dark energy as there is energy in normal matter (neutrons, protons, etc.)  Not much else is known about dark energy; dark energy is very hard to study because its effects as so small.  Astronomers have to look at very large things (clusters and super clusters of galaxies) to see effects of dark energy.

Thinking about the ether an dark energy makes me wonder if we are just being stupid (i.e. dull, lacking keenness of mind) in our thinking about dark energy like the physicists of 1880 thinking about the ether.  I expect there is some wonderful new picture that will emerge and in twenty, fifty or a hundred years, our descendents will wonder why we couldn’t see something so obvious.

Lecture 3


Oh, ouch! Ouch! Ouch!  Acid tummy, trouble sleeping, cranky colleagues and proposals due.  It must be September!

This year starts unusually badly: our DOE guy visits tomorrow, my NSF proposal is due next week, my daughter’s birthday is next week too and EVERYONE is cranky and one edge.

Tali is back and I guess everything came out alright (pun intended). 8.033 is launching into relativity with the Michelson-Morley experiment.  Historically, it is not clear how much Einstein knew about M-M.  He certainly had the idea c is the same in all interial frames, but I’ve never really been clear how much Einstein was up on the scientific literature.  Certainly, MM has a large impact on Lorentz, Fitzgerald, Poincare, etc. who did a great deal to develop the theory prior to Einstein’s great synthesis.

The MM experiment itself is quite amazing.  Their limit on the shift from aether drift in 1887 was 0.0005 of a fringe.  This corresponds to a a change of travel distance of 0.2 nm (200 trillionths of a meter).  This in the era when experimental apparatus was made from wood, leather, candles, glass mirrors, and, of course, mercury.  These days, Michelson interferometers like LIGO (Laser Interferometer Gravity wave Observatory, http://www.ligo.caltech.edu/) look for distance shifts a million times smaller from passing gravity waves.  MIT has a big group working on LIGO and its a great place to do a UROP (talk to Nergis Mavalvala (nergis@ligo.mit.edu) or Erik Katsavounidis (kats@mit.edu)).

Next year in 8.13 or 8.14 (Junior Lab), you may have a chance to use a Michelson interferometer yourselves in the Moessbauer experiment (http://web.mit.edu/8.13/www/13.shtml) where it is used to calibrate an apparatus capable of measuring energies to a part in a trillion.  Inventing this technique got my pal Rudolf Moessbauer a quick Nobel when he was, like, 25 years old.

Lecture 2 - Symmetries


I always forget how hard lecturing is.  When preparing, it seems so daunting to have enough material to fill and hour and a half and then, during the lecture, the time just flies by.  Today was the first time I lecturing power point (well, actually keynote) slides.  I’m nost sure they added much as my talking always seems to run aehad of the slides.

Symmetries is a tough topic: it is really the first major abstraction of the course.  In 8.01 and 8.02, we keep things very specific and very physical.  Understanding symmetries means you really have to step back from the specific and think in general terms.  It is very to convey how imporant they are: I can see the students thinking, “Yeah, yeah” for foruth time I tell them.

There have been several questions about Problem 1 on the pset.  Operationally, it is a very simple problem: the measurment of the speed of light comes from taking into account how long it takes the ligth to cross the diameter of the Earth’s orbit.  The specifics of the problem are tough to state: the orbit period of Io is 1.8 days, of the Earth is 360 days.  Say we call t=0 when the Earth is closest to Jupiter and Io goes behind Jupiter.  An astronomer on Earth 1.8 days later will see Io go behind Jupiter a little later; over the 1.8 days since the first occlusion, the Earth has moved 1% along it’s orbit and is a little further alway, so it takes the light a little longer to get there. The next day, a little longer, etc.

Wait 180 days (six months).  Now the Earth is furtherest away from Jupiter, about 300 million km further than t=0.  Io has gone through 100 orbits of Jupiter (180 days/1.8 days/orbit).  Now, it takes the light from Io just as is goes behind Jupiter 22 minutes longer to reach Earth, so the 100th occlusion occurs at t=180 days + 22 minutes.

I hope this helps.  Astronomers to really amazing things like this all the time using the Earth’s orbits to measure distances and time things.

Second post.


Okay, it looks like it works.  Man, what a stressful week.  I always make a mistake and worry about the first week of the term and it is really the second week that is murder.

However, with 55 minutes before my first 8.033 recitation, it looks like I’m ready to go.  This blog  will be devoted to subjects relating to relativity.  While it will be centered around the course I am recitation instructor for (8.033, “Relativity” at MIT), I plan to make this more generally about ideas and thoughts related to relativity.  You can find the official place for course announcements here.

In preparing for today’s first recitation, I’ve been reviewing Newton’s Laws and Einstein’s Postulates.  I am always struck by how profound these simple statements are, but today I am even more struck by how easy it is to miss seeing how important they are.  In the 321 years since Newton, his ideas have become so embedded in our thinking that it is hard to imagine there was a time before the three laws.  Of course, when we first teach mechanics, we only mention the parts where Newton had it right and tend not to go into absolute space and time so much.